CTPLQT computes a blocked LQ factorization of a complex
 "triangular-pentagonal" matrix C, which is composed of a
 triangular block A and pentagonal block B, using the compact
 WY representation for Q.

M

          M is INTEGER
          The number of rows of the matrix B, and the order of the
          triangular matrix A.
          M >= 0.

N

          N is INTEGER
          The number of columns of the matrix B.
          N >= 0.

L

          L is INTEGER
          The number of rows of the lower trapezoidal part of B.
          MIN(M,N) >= L >= 0.  See Further Details.

MB

          MB is INTEGER
          The block size to be used in the blocked QR.  M >= MB >= 1.

A

          A is COMPLEX array, dimension (LDA,M)
          On entry, the lower triangular M-by-M matrix A.
          On exit, the elements on and below the diagonal of the array
          contain the lower triangular matrix L.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

B

          B is COMPLEX array, dimension (LDB,N)
          On entry, the pentagonal M-by-N matrix B.  The first N-L columns
          are rectangular, and the last L columns are lower trapezoidal.
          On exit, B contains the pentagonal matrix V.  See Further Details.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,M).

T

          T is COMPLEX array, dimension (LDT,N)
          The lower triangular block reflectors stored in compact form
          as a sequence of upper triangular blocks.  See Further Details.

LDT

          LDT is INTEGER
          The leading dimension of the array T.  LDT >= MB.

WORK

          WORK is COMPLEX array, dimension (MB*M)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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  The input matrix C is a M-by-(M+N) matrix
               C = [ A ] [ B ]
  where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
  matrix consisting of a M-by-(N-L) rectangular matrix B1 on left of a M-by-L
  upper trapezoidal matrix B2:
          [ B ] = [ B1 ] [ B2 ]
                   [ B1 ]  <- M-by-(N-L) rectangular
                   [ B2 ]  <-     M-by-L lower trapezoidal.
  The lower trapezoidal matrix B2 consists of the first L columns of a
  M-by-M lower triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
  B is rectangular M-by-N; if M=L=N, B is lower triangular.
  The matrix W stores the elementary reflectors H(i) in the i-th row
  above the diagonal (of A) in the M-by-(M+N) input matrix C
            [ C ] = [ A ] [ B ]
                   [ A ]  <- lower triangular M-by-M
                   [ B ]  <- M-by-N pentagonal
  so that W can be represented as
            [ W ] = [ I ] [ V ]
                   [ I ]  <- identity, M-by-M
                   [ V ]  <- M-by-N, same form as B.
  Thus, all of information needed for W is contained on exit in B, which
  we call V above.  Note that V has the same form as B; that is,
            [ V ] = [ V1 ] [ V2 ]
                   [ V1 ] <- M-by-(N-L) rectangular
                   [ V2 ] <-     M-by-L lower trapezoidal.
  The rows of V represent the vectors which define the H(i)'s.
  The number of blocks is B = ceiling(M/MB), where each
  block is of order MB except for the last block, which is of order
  IB = M - (M-1)*MB.  For each of the B blocks, a upper triangular block
  reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
  for the last block) T's are stored in the MB-by-N matrix T as
               T = [T1 T2 ... TB].

 CTPLQT2 computes a LQ a factorization of a complex "triangular-pentagonal"
 matrix C, which is composed of a triangular block A and pentagonal block B,
 using the compact WY representation for Q.

M

          M is INTEGER
          The total number of rows of the matrix B.
          M >= 0.

N

          N is INTEGER
          The number of columns of the matrix B, and the order of
          the triangular matrix A.
          N >= 0.

L

          L is INTEGER
          The number of rows of the lower trapezoidal part of B.
          MIN(M,N) >= L >= 0.  See Further Details.

A

          A is COMPLEX array, dimension (LDA,M)
          On entry, the lower triangular M-by-M matrix A.
          On exit, the elements on and below the diagonal of the array
          contain the lower triangular matrix L.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

B

          B is COMPLEX array, dimension (LDB,N)
          On entry, the pentagonal M-by-N matrix B.  The first N-L columns
          are rectangular, and the last L columns are lower trapezoidal.
          On exit, B contains the pentagonal matrix V.  See Further Details.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,M).

T

          T is COMPLEX array, dimension (LDT,M)
          The N-by-N upper triangular factor T of the block reflector.
          See Further Details.

LDT

          LDT is INTEGER
          The leading dimension of the array T.  LDT >= max(1,M)

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

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  The input matrix C is a M-by-(M+N) matrix
               C = [ A ][ B ]
  where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
  matrix consisting of a M-by-(N-L) rectangular matrix B1 left of a M-by-L
  upper trapezoidal matrix B2:
               B = [ B1 ][ B2 ]
                   [ B1 ]  <-     M-by-(N-L) rectangular
                   [ B2 ]  <-     M-by-L lower trapezoidal.
  The lower trapezoidal matrix B2 consists of the first L columns of a
  N-by-N lower triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
  B is rectangular M-by-N; if M=L=N, B is lower triangular.
  The matrix W stores the elementary reflectors H(i) in the i-th row
  above the diagonal (of A) in the M-by-(M+N) input matrix C
               C = [ A ][ B ]
                   [ A ]  <- lower triangular M-by-M
                   [ B ]  <- M-by-N pentagonal
  so that W can be represented as
               W = [ I ][ V ]
                   [ I ]  <- identity, M-by-M
                   [ V ]  <- M-by-N, same form as B.
  Thus, all of information needed for W is contained on exit in B, which
  we call V above.  Note that V has the same form as B; that is,
               W = [ V1 ][ V2 ]
                   [ V1 ] <-     M-by-(N-L) rectangular
                   [ V2 ] <-     M-by-L lower trapezoidal.
  The rows of V represent the vectors which define the H(i)'s.
  The (M+N)-by-(M+N) block reflector H is then given by
               H = I - W**T * T * W
  where W^H is the conjugate transpose of W and T is the upper triangular
  factor of the block reflector.

 CTPMLQT applies a complex orthogonal matrix Q obtained from a
 "triangular-pentagonal" complex block reflector H to a general
 complex matrix C, which consists of two blocks A and B.

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**H from the Left;
          = 'R': apply Q or Q**H from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'C':  Conjugate transpose, apply Q**H.

M

          M is INTEGER
          The number of rows of the matrix B. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix B. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.

L

          L is INTEGER
          The order of the trapezoidal part of V.
          K >= L >= 0.  See Further Details.

MB

          MB is INTEGER
          The block size used for the storage of T.  K >= MB >= 1.
          This must be the same value of MB used to generate T
          in CTPLQT.

V

          V is COMPLEX array, dimension (LDV,K)
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          CTPLQT in B.  See Further Details.

LDV

          LDV is INTEGER
          The leading dimension of the array V.
          If SIDE = 'L', LDV >= max(1,M);
          if SIDE = 'R', LDV >= max(1,N).

T

          T is COMPLEX array, dimension (LDT,K)
          The upper triangular factors of the block reflectors
          as returned by CTPLQT, stored as a MB-by-K matrix.

LDT

          LDT is INTEGER
          The leading dimension of the array T.  LDT >= MB.

A

          A is COMPLEX array, dimension
          (LDA,N) if SIDE = 'L' or
          (LDA,K) if SIDE = 'R'
          On entry, the K-by-N or M-by-K matrix A.
          On exit, A is overwritten by the corresponding block of
          Q*C or Q**H*C or C*Q or C*Q**H.  See Further Details.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDC >= max(1,K);
          If SIDE = 'R', LDC >= max(1,M).

B

          B is COMPLEX array, dimension (LDB,N)
          On entry, the M-by-N matrix B.
          On exit, B is overwritten by the corresponding block of
          Q*C or Q**H*C or C*Q or C*Q**H.  See Further Details.

LDB

          LDB is INTEGER
          The leading dimension of the array B.
          LDB >= max(1,M).

WORK

          WORK is COMPLEX array. The dimension of WORK is
           N*MB if SIDE = 'L', or  M*MB if SIDE = 'R'.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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  The columns of the pentagonal matrix V contain the elementary reflectors
  H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a
  trapezoidal block V2:
        V = [V1] [V2].
  The size of the trapezoidal block V2 is determined by the parameter L,
  where 0 <= L <= K; V2 is lower trapezoidal, consisting of the first L
  rows of a K-by-K upper triangular matrix.  If L=K, V2 is lower triangular;
  if L=0, there is no trapezoidal block, hence V = V1 is rectangular.
  If SIDE = 'L':  C = [A]  where A is K-by-N,  B is M-by-N and V is K-by-M.
                      [B]
  If SIDE = 'R':  C = [A B]  where A is M-by-K, B is M-by-N and V is K-by-N.
  The real orthogonal matrix Q is formed from V and T.
  If TRANS='N' and SIDE='L', C is on exit replaced with Q * C.
  If TRANS='C' and SIDE='L', C is on exit replaced with Q**H * C.
  If TRANS='N' and SIDE='R', C is on exit replaced with C * Q.
  If TRANS='C' and SIDE='R', C is on exit replaced with C * Q**H.

 DBBCSD computes the CS decomposition of an orthogonal matrix in
 bidiagonal-block form,
     [ B11 | B12 0  0 ]
     [  0  |  0 -I  0 ]
 X = [----------------]
     [ B21 | B22 0  0 ]
     [  0  |  0  0  I ]
                               [  C | -S  0  0 ]
                   [ U1 |    ] [  0 |  0 -I  0 ] [ V1 |    ]**T
                 = [---------] [---------------] [---------]   .
                   [    | U2 ] [  S |  C  0  0 ] [    | V2 ]
                               [  0 |  0  0  I ]
 X is M-by-M, its top-left block is P-by-Q, and Q must be no larger
 than P, M-P, or M-Q. (If Q is not the smallest index, then X must be
 transposed and/or permuted. This can be done in constant time using
 the TRANS and SIGNS options. See DORCSD for details.)
 The bidiagonal matrices B11, B12, B21, and B22 are represented
 implicitly by angles THETA(1:Q) and PHI(1:Q-1).
 The orthogonal matrices U1, U2, V1T, and V2T are input/output.
 The input matrices are pre- or post-multiplied by the appropriate
 singular vector matrices.

JOBU1

          JOBU1 is CHARACTER
          = 'Y':      U1 is updated;
          otherwise:  U1 is not updated.

JOBU2

          JOBU2 is CHARACTER
          = 'Y':      U2 is updated;
          otherwise:  U2 is not updated.

JOBV1T

          JOBV1T is CHARACTER
          = 'Y':      V1T is updated;
          otherwise:  V1T is not updated.

JOBV2T

          JOBV2T is CHARACTER
          = 'Y':      V2T is updated;
          otherwise:  V2T is not updated.

TRANS

          TRANS is CHARACTER
          = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
                      order;
          otherwise:  X, U1, U2, V1T, and V2T are stored in column-
                      major order.

M

          M is INTEGER
          The number of rows and columns in X, the orthogonal matrix in
          bidiagonal-block form.

P

          P is INTEGER
          The number of rows in the top-left block of X. 0 <= P <= M.

Q

          Q is INTEGER
          The number of columns in the top-left block of X.
          0 <= Q <= MIN(P,M-P,M-Q).

THETA

          THETA is DOUBLE PRECISION array, dimension (Q)
          On entry, the angles THETA(1),...,THETA(Q) that, along with
          PHI(1), ...,PHI(Q-1), define the matrix in bidiagonal-block
          form. On exit, the angles whose cosines and sines define the
          diagonal blocks in the CS decomposition.

PHI

          PHI is DOUBLE PRECISION array, dimension (Q-1)
          The angles PHI(1),...,PHI(Q-1) that, along with THETA(1),...,
          THETA(Q), define the matrix in bidiagonal-block form.

U1

          U1 is DOUBLE PRECISION array, dimension (LDU1,P)
          On entry, a P-by-P matrix. On exit, U1 is postmultiplied
          by the left singular vector matrix common to [ B11 ; 0 ] and
          [ B12 0 0 ; 0 -I 0 0 ].

LDU1

          LDU1 is INTEGER
          The leading dimension of the array U1, LDU1 >= MAX(1,P).

U2

          U2 is DOUBLE PRECISION array, dimension (LDU2,M-P)
          On entry, an (M-P)-by-(M-P) matrix. On exit, U2 is
          postmultiplied by the left singular vector matrix common to
          [ B21 ; 0 ] and [ B22 0 0 ; 0 0 I ].

LDU2

          LDU2 is INTEGER
          The leading dimension of the array U2, LDU2 >= MAX(1,M-P).

V1T

          V1T is DOUBLE PRECISION array, dimension (LDV1T,Q)
          On entry, a Q-by-Q matrix. On exit, V1T is premultiplied
          by the transpose of the right singular vector
          matrix common to [ B11 ; 0 ] and [ B21 ; 0 ].

LDV1T

          LDV1T is INTEGER
          The leading dimension of the array V1T, LDV1T >= MAX(1,Q).

V2T

          V2T is DOUBLE PRECISION array, dimension (LDV2T,M-Q)
          On entry, an (M-Q)-by-(M-Q) matrix. On exit, V2T is
          premultiplied by the transpose of the right
          singular vector matrix common to [ B12 0 0 ; 0 -I 0 ] and
          [ B22 0 0 ; 0 0 I ].

LDV2T

          LDV2T is INTEGER
          The leading dimension of the array V2T, LDV2T >= MAX(1,M-Q).

B11D

          B11D is DOUBLE PRECISION array, dimension (Q)
          When DBBCSD converges, B11D contains the cosines of THETA(1),
          ..., THETA(Q). If DBBCSD fails to converge, then B11D
          contains the diagonal of the partially reduced top-left
          block.

B11E

          B11E is DOUBLE PRECISION array, dimension (Q-1)
          When DBBCSD converges, B11E contains zeros. If DBBCSD fails
          to converge, then B11E contains the superdiagonal of the
          partially reduced top-left block.

B12D

          B12D is DOUBLE PRECISION array, dimension (Q)
          When DBBCSD converges, B12D contains the negative sines of
          THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
          B12D contains the diagonal of the partially reduced top-right
          block.

B12E

          B12E is DOUBLE PRECISION array, dimension (Q-1)
          When DBBCSD converges, B12E contains zeros. If DBBCSD fails
          to converge, then B12E contains the subdiagonal of the
          partially reduced top-right block.

B21D

          B21D is DOUBLE PRECISION  array, dimension (Q)
          When DBBCSD converges, B21D contains the negative sines of
          THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
          B21D contains the diagonal of the partially reduced bottom-left
          block.

B21E

          B21E is DOUBLE PRECISION  array, dimension (Q-1)
          When DBBCSD converges, B21E contains zeros. If DBBCSD fails
          to converge, then B21E contains the subdiagonal of the
          partially reduced bottom-left block.

B22D

          B22D is DOUBLE PRECISION  array, dimension (Q)
          When DBBCSD converges, B22D contains the negative sines of
          THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
          B22D contains the diagonal of the partially reduced bottom-right
          block.

B22E

          B22E is DOUBLE PRECISION  array, dimension (Q-1)
          When DBBCSD converges, B22E contains zeros. If DBBCSD fails
          to converge, then B22E contains the subdiagonal of the
          partially reduced bottom-right block.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= MAX(1,8*Q).
          If LWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the WORK array,
          returns this value as the first entry of the work array, and
          no error message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if DBBCSD did not converge, INFO specifies the number
                of nonzero entries in PHI, and B11D, B11E, etc.,
                contain the partially reduced matrix.

  TOLMUL  DOUBLE PRECISION, default = MAX(10,MIN(100,EPS**(-1/8)))
          TOLMUL controls the convergence criterion of the QR loop.
          Angles THETA(i), PHI(i) are rounded to 0 or PI/2 when they
          are within TOLMUL*EPS of either bound.

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

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 DGETSQRHRT computes a NB2-sized column blocked QR-factorization
 of a real M-by-N matrix A with M >= N,
    A = Q * R.
 The routine uses internally a NB1-sized column blocked and MB1-sized
 row blocked TSQR-factorization and perfors the reconstruction
 of the Householder vectors from the TSQR output. The routine also
 converts the R_tsqr factor from the TSQR-factorization output into
 the R factor that corresponds to the Householder QR-factorization,
    A = Q_tsqr * R_tsqr = Q * R.
 The output Q and R factors are stored in the same format as in DGEQRT
 (Q is in blocked compact WY-representation). See the documentation
 of DGEQRT for more details on the format.

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

N

          N is INTEGER
          The number of columns of the matrix A. M >= N >= 0.

MB1

          MB1 is INTEGER
          The row block size to be used in the blocked TSQR.
          MB1 > N.

NB1

          NB1 is INTEGER
          The column block size to be used in the blocked TSQR.
          N >= NB1 >= 1.

NB2

          NB2 is INTEGER
          The block size to be used in the blocked QR that is
          output. NB2 >= 1.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry: an M-by-N matrix A.
          On exit:
           a) the elements on and above the diagonal
              of the array contain the N-by-N upper-triangular
              matrix R corresponding to the Householder QR;
           b) the elements below the diagonal represent Q by
              the columns of blocked V (compact WY-representation).

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

T

          T is DOUBLE PRECISION array, dimension (LDT,N))
          The upper triangular block reflectors stored in compact form
          as a sequence of upper triangular blocks.

LDT

          LDT is INTEGER
          The leading dimension of the array T.  LDT >= NB2.

WORK

          (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          The dimension of the array WORK.
          LWORK >= MAX( LWT + LW1, MAX( LWT+N*N+LW2, LWT+N*N+N ) ),
          where
             NUM_ALL_ROW_BLOCKS = CEIL((M-N)/(MB1-N)),
             NB1LOCAL = MIN(NB1,N).
             LWT = NUM_ALL_ROW_BLOCKS * N * NB1LOCAL,
             LW1 = NB1LOCAL * N,
             LW2 = NB1LOCAL * MAX( NB1LOCAL, ( N - NB1LOCAL ) ),
          If LWORK = -1, then a workspace query is assumed.
          The routine only calculates the optimal size of the WORK
          array, returns this value as the first entry of the WORK
          array, and no error message related to LWORK is issued
          by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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 November 2020, Igor Kozachenko,
                Computer Science Division,
                University of California, Berkeley

 DGGHD3 reduces a pair of real matrices (A,B) to generalized upper
 Hessenberg form using orthogonal transformations, where A is a
 general matrix and B is upper triangular.  The form of the
 generalized eigenvalue problem is
    A*x = lambda*B*x,
 and B is typically made upper triangular by computing its QR
 factorization and moving the orthogonal matrix Q to the left side
 of the equation.
 This subroutine simultaneously reduces A to a Hessenberg matrix H:
    Q**T*A*Z = H
 and transforms B to another upper triangular matrix T:
    Q**T*B*Z = T
 in order to reduce the problem to its standard form
    H*y = lambda*T*y
 where y = Z**T*x.
 The orthogonal matrices Q and Z are determined as products of Givens
 rotations.  They may either be formed explicitly, or they may be
 postmultiplied into input matrices Q1 and Z1, so that
      Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
      Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
 If Q1 is the orthogonal matrix from the QR factorization of B in the
 original equation A*x = lambda*B*x, then DGGHD3 reduces the original
 problem to generalized Hessenberg form.
 This is a blocked variant of DGGHRD, using matrix-matrix
 multiplications for parts of the computation to enhance performance.

COMPQ

          COMPQ is CHARACTER*1
          = 'N': do not compute Q;
          = 'I': Q is initialized to the unit matrix, and the
                 orthogonal matrix Q is returned;
          = 'V': Q must contain an orthogonal matrix Q1 on entry,
                 and the product Q1*Q is returned.

COMPZ

          COMPZ is CHARACTER*1
          = 'N': do not compute Z;
          = 'I': Z is initialized to the unit matrix, and the
                 orthogonal matrix Z is returned;
          = 'V': Z must contain an orthogonal matrix Z1 on entry,
                 and the product Z1*Z is returned.

N

          N is INTEGER
          The order of the matrices A and B.  N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
          ILO and IHI mark the rows and columns of A which are to be
          reduced.  It is assumed that A is already upper triangular
          in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are
          normally set by a previous call to DGGBAL; otherwise they
          should be set to 1 and N respectively.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

A

          A is DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the N-by-N general matrix to be reduced.
          On exit, the upper triangle and the first subdiagonal of A
          are overwritten with the upper Hessenberg matrix H, and the
          rest is set to zero.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

B

          B is DOUBLE PRECISION array, dimension (LDB, N)
          On entry, the N-by-N upper triangular matrix B.
          On exit, the upper triangular matrix T = Q**T B Z.  The
          elements below the diagonal are set to zero.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

Q

          Q is DOUBLE PRECISION array, dimension (LDQ, N)
          On entry, if COMPQ = 'V', the orthogonal matrix Q1,
          typically from the QR factorization of B.
          On exit, if COMPQ='I', the orthogonal matrix Q, and if
          COMPQ = 'V', the product Q1*Q.
          Not referenced if COMPQ='N'.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.
          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          On entry, if COMPZ = 'V', the orthogonal matrix Z1.
          On exit, if COMPZ='I', the orthogonal matrix Z, and if
          COMPZ = 'V', the product Z1*Z.
          Not referenced if COMPZ='N'.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.
          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The length of the array WORK.  LWORK >= 1.
          For optimum performance LWORK >= 6*N*NB, where NB is the
          optimal blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

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  This routine reduces A to Hessenberg form and maintains B in triangular form
  using a blocked variant of Moler and Stewart's original algorithm,
  as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti
  (BIT 2008).

 DGGHRD reduces a pair of real matrices (A,B) to generalized upper
 Hessenberg form using orthogonal transformations, where A is a
 general matrix and B is upper triangular.  The form of the
 generalized eigenvalue problem is
    A*x = lambda*B*x,
 and B is typically made upper triangular by computing its QR
 factorization and moving the orthogonal matrix Q to the left side
 of the equation.
 This subroutine simultaneously reduces A to a Hessenberg matrix H:
    Q**T*A*Z = H
 and transforms B to another upper triangular matrix T:
    Q**T*B*Z = T
 in order to reduce the problem to its standard form
    H*y = lambda*T*y
 where y = Z**T*x.
 The orthogonal matrices Q and Z are determined as products of Givens
 rotations.  They may either be formed explicitly, or they may be
 postmultiplied into input matrices Q1 and Z1, so that
      Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
      Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
 If Q1 is the orthogonal matrix from the QR factorization of B in the
 original equation A*x = lambda*B*x, then DGGHRD reduces the original
 problem to generalized Hessenberg form.

COMPQ

          COMPQ is CHARACTER*1
          = 'N': do not compute Q;
          = 'I': Q is initialized to the unit matrix, and the
                 orthogonal matrix Q is returned;
          = 'V': Q must contain an orthogonal matrix Q1 on entry,
                 and the product Q1*Q is returned.

COMPZ

          COMPZ is CHARACTER*1
          = 'N': do not compute Z;
          = 'I': Z is initialized to the unit matrix, and the
                 orthogonal matrix Z is returned;
          = 'V': Z must contain an orthogonal matrix Z1 on entry,
                 and the product Z1*Z is returned.

N

          N is INTEGER
          The order of the matrices A and B.  N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
          ILO and IHI mark the rows and columns of A which are to be
          reduced.  It is assumed that A is already upper triangular
          in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are
          normally set by a previous call to DGGBAL; otherwise they
          should be set to 1 and N respectively.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

A

          A is DOUBLE PRECISION array, dimension (LDA, N)
          On entry, the N-by-N general matrix to be reduced.
          On exit, the upper triangle and the first subdiagonal of A
          are overwritten with the upper Hessenberg matrix H, and the
          rest is set to zero.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

B

          B is DOUBLE PRECISION array, dimension (LDB, N)
          On entry, the N-by-N upper triangular matrix B.
          On exit, the upper triangular matrix T = Q**T B Z.  The
          elements below the diagonal are set to zero.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

Q

          Q is DOUBLE PRECISION array, dimension (LDQ, N)
          On entry, if COMPQ = 'V', the orthogonal matrix Q1,
          typically from the QR factorization of B.
          On exit, if COMPQ='I', the orthogonal matrix Q, and if
          COMPQ = 'V', the product Q1*Q.
          Not referenced if COMPQ='N'.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.
          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          On entry, if COMPZ = 'V', the orthogonal matrix Z1.
          On exit, if COMPZ='I', the orthogonal matrix Z, and if
          COMPZ = 'V', the product Z1*Z.
          Not referenced if COMPZ='N'.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.
          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

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  This routine reduces A to Hessenberg and B to triangular form by
  an unblocked reduction, as described in _Matrix_Computations_,
  by Golub and Van Loan (Johns Hopkins Press.)

 DGGQRF computes a generalized QR factorization of an N-by-M matrix A
 and an N-by-P matrix B:
             A = Q*R,        B = Q*T*Z,
 where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
 matrix, and R and T assume one of the forms:
 if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
                 (  0  ) N-M                         N   M-N
                    M
 where R11 is upper triangular, and
 if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
                  P-N  N                           ( T21 ) P
                                                      P
 where T12 or T21 is upper triangular.
 In particular, if B is square and nonsingular, the GQR factorization
 of A and B implicitly gives the QR factorization of inv(B)*A:
              inv(B)*A = Z**T*(inv(T)*R)
 where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
 transpose of the matrix Z.

N

          N is INTEGER
          The number of rows of the matrices A and B. N >= 0.

M

          M is INTEGER
          The number of columns of the matrix A.  M >= 0.

P

          P is INTEGER
          The number of columns of the matrix B.  P >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,M)
          On entry, the N-by-M matrix A.
          On exit, the elements on and above the diagonal of the array
          contain the min(N,M)-by-M upper trapezoidal matrix R (R is
          upper triangular if N >= M); the elements below the diagonal,
          with the array TAUA, represent the orthogonal matrix Q as a
          product of min(N,M) elementary reflectors (see Further
          Details).

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

TAUA

          TAUA is DOUBLE PRECISION array, dimension (min(N,M))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Q (see Further Details).

B

          B is DOUBLE PRECISION array, dimension (LDB,P)
          On entry, the N-by-P matrix B.
          On exit, if N <= P, the upper triangle of the subarray
          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
          if N > P, the elements on and above the (N-P)-th subdiagonal
          contain the N-by-P upper trapezoidal matrix T; the remaining
          elements, with the array TAUB, represent the orthogonal
          matrix Z as a product of elementary reflectors (see Further
          Details).

LDB

          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

TAUB

          TAUB is DOUBLE PRECISION array, dimension (min(N,P))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Z (see Further Details).

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,N,M,P).
          For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
          where NB1 is the optimal blocksize for the QR factorization
          of an N-by-M matrix, NB2 is the optimal blocksize for the
          RQ factorization of an N-by-P matrix, and NB3 is the optimal
          blocksize for a call of DORMQR.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

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  The matrix Q is represented as a product of elementary reflectors
     Q = H(1) H(2) . . . H(k), where k = min(n,m).
  Each H(i) has the form
     H(i) = I - taua * v * v**T
  where taua is a real scalar, and v is a real vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
  and taua in TAUA(i).
  To form Q explicitly, use LAPACK subroutine DORGQR.
  To use Q to update another matrix, use LAPACK subroutine DORMQR.
  The matrix Z is represented as a product of elementary reflectors
     Z = H(1) H(2) . . . H(k), where k = min(n,p).
  Each H(i) has the form
     H(i) = I - taub * v * v**T
  where taub is a real scalar, and v is a real vector with
  v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
  B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
  To form Z explicitly, use LAPACK subroutine DORGRQ.
  To use Z to update another matrix, use LAPACK subroutine DORMRQ.

 DGGRQF computes a generalized RQ factorization of an M-by-N matrix A
 and a P-by-N matrix B:
             A = R*Q,        B = Z*T*Q,
 where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
 matrix, and R and T assume one of the forms:
 if M <= N,  R = ( 0  R12 ) M,   or if M > N,  R = ( R11 ) M-N,
                  N-M  M                           ( R21 ) N
                                                      N
 where R12 or R21 is upper triangular, and
 if P >= N,  T = ( T11 ) N  ,   or if P < N,  T = ( T11  T12 ) P,
                 (  0  ) P-N                         P   N-P
                    N
 where T11 is upper triangular.
 In particular, if B is square and nonsingular, the GRQ factorization
 of A and B implicitly gives the RQ factorization of A*inv(B):
              A*inv(B) = (R*inv(T))*Z**T
 where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
 transpose of the matrix Z.

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

P

          P is INTEGER
          The number of rows of the matrix B.  P >= 0.

N

          N is INTEGER
          The number of columns of the matrices A and B. N >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, if M <= N, the upper triangle of the subarray
          A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
          if M > N, the elements on and above the (M-N)-th subdiagonal
          contain the M-by-N upper trapezoidal matrix R; the remaining
          elements, with the array TAUA, represent the orthogonal
          matrix Q as a product of elementary reflectors (see Further
          Details).

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

TAUA

          TAUA is DOUBLE PRECISION array, dimension (min(M,N))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Q (see Further Details).

B

          B is DOUBLE PRECISION array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, the elements on and above the diagonal of the array
          contain the min(P,N)-by-N upper trapezoidal matrix T (T is
          upper triangular if P >= N); the elements below the diagonal,
          with the array TAUB, represent the orthogonal matrix Z as a
          product of elementary reflectors (see Further Details).

LDB

          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

TAUB

          TAUB is DOUBLE PRECISION array, dimension (min(P,N))
          The scalar factors of the elementary reflectors which
          represent the orthogonal matrix Z (see Further Details).

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,N,M,P).
          For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
          where NB1 is the optimal blocksize for the RQ factorization
          of an M-by-N matrix, NB2 is the optimal blocksize for the
          QR factorization of a P-by-N matrix, and NB3 is the optimal
          blocksize for a call of DORMRQ.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INF0= -i, the i-th argument had an illegal value.

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  The matrix Q is represented as a product of elementary reflectors
     Q = H(1) H(2) . . . H(k), where k = min(m,n).
  Each H(i) has the form
     H(i) = I - taua * v * v**T
  where taua is a real scalar, and v is a real vector with
  v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
  A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
  To form Q explicitly, use LAPACK subroutine DORGRQ.
  To use Q to update another matrix, use LAPACK subroutine DORMRQ.
  The matrix Z is represented as a product of elementary reflectors
     Z = H(1) H(2) . . . H(k), where k = min(p,n).
  Each H(i) has the form
     H(i) = I - taub * v * v**T
  where taub is a real scalar, and v is a real vector with
  v(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
  and taub in TAUB(i).
  To form Z explicitly, use LAPACK subroutine DORGQR.
  To use Z to update another matrix, use LAPACK subroutine DORMQR.

 DGGSVP3 computes orthogonal matrices U, V and Q such that
                    N-K-L  K    L
  U**T*A*Q =     K ( 0    A12  A13 )  if M-K-L >= 0;
                 L ( 0     0   A23 )
             M-K-L ( 0     0    0  )
                  N-K-L  K    L
         =     K ( 0    A12  A13 )  if M-K-L < 0;
             M-K ( 0     0   A23 )
                  N-K-L  K    L
  V**T*B*Q =   L ( 0     0   B13 )
             P-L ( 0     0    0  )
 where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
 upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
 otherwise A23 is (M-K)-by-L upper trapezoidal.  K+L = the effective
 numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T.
 This decomposition is the preprocessing step for computing the
 Generalized Singular Value Decomposition (GSVD), see subroutine
 DGGSVD3.

JOBU

          JOBU is CHARACTER*1
          = 'U':  Orthogonal matrix U is computed;
          = 'N':  U is not computed.

JOBV

          JOBV is CHARACTER*1
          = 'V':  Orthogonal matrix V is computed;
          = 'N':  V is not computed.

JOBQ

          JOBQ is CHARACTER*1
          = 'Q':  Orthogonal matrix Q is computed;
          = 'N':  Q is not computed.

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

P

          P is INTEGER
          The number of rows of the matrix B.  P >= 0.

N

          N is INTEGER
          The number of columns of the matrices A and B.  N >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A contains the triangular (or trapezoidal) matrix
          described in the Purpose section.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

B

          B is DOUBLE PRECISION array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, B contains the triangular matrix described in
          the Purpose section.

LDB

          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

TOLA

          TOLA is DOUBLE PRECISION

TOLB

          TOLB is DOUBLE PRECISION
          TOLA and TOLB are the thresholds to determine the effective
          numerical rank of matrix B and a subblock of A. Generally,
          they are set to
             TOLA = MAX(M,N)*norm(A)*MACHEPS,
             TOLB = MAX(P,N)*norm(B)*MACHEPS.
          The size of TOLA and TOLB may affect the size of backward
          errors of the decomposition.

K

          K is INTEGER

L

          L is INTEGER
          On exit, K and L specify the dimension of the subblocks
          described in Purpose section.
          K + L = effective numerical rank of (A**T,B**T)**T.

U

          U is DOUBLE PRECISION array, dimension (LDU,M)
          If JOBU = 'U', U contains the orthogonal matrix U.
          If JOBU = 'N', U is not referenced.

LDU

          LDU is INTEGER
          The leading dimension of the array U. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise.

V

          V is DOUBLE PRECISION array, dimension (LDV,P)
          If JOBV = 'V', V contains the orthogonal matrix V.
          If JOBV = 'N', V is not referenced.

LDV

          LDV is INTEGER
          The leading dimension of the array V. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise.

Q

          Q is DOUBLE PRECISION array, dimension (LDQ,N)
          If JOBQ = 'Q', Q contains the orthogonal matrix Q.
          If JOBQ = 'N', Q is not referenced.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise.

IWORK

          IWORK is INTEGER array, dimension (N)

TAU

          TAU is DOUBLE PRECISION array, dimension (N)

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

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  The subroutine uses LAPACK subroutine DGEQP3 for the QR factorization
  with column pivoting to detect the effective numerical rank of the
  a matrix. It may be replaced by a better rank determination strategy.
  DGGSVP3 replaces the deprecated subroutine DGGSVP.

 DGSVJ0 is called from DGESVJ as a pre-processor and that is its main
 purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
 it does not check convergence (stopping criterion). Few tuning
 parameters (marked by [TP]) are available for the implementer.

JOBV

          JOBV is CHARACTER*1
          Specifies whether the output from this procedure is used
          to compute the matrix V:
          = 'V': the product of the Jacobi rotations is accumulated
                 by postmulyiplying the N-by-N array V.
                (See the description of V.)
          = 'A': the product of the Jacobi rotations is accumulated
                 by postmulyiplying the MV-by-N array V.
                (See the descriptions of MV and V.)
          = 'N': the Jacobi rotations are not accumulated.

M

          M is INTEGER
          The number of rows of the input matrix A.  M >= 0.

N

          N is INTEGER
          The number of columns of the input matrix A.
          M >= N >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, M-by-N matrix A, such that A*diag(D) represents
          the input matrix.
          On exit,
          A_onexit * D_onexit represents the input matrix A*diag(D)
          post-multiplied by a sequence of Jacobi rotations, where the
          rotation threshold and the total number of sweeps are given in
          TOL and NSWEEP, respectively.
          (See the descriptions of D, TOL and NSWEEP.)

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

D

          D is DOUBLE PRECISION array, dimension (N)
          The array D accumulates the scaling factors from the fast scaled
          Jacobi rotations.
          On entry, A*diag(D) represents the input matrix.
          On exit, A_onexit*diag(D_onexit) represents the input matrix
          post-multiplied by a sequence of Jacobi rotations, where the
          rotation threshold and the total number of sweeps are given in
          TOL and NSWEEP, respectively.
          (See the descriptions of A, TOL and NSWEEP.)

SVA

          SVA is DOUBLE PRECISION array, dimension (N)
          On entry, SVA contains the Euclidean norms of the columns of
          the matrix A*diag(D).
          On exit, SVA contains the Euclidean norms of the columns of
          the matrix onexit*diag(D_onexit).

MV

          MV is INTEGER
          If JOBV = 'A', then MV rows of V are post-multipled by a
                           sequence of Jacobi rotations.
          If JOBV = 'N',   then MV is not referenced.

V

          V is DOUBLE PRECISION array, dimension (LDV,N)
          If JOBV = 'V' then N rows of V are post-multipled by a
                           sequence of Jacobi rotations.
          If JOBV = 'A' then MV rows of V are post-multipled by a
                           sequence of Jacobi rotations.
          If JOBV = 'N',   then V is not referenced.

LDV

          LDV is INTEGER
          The leading dimension of the array V,  LDV >= 1.
          If JOBV = 'V', LDV >= N.
          If JOBV = 'A', LDV >= MV.

EPS

          EPS is DOUBLE PRECISION
          EPS = DLAMCH('Epsilon')

SFMIN

          SFMIN is DOUBLE PRECISION
          SFMIN = DLAMCH('Safe Minimum')

TOL

          TOL is DOUBLE PRECISION
          TOL is the threshold for Jacobi rotations. For a pair
          A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
          applied only if DABS(COS(angle(A(:,p),A(:,q)))) > TOL.

NSWEEP

          NSWEEP is INTEGER
          NSWEEP is the number of sweeps of Jacobi rotations to be
          performed.

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

          LWORK is INTEGER
          LWORK is the dimension of WORK. LWORK >= M.

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, then the i-th argument had an illegal value

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DGSVJ0 is used just to enable DGESVJ to call a simplified version of itself to work on a submatrix of the original matrix.

Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)

Please report all bugs and send interesting test examples and comments to drmac@math.hr. Thank you.

 DGSVJ1 is called from DGESVJ as a pre-processor and that is its main
 purpose. It applies Jacobi rotations in the same way as DGESVJ does, but
 it targets only particular pivots and it does not check convergence
 (stopping criterion). Few tuning parameters (marked by [TP]) are
 available for the implementer.
 Further Details
 ~~~~~~~~~~~~~~~
 DGSVJ1 applies few sweeps of Jacobi rotations in the column space of
 the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
 off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
 block-entries (tiles) of the (1,2) off-diagonal block are marked by the
 [x]'s in the following scheme:
    | *  *  * [x] [x] [x]|
    | *  *  * [x] [x] [x]|    Row-cycling in the nblr-by-nblc [x] blocks.
    | *  *  * [x] [x] [x]|    Row-cyclic pivoting inside each [x] block.
    |[x] [x] [x] *  *  * |
    |[x] [x] [x] *  *  * |
    |[x] [x] [x] *  *  * |
 In terms of the columns of A, the first N1 columns are rotated 'against'
 the remaining N-N1 columns, trying to increase the angle between the
 corresponding subspaces. The off-diagonal block is N1-by(N-N1) and it is
 tiled using quadratic tiles of side KBL. Here, KBL is a tuning parameter.
 The number of sweeps is given in NSWEEP and the orthogonality threshold
 is given in TOL.

JOBV

          JOBV is CHARACTER*1
          Specifies whether the output from this procedure is used
          to compute the matrix V:
          = 'V': the product of the Jacobi rotations is accumulated
                 by postmulyiplying the N-by-N array V.
                (See the description of V.)
          = 'A': the product of the Jacobi rotations is accumulated
                 by postmulyiplying the MV-by-N array V.
                (See the descriptions of MV and V.)
          = 'N': the Jacobi rotations are not accumulated.

M

          M is INTEGER
          The number of rows of the input matrix A.  M >= 0.

N

          N is INTEGER
          The number of columns of the input matrix A.
          M >= N >= 0.

N1

          N1 is INTEGER
          N1 specifies the 2 x 2 block partition, the first N1 columns are
          rotated 'against' the remaining N-N1 columns of A.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, M-by-N matrix A, such that A*diag(D) represents
          the input matrix.
          On exit,
          A_onexit * D_onexit represents the input matrix A*diag(D)
          post-multiplied by a sequence of Jacobi rotations, where the
          rotation threshold and the total number of sweeps are given in
          TOL and NSWEEP, respectively.
          (See the descriptions of N1, D, TOL and NSWEEP.)

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

D

          D is DOUBLE PRECISION array, dimension (N)
          The array D accumulates the scaling factors from the fast scaled
          Jacobi rotations.
          On entry, A*diag(D) represents the input matrix.
          On exit, A_onexit*diag(D_onexit) represents the input matrix
          post-multiplied by a sequence of Jacobi rotations, where the
          rotation threshold and the total number of sweeps are given in
          TOL and NSWEEP, respectively.
          (See the descriptions of N1, A, TOL and NSWEEP.)

SVA

          SVA is DOUBLE PRECISION array, dimension (N)
          On entry, SVA contains the Euclidean norms of the columns of
          the matrix A*diag(D).
          On exit, SVA contains the Euclidean norms of the columns of
          the matrix onexit*diag(D_onexit).

MV

          MV is INTEGER
          If JOBV = 'A', then MV rows of V are post-multipled by a
                         sequence of Jacobi rotations.
          If JOBV = 'N', then MV is not referenced.

V

          V is DOUBLE PRECISION array, dimension (LDV,N)
          If JOBV = 'V', then N rows of V are post-multipled by a
                         sequence of Jacobi rotations.
          If JOBV = 'A', then MV rows of V are post-multipled by a
                         sequence of Jacobi rotations.
          If JOBV = 'N', then V is not referenced.

LDV

          LDV is INTEGER
          The leading dimension of the array V,  LDV >= 1.
          If JOBV = 'V', LDV >= N.
          If JOBV = 'A', LDV >= MV.

EPS

          EPS is DOUBLE PRECISION
          EPS = DLAMCH('Epsilon')

SFMIN

          SFMIN is DOUBLE PRECISION
          SFMIN = DLAMCH('Safe Minimum')

TOL

          TOL is DOUBLE PRECISION
          TOL is the threshold for Jacobi rotations. For a pair
          A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
          applied only if DABS(COS(angle(A(:,p),A(:,q)))) > TOL.

NSWEEP

          NSWEEP is INTEGER
          NSWEEP is the number of sweeps of Jacobi rotations to be
          performed.

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

          LWORK is INTEGER
          LWORK is the dimension of WORK. LWORK >= M.

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, then the i-th argument had an illegal value

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 DHSEIN uses inverse iteration to find specified right and/or left
 eigenvectors of a real upper Hessenberg matrix H.
 The right eigenvector x and the left eigenvector y of the matrix H
 corresponding to an eigenvalue w are defined by:
              H * x = w * x,     y**h * H = w * y**h
 where y**h denotes the conjugate transpose of the vector y.

SIDE

          SIDE is CHARACTER*1
          = 'R': compute right eigenvectors only;
          = 'L': compute left eigenvectors only;
          = 'B': compute both right and left eigenvectors.

EIGSRC

          EIGSRC is CHARACTER*1
          Specifies the source of eigenvalues supplied in (WR,WI):
          = 'Q': the eigenvalues were found using DHSEQR; thus, if
                 H has zero subdiagonal elements, and so is
                 block-triangular, then the j-th eigenvalue can be
                 assumed to be an eigenvalue of the block containing
                 the j-th row/column.  This property allows DHSEIN to
                 perform inverse iteration on just one diagonal block.
          = 'N': no assumptions are made on the correspondence
                 between eigenvalues and diagonal blocks.  In this
                 case, DHSEIN must always perform inverse iteration
                 using the whole matrix H.

INITV

          INITV is CHARACTER*1
          = 'N': no initial vectors are supplied;
          = 'U': user-supplied initial vectors are stored in the arrays
                 VL and/or VR.

SELECT

          SELECT is LOGICAL array, dimension (N)
          Specifies the eigenvectors to be computed. To select the
          real eigenvector corresponding to a real eigenvalue WR(j),
          SELECT(j) must be set to .TRUE.. To select the complex
          eigenvector corresponding to a complex eigenvalue
          (WR(j),WI(j)), with complex conjugate (WR(j+1),WI(j+1)),
          either SELECT(j) or SELECT(j+1) or both must be set to
          .TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is
          .FALSE..

N

          N is INTEGER
          The order of the matrix H.  N >= 0.

H

          H is DOUBLE PRECISION array, dimension (LDH,N)
          The upper Hessenberg matrix H.
          If a NaN is detected in H, the routine will return with INFO=-6.

LDH

          LDH is INTEGER
          The leading dimension of the array H.  LDH >= max(1,N).

WR

          WR is DOUBLE PRECISION array, dimension (N)

WI

          WI is DOUBLE PRECISION array, dimension (N)
          On entry, the real and imaginary parts of the eigenvalues of
          H; a complex conjugate pair of eigenvalues must be stored in
          consecutive elements of WR and WI.
          On exit, WR may have been altered since close eigenvalues
          are perturbed slightly in searching for independent
          eigenvectors.

VL

          VL is DOUBLE PRECISION array, dimension (LDVL,MM)
          On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must
          contain starting vectors for the inverse iteration for the
          left eigenvectors; the starting vector for each eigenvector
          must be in the same column(s) in which the eigenvector will
          be stored.
          On exit, if SIDE = 'L' or 'B', the left eigenvectors
          specified by SELECT will be stored consecutively in the
          columns of VL, in the same order as their eigenvalues. A
          complex eigenvector corresponding to a complex eigenvalue is
          stored in two consecutive columns, the first holding the real
          part and the second the imaginary part.
          If SIDE = 'R', VL is not referenced.

LDVL

          LDVL is INTEGER
          The leading dimension of the array VL.
          LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.

VR

          VR is DOUBLE PRECISION array, dimension (LDVR,MM)
          On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must
          contain starting vectors for the inverse iteration for the
          right eigenvectors; the starting vector for each eigenvector
          must be in the same column(s) in which the eigenvector will
          be stored.
          On exit, if SIDE = 'R' or 'B', the right eigenvectors
          specified by SELECT will be stored consecutively in the
          columns of VR, in the same order as their eigenvalues. A
          complex eigenvector corresponding to a complex eigenvalue is
          stored in two consecutive columns, the first holding the real
          part and the second the imaginary part.
          If SIDE = 'L', VR is not referenced.

LDVR

          LDVR is INTEGER
          The leading dimension of the array VR.
          LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.

MM

          MM is INTEGER
          The number of columns in the arrays VL and/or VR. MM >= M.

M

          M is INTEGER
          The number of columns in the arrays VL and/or VR required to
          store the eigenvectors; each selected real eigenvector
          occupies one column and each selected complex eigenvector
          occupies two columns.

WORK

          WORK is DOUBLE PRECISION array, dimension ((N+2)*N)

IFAILL

          IFAILL is INTEGER array, dimension (MM)
          If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left
          eigenvector in the i-th column of VL (corresponding to the
          eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
          eigenvector converged satisfactorily. If the i-th and (i+1)th
          columns of VL hold a complex eigenvector, then IFAILL(i) and
          IFAILL(i+1) are set to the same value.
          If SIDE = 'R', IFAILL is not referenced.

IFAILR

          IFAILR is INTEGER array, dimension (MM)
          If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right
          eigenvector in the i-th column of VR (corresponding to the
          eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
          eigenvector converged satisfactorily. If the i-th and (i+1)th
          columns of VR hold a complex eigenvector, then IFAILR(i) and
          IFAILR(i+1) are set to the same value.
          If SIDE = 'L', IFAILR is not referenced.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, i is the number of eigenvectors which
                failed to converge; see IFAILL and IFAILR for further
                details.

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  Each eigenvector is normalized so that the element of largest
  magnitude has magnitude 1; here the magnitude of a complex number
  (x,y) is taken to be |x|+|y|.

    DHSEQR computes the eigenvalues of a Hessenberg matrix H
    and, optionally, the matrices T and Z from the Schur decomposition
    H = Z T Z**T, where T is an upper quasi-triangular matrix (the
    Schur form), and Z is the orthogonal matrix of Schur vectors.
    Optionally Z may be postmultiplied into an input orthogonal
    matrix Q so that this routine can give the Schur factorization
    of a matrix A which has been reduced to the Hessenberg form H
    by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

JOB

          JOB is CHARACTER*1
           = 'E':  compute eigenvalues only;
           = 'S':  compute eigenvalues and the Schur form T.

COMPZ

          COMPZ is CHARACTER*1
           = 'N':  no Schur vectors are computed;
           = 'I':  Z is initialized to the unit matrix and the matrix Z
                   of Schur vectors of H is returned;
           = 'V':  Z must contain an orthogonal matrix Q on entry, and
                   the product Q*Z is returned.

N

          N is INTEGER
           The order of the matrix H.  N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
           It is assumed that H is already upper triangular in rows
           and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
           set by a previous call to DGEBAL, and then passed to ZGEHRD
           when the matrix output by DGEBAL is reduced to Hessenberg
           form. Otherwise ILO and IHI should be set to 1 and N
           respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
           If N = 0, then ILO = 1 and IHI = 0.

H

          H is DOUBLE PRECISION array, dimension (LDH,N)
           On entry, the upper Hessenberg matrix H.
           On exit, if INFO = 0 and JOB = 'S', then H contains the
           upper quasi-triangular matrix T from the Schur decomposition
           (the Schur form); 2-by-2 diagonal blocks (corresponding to
           complex conjugate pairs of eigenvalues) are returned in
           standard form, with H(i,i) = H(i+1,i+1) and
           H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and JOB = 'E', the
           contents of H are unspecified on exit.  (The output value of
           H when INFO > 0 is given under the description of INFO
           below.)
           Unlike earlier versions of DHSEQR, this subroutine may
           explicitly H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1
           or j = IHI+1, IHI+2, ... N.

LDH

          LDH is INTEGER
           The leading dimension of the array H. LDH >= max(1,N).

WR

          WR is DOUBLE PRECISION array, dimension (N)

WI

          WI is DOUBLE PRECISION array, dimension (N)
           The real and imaginary parts, respectively, of the computed
           eigenvalues. If two eigenvalues are computed as a complex
           conjugate pair, they are stored in consecutive elements of
           WR and WI, say the i-th and (i+1)th, with WI(i) > 0 and
           WI(i+1) < 0. If JOB = 'S', the eigenvalues are stored in
           the same order as on the diagonal of the Schur form returned
           in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2
           diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
           WI(i+1) = -WI(i).

Z

          Z is DOUBLE PRECISION array, dimension (LDZ,N)
           If COMPZ = 'N', Z is not referenced.
           If COMPZ = 'I', on entry Z need not be set and on exit,
           if INFO = 0, Z contains the orthogonal matrix Z of the Schur
           vectors of H.  If COMPZ = 'V', on entry Z must contain an
           N-by-N matrix Q, which is assumed to be equal to the unit
           matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
           if INFO = 0, Z contains Q*Z.
           Normally Q is the orthogonal matrix generated by DORGHR
           after the call to DGEHRD which formed the Hessenberg matrix
           H. (The output value of Z when INFO > 0 is given under
           the description of INFO below.)

LDZ

          LDZ is INTEGER
           The leading dimension of the array Z.  if COMPZ = 'I' or
           COMPZ = 'V', then LDZ >= MAX(1,N).  Otherwise, LDZ >= 1.

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)
           On exit, if INFO = 0, WORK(1) returns an estimate of
           the optimal value for LWORK.

LWORK

          LWORK is INTEGER
           The dimension of the array WORK.  LWORK >= max(1,N)
           is sufficient and delivers very good and sometimes
           optimal performance.  However, LWORK as large as 11*N
           may be required for optimal performance.  A workspace
           query is recommended to determine the optimal workspace
           size.
           If LWORK = -1, then DHSEQR does a workspace query.
           In this case, DHSEQR checks the input parameters and
           estimates the optimal workspace size for the given
           values of N, ILO and IHI.  The estimate is returned
           in WORK(1).  No error message related to LWORK is
           issued by XERBLA.  Neither H nor Z are accessed.

INFO

          INFO is INTEGER
             = 0:  successful exit
             < 0:  if INFO = -i, the i-th argument had an illegal
                    value
             > 0:  if INFO = i, DHSEQR failed to compute all of
                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
                and WI contain those eigenvalues which have been
                successfully computed.  (Failures are rare.)
                If INFO > 0 and JOB = 'E', then on exit, the
                remaining unconverged eigenvalues are the eigen-
                values of the upper Hessenberg matrix rows and
                columns ILO through INFO of the final, output
                value of H.
                If INFO > 0 and JOB   = 'S', then on exit
           (*)  (initial value of H)*U  = U*(final value of H)
                where U is an orthogonal matrix.  The final
                value of H is upper Hessenberg and quasi-triangular
                in rows and columns INFO+1 through IHI.
                If INFO > 0 and COMPZ = 'V', then on exit
                  (final value of Z)  =  (initial value of Z)*U
                where U is the orthogonal matrix in (*) (regard-
                less of the value of JOB.)
                If INFO > 0 and COMPZ = 'I', then on exit
                      (final value of Z)  = U
                where U is the orthogonal matrix in (*) (regard-
                less of the value of JOB.)
                If INFO > 0 and COMPZ = 'N', then Z is not
                accessed.

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             Default values supplied by
             ILAENV(ISPEC,'DHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
             It is suggested that these defaults be adjusted in order
             to attain best performance in each particular
             computational environment.
            ISPEC=12: The DLAHQR vs DLAQR0 crossover point.
                      Default: 75. (Must be at least 11.)
            ISPEC=13: Recommended deflation window size.
                      This depends on ILO, IHI and NS.  NS is the
                      number of simultaneous shifts returned
                      by ILAENV(ISPEC=15).  (See ISPEC=15 below.)
                      The default for (IHI-ILO+1) <= 500 is NS.
                      The default for (IHI-ILO+1) >  500 is 3*NS/2.
            ISPEC=14: Nibble crossover point. (See IPARMQ for
                      details.)  Default: 14% of deflation window
                      size.
            ISPEC=15: Number of simultaneous shifts in a multishift
                      QR iteration.
                      If IHI-ILO+1 is ...
                      greater than      ...but less    ... the
                      or equal to ...      than        default is
                           1               30          NS =   2(+)
                          30               60          NS =   4(+)
                          60              150          NS =  10(+)
                         150              590          NS =  **
                         590             3000          NS =  64
                        3000             6000          NS = 128
                        6000             infinity      NS = 256
                  (+)  By default some or all matrices of this order
                       are passed to the implicit double shift routine
                       DLAHQR and this parameter is ignored.  See
                       ISPEC=12 above and comments in IPARMQ for
                       details.
                 (**)  The asterisks (**) indicate an ad-hoc
                       function of N increasing from 10 to 64.
            ISPEC=16: Select structured matrix multiply.
                      If the number of simultaneous shifts (specified
                      by ISPEC=15) is less than 14, then the default
                      for ISPEC=16 is 0.  Otherwise the default for
                      ISPEC=16 is 2.

  K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
  Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
  Performance, SIAM Journal of Matrix Analysis, volume 23, pages
  929--947, 2002.

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

    DLA_LIN_BERR computes component-wise relative backward error from
    the formula
        max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
    where abs(Z) is the component-wise absolute value of the matrix
    or vector Z.

N

          N is INTEGER
     The number of linear equations, i.e., the order of the
     matrix A.  N >= 0.

NZ

          NZ is INTEGER
     We add (NZ+1)*SLAMCH( 'Safe minimum' ) to R(i) in the numerator to
     guard against spuriously zero residuals. Default value is N.

NRHS

          NRHS is INTEGER
     The number of right hand sides, i.e., the number of columns
     of the matrices AYB, RES, and BERR.  NRHS >= 0.

RES

          RES is DOUBLE PRECISION array, dimension (N,NRHS)
     The residual matrix, i.e., the matrix R in the relative backward
     error formula above.

AYB

          AYB is DOUBLE PRECISION array, dimension (N, NRHS)
     The denominator in the relative backward error formula above, i.e.,
     the matrix abs(op(A_s))*abs(Y) + abs(B_s). The matrices A, Y, and B
     are from iterative refinement (see dla_gerfsx_extended.f).

BERR

          BERR is DOUBLE PRECISION array, dimension (NRHS)
     The component-wise relative backward error from the formula above.

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    DLA_WWADDW adds a vector W into a doubled-single vector (X, Y).
    This works for all extant IBM's hex and binary floating point
    arithmetic, but not for decimal.

N

          N is INTEGER
            The length of vectors X, Y, and W.

X

          X is DOUBLE PRECISION array, dimension (N)
            The first part of the doubled-single accumulation vector.

Y

          Y is DOUBLE PRECISION array, dimension (N)
            The second part of the doubled-single accumulation vector.

W

          W is DOUBLE PRECISION array, dimension (N)
            The vector to be added.

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 DLALS0 applies back the multiplying factors of either the left or the
 right singular vector matrix of a diagonal matrix appended by a row
 to the right hand side matrix B in solving the least squares problem
 using the divide-and-conquer SVD approach.
 For the left singular vector matrix, three types of orthogonal
 matrices are involved:
 (1L) Givens rotations: the number of such rotations is GIVPTR; the
      pairs of columns/rows they were applied to are stored in GIVCOL;
      and the C- and S-values of these rotations are stored in GIVNUM.
 (2L) Permutation. The (NL+1)-st row of B is to be moved to the first
      row, and for J=2:N, PERM(J)-th row of B is to be moved to the
      J-th row.
 (3L) The left singular vector matrix of the remaining matrix.
 For the right singular vector matrix, four types of orthogonal
 matrices are involved:
 (1R) The right singular vector matrix of the remaining matrix.
 (2R) If SQRE = 1, one extra Givens rotation to generate the right
      null space.
 (3R) The inverse transformation of (2L).
 (4R) The inverse transformation of (1L).

ICOMPQ

          ICOMPQ is INTEGER
         Specifies whether singular vectors are to be computed in
         factored form:
         = 0: Left singular vector matrix.
         = 1: Right singular vector matrix.

NL

          NL is INTEGER
         The row dimension of the upper block. NL >= 1.

NR

          NR is INTEGER
         The row dimension of the lower block. NR >= 1.

SQRE

          SQRE is INTEGER
         = 0: the lower block is an NR-by-NR square matrix.
         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
         The bidiagonal matrix has row dimension N = NL + NR + 1,
         and column dimension M = N + SQRE.

NRHS

          NRHS is INTEGER
         The number of columns of B and BX. NRHS must be at least 1.

B

          B is DOUBLE PRECISION array, dimension ( LDB, NRHS )
         On input, B contains the right hand sides of the least
         squares problem in rows 1 through M. On output, B contains
         the solution X in rows 1 through N.

LDB

          LDB is INTEGER
         The leading dimension of B. LDB must be at least
         max(1,MAX( M, N ) ).

BX

          BX is DOUBLE PRECISION array, dimension ( LDBX, NRHS )

LDBX

          LDBX is INTEGER
         The leading dimension of BX.

PERM

          PERM is INTEGER array, dimension ( N )
         The permutations (from deflation and sorting) applied
         to the two blocks.

GIVPTR

          GIVPTR is INTEGER
         The number of Givens rotations which took place in this
         subproblem.

GIVCOL

          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 )
         Each pair of numbers indicates a pair of rows/columns
         involved in a Givens rotation.

LDGCOL

          LDGCOL is INTEGER
         The leading dimension of GIVCOL, must be at least N.

GIVNUM

          GIVNUM is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
         Each number indicates the C or S value used in the
         corresponding Givens rotation.

LDGNUM

          LDGNUM is INTEGER
         The leading dimension of arrays DIFR, POLES and
         GIVNUM, must be at least K.

POLES

          POLES is DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
         On entry, POLES(1:K, 1) contains the new singular
         values obtained from solving the secular equation, and
         POLES(1:K, 2) is an array containing the poles in the secular
         equation.

DIFL

          DIFL is DOUBLE PRECISION array, dimension ( K ).
         On entry, DIFL(I) is the distance between I-th updated
         (undeflated) singular value and the I-th (undeflated) old
         singular value.

DIFR

          DIFR is DOUBLE PRECISION array, dimension ( LDGNUM, 2 ).
         On entry, DIFR(I, 1) contains the distances between I-th
         updated (undeflated) singular value and the I+1-th
         (undeflated) old singular value. And DIFR(I, 2) is the
         normalizing factor for the I-th right singular vector.

Z

          Z is DOUBLE PRECISION array, dimension ( K )
         Contain the components of the deflation-adjusted updating row
         vector.

K

          K is INTEGER
         Contains the dimension of the non-deflated matrix,
         This is the order of the related secular equation. 1 <= K <=N.

C

          C is DOUBLE PRECISION
         C contains garbage if SQRE =0 and the C-value of a Givens
         rotation related to the right null space if SQRE = 1.

S

          S is DOUBLE PRECISION
         S contains garbage if SQRE =0 and the S-value of a Givens
         rotation related to the right null space if SQRE = 1.

WORK

          WORK is DOUBLE PRECISION array, dimension ( K )

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

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 DLALSA is an itermediate step in solving the least squares problem
 by computing the SVD of the coefficient matrix in compact form (The
 singular vectors are computed as products of simple orthorgonal
 matrices.).
 If ICOMPQ = 0, DLALSA applies the inverse of the left singular vector
 matrix of an upper bidiagonal matrix to the right hand side; and if
 ICOMPQ = 1, DLALSA applies the right singular vector matrix to the
 right hand side. The singular vector matrices were generated in
 compact form by DLALSA.

ICOMPQ

          ICOMPQ is INTEGER
         Specifies whether the left or the right singular vector
         matrix is involved.
         = 0: Left singular vector matrix
         = 1: Right singular vector matrix

SMLSIZ

          SMLSIZ is INTEGER
         The maximum size of the subproblems at the bottom of the
         computation tree.

N

          N is INTEGER
         The row and column dimensions of the upper bidiagonal matrix.

NRHS

          NRHS is INTEGER
         The number of columns of B and BX. NRHS must be at least 1.

B

          B is DOUBLE PRECISION array, dimension ( LDB, NRHS )
         On input, B contains the right hand sides of the least
         squares problem in rows 1 through M.
         On output, B contains the solution X in rows 1 through N.

LDB

          LDB is INTEGER
         The leading dimension of B in the calling subprogram.
         LDB must be at least max(1,MAX( M, N ) ).

BX

          BX is DOUBLE PRECISION array, dimension ( LDBX, NRHS )
         On exit, the result of applying the left or right singular
         vector matrix to B.

LDBX

          LDBX is INTEGER
         The leading dimension of BX.

U

          U is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ ).
         On entry, U contains the left singular vector matrices of all
         subproblems at the bottom level.

LDU

          LDU is INTEGER, LDU = > N.
         The leading dimension of arrays U, VT, DIFL, DIFR,
         POLES, GIVNUM, and Z.

VT

          VT is DOUBLE PRECISION array, dimension ( LDU, SMLSIZ+1 ).
         On entry, VT**T contains the right singular vector matrices of
         all subproblems at the bottom level.

K

          K is INTEGER array, dimension ( N ).

DIFL

          DIFL is DOUBLE PRECISION array, dimension ( LDU, NLVL ).
         where NLVL = INT(log_2 (N/(SMLSIZ+1))) + 1.

DIFR

          DIFR is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
         On entry, DIFL(*, I) and DIFR(*, 2 * I -1) record
         distances between singular values on the I-th level and
         singular values on the (I -1)-th level, and DIFR(*, 2 * I)
         record the normalizing factors of the right singular vectors
         matrices of subproblems on I-th level.

Z

          Z is DOUBLE PRECISION array, dimension ( LDU, NLVL ).
         On entry, Z(1, I) contains the components of the deflation-
         adjusted updating row vector for subproblems on the I-th
         level.

POLES

          POLES is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
         On entry, POLES(*, 2 * I -1: 2 * I) contains the new and old
         singular values involved in the secular equations on the I-th
         level.

GIVPTR

          GIVPTR is INTEGER array, dimension ( N ).
         On entry, GIVPTR( I ) records the number of Givens
         rotations performed on the I-th problem on the computation
         tree.

GIVCOL

          GIVCOL is INTEGER array, dimension ( LDGCOL, 2 * NLVL ).
         On entry, for each I, GIVCOL(*, 2 * I - 1: 2 * I) records the
         locations of Givens rotations performed on the I-th level on
         the computation tree.

LDGCOL

          LDGCOL is INTEGER, LDGCOL = > N.
         The leading dimension of arrays GIVCOL and PERM.

PERM

          PERM is INTEGER array, dimension ( LDGCOL, NLVL ).
         On entry, PERM(*, I) records permutations done on the I-th
         level of the computation tree.

GIVNUM

          GIVNUM is DOUBLE PRECISION array, dimension ( LDU, 2 * NLVL ).
         On entry, GIVNUM(*, 2 *I -1 : 2 * I) records the C- and S-
         values of Givens rotations performed on the I-th level on the
         computation tree.

C

          C is DOUBLE PRECISION array, dimension ( N ).
         On entry, if the I-th subproblem is not square,
         C( I ) contains the C-value of a Givens rotation related to
         the right null space of the I-th subproblem.

S

          S is DOUBLE PRECISION array, dimension ( N ).
         On entry, if the I-th subproblem is not square,
         S( I ) contains the S-value of a Givens rotation related to
         the right null space of the I-th subproblem.

WORK

          WORK is DOUBLE PRECISION array, dimension (N)

IWORK

          IWORK is INTEGER array, dimension (3*N)

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

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Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA Osni Marques, LBNL/NERSC, USA

 DLALSD uses the singular value decomposition of A to solve the least
 squares problem of finding X to minimize the Euclidean norm of each
 column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
 are N-by-NRHS. The solution X overwrites B.
 The singular values of A smaller than RCOND times the largest
 singular value are treated as zero in solving the least squares
 problem; in this case a minimum norm solution is returned.
 The actual singular values are returned in D in ascending order.
 This code makes very mild assumptions about floating point
 arithmetic. It will work on machines with a guard digit in
 add/subtract, or on those binary machines without guard digits
 which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
 It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.

UPLO

          UPLO is CHARACTER*1
         = 'U': D and E define an upper bidiagonal matrix.
         = 'L': D and E define a  lower bidiagonal matrix.

SMLSIZ

          SMLSIZ is INTEGER
         The maximum size of the subproblems at the bottom of the
         computation tree.

N

          N is INTEGER
         The dimension of the  bidiagonal matrix.  N >= 0.

NRHS

          NRHS is INTEGER
         The number of columns of B. NRHS must be at least 1.

D

          D is DOUBLE PRECISION array, dimension (N)
         On entry D contains the main diagonal of the bidiagonal
         matrix. On exit, if INFO = 0, D contains its singular values.

E

          E is DOUBLE PRECISION array, dimension (N-1)
         Contains the super-diagonal entries of the bidiagonal matrix.
         On exit, E has been destroyed.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
         On input, B contains the right hand sides of the least
         squares problem. On output, B contains the solution X.

LDB

          LDB is INTEGER
         The leading dimension of B in the calling subprogram.
         LDB must be at least max(1,N).

RCOND

          RCOND is DOUBLE PRECISION
         The singular values of A less than or equal to RCOND times
         the largest singular value are treated as zero in solving
         the least squares problem. If RCOND is negative,
         machine precision is used instead.
         For example, if diag(S)*X=B were the least squares problem,
         where diag(S) is a diagonal matrix of singular values, the
         solution would be X(i) = B(i) / S(i) if S(i) is greater than
         RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
         RCOND*max(S).

RANK

          RANK is INTEGER
         The number of singular values of A greater than RCOND times
         the largest singular value.

WORK

          WORK is DOUBLE PRECISION array, dimension at least
         (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
         where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).

IWORK

          IWORK is INTEGER array, dimension at least
         (3*N*NLVL + 11*N)

INFO

          INFO is INTEGER
         = 0:  successful exit.
         < 0:  if INFO = -i, the i-th argument had an illegal value.
         > 0:  The algorithm failed to compute a singular value while
               working on the submatrix lying in rows and columns
               INFO/(N+1) through MOD(INFO,N+1).

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Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA Osni Marques, LBNL/NERSC, USA

 DLANSF returns the value of the one norm, or the Frobenius norm, or
 the infinity norm, or the element of largest absolute value of a
 real symmetric matrix A in RFP format.

DLANSF

    DLANSF = ( max(abs(A(i,j))), NORM = 'M' or 'm'
             (
             ( norm1(A),         NORM = '1', 'O' or 'o'
             (
             ( normI(A),         NORM = 'I' or 'i'
             (
             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
 where  norm1  denotes the  one norm of a matrix (maximum column sum),
 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
 normF  denotes the  Frobenius norm of a matrix (square root of sum of
 squares).  Note that  max(abs(A(i,j)))  is not a  matrix norm.

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in DLANSF as described
          above.

TRANSR

          TRANSR is CHARACTER*1
          Specifies whether the RFP format of A is normal or
          transposed format.
          = 'N':  RFP format is Normal;
          = 'T':  RFP format is Transpose.

UPLO

          UPLO is CHARACTER*1
           On entry, UPLO specifies whether the RFP matrix A came from
           an upper or lower triangular matrix as follows:
           = 'U': RFP A came from an upper triangular matrix;
           = 'L': RFP A came from a lower triangular matrix.

N

          N is INTEGER
          The order of the matrix A. N >= 0. When N = 0, DLANSF is
          set to zero.

A

          A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 );
          On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
          part of the symmetric matrix A stored in RFP format. See the
          "Notes" below for more details.
          Unchanged on exit.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
          WORK is not referenced.

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  We first consider Rectangular Full Packed (RFP) Format when N is
  even. We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  the transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  the transpose of the last three columns of AP lower.
  This covers the case N even and TRANSR = 'N'.
         RFP A                   RFP A
        03 04 05                33 43 53
        13 14 15                00 44 54
        23 24 25                10 11 55
        33 34 35                20 21 22
        00 44 45                30 31 32
        01 11 55                40 41 42
        02 12 22                50 51 52
  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We then consider Rectangular Full Packed (RFP) Format when N is
  odd. We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  the transpose of the first two columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  the transpose of the last two columns of AP lower.
  This covers the case N odd and TRANSR = 'N'.
         RFP A                   RFP A
        02 03 04                00 33 43
        12 13 14                10 11 44
        22 23 24                20 21 22
        00 33 34                30 31 32
        01 11 44                40 41 42
  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     02 12 22 00 01             00 10 20 30 40 50
     03 13 23 33 11             33 11 21 31 41 51
     04 14 24 34 44             43 44 22 32 42 52

 DLARSCL2 performs a reciprocal diagonal scaling on an vector:
   x <-- inv(D) * x
 where the diagonal matrix D is stored as a vector.
 Eventually to be replaced by BLAS_dge_diag_scale in the new BLAS
 standard.

M

          M is INTEGER
     The number of rows of D and X. M >= 0.

N

          N is INTEGER
     The number of columns of X. N >= 0.

D

          D is DOUBLE PRECISION array, dimension (M)
     Diagonal matrix D, stored as a vector of length M.

X

          X is DOUBLE PRECISION array, dimension (LDX,N)
     On entry, the vector X to be scaled by D.
     On exit, the scaled vector.

LDX

          LDX is INTEGER
     The leading dimension of the vector X. LDX >= M.

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 DLARZ applies a real elementary reflector H to a real M-by-N
 matrix C, from either the left or the right. H is represented in the
 form
       H = I - tau * v * v**T
 where tau is a real scalar and v is a real vector.
 If tau = 0, then H is taken to be the unit matrix.
 H is a product of k elementary reflectors as returned by DTZRZF.

SIDE

          SIDE is CHARACTER*1
          = 'L': form  H * C
          = 'R': form  C * H

M

          M is INTEGER
          The number of rows of the matrix C.

N

          N is INTEGER
          The number of columns of the matrix C.

L

          L is INTEGER
          The number of entries of the vector V containing
          the meaningful part of the Householder vectors.
          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

V

          V is DOUBLE PRECISION array, dimension (1+(L-1)*abs(INCV))
          The vector v in the representation of H as returned by
          DTZRZF. V is not used if TAU = 0.

INCV

          INCV is INTEGER
          The increment between elements of v. INCV <> 0.

TAU

          TAU is DOUBLE PRECISION
          The value tau in the representation of H.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by the matrix H * C if SIDE = 'L',
          or C * H if SIDE = 'R'.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension
                         (N) if SIDE = 'L'
                      or (M) if SIDE = 'R'

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 DLARZB applies a real block reflector H or its transpose H**T to
 a real distributed M-by-N  C from the left or the right.
 Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

SIDE

          SIDE is CHARACTER*1
          = 'L': apply H or H**T from the Left
          = 'R': apply H or H**T from the Right

TRANS

          TRANS is CHARACTER*1
          = 'N': apply H (No transpose)
          = 'C': apply H**T (Transpose)

DIRECT

          DIRECT is CHARACTER*1
          Indicates how H is formed from a product of elementary
          reflectors
          = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
          = 'B': H = H(k) . . . H(2) H(1) (Backward)

STOREV

          STOREV is CHARACTER*1
          Indicates how the vectors which define the elementary
          reflectors are stored:
          = 'C': Columnwise                        (not supported yet)
          = 'R': Rowwise

M

          M is INTEGER
          The number of rows of the matrix C.

N

          N is INTEGER
          The number of columns of the matrix C.

K

          K is INTEGER
          The order of the matrix T (= the number of elementary
          reflectors whose product defines the block reflector).

L

          L is INTEGER
          The number of columns of the matrix V containing the
          meaningful part of the Householder reflectors.
          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

V

          V is DOUBLE PRECISION array, dimension (LDV,NV).
          If STOREV = 'C', NV = K; if STOREV = 'R', NV = L.

LDV

          LDV is INTEGER
          The leading dimension of the array V.
          If STOREV = 'C', LDV >= L; if STOREV = 'R', LDV >= K.

T

          T is DOUBLE PRECISION array, dimension (LDT,K)
          The triangular K-by-K matrix T in the representation of the
          block reflector.

LDT

          LDT is INTEGER
          The leading dimension of the array T. LDT >= K.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension (LDWORK,K)

LDWORK

          LDWORK is INTEGER
          The leading dimension of the array WORK.
          If SIDE = 'L', LDWORK >= max(1,N);
          if SIDE = 'R', LDWORK >= max(1,M).

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 DLARZT forms the triangular factor T of a real block reflector
 H of order > n, which is defined as a product of k elementary
 reflectors.
 If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
 If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
 If STOREV = 'C', the vector which defines the elementary reflector
 H(i) is stored in the i-th column of the array V, and
    H  =  I - V * T * V**T
 If STOREV = 'R', the vector which defines the elementary reflector
 H(i) is stored in the i-th row of the array V, and
    H  =  I - V**T * T * V
 Currently, only STOREV = 'R' and DIRECT = 'B' are supported.

DIRECT

          DIRECT is CHARACTER*1
          Specifies the order in which the elementary reflectors are
          multiplied to form the block reflector:
          = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
          = 'B': H = H(k) . . . H(2) H(1) (Backward)

STOREV

          STOREV is CHARACTER*1
          Specifies how the vectors which define the elementary
          reflectors are stored (see also Further Details):
          = 'C': columnwise                        (not supported yet)
          = 'R': rowwise

N

          N is INTEGER
          The order of the block reflector H. N >= 0.

K

          K is INTEGER
          The order of the triangular factor T (= the number of
          elementary reflectors). K >= 1.

V

          V is DOUBLE PRECISION array, dimension
                               (LDV,K) if STOREV = 'C'
                               (LDV,N) if STOREV = 'R'
          The matrix V. See further details.

LDV

          LDV is INTEGER
          The leading dimension of the array V.
          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i).

T

          T is DOUBLE PRECISION array, dimension (LDT,K)
          The k by k triangular factor T of the block reflector.
          If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
          lower triangular. The rest of the array is not used.

LDT

          LDT is INTEGER
          The leading dimension of the array T. LDT >= K.

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  The shape of the matrix V and the storage of the vectors which define
  the H(i) is best illustrated by the following example with n = 5 and
  k = 3. The elements equal to 1 are not stored; the corresponding
  array elements are modified but restored on exit. The rest of the
  array is not used.
  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
                                              ______V_____
         ( v1 v2 v3 )                        /                     ( v1 v2 v3 )                      ( v1 v1 v1 v1 v1 . . . . 1 )
     V = ( v1 v2 v3 )                      ( v2 v2 v2 v2 v2 . . . 1   )
         ( v1 v2 v3 )                      ( v3 v3 v3 v3 v3 . . 1     )
         ( v1 v2 v3 )
            .  .  .
            .  .  .
            1  .  .
               1  .
                  1
  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
                                                        ______V_____
            1                                          /                        .  1                           ( 1 . . . . v1 v1 v1 v1 v1 )
            .  .  1                        ( . 1 . . . v2 v2 v2 v2 v2 )
            .  .  .                        ( . . 1 . . v3 v3 v3 v3 v3 )
            .  .  .
         ( v1 v2 v3 )
         ( v1 v2 v3 )
     V = ( v1 v2 v3 )
         ( v1 v2 v3 )
         ( v1 v2 v3 )

 DLASCL2 performs a diagonal scaling on a vector:
   x <-- D * x
 where the diagonal matrix D is stored as a vector.
 Eventually to be replaced by BLAS_dge_diag_scale in the new BLAS
 standard.

M

          M is INTEGER
     The number of rows of D and X. M >= 0.

N

          N is INTEGER
     The number of columns of X. N >= 0.

D

          D is DOUBLE PRECISION array, length M
     Diagonal matrix D, stored as a vector of length M.

X

          X is DOUBLE PRECISION array, dimension (LDX,N)
     On entry, the vector X to be scaled by D.
     On exit, the scaled vector.

LDX

          LDX is INTEGER
     The leading dimension of the vector X. LDX >= M.

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 DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
 [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z, by means
 of orthogonal transformations.  Z is an (M+L)-by-(M+L) orthogonal
 matrix and, R and A1 are M-by-M upper triangular matrices.

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

N

          N is INTEGER
          The number of columns of the matrix A.  N >= 0.

L

          L is INTEGER
          The number of columns of the matrix A containing the
          meaningful part of the Householder vectors. N-M >= L >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading M-by-N upper trapezoidal part of the
          array A must contain the matrix to be factorized.
          On exit, the leading M-by-M upper triangular part of A
          contains the upper triangular matrix R, and elements N-L+1 to
          N of the first M rows of A, with the array TAU, represent the
          orthogonal matrix Z as a product of M elementary reflectors.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

TAU

          TAU is DOUBLE PRECISION array, dimension (M)
          The scalar factors of the elementary reflectors.

WORK

          WORK is DOUBLE PRECISION array, dimension (M)

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  The factorization is obtained by Householder's method.  The kth
  transformation matrix, Z( k ), which is used to introduce zeros into
  the ( m - k + 1 )th row of A, is given in the form
     Z( k ) = ( I     0   ),
              ( 0  T( k ) )
  where
     T( k ) = I - tau*u( k )*u( k )**T,   u( k ) = (   1    ),
                                                 (   0    )
                                                 ( z( k ) )
  tau is a scalar and z( k ) is an l element vector. tau and z( k )
  are chosen to annihilate the elements of the kth row of A2.
  The scalar tau is returned in the kth element of TAU and the vector
  u( k ) in the kth row of A2, such that the elements of z( k ) are
  in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
  the upper triangular part of A1.
  Z is given by
     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).

 DOPGTR generates a real orthogonal matrix Q which is defined as the
 product of n-1 elementary reflectors H(i) of order n, as returned by
 DSPTRD using packed storage:
 if UPLO = 'U', Q = H(n-1) . . . H(2) H(1),
 if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).

UPLO

          UPLO is CHARACTER*1
          = 'U': Upper triangular packed storage used in previous
                 call to DSPTRD;
          = 'L': Lower triangular packed storage used in previous
                 call to DSPTRD.

N

          N is INTEGER
          The order of the matrix Q. N >= 0.

AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The vectors which define the elementary reflectors, as
          returned by DSPTRD.

TAU

          TAU is DOUBLE PRECISION array, dimension (N-1)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DSPTRD.

Q

          Q is DOUBLE PRECISION array, dimension (LDQ,N)
          The N-by-N orthogonal matrix Q.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N).

WORK

          WORK is DOUBLE PRECISION array, dimension (N-1)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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 DOPMTR overwrites the general real M-by-N matrix C with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'T':      Q**T * C       C * Q**T
 where Q is a real orthogonal matrix of order nq, with nq = m if
 SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
 nq-1 elementary reflectors, as returned by DSPTRD using packed
 storage:
 if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);
 if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

UPLO

          UPLO is CHARACTER*1
          = 'U': Upper triangular packed storage used in previous
                 call to DSPTRD;
          = 'L': Lower triangular packed storage used in previous
                 call to DSPTRD.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

AP

          AP is DOUBLE PRECISION array, dimension
                               (M*(M+1)/2) if SIDE = 'L'
                               (N*(N+1)/2) if SIDE = 'R'
          The vectors which define the elementary reflectors, as
          returned by DSPTRD.  AP is modified by the routine but
          restored on exit.

TAU

          TAU is DOUBLE PRECISION array, dimension (M-1) if SIDE = 'L'
                                     or (N-1) if SIDE = 'R'
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DSPTRD.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension
                                   (N) if SIDE = 'L'
                                   (M) if SIDE = 'R'

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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 DORBDB simultaneously bidiagonalizes the blocks of an M-by-M
 partitioned orthogonal matrix X:
                                 [ B11 | B12 0  0 ]
     [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**T
 X = [-----------] = [---------] [----------------] [---------]   .
     [ X21 | X22 ]   [    | P2 ] [ B21 | B22 0  0 ] [    | Q2 ]
                                 [  0  |  0  0  I ]
 X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
 not the case, then X must be transposed and/or permuted. This can be
 done in constant time using the TRANS and SIGNS options. See DORCSD
 for details.)
 The orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
 (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
 represented implicitly by Householder vectors.
 B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
 implicitly by angles THETA, PHI.

TRANS

          TRANS is CHARACTER
          = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
                      order;
          otherwise:  X, U1, U2, V1T, and V2T are stored in column-
                      major order.

SIGNS

          SIGNS is CHARACTER
          = 'O':      The lower-left block is made nonpositive (the
                      "other" convention);
          otherwise:  The upper-right block is made nonpositive (the
                      "default" convention).

M

          M is INTEGER
          The number of rows and columns in X.

P

          P is INTEGER
          The number of rows in X11 and X12. 0 <= P <= M.

Q

          Q is INTEGER
          The number of columns in X11 and X21. 0 <= Q <=
          MIN(P,M-P,M-Q).

X11

          X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
          On entry, the top-left block of the orthogonal matrix to be
          reduced. On exit, the form depends on TRANS:
          If TRANS = 'N', then
             the columns of tril(X11) specify reflectors for P1,
             the rows of triu(X11,1) specify reflectors for Q1;
          else TRANS = 'T', and
             the rows of triu(X11) specify reflectors for P1,
             the columns of tril(X11,-1) specify reflectors for Q1.

LDX11

          LDX11 is INTEGER
          The leading dimension of X11. If TRANS = 'N', then LDX11 >=
          P; else LDX11 >= Q.

X12

          X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
          On entry, the top-right block of the orthogonal matrix to
          be reduced. On exit, the form depends on TRANS:
          If TRANS = 'N', then
             the rows of triu(X12) specify the first P reflectors for
             Q2;
          else TRANS = 'T', and
             the columns of tril(X12) specify the first P reflectors
             for Q2.

LDX12

          LDX12 is INTEGER
          The leading dimension of X12. If TRANS = 'N', then LDX12 >=
          P; else LDX11 >= M-Q.

X21

          X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
          On entry, the bottom-left block of the orthogonal matrix to
          be reduced. On exit, the form depends on TRANS:
          If TRANS = 'N', then
             the columns of tril(X21) specify reflectors for P2;
          else TRANS = 'T', and
             the rows of triu(X21) specify reflectors for P2.

LDX21

          LDX21 is INTEGER
          The leading dimension of X21. If TRANS = 'N', then LDX21 >=
          M-P; else LDX21 >= Q.

X22

          X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
          On entry, the bottom-right block of the orthogonal matrix to
          be reduced. On exit, the form depends on TRANS:
          If TRANS = 'N', then
             the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
             M-P-Q reflectors for Q2,
          else TRANS = 'T', and
             the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
             M-P-Q reflectors for P2.

LDX22

          LDX22 is INTEGER
          The leading dimension of X22. If TRANS = 'N', then LDX22 >=
          M-P; else LDX22 >= M-Q.

THETA

          THETA is DOUBLE PRECISION array, dimension (Q)
          The entries of the bidiagonal blocks B11, B12, B21, B22 can
          be computed from the angles THETA and PHI. See Further
          Details.

PHI

          PHI is DOUBLE PRECISION array, dimension (Q-1)
          The entries of the bidiagonal blocks B11, B12, B21, B22 can
          be computed from the angles THETA and PHI. See Further
          Details.

TAUP1

          TAUP1 is DOUBLE PRECISION array, dimension (P)
          The scalar factors of the elementary reflectors that define
          P1.

TAUP2

          TAUP2 is DOUBLE PRECISION array, dimension (M-P)
          The scalar factors of the elementary reflectors that define
          P2.

TAUQ1

          TAUQ1 is DOUBLE PRECISION array, dimension (Q)
          The scalar factors of the elementary reflectors that define
          Q1.

TAUQ2

          TAUQ2 is DOUBLE PRECISION array, dimension (M-Q)
          The scalar factors of the elementary reflectors that define
          Q2.

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= M-Q.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.

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  The bidiagonal blocks B11, B12, B21, and B22 are represented
  implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
  PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
  lower bidiagonal. Every entry in each bidiagonal band is a product
  of a sine or cosine of a THETA with a sine or cosine of a PHI. See
  [1] or DORCSD for details.
  P1, P2, Q1, and Q2 are represented as products of elementary
  reflectors. See DORCSD for details on generating P1, P2, Q1, and Q2
  using DORGQR and DORGLQ.

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

 DORBDB1 simultaneously bidiagonalizes the blocks of a tall and skinny
 matrix X with orthonomal columns:
                            [ B11 ]
      [ X11 ]   [ P1 |    ] [  0  ]
      [-----] = [---------] [-----] Q1**T .
      [ X21 ]   [    | P2 ] [ B21 ]
                            [  0  ]
 X11 is P-by-Q, and X21 is (M-P)-by-Q. Q must be no larger than P,
 M-P, or M-Q. Routines DORBDB2, DORBDB3, and DORBDB4 handle cases in
 which Q is not the minimum dimension.
 The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
 and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
 Householder vectors.
 B11 and B12 are Q-by-Q bidiagonal matrices represented implicitly by
 angles THETA, PHI.

M

          M is INTEGER
           The number of rows X11 plus the number of rows in X21.

P

          P is INTEGER
           The number of rows in X11. 0 <= P <= M.

Q

          Q is INTEGER
           The number of columns in X11 and X21. 0 <= Q <=
           MIN(P,M-P,M-Q).

X11

          X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
           On entry, the top block of the matrix X to be reduced. On
           exit, the columns of tril(X11) specify reflectors for P1 and
           the rows of triu(X11,1) specify reflectors for Q1.

LDX11

          LDX11 is INTEGER
           The leading dimension of X11. LDX11 >= P.

X21

          X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
           On entry, the bottom block of the matrix X to be reduced. On
           exit, the columns of tril(X21) specify reflectors for P2.

LDX21

          LDX21 is INTEGER
           The leading dimension of X21. LDX21 >= M-P.

THETA

          THETA is DOUBLE PRECISION array, dimension (Q)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.

PHI

          PHI is DOUBLE PRECISION array, dimension (Q-1)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.

TAUP1

          TAUP1 is DOUBLE PRECISION array, dimension (P)
           The scalar factors of the elementary reflectors that define
           P1.

TAUP2

          TAUP2 is DOUBLE PRECISION array, dimension (M-P)
           The scalar factors of the elementary reflectors that define
           P2.

TAUQ1

          TAUQ1 is DOUBLE PRECISION array, dimension (Q)
           The scalar factors of the elementary reflectors that define
           Q1.

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

          LWORK is INTEGER
           The dimension of the array WORK. LWORK >= M-Q.
           If LWORK = -1, then a workspace query is assumed; the routine
           only calculates the optimal size of the WORK array, returns
           this value as the first entry of the WORK array, and no error
           message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
           = 0:  successful exit.
           < 0:  if INFO = -i, the i-th argument had an illegal value.

Univ. of Tennessee

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NAG Ltd.

  The upper-bidiagonal blocks B11, B21 are represented implicitly by
  angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
  in each bidiagonal band is a product of a sine or cosine of a THETA
  with a sine or cosine of a PHI. See [1] or DORCSD for details.
  P1, P2, and Q1 are represented as products of elementary reflectors.
  See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
  and DORGLQ.

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

 DORBDB2 simultaneously bidiagonalizes the blocks of a tall and skinny
 matrix X with orthonomal columns:
                            [ B11 ]
      [ X11 ]   [ P1 |    ] [  0  ]
      [-----] = [---------] [-----] Q1**T .
      [ X21 ]   [    | P2 ] [ B21 ]
                            [  0  ]
 X11 is P-by-Q, and X21 is (M-P)-by-Q. P must be no larger than M-P,
 Q, or M-Q. Routines DORBDB1, DORBDB3, and DORBDB4 handle cases in
 which P is not the minimum dimension.
 The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
 and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
 Householder vectors.
 B11 and B12 are P-by-P bidiagonal matrices represented implicitly by
 angles THETA, PHI.

M

          M is INTEGER
           The number of rows X11 plus the number of rows in X21.

P

          P is INTEGER
           The number of rows in X11. 0 <= P <= min(M-P,Q,M-Q).

Q

          Q is INTEGER
           The number of columns in X11 and X21. 0 <= Q <= M.

X11

          X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
           On entry, the top block of the matrix X to be reduced. On
           exit, the columns of tril(X11) specify reflectors for P1 and
           the rows of triu(X11,1) specify reflectors for Q1.

LDX11

          LDX11 is INTEGER
           The leading dimension of X11. LDX11 >= P.

X21

          X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
           On entry, the bottom block of the matrix X to be reduced. On
           exit, the columns of tril(X21) specify reflectors for P2.

LDX21

          LDX21 is INTEGER
           The leading dimension of X21. LDX21 >= M-P.

THETA

          THETA is DOUBLE PRECISION array, dimension (Q)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.

PHI

          PHI is DOUBLE PRECISION array, dimension (Q-1)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.

TAUP1

          TAUP1 is DOUBLE PRECISION array, dimension (P)
           The scalar factors of the elementary reflectors that define
           P1.

TAUP2

          TAUP2 is DOUBLE PRECISION array, dimension (M-P)
           The scalar factors of the elementary reflectors that define
           P2.

TAUQ1

          TAUQ1 is DOUBLE PRECISION array, dimension (Q)
           The scalar factors of the elementary reflectors that define
           Q1.

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

          LWORK is INTEGER
           The dimension of the array WORK. LWORK >= M-Q.
           If LWORK = -1, then a workspace query is assumed; the routine
           only calculates the optimal size of the WORK array, returns
           this value as the first entry of the WORK array, and no error
           message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
           = 0:  successful exit.
           < 0:  if INFO = -i, the i-th argument had an illegal value.

Univ. of Tennessee

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NAG Ltd.

  The upper-bidiagonal blocks B11, B21 are represented implicitly by
  angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
  in each bidiagonal band is a product of a sine or cosine of a THETA
  with a sine or cosine of a PHI. See [1] or DORCSD for details.
  P1, P2, and Q1 are represented as products of elementary reflectors.
  See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
  and DORGLQ.

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

 DORBDB3 simultaneously bidiagonalizes the blocks of a tall and skinny
 matrix X with orthonomal columns:
                            [ B11 ]
      [ X11 ]   [ P1 |    ] [  0  ]
      [-----] = [---------] [-----] Q1**T .
      [ X21 ]   [    | P2 ] [ B21 ]
                            [  0  ]
 X11 is P-by-Q, and X21 is (M-P)-by-Q. M-P must be no larger than P,
 Q, or M-Q. Routines DORBDB1, DORBDB2, and DORBDB4 handle cases in
 which M-P is not the minimum dimension.
 The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
 and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
 Householder vectors.
 B11 and B12 are (M-P)-by-(M-P) bidiagonal matrices represented
 implicitly by angles THETA, PHI.

M

          M is INTEGER
           The number of rows X11 plus the number of rows in X21.

P

          P is INTEGER
           The number of rows in X11. 0 <= P <= M. M-P <= min(P,Q,M-Q).

Q

          Q is INTEGER
           The number of columns in X11 and X21. 0 <= Q <= M.

X11

          X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
           On entry, the top block of the matrix X to be reduced. On
           exit, the columns of tril(X11) specify reflectors for P1 and
           the rows of triu(X11,1) specify reflectors for Q1.

LDX11

          LDX11 is INTEGER
           The leading dimension of X11. LDX11 >= P.

X21

          X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
           On entry, the bottom block of the matrix X to be reduced. On
           exit, the columns of tril(X21) specify reflectors for P2.

LDX21

          LDX21 is INTEGER
           The leading dimension of X21. LDX21 >= M-P.

THETA

          THETA is DOUBLE PRECISION array, dimension (Q)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.

PHI

          PHI is DOUBLE PRECISION array, dimension (Q-1)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.

TAUP1

          TAUP1 is DOUBLE PRECISION array, dimension (P)
           The scalar factors of the elementary reflectors that define
           P1.

TAUP2

          TAUP2 is DOUBLE PRECISION array, dimension (M-P)
           The scalar factors of the elementary reflectors that define
           P2.

TAUQ1

          TAUQ1 is DOUBLE PRECISION array, dimension (Q)
           The scalar factors of the elementary reflectors that define
           Q1.

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

          LWORK is INTEGER
           The dimension of the array WORK. LWORK >= M-Q.
           If LWORK = -1, then a workspace query is assumed; the routine
           only calculates the optimal size of the WORK array, returns
           this value as the first entry of the WORK array, and no error
           message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
           = 0:  successful exit.
           < 0:  if INFO = -i, the i-th argument had an illegal value.

Univ. of Tennessee

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NAG Ltd.

  The upper-bidiagonal blocks B11, B21 are represented implicitly by
  angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
  in each bidiagonal band is a product of a sine or cosine of a THETA
  with a sine or cosine of a PHI. See [1] or DORCSD for details.
  P1, P2, and Q1 are represented as products of elementary reflectors.
  See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
  and DORGLQ.

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

 DORBDB4 simultaneously bidiagonalizes the blocks of a tall and skinny
 matrix X with orthonomal columns:
                            [ B11 ]
      [ X11 ]   [ P1 |    ] [  0  ]
      [-----] = [---------] [-----] Q1**T .
      [ X21 ]   [    | P2 ] [ B21 ]
                            [  0  ]
 X11 is P-by-Q, and X21 is (M-P)-by-Q. M-Q must be no larger than P,
 M-P, or Q. Routines DORBDB1, DORBDB2, and DORBDB3 handle cases in
 which M-Q is not the minimum dimension.
 The orthogonal matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
 and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
 Householder vectors.
 B11 and B12 are (M-Q)-by-(M-Q) bidiagonal matrices represented
 implicitly by angles THETA, PHI.

M

          M is INTEGER
           The number of rows X11 plus the number of rows in X21.

P

          P is INTEGER
           The number of rows in X11. 0 <= P <= M.

Q

          Q is INTEGER
           The number of columns in X11 and X21. 0 <= Q <= M and
           M-Q <= min(P,M-P,Q).

X11

          X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
           On entry, the top block of the matrix X to be reduced. On
           exit, the columns of tril(X11) specify reflectors for P1 and
           the rows of triu(X11,1) specify reflectors for Q1.

LDX11

          LDX11 is INTEGER
           The leading dimension of X11. LDX11 >= P.

X21

          X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
           On entry, the bottom block of the matrix X to be reduced. On
           exit, the columns of tril(X21) specify reflectors for P2.

LDX21

          LDX21 is INTEGER
           The leading dimension of X21. LDX21 >= M-P.

THETA

          THETA is DOUBLE PRECISION array, dimension (Q)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.

PHI

          PHI is DOUBLE PRECISION array, dimension (Q-1)
           The entries of the bidiagonal blocks B11, B21 are defined by
           THETA and PHI. See Further Details.

TAUP1

          TAUP1 is DOUBLE PRECISION array, dimension (P)
           The scalar factors of the elementary reflectors that define
           P1.

TAUP2

          TAUP2 is DOUBLE PRECISION array, dimension (M-P)
           The scalar factors of the elementary reflectors that define
           P2.

TAUQ1

          TAUQ1 is DOUBLE PRECISION array, dimension (Q)
           The scalar factors of the elementary reflectors that define
           Q1.

PHANTOM

          PHANTOM is DOUBLE PRECISION array, dimension (M)
           The routine computes an M-by-1 column vector Y that is
           orthogonal to the columns of [ X11; X21 ]. PHANTOM(1:P) and
           PHANTOM(P+1:M) contain Householder vectors for Y(1:P) and
           Y(P+1:M), respectively.

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

          LWORK is INTEGER
           The dimension of the array WORK. LWORK >= M-Q.
           If LWORK = -1, then a workspace query is assumed; the routine
           only calculates the optimal size of the WORK array, returns
           this value as the first entry of the WORK array, and no error
           message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
           = 0:  successful exit.
           < 0:  if INFO = -i, the i-th argument had an illegal value.

Univ. of Tennessee

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NAG Ltd.

  The upper-bidiagonal blocks B11, B21 are represented implicitly by
  angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
  in each bidiagonal band is a product of a sine or cosine of a THETA
  with a sine or cosine of a PHI. See [1] or DORCSD for details.
  P1, P2, and Q1 are represented as products of elementary reflectors.
  See DORCSD2BY1 for details on generating P1, P2, and Q1 using DORGQR
  and DORGLQ.

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

 DORBDB5 orthogonalizes the column vector
      X = [ X1 ]
          [ X2 ]
 with respect to the columns of
      Q = [ Q1 ] .
          [ Q2 ]
 The columns of Q must be orthonormal.
 If the projection is zero according to Kahan's "twice is enough"
 criterion, then some other vector from the orthogonal complement
 is returned. This vector is chosen in an arbitrary but deterministic
 way.

M1

          M1 is INTEGER
           The dimension of X1 and the number of rows in Q1. 0 <= M1.

M2

          M2 is INTEGER
           The dimension of X2 and the number of rows in Q2. 0 <= M2.

N

          N is INTEGER
           The number of columns in Q1 and Q2. 0 <= N.

X1

          X1 is DOUBLE PRECISION array, dimension (M1)
           On entry, the top part of the vector to be orthogonalized.
           On exit, the top part of the projected vector.

INCX1

          INCX1 is INTEGER
           Increment for entries of X1.

X2

          X2 is DOUBLE PRECISION array, dimension (M2)
           On entry, the bottom part of the vector to be
           orthogonalized. On exit, the bottom part of the projected
           vector.

INCX2

          INCX2 is INTEGER
           Increment for entries of X2.

Q1

          Q1 is DOUBLE PRECISION array, dimension (LDQ1, N)
           The top part of the orthonormal basis matrix.

LDQ1

          LDQ1 is INTEGER
           The leading dimension of Q1. LDQ1 >= M1.

Q2

          Q2 is DOUBLE PRECISION array, dimension (LDQ2, N)
           The bottom part of the orthonormal basis matrix.

LDQ2

          LDQ2 is INTEGER
           The leading dimension of Q2. LDQ2 >= M2.

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

          LWORK is INTEGER
           The dimension of the array WORK. LWORK >= N.

INFO

          INFO is INTEGER
           = 0:  successful exit.
           < 0:  if INFO = -i, the i-th argument had an illegal value.

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NAG Ltd.

 DORBDB6 orthogonalizes the column vector
      X = [ X1 ]
          [ X2 ]
 with respect to the columns of
      Q = [ Q1 ] .
          [ Q2 ]
 The columns of Q must be orthonormal.
 If the projection is zero according to Kahan's "twice is enough"
 criterion, then the zero vector is returned.

M1

          M1 is INTEGER
           The dimension of X1 and the number of rows in Q1. 0 <= M1.

M2

          M2 is INTEGER
           The dimension of X2 and the number of rows in Q2. 0 <= M2.

N

          N is INTEGER
           The number of columns in Q1 and Q2. 0 <= N.

X1

          X1 is DOUBLE PRECISION array, dimension (M1)
           On entry, the top part of the vector to be orthogonalized.
           On exit, the top part of the projected vector.

INCX1

          INCX1 is INTEGER
           Increment for entries of X1.

X2

          X2 is DOUBLE PRECISION array, dimension (M2)
           On entry, the bottom part of the vector to be
           orthogonalized. On exit, the bottom part of the projected
           vector.

INCX2

          INCX2 is INTEGER
           Increment for entries of X2.

Q1

          Q1 is DOUBLE PRECISION array, dimension (LDQ1, N)
           The top part of the orthonormal basis matrix.

LDQ1

          LDQ1 is INTEGER
           The leading dimension of Q1. LDQ1 >= M1.

Q2

          Q2 is DOUBLE PRECISION array, dimension (LDQ2, N)
           The bottom part of the orthonormal basis matrix.

LDQ2

          LDQ2 is INTEGER
           The leading dimension of Q2. LDQ2 >= M2.

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

          LWORK is INTEGER
           The dimension of the array WORK. LWORK >= N.

INFO

          INFO is INTEGER
           = 0:  successful exit.
           < 0:  if INFO = -i, the i-th argument had an illegal value.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORCSD computes the CS decomposition of an M-by-M partitioned
 orthogonal matrix X:
                                 [  I  0  0 |  0  0  0 ]
                                 [  0  C  0 |  0 -S  0 ]
     [ X11 | X12 ]   [ U1 |    ] [  0  0  0 |  0  0 -I ] [ V1 |    ]**T
 X = [-----------] = [---------] [---------------------] [---------]   .
     [ X21 | X22 ]   [    | U2 ] [  0  0  0 |  I  0  0 ] [    | V2 ]
                                 [  0  S  0 |  0  C  0 ]
                                 [  0  0  I |  0  0  0 ]
 X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P,
 (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are
 R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in
 which R = MIN(P,M-P,Q,M-Q).

JOBU1

          JOBU1 is CHARACTER
          = 'Y':      U1 is computed;
          otherwise:  U1 is not computed.

JOBU2

          JOBU2 is CHARACTER
          = 'Y':      U2 is computed;
          otherwise:  U2 is not computed.

JOBV1T

          JOBV1T is CHARACTER
          = 'Y':      V1T is computed;
          otherwise:  V1T is not computed.

JOBV2T

          JOBV2T is CHARACTER
          = 'Y':      V2T is computed;
          otherwise:  V2T is not computed.

TRANS

          TRANS is CHARACTER
          = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
                      order;
          otherwise:  X, U1, U2, V1T, and V2T are stored in column-
                      major order.

SIGNS

          SIGNS is CHARACTER
          = 'O':      The lower-left block is made nonpositive (the
                      "other" convention);
          otherwise:  The upper-right block is made nonpositive (the
                      "default" convention).

M

          M is INTEGER
          The number of rows and columns in X.

P

          P is INTEGER
          The number of rows in X11 and X12. 0 <= P <= M.

Q

          Q is INTEGER
          The number of columns in X11 and X21. 0 <= Q <= M.

X11

          X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
          On entry, part of the orthogonal matrix whose CSD is desired.

LDX11

          LDX11 is INTEGER
          The leading dimension of X11. LDX11 >= MAX(1,P).

X12

          X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
          On entry, part of the orthogonal matrix whose CSD is desired.

LDX12

          LDX12 is INTEGER
          The leading dimension of X12. LDX12 >= MAX(1,P).

X21

          X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
          On entry, part of the orthogonal matrix whose CSD is desired.

LDX21

          LDX21 is INTEGER
          The leading dimension of X11. LDX21 >= MAX(1,M-P).

X22

          X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
          On entry, part of the orthogonal matrix whose CSD is desired.

LDX22

          LDX22 is INTEGER
          The leading dimension of X11. LDX22 >= MAX(1,M-P).

THETA

          THETA is DOUBLE PRECISION array, dimension (R), in which R =
          MIN(P,M-P,Q,M-Q).
          C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
          S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).

U1

          U1 is DOUBLE PRECISION array, dimension (LDU1,P)
          If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.

LDU1

          LDU1 is INTEGER
          The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
          MAX(1,P).

U2

          U2 is DOUBLE PRECISION array, dimension (LDU2,M-P)
          If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
          matrix U2.

LDU2

          LDU2 is INTEGER
          The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
          MAX(1,M-P).

V1T

          V1T is DOUBLE PRECISION array, dimension (LDV1T,Q)
          If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
          matrix V1**T.

LDV1T

          LDV1T is INTEGER
          The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
          MAX(1,Q).

V2T

          V2T is DOUBLE PRECISION array, dimension (LDV2T,M-Q)
          If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal
          matrix V2**T.

LDV2T

          LDV2T is INTEGER
          The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=
          MAX(1,M-Q).

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
          If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),
          ..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
          define the matrix in intermediate bidiagonal-block form
          remaining after nonconvergence. INFO specifies the number
          of nonzero PHI's.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the work array, and no error
          message related to LWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q))

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  DBBCSD did not converge. See the description of WORK
                above for details.

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORCSD2BY1 computes the CS decomposition of an M-by-Q matrix X with
 orthonormal columns that has been partitioned into a 2-by-1 block
 structure:
                                [  I1 0  0 ]
                                [  0  C  0 ]
          [ X11 ]   [ U1 |    ] [  0  0  0 ]
      X = [-----] = [---------] [----------] V1**T .
          [ X21 ]   [    | U2 ] [  0  0  0 ]
                                [  0  S  0 ]
                                [  0  0  I2]
 X11 is P-by-Q. The orthogonal matrices U1, U2, and V1 are P-by-P,
 (M-P)-by-(M-P), and Q-by-Q, respectively. C and S are R-by-R
 nonnegative diagonal matrices satisfying C^2 + S^2 = I, in which
 R = MIN(P,M-P,Q,M-Q). I1 is a K1-by-K1 identity matrix and I2 is a
 K2-by-K2 identity matrix, where K1 = MAX(Q+P-M,0), K2 = MAX(Q-P,0).

JOBU1

          JOBU1 is CHARACTER
          = 'Y':      U1 is computed;
          otherwise:  U1 is not computed.

JOBU2

          JOBU2 is CHARACTER
          = 'Y':      U2 is computed;
          otherwise:  U2 is not computed.

JOBV1T

          JOBV1T is CHARACTER
          = 'Y':      V1T is computed;
          otherwise:  V1T is not computed.

M

          M is INTEGER
          The number of rows in X.

P

          P is INTEGER
          The number of rows in X11. 0 <= P <= M.

Q

          Q is INTEGER
          The number of columns in X11 and X21. 0 <= Q <= M.

X11

          X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
          On entry, part of the orthogonal matrix whose CSD is desired.

LDX11

          LDX11 is INTEGER
          The leading dimension of X11. LDX11 >= MAX(1,P).

X21

          X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
          On entry, part of the orthogonal matrix whose CSD is desired.

LDX21

          LDX21 is INTEGER
          The leading dimension of X21. LDX21 >= MAX(1,M-P).

THETA

          THETA is DOUBLE PRECISION array, dimension (R), in which R =
          MIN(P,M-P,Q,M-Q).
          C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
          S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).

U1

          U1 is DOUBLE PRECISION array, dimension (P)
          If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.

LDU1

          LDU1 is INTEGER
          The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
          MAX(1,P).

U2

          U2 is DOUBLE PRECISION array, dimension (M-P)
          If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
          matrix U2.

LDU2

          LDU2 is INTEGER
          The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
          MAX(1,M-P).

V1T

          V1T is DOUBLE PRECISION array, dimension (Q)
          If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
          matrix V1**T.

LDV1T

          LDV1T is INTEGER
          The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
          MAX(1,Q).

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
          If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),
          ..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
          define the matrix in intermediate bidiagonal-block form
          remaining after nonconvergence. INFO specifies the number
          of nonzero PHI's.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the work array, and no error
          message related to LWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (M-MIN(P,M-P,Q,M-Q))

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  DBBCSD did not converge. See the description of WORK
                above for details.

[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORG2L generates an m by n real matrix Q with orthonormal columns,
 which is defined as the last n columns of a product of k elementary
 reflectors of order m
       Q  =  H(k) . . . H(2) H(1)
 as returned by DGEQLF.

M

          M is INTEGER
          The number of rows of the matrix Q. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix Q. M >= N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. N >= K >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the (n-k+i)-th column must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as
          returned by DGEQLF in the last k columns of its array
          argument A.
          On exit, the m by n matrix Q.

LDA

          LDA is INTEGER
          The first dimension of the array A. LDA >= max(1,M).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEQLF.

WORK

          WORK is DOUBLE PRECISION array, dimension (N)

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument has an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORG2R generates an m by n real matrix Q with orthonormal columns,
 which is defined as the first n columns of a product of k elementary
 reflectors of order m
       Q  =  H(1) H(2) . . . H(k)
 as returned by DGEQRF.

M

          M is INTEGER
          The number of rows of the matrix Q. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix Q. M >= N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. N >= K >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the i-th column must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as
          returned by DGEQRF in the first k columns of its array
          argument A.
          On exit, the m-by-n matrix Q.

LDA

          LDA is INTEGER
          The first dimension of the array A. LDA >= max(1,M).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEQRF.

WORK

          WORK is DOUBLE PRECISION array, dimension (N)

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument has an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORGHR generates a real orthogonal matrix Q which is defined as the
 product of IHI-ILO elementary reflectors of order N, as returned by
 DGEHRD:
 Q = H(ilo) H(ilo+1) . . . H(ihi-1).

N

          N is INTEGER
          The order of the matrix Q. N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
          ILO and IHI must have the same values as in the previous call
          of DGEHRD. Q is equal to the unit matrix except in the
          submatrix Q(ilo+1:ihi,ilo+1:ihi).
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the vectors which define the elementary reflectors,
          as returned by DGEHRD.
          On exit, the N-by-N orthogonal matrix Q.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

TAU

          TAU is DOUBLE PRECISION array, dimension (N-1)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEHRD.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= IHI-ILO.
          For optimum performance LWORK >= (IHI-ILO)*NB, where NB is
          the optimal blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORGL2 generates an m by n real matrix Q with orthonormal rows,
 which is defined as the first m rows of a product of k elementary
 reflectors of order n
       Q  =  H(k) . . . H(2) H(1)
 as returned by DGELQF.

M

          M is INTEGER
          The number of rows of the matrix Q. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix Q. N >= M.

K

          K is INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. M >= K >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the i-th row must contain the vector which defines
          the elementary reflector H(i), for i = 1,2,...,k, as returned
          by DGELQF in the first k rows of its array argument A.
          On exit, the m-by-n matrix Q.

LDA

          LDA is INTEGER
          The first dimension of the array A. LDA >= max(1,M).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGELQF.

WORK

          WORK is DOUBLE PRECISION array, dimension (M)

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument has an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORGLQ generates an M-by-N real matrix Q with orthonormal rows,
 which is defined as the first M rows of a product of K elementary
 reflectors of order N
       Q  =  H(k) . . . H(2) H(1)
 as returned by DGELQF.

M

          M is INTEGER
          The number of rows of the matrix Q. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix Q. N >= M.

K

          K is INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. M >= K >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the i-th row must contain the vector which defines
          the elementary reflector H(i), for i = 1,2,...,k, as returned
          by DGELQF in the first k rows of its array argument A.
          On exit, the M-by-N matrix Q.

LDA

          LDA is INTEGER
          The first dimension of the array A. LDA >= max(1,M).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGELQF.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,M).
          For optimum performance LWORK >= M*NB, where NB is
          the optimal blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument has an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORGQL generates an M-by-N real matrix Q with orthonormal columns,
 which is defined as the last N columns of a product of K elementary
 reflectors of order M
       Q  =  H(k) . . . H(2) H(1)
 as returned by DGEQLF.

M

          M is INTEGER
          The number of rows of the matrix Q. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix Q. M >= N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. N >= K >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the (n-k+i)-th column must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as
          returned by DGEQLF in the last k columns of its array
          argument A.
          On exit, the M-by-N matrix Q.

LDA

          LDA is INTEGER
          The first dimension of the array A. LDA >= max(1,M).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEQLF.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,N).
          For optimum performance LWORK >= N*NB, where NB is the
          optimal blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument has an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORGQR generates an M-by-N real matrix Q with orthonormal columns,
 which is defined as the first N columns of a product of K elementary
 reflectors of order M
       Q  =  H(1) H(2) . . . H(k)
 as returned by DGEQRF.

M

          M is INTEGER
          The number of rows of the matrix Q. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix Q. M >= N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. N >= K >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the i-th column must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as
          returned by DGEQRF in the first k columns of its array
          argument A.
          On exit, the M-by-N matrix Q.

LDA

          LDA is INTEGER
          The first dimension of the array A. LDA >= max(1,M).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEQRF.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,N).
          For optimum performance LWORK >= N*NB, where NB is the
          optimal blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument has an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORGR2 generates an m by n real matrix Q with orthonormal rows,
 which is defined as the last m rows of a product of k elementary
 reflectors of order n
       Q  =  H(1) H(2) . . . H(k)
 as returned by DGERQF.

M

          M is INTEGER
          The number of rows of the matrix Q. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix Q. N >= M.

K

          K is INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. M >= K >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the (m-k+i)-th row must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as
          returned by DGERQF in the last k rows of its array argument
          A.
          On exit, the m by n matrix Q.

LDA

          LDA is INTEGER
          The first dimension of the array A. LDA >= max(1,M).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGERQF.

WORK

          WORK is DOUBLE PRECISION array, dimension (M)

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument has an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORGRQ generates an M-by-N real matrix Q with orthonormal rows,
 which is defined as the last M rows of a product of K elementary
 reflectors of order N
       Q  =  H(1) H(2) . . . H(k)
 as returned by DGERQF.

M

          M is INTEGER
          The number of rows of the matrix Q. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix Q. N >= M.

K

          K is INTEGER
          The number of elementary reflectors whose product defines the
          matrix Q. M >= K >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the (m-k+i)-th row must contain the vector which
          defines the elementary reflector H(i), for i = 1,2,...,k, as
          returned by DGERQF in the last k rows of its array argument
          A.
          On exit, the M-by-N matrix Q.

LDA

          LDA is INTEGER
          The first dimension of the array A. LDA >= max(1,M).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGERQF.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,M).
          For optimum performance LWORK >= M*NB, where NB is the
          optimal blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument has an illegal value

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 DORGTR generates a real orthogonal matrix Q which is defined as the
 product of n-1 elementary reflectors of order N, as returned by
 DSYTRD:
 if UPLO = 'U', Q = H(n-1) . . . H(2) H(1),
 if UPLO = 'L', Q = H(1) H(2) . . . H(n-1).

UPLO

          UPLO is CHARACTER*1
          = 'U': Upper triangle of A contains elementary reflectors
                 from DSYTRD;
          = 'L': Lower triangle of A contains elementary reflectors
                 from DSYTRD.

N

          N is INTEGER
          The order of the matrix Q. N >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the vectors which define the elementary reflectors,
          as returned by DSYTRD.
          On exit, the N-by-N orthogonal matrix Q.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

TAU

          TAU is DOUBLE PRECISION array, dimension (N-1)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DSYTRD.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,N-1).
          For optimum performance LWORK >= (N-1)*NB, where NB is
          the optimal blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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 DORGTSQR generates an M-by-N real matrix Q_out with orthonormal columns,
 which are the first N columns of a product of real orthogonal
 matrices of order M which are returned by DLATSQR
      Q_out = first_N_columns_of( Q(1)_in * Q(2)_in * ... * Q(k)_in ).
 See the documentation for DLATSQR.

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

N

          N is INTEGER
          The number of columns of the matrix A. M >= N >= 0.

MB

          MB is INTEGER
          The row block size used by DLATSQR to return
          arrays A and T. MB > N.
          (Note that if MB > M, then M is used instead of MB
          as the row block size).

NB

          NB is INTEGER
          The column block size used by DLATSQR to return
          arrays A and T. NB >= 1.
          (Note that if NB > N, then N is used instead of NB
          as the column block size).

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry:
             The elements on and above the diagonal are not accessed.
             The elements below the diagonal represent the unit
             lower-trapezoidal blocked matrix V computed by DLATSQR
             that defines the input matrices Q_in(k) (ones on the
             diagonal are not stored) (same format as the output A
             below the diagonal in DLATSQR).
          On exit:
             The array A contains an M-by-N orthonormal matrix Q_out,
             i.e the columns of A are orthogonal unit vectors.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

T

          T is DOUBLE PRECISION array,
          dimension (LDT, N * NIRB)
          where NIRB = Number_of_input_row_blocks
                     = MAX( 1, CEIL((M-N)/(MB-N)) )
          Let NICB = Number_of_input_col_blocks
                   = CEIL(N/NB)
          The upper-triangular block reflectors used to define the
          input matrices Q_in(k), k=(1:NIRB*NICB). The block
          reflectors are stored in compact form in NIRB block
          reflector sequences. Each of NIRB block reflector sequences
          is stored in a larger NB-by-N column block of T and consists
          of NICB smaller NB-by-NB upper-triangular column blocks.
          (same format as the output T in DLATSQR).

LDT

          LDT is INTEGER
          The leading dimension of the array T.
          LDT >= max(1,min(NB1,N)).

WORK

          (workspace) DOUBLE PRECISION array, dimension (MAX(2,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          The dimension of the array WORK.  LWORK >= (M+NB)*N.
          If LWORK = -1, then a workspace query is assumed.
          The routine only calculates the optimal size of the WORK
          array, returns this value as the first entry of the WORK
          array, and no error message related to LWORK is issued
          by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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 November 2019, Igor Kozachenko,
                Computer Science Division,
                University of California, Berkeley

 DORGTSQR_ROW generates an M-by-N real matrix Q_out with
 orthonormal columns from the output of DLATSQR. These N orthonormal
 columns are the first N columns of a product of complex unitary
 matrices Q(k)_in of order M, which are returned by DLATSQR in
 a special format.
      Q_out = first_N_columns_of( Q(1)_in * Q(2)_in * ... * Q(k)_in ).
 The input matrices Q(k)_in are stored in row and column blocks in A.
 See the documentation of DLATSQR for more details on the format of
 Q(k)_in, where each Q(k)_in is represented by block Householder
 transformations. This routine calls an auxiliary routine DLARFB_GETT,
 where the computation is performed on each individual block. The
 algorithm first sweeps NB-sized column blocks from the right to left
 starting in the bottom row block and continues to the top row block
 (hence _ROW in the routine name). This sweep is in reverse order of
 the order in which DLATSQR generates the output blocks.

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

N

          N is INTEGER
          The number of columns of the matrix A. M >= N >= 0.

MB

          MB is INTEGER
          The row block size used by DLATSQR to return
          arrays A and T. MB > N.
          (Note that if MB > M, then M is used instead of MB
          as the row block size).

NB

          NB is INTEGER
          The column block size used by DLATSQR to return
          arrays A and T. NB >= 1.
          (Note that if NB > N, then N is used instead of NB
          as the column block size).

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry:
             The elements on and above the diagonal are not used as
             input. The elements below the diagonal represent the unit
             lower-trapezoidal blocked matrix V computed by DLATSQR
             that defines the input matrices Q_in(k) (ones on the
             diagonal are not stored). See DLATSQR for more details.
          On exit:
             The array A contains an M-by-N orthonormal matrix Q_out,
             i.e the columns of A are orthogonal unit vectors.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

T

          T is DOUBLE PRECISION array,
          dimension (LDT, N * NIRB)
          where NIRB = Number_of_input_row_blocks
                     = MAX( 1, CEIL((M-N)/(MB-N)) )
          Let NICB = Number_of_input_col_blocks
                   = CEIL(N/NB)
          The upper-triangular block reflectors used to define the
          input matrices Q_in(k), k=(1:NIRB*NICB). The block
          reflectors are stored in compact form in NIRB block
          reflector sequences. Each of the NIRB block reflector
          sequences is stored in a larger NB-by-N column block of T
          and consists of NICB smaller NB-by-NB upper-triangular
          column blocks. See DLATSQR for more details on the format
          of T.

LDT

          LDT is INTEGER
          The leading dimension of the array T.
          LDT >= max(1,min(NB,N)).

WORK

          (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          The dimension of the array WORK.
          LWORK >= NBLOCAL * MAX(NBLOCAL,(N-NBLOCAL)),
          where NBLOCAL=MIN(NB,N).
          If LWORK = -1, then a workspace query is assumed.
          The routine only calculates the optimal size of the WORK
          array, returns this value as the first entry of the WORK
          array, and no error message related to LWORK is issued
          by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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 November 2020, Igor Kozachenko,
                Computer Science Division,
                University of California, Berkeley

  DORHR_COL takes an M-by-N real matrix Q_in with orthonormal columns
  as input, stored in A, and performs Householder Reconstruction (HR),
  i.e. reconstructs Householder vectors V(i) implicitly representing
  another M-by-N matrix Q_out, with the property that Q_in = Q_out*S,
  where S is an N-by-N diagonal matrix with diagonal entries
  equal to +1 or -1. The Householder vectors (columns V(i) of V) are
  stored in A on output, and the diagonal entries of S are stored in D.
  Block reflectors are also returned in T
  (same output format as DGEQRT).

M

          M is INTEGER
          The number of rows of the matrix A. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix A. M >= N >= 0.

NB

          NB is INTEGER
          The column block size to be used in the reconstruction
          of Householder column vector blocks in the array A and
          corresponding block reflectors in the array T. NB >= 1.
          (Note that if NB > N, then N is used instead of NB
          as the column block size.)

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry:
             The array A contains an M-by-N orthonormal matrix Q_in,
             i.e the columns of A are orthogonal unit vectors.
          On exit:
             The elements below the diagonal of A represent the unit
             lower-trapezoidal matrix V of Householder column vectors
             V(i). The unit diagonal entries of V are not stored
             (same format as the output below the diagonal in A from
             DGEQRT). The matrix T and the matrix V stored on output
             in A implicitly define Q_out.
             The elements above the diagonal contain the factor U
             of the "modified" LU-decomposition:
                Q_in - ( S ) = V * U
                       ( 0 )
             where 0 is a (M-N)-by-(M-N) zero matrix.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

T

          T is DOUBLE PRECISION array,
          dimension (LDT, N)
          Let NOCB = Number_of_output_col_blocks
                   = CEIL(N/NB)
          On exit, T(1:NB, 1:N) contains NOCB upper-triangular
          block reflectors used to define Q_out stored in compact
          form as a sequence of upper-triangular NB-by-NB column
          blocks (same format as the output T in DGEQRT).
          The matrix T and the matrix V stored on output in A
          implicitly define Q_out. NOTE: The lower triangles
          below the upper-triangular blocks will be filled with
          zeros. See Further Details.

LDT

          LDT is INTEGER
          The leading dimension of the array T.
          LDT >= max(1,min(NB,N)).

D

          D is DOUBLE PRECISION array, dimension min(M,N).
          The elements can be only plus or minus one.
          D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where
          1 <= i <= min(M,N), and Q_in_i is Q_in after performing
          i-1 steps of “modified” Gaussian elimination.
          See Further Details.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

 The computed M-by-M orthogonal factor Q_out is defined implicitly as
 a product of orthogonal matrices Q_out(i). Each Q_out(i) is stored in
 the compact WY-representation format in the corresponding blocks of
 matrices V (stored in A) and T.
 The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N
 matrix A contains the column vectors V(i) in NB-size column
 blocks VB(j). For example, VB(1) contains the columns
 V(1), V(2), ... V(NB). NOTE: The unit entries on
 the diagonal of Y are not stored in A.
 The number of column blocks is
     NOCB = Number_of_output_col_blocks = CEIL(N/NB)
 where each block is of order NB except for the last block, which
 is of order LAST_NB = N - (NOCB-1)*NB.
 For example, if M=6,  N=5 and NB=2, the matrix V is
     V = (    VB(1),   VB(2), VB(3) ) =
       = (   1                      )
         ( v21    1                 )
         ( v31  v32    1            )
         ( v41  v42  v43   1        )
         ( v51  v52  v53  v54    1  )
         ( v61  v62  v63  v54   v65 )
 For each of the column blocks VB(i), an upper-triangular block
 reflector TB(i) is computed. These blocks are stored as
 a sequence of upper-triangular column blocks in the NB-by-N
 matrix T. The size of each TB(i) block is NB-by-NB, except
 for the last block, whose size is LAST_NB-by-LAST_NB.
 For example, if M=6,  N=5 and NB=2, the matrix T is
     T  = (    TB(1),    TB(2), TB(3) ) =
        = ( t11  t12  t13  t14   t15  )
          (      t22       t24        )
 The M-by-M factor Q_out is given as a product of NOCB
 orthogonal M-by-M matrices Q_out(i).
     Q_out = Q_out(1) * Q_out(2) * ... * Q_out(NOCB),
 where each matrix Q_out(i) is given by the WY-representation
 using corresponding blocks from the matrices V and T:
     Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T,
 where I is the identity matrix. Here is the formula with matrix
 dimensions:
  Q(i){M-by-M} = I{M-by-M} -
    VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M},
 where INB = NB, except for the last block NOCB
 for which INB=LAST_NB.
 =====
 NOTE:
 =====
 If Q_in is the result of doing a QR factorization
 B = Q_in * R_in, then:
 B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out.
 So if one wants to interpret Q_out as the result
 of the QR factorization of B, then the corresponding R_out
 should be equal to R_out = S * R_in, i.e. some rows of R_in
 should be multiplied by -1.
 For the details of the algorithm, see [1].
 [1] "Reconstructing Householder vectors from tall-skinny QR",
     G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
     E. Solomonik, J. Parallel Distrib. Comput.,
     vol. 85, pp. 3-31, 2015.

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 November   2019, Igor Kozachenko,
            Computer Science Division,
            University of California, Berkeley

 DORM2L overwrites the general real m by n matrix C with
       Q * C  if SIDE = 'L' and TRANS = 'N', or
       Q**T * C  if SIDE = 'L' and TRANS = 'T', or
       C * Q  if SIDE = 'R' and TRANS = 'N', or
       C * Q**T if SIDE = 'R' and TRANS = 'T',
 where Q is a real orthogonal matrix defined as the product of k
 elementary reflectors
       Q = H(k) . . . H(2) H(1)
 as returned by DGEQLF. Q is of order m if SIDE = 'L' and of order n
 if SIDE = 'R'.

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**T from the Left
          = 'R': apply Q or Q**T from the Right

TRANS

          TRANS is CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'T': apply Q**T (Transpose)

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          DGEQLF in the last k columns of its array argument A.
          A is modified by the routine but restored on exit.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,M);
          if SIDE = 'R', LDA >= max(1,N).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEQLF.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

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 DORM2R overwrites the general real m by n matrix C with
       Q * C  if SIDE = 'L' and TRANS = 'N', or
       Q**T* C  if SIDE = 'L' and TRANS = 'T', or
       C * Q  if SIDE = 'R' and TRANS = 'N', or
       C * Q**T if SIDE = 'R' and TRANS = 'T',
 where Q is a real orthogonal matrix defined as the product of k
 elementary reflectors
       Q = H(1) H(2) . . . H(k)
 as returned by DGEQRF. Q is of order m if SIDE = 'L' and of order n
 if SIDE = 'R'.

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**T from the Left
          = 'R': apply Q or Q**T from the Right

TRANS

          TRANS is CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'T': apply Q**T (Transpose)

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          DGEQRF in the first k columns of its array argument A.
          A is modified by the routine but restored on exit.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,M);
          if SIDE = 'R', LDA >= max(1,N).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEQRF.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

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 If VECT = 'Q', DORMBR overwrites the general real M-by-N matrix C
 with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'T':      Q**T * C       C * Q**T
 If VECT = 'P', DORMBR overwrites the general real M-by-N matrix C
 with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      P * C          C * P
 TRANS = 'T':      P**T * C       C * P**T
 Here Q and P**T are the orthogonal matrices determined by DGEBRD when
 reducing a real matrix A to bidiagonal form: A = Q * B * P**T. Q and
 P**T are defined as products of elementary reflectors H(i) and G(i)
 respectively.
 Let nq = m if SIDE = 'L' and nq = n if SIDE = 'R'. Thus nq is the
 order of the orthogonal matrix Q or P**T that is applied.
 If VECT = 'Q', A is assumed to have been an NQ-by-K matrix:
 if nq >= k, Q = H(1) H(2) . . . H(k);
 if nq < k, Q = H(1) H(2) . . . H(nq-1).
 If VECT = 'P', A is assumed to have been a K-by-NQ matrix:
 if k < nq, P = G(1) G(2) . . . G(k);
 if k >= nq, P = G(1) G(2) . . . G(nq-1).

VECT

          VECT is CHARACTER*1
          = 'Q': apply Q or Q**T;
          = 'P': apply P or P**T.

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q, Q**T, P or P**T from the Left;
          = 'R': apply Q, Q**T, P or P**T from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q  or P;
          = 'T':  Transpose, apply Q**T or P**T.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          If VECT = 'Q', the number of columns in the original
          matrix reduced by DGEBRD.
          If VECT = 'P', the number of rows in the original
          matrix reduced by DGEBRD.
          K >= 0.

A

          A is DOUBLE PRECISION array, dimension
                                (LDA,min(nq,K)) if VECT = 'Q'
                                (LDA,nq)        if VECT = 'P'
          The vectors which define the elementary reflectors H(i) and
          G(i), whose products determine the matrices Q and P, as
          returned by DGEBRD.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          If VECT = 'Q', LDA >= max(1,nq);
          if VECT = 'P', LDA >= max(1,min(nq,K)).

TAU

          TAU is DOUBLE PRECISION array, dimension (min(nq,K))
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i) or G(i) which determines Q or P, as returned
          by DGEBRD in the array argument TAUQ or TAUP.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q
          or P*C or P**T*C or C*P or C*P**T.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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 DORMHR overwrites the general real M-by-N matrix C with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'T':      Q**T * C       C * Q**T
 where Q is a real orthogonal matrix of order nq, with nq = m if
 SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
 IHI-ILO elementary reflectors, as returned by DGEHRD:
 Q = H(ilo) H(ilo+1) . . . H(ihi-1).

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
          ILO and IHI must have the same values as in the previous call
          of DGEHRD. Q is equal to the unit matrix except in the
          submatrix Q(ilo+1:ihi,ilo+1:ihi).
          If SIDE = 'L', then 1 <= ILO <= IHI <= M, if M > 0, and
          ILO = 1 and IHI = 0, if M = 0;
          if SIDE = 'R', then 1 <= ILO <= IHI <= N, if N > 0, and
          ILO = 1 and IHI = 0, if N = 0.

A

          A is DOUBLE PRECISION array, dimension
                               (LDA,M) if SIDE = 'L'
                               (LDA,N) if SIDE = 'R'
          The vectors which define the elementary reflectors, as
          returned by DGEHRD.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.

TAU

          TAU is DOUBLE PRECISION array, dimension
                               (M-1) if SIDE = 'L'
                               (N-1) if SIDE = 'R'
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEHRD.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORML2 overwrites the general real m by n matrix C with
       Q * C  if SIDE = 'L' and TRANS = 'N', or
       Q**T* C  if SIDE = 'L' and TRANS = 'T', or
       C * Q  if SIDE = 'R' and TRANS = 'N', or
       C * Q**T if SIDE = 'R' and TRANS = 'T',
 where Q is a real orthogonal matrix defined as the product of k
 elementary reflectors
       Q = H(k) . . . H(2) H(1)
 as returned by DGELQF. Q is of order m if SIDE = 'L' and of order n
 if SIDE = 'R'.

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**T from the Left
          = 'R': apply Q or Q**T from the Right

TRANS

          TRANS is CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'T': apply Q**T (Transpose)

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

A

          A is DOUBLE PRECISION array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          DGELQF in the first k rows of its array argument A.
          A is modified by the routine but restored on exit.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGELQF.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORMLQ overwrites the general real M-by-N matrix C with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'T':      Q**T * C       C * Q**T
 where Q is a real orthogonal matrix defined as the product of k
 elementary reflectors
       Q = H(k) . . . H(2) H(1)
 as returned by DGELQF. Q is of order M if SIDE = 'L' and of order N
 if SIDE = 'R'.

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

A

          A is DOUBLE PRECISION array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          DGELQF in the first k rows of its array argument A.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGELQF.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For good performance, LWORK should generally be larger.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORMQL overwrites the general real M-by-N matrix C with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'T':      Q**T * C       C * Q**T
 where Q is a real orthogonal matrix defined as the product of k
 elementary reflectors
       Q = H(k) . . . H(2) H(1)
 as returned by DGEQLF. Q is of order M if SIDE = 'L' and of order N
 if SIDE = 'R'.

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          DGEQLF in the last k columns of its array argument A.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,M);
          if SIDE = 'R', LDA >= max(1,N).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEQLF.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For good performance, LWORK should generally be larger.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORMQR overwrites the general real M-by-N matrix C with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'T':      Q**T * C       C * Q**T
 where Q is a real orthogonal matrix defined as the product of k
 elementary reflectors
       Q = H(1) H(2) . . . H(k)
 as returned by DGEQRF. Q is of order M if SIDE = 'L' and of order N
 if SIDE = 'R'.

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,K)
          The i-th column must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          DGEQRF in the first k columns of its array argument A.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,M);
          if SIDE = 'R', LDA >= max(1,N).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGEQRF.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For good performance, LWORK should generally be larger.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORMR2 overwrites the general real m by n matrix C with
       Q * C  if SIDE = 'L' and TRANS = 'N', or
       Q**T* C  if SIDE = 'L' and TRANS = 'T', or
       C * Q  if SIDE = 'R' and TRANS = 'N', or
       C * Q**T if SIDE = 'R' and TRANS = 'T',
 where Q is a real orthogonal matrix defined as the product of k
 elementary reflectors
       Q = H(1) H(2) . . . H(k)
 as returned by DGERQF. Q is of order m if SIDE = 'L' and of order n
 if SIDE = 'R'.

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**T from the Left
          = 'R': apply Q or Q**T from the Right

TRANS

          TRANS is CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'T': apply Q' (Transpose)

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

A

          A is DOUBLE PRECISION array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          DGERQF in the last k rows of its array argument A.
          A is modified by the routine but restored on exit.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGERQF.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORMR3 overwrites the general real m by n matrix C with
       Q * C  if SIDE = 'L' and TRANS = 'N', or
       Q**T* C  if SIDE = 'L' and TRANS = 'C', or
       C * Q  if SIDE = 'R' and TRANS = 'N', or
       C * Q**T if SIDE = 'R' and TRANS = 'C',
 where Q is a real orthogonal matrix defined as the product of k
 elementary reflectors
       Q = H(1) H(2) . . . H(k)
 as returned by DTZRZF. Q is of order m if SIDE = 'L' and of order n
 if SIDE = 'R'.

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**T from the Left
          = 'R': apply Q or Q**T from the Right

TRANS

          TRANS is CHARACTER*1
          = 'N': apply Q  (No transpose)
          = 'T': apply Q**T (Transpose)

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

L

          L is INTEGER
          The number of columns of the matrix A containing
          the meaningful part of the Householder reflectors.
          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

A

          A is DOUBLE PRECISION array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          DTZRZF in the last k rows of its array argument A.
          A is modified by the routine but restored on exit.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DTZRZF.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the m-by-n matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension
                                   (N) if SIDE = 'L',
                                   (M) if SIDE = 'R'

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

 DORMRQ overwrites the general real M-by-N matrix C with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'T':      Q**T * C       C * Q**T
 where Q is a real orthogonal matrix defined as the product of k
 elementary reflectors
       Q = H(1) H(2) . . . H(k)
 as returned by DGERQF. Q is of order M if SIDE = 'L' and of order N
 if SIDE = 'R'.

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

A

          A is DOUBLE PRECISION array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          DGERQF in the last k rows of its array argument A.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DGERQF.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For good performance, LWORK should generally be larger.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DORMRZ overwrites the general real M-by-N matrix C with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'T':      Q**T * C       C * Q**T
 where Q is a real orthogonal matrix defined as the product of k
 elementary reflectors
       Q = H(1) H(2) . . . H(k)
 as returned by DTZRZF. Q is of order M if SIDE = 'L' and of order N
 if SIDE = 'R'.

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

K

          K is INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

L

          L is INTEGER
          The number of columns of the matrix A containing
          the meaningful part of the Householder reflectors.
          If SIDE = 'L', M >= L >= 0, if SIDE = 'R', N >= L >= 0.

A

          A is DOUBLE PRECISION array, dimension
                               (LDA,M) if SIDE = 'L',
                               (LDA,N) if SIDE = 'R'
          The i-th row must contain the vector which defines the
          elementary reflector H(i), for i = 1,2,...,k, as returned by
          DTZRZF in the last k rows of its array argument A.
          A is modified by the routine but restored on exit.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

TAU

          TAU is DOUBLE PRECISION array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DTZRZF.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For good performance, LWORK should generally be larger.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

 DORMTR overwrites the general real M-by-N matrix C with
                 SIDE = 'L'     SIDE = 'R'
 TRANS = 'N':      Q * C          C * Q
 TRANS = 'T':      Q**T * C       C * Q**T
 where Q is a real orthogonal matrix of order nq, with nq = m if
 SIDE = 'L' and nq = n if SIDE = 'R'. Q is defined as the product of
 nq-1 elementary reflectors, as returned by DSYTRD:
 if UPLO = 'U', Q = H(nq-1) . . . H(2) H(1);
 if UPLO = 'L', Q = H(1) H(2) . . . H(nq-1).

SIDE

          SIDE is CHARACTER*1
          = 'L': apply Q or Q**T from the Left;
          = 'R': apply Q or Q**T from the Right.

UPLO

          UPLO is CHARACTER*1
          = 'U': Upper triangle of A contains elementary reflectors
                 from DSYTRD;
          = 'L': Lower triangle of A contains elementary reflectors
                 from DSYTRD.

TRANS

          TRANS is CHARACTER*1
          = 'N':  No transpose, apply Q;
          = 'T':  Transpose, apply Q**T.

M

          M is INTEGER
          The number of rows of the matrix C. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix C. N >= 0.

A

          A is DOUBLE PRECISION array, dimension
                               (LDA,M) if SIDE = 'L'
                               (LDA,N) if SIDE = 'R'
          The vectors which define the elementary reflectors, as
          returned by DSYTRD.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          LDA >= max(1,M) if SIDE = 'L'; LDA >= max(1,N) if SIDE = 'R'.

TAU

          TAU is DOUBLE PRECISION array, dimension
                               (M-1) if SIDE = 'L'
                               (N-1) if SIDE = 'R'
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i), as returned by DSYTRD.

C

          C is DOUBLE PRECISION array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK.
          If SIDE = 'L', LWORK >= max(1,N);
          if SIDE = 'R', LWORK >= max(1,M).
          For optimum performance LWORK >= N*NB if SIDE = 'L', and
          LWORK >= M*NB if SIDE = 'R', where NB is the optimal
          blocksize.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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 DPBCON estimates the reciprocal of the condition number (in the
 1-norm) of a real symmetric positive definite band matrix using the
 Cholesky factorization A = U**T*U or A = L*L**T computed by DPBTRF.
 An estimate is obtained for norm(inv(A)), and the reciprocal of the
 condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangular factor stored in AB;
          = 'L':  Lower triangular factor stored in AB.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          The triangular factor U or L from the Cholesky factorization
          A = U**T*U or A = L*L**T of the band matrix A, stored in the
          first KD+1 rows of the array.  The j-th column of U or L is
          stored in the j-th column of the array AB as follows:
          if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
          if UPLO ='L', AB(1+i-j,j)    = L(i,j) for j<=i<=min(n,j+kd).

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

ANORM

          ANORM is DOUBLE PRECISION
          The 1-norm (or infinity-norm) of the symmetric band matrix A.

RCOND

          RCOND is DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

WORK

          WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

          IWORK is INTEGER array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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 DPBEQU computes row and column scalings intended to equilibrate a
 symmetric positive definite band matrix A and reduce its condition
 number (with respect to the two-norm).  S contains the scale factors,
 S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
 elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This
 choice of S puts the condition number of B within a factor N of the
 smallest possible condition number over all possible diagonal
 scalings.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangular of A is stored;
          = 'L':  Lower triangular of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangle of the symmetric band matrix A,
          stored in the first KD+1 rows of the array.  The j-th column
          of A is stored in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

LDAB

          LDAB is INTEGER
          The leading dimension of the array A.  LDAB >= KD+1.

S

          S is DOUBLE PRECISION array, dimension (N)
          If INFO = 0, S contains the scale factors for A.

SCOND

          SCOND is DOUBLE PRECISION
          If INFO = 0, S contains the ratio of the smallest S(i) to
          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
          large nor too small, it is not worth scaling by S.

AMAX

          AMAX is DOUBLE PRECISION
          Absolute value of largest matrix element.  If AMAX is very
          close to overflow or very close to underflow, the matrix
          should be scaled.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = i, the i-th diagonal element is nonpositive.

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 DPBRFS improves the computed solution to a system of linear
 equations when the coefficient matrix is symmetric positive definite
 and banded, and provides error bounds and backward error estimates
 for the solution.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangle of the symmetric band matrix A,
          stored in the first KD+1 rows of the array.  The j-th column
          of A is stored in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

AFB

          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
          The triangular factor U or L from the Cholesky factorization
          A = U**T*U or A = L*L**T of the band matrix A as computed by
          DPBTRF, in the same storage format as A (see AB).

LDAFB

          LDAFB is INTEGER
          The leading dimension of the array AFB.  LDAFB >= KD+1.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by DPBTRS.
          On exit, the improved solution matrix X.

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

FERR

          FERR is DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)
          is an estimated upper bound for the magnitude of the largest
          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

BERR

          BERR is DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

WORK

          WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

          IWORK is INTEGER array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  ITMAX is the maximum number of steps of iterative refinement.

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 DPBSTF computes a split Cholesky factorization of a real
 symmetric positive definite band matrix A.
 This routine is designed to be used in conjunction with DSBGST.
 The factorization has the form  A = S**T*S  where S is a band matrix
 of the same bandwidth as A and the following structure:
   S = ( U    )
       ( M  L )
 where U is upper triangular of order m = (n+kd)/2, and L is lower
 triangular of order n-m.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first kd+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          On exit, if INFO = 0, the factor S from the split Cholesky
          factorization A = S**T*S. See Further Details.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, the factorization could not be completed,
               because the updated element a(i,i) was negative; the
               matrix A is not positive definite.

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  The band storage scheme is illustrated by the following example, when
  N = 7, KD = 2:
  S = ( s11  s12  s13                     )
      (      s22  s23  s24                )
      (           s33  s34                )
      (                s44                )
      (           s53  s54  s55           )
      (                s64  s65  s66      )
      (                     s75  s76  s77 )
  If UPLO = 'U', the array AB holds:
  on entry:                          on exit:
   *    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53  s64  s75
   *   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54  s65  s76
  a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77
  If UPLO = 'L', the array AB holds:
  on entry:                          on exit:
  a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55  s66  s77
  a21  a32  a43  a54  a65  a76   *   s12  s23  s34  s54  s65  s76   *
  a31  a42  a53  a64  a64   *    *   s13  s24  s53  s64  s75   *    *
  Array elements marked * are not used by the routine.

 DPBTF2 computes the Cholesky factorization of a real symmetric
 positive definite band matrix A.
 The factorization has the form
    A = U**T * U ,  if UPLO = 'U', or
    A = L  * L**T,  if UPLO = 'L',
 where U is an upper triangular matrix, U**T is the transpose of U, and
 L is lower triangular.
 This is the unblocked version of the algorithm, calling Level 2 BLAS.

UPLO

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of super-diagonals of the matrix A if UPLO = 'U',
          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U**T*U or A = L*L**T of the band
          matrix A, in the same storage format as A.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, the leading minor of order k is not
               positive definite, and the factorization could not be
               completed.

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  The band storage scheme is illustrated by the following example, when
  N = 6, KD = 2, and UPLO = 'U':
  On entry:                       On exit:
      *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
  Similarly, if UPLO = 'L' the format of A is as follows:
  On entry:                       On exit:
     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *
  Array elements marked * are not used by the routine.

 DPBTRF computes the Cholesky factorization of a real symmetric
 positive definite band matrix A.
 The factorization has the form
    A = U**T * U,  if UPLO = 'U', or
    A = L  * L**T,  if UPLO = 'L',
 where U is an upper triangular matrix and L is lower triangular.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U**T*U or A = L*L**T of the band
          matrix A, in the same storage format as A.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i is not
                positive definite, and the factorization could not be
                completed.

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  The band storage scheme is illustrated by the following example, when
  N = 6, KD = 2, and UPLO = 'U':
  On entry:                       On exit:
      *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
  Similarly, if UPLO = 'L' the format of A is as follows:
  On entry:                       On exit:
     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *
  Array elements marked * are not used by the routine.

Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989

 DPBTRS solves a system of linear equations A*X = B with a symmetric
 positive definite band matrix A using the Cholesky factorization
 A = U**T*U or A = L*L**T computed by DPBTRF.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangular factor stored in AB;
          = 'L':  Lower triangular factor stored in AB.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          The triangular factor U or L from the Cholesky factorization
          A = U**T*U or A = L*L**T of the band matrix A, stored in the
          first KD+1 rows of the array.  The j-th column of U or L is
          stored in the j-th column of the array AB as follows:
          if UPLO ='U', AB(kd+1+i-j,j) = U(i,j) for max(1,j-kd)<=i<=j;
          if UPLO ='L', AB(1+i-j,j)    = L(i,j) for j<=i<=min(n,j+kd).

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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 DPFTRF computes the Cholesky factorization of a real symmetric
 positive definite matrix A.
 The factorization has the form
    A = U**T * U,  if UPLO = 'U', or
    A = L  * L**T,  if UPLO = 'L',
 where U is an upper triangular matrix and L is lower triangular.
 This is the block version of the algorithm, calling Level 3 BLAS.

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  The Normal TRANSR of RFP A is stored;
          = 'T':  The Transpose TRANSR of RFP A is stored.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of RFP A is stored;
          = 'L':  Lower triangle of RFP A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 );
          On entry, the symmetric matrix A in RFP format. RFP format is
          described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
          then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
          (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
          the transpose of RFP A as defined when
          TRANSR = 'N'. The contents of RFP A are defined by UPLO as
          follows: If UPLO = 'U' the RFP A contains the NT elements of
          upper packed A. If UPLO = 'L' the RFP A contains the elements
          of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
          'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N
          is odd. See the Note below for more details.
          On exit, if INFO = 0, the factor U or L from the Cholesky
          factorization RFP A = U**T*U or RFP A = L*L**T.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i is not
                positive definite, and the factorization could not be
                completed.

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  We first consider Rectangular Full Packed (RFP) Format when N is
  even. We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  the transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  the transpose of the last three columns of AP lower.
  This covers the case N even and TRANSR = 'N'.
         RFP A                   RFP A
        03 04 05                33 43 53
        13 14 15                00 44 54
        23 24 25                10 11 55
        33 34 35                20 21 22
        00 44 45                30 31 32
        01 11 55                40 41 42
        02 12 22                50 51 52
  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We then consider Rectangular Full Packed (RFP) Format when N is
  odd. We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  the transpose of the first two columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  the transpose of the last two columns of AP lower.
  This covers the case N odd and TRANSR = 'N'.
         RFP A                   RFP A
        02 03 04                00 33 43
        12 13 14                10 11 44
        22 23 24                20 21 22
        00 33 34                30 31 32
        01 11 44                40 41 42
  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     02 12 22 00 01             00 10 20 30 40 50
     03 13 23 33 11             33 11 21 31 41 51
     04 14 24 34 44             43 44 22 32 42 52

 DPFTRI computes the inverse of a (real) symmetric positive definite
 matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
 computed by DPFTRF.

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  The Normal TRANSR of RFP A is stored;
          = 'T':  The Transpose TRANSR of RFP A is stored.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 )
          On entry, the symmetric matrix A in RFP format. RFP format is
          described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
          then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
          (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
          the transpose of RFP A as defined when
          TRANSR = 'N'. The contents of RFP A are defined by UPLO as
          follows: If UPLO = 'U' the RFP A contains the nt elements of
          upper packed A. If UPLO = 'L' the RFP A contains the elements
          of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
          'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N
          is odd. See the Note below for more details.
          On exit, the symmetric inverse of the original matrix, in the
          same storage format.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the (i,i) element of the factor U or L is
                zero, and the inverse could not be computed.

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  We first consider Rectangular Full Packed (RFP) Format when N is
  even. We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  the transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  the transpose of the last three columns of AP lower.
  This covers the case N even and TRANSR = 'N'.
         RFP A                   RFP A
        03 04 05                33 43 53
        13 14 15                00 44 54
        23 24 25                10 11 55
        33 34 35                20 21 22
        00 44 45                30 31 32
        01 11 55                40 41 42
        02 12 22                50 51 52
  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We then consider Rectangular Full Packed (RFP) Format when N is
  odd. We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  the transpose of the first two columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  the transpose of the last two columns of AP lower.
  This covers the case N odd and TRANSR = 'N'.
         RFP A                   RFP A
        02 03 04                00 33 43
        12 13 14                10 11 44
        22 23 24                20 21 22
        00 33 34                30 31 32
        01 11 44                40 41 42
  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     02 12 22 00 01             00 10 20 30 40 50
     03 13 23 33 11             33 11 21 31 41 51
     04 14 24 34 44             43 44 22 32 42 52

 DPFTRS solves a system of linear equations A*X = B with a symmetric
 positive definite matrix A using the Cholesky factorization
 A = U**T*U or A = L*L**T computed by DPFTRF.

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  The Normal TRANSR of RFP A is stored;
          = 'T':  The Transpose TRANSR of RFP A is stored.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of RFP A is stored;
          = 'L':  Lower triangle of RFP A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

A

          A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ).
          The triangular factor U or L from the Cholesky factorization
          of RFP A = U**T*U or RFP A = L*L**T, as computed by DPFTRF.
          See note below for more details about RFP A.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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  We first consider Rectangular Full Packed (RFP) Format when N is
  even. We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  the transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  the transpose of the last three columns of AP lower.
  This covers the case N even and TRANSR = 'N'.
         RFP A                   RFP A
        03 04 05                33 43 53
        13 14 15                00 44 54
        23 24 25                10 11 55
        33 34 35                20 21 22
        00 44 45                30 31 32
        01 11 55                40 41 42
        02 12 22                50 51 52
  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We then consider Rectangular Full Packed (RFP) Format when N is
  odd. We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  the transpose of the first two columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  the transpose of the last two columns of AP lower.
  This covers the case N odd and TRANSR = 'N'.
         RFP A                   RFP A
        02 03 04                00 33 43
        12 13 14                10 11 44
        22 23 24                20 21 22
        00 33 34                30 31 32
        01 11 44                40 41 42
  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     02 12 22 00 01             00 10 20 30 40 50
     03 13 23 33 11             33 11 21 31 41 51
     04 14 24 34 44             43 44 22 32 42 52

 DPPCON estimates the reciprocal of the condition number (in the
 1-norm) of a real symmetric positive definite packed matrix using
 the Cholesky factorization A = U**T*U or A = L*L**T computed by
 DPPTRF.
 An estimate is obtained for norm(inv(A)), and the reciprocal of the
 condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The triangular factor U or L from the Cholesky factorization
          A = U**T*U or A = L*L**T, packed columnwise in a linear
          array.  The j-th column of U or L is stored in the array AP
          as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

ANORM

          ANORM is DOUBLE PRECISION
          The 1-norm (or infinity-norm) of the symmetric matrix A.

RCOND

          RCOND is DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

WORK

          WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

          IWORK is INTEGER array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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 DPPEQU computes row and column scalings intended to equilibrate a
 symmetric positive definite matrix A in packed storage and reduce
 its condition number (with respect to the two-norm).  S contains the
 scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
 B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
 This choice of S puts the condition number of B within a factor N of
 the smallest possible condition number over all possible diagonal
 scalings.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The upper or lower triangle of the symmetric matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

S

          S is DOUBLE PRECISION array, dimension (N)
          If INFO = 0, S contains the scale factors for A.

SCOND

          SCOND is DOUBLE PRECISION
          If INFO = 0, S contains the ratio of the smallest S(i) to
          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
          large nor too small, it is not worth scaling by S.

AMAX

          AMAX is DOUBLE PRECISION
          Absolute value of largest matrix element.  If AMAX is very
          close to overflow or very close to underflow, the matrix
          should be scaled.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the i-th diagonal element is nonpositive.

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 DPPRFS improves the computed solution to a system of linear
 equations when the coefficient matrix is symmetric positive definite
 and packed, and provides error bounds and backward error estimates
 for the solution.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The upper or lower triangle of the symmetric matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

AFP

          AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The triangular factor U or L from the Cholesky factorization
          A = U**T*U or A = L*L**T, as computed by DPPTRF/ZPPTRF,
          packed columnwise in a linear array in the same format as A
          (see AP).

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by DPPTRS.
          On exit, the improved solution matrix X.

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

FERR

          FERR is DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)
          is an estimated upper bound for the magnitude of the largest
          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

BERR

          BERR is DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

WORK

          WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

          IWORK is INTEGER array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  ITMAX is the maximum number of steps of iterative refinement.

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 DPPTRF computes the Cholesky factorization of a real symmetric
 positive definite matrix A stored in packed format.
 The factorization has the form
    A = U**T * U,  if UPLO = 'U', or
    A = L  * L**T,  if UPLO = 'L',
 where U is an upper triangular matrix and L is lower triangular.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          See below for further details.
          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U**T*U or A = L*L**T, in the same
          storage format as A.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading minor of order i is not
                positive definite, and the factorization could not be
                completed.

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  The packed storage scheme is illustrated by the following example
  when N = 4, UPLO = 'U':
  Two-dimensional storage of the symmetric matrix A:
     a11 a12 a13 a14
         a22 a23 a24
             a33 a34     (aij = aji)
                 a44
  Packed storage of the upper triangle of A:
  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

 DPPTRI computes the inverse of a real symmetric positive definite
 matrix A using the Cholesky factorization A = U**T*U or A = L*L**T
 computed by DPPTRF.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangular factor is stored in AP;
          = 'L':  Lower triangular factor is stored in AP.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          On entry, the triangular factor U or L from the Cholesky
          factorization A = U**T*U or A = L*L**T, packed columnwise as
          a linear array.  The j-th column of U or L is stored in the
          array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.
          On exit, the upper or lower triangle of the (symmetric)
          inverse of A, overwriting the input factor U or L.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the (i,i) element of the factor U or L is
                zero, and the inverse could not be computed.

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 DPPTRS solves a system of linear equations A*X = B with a symmetric
 positive definite matrix A in packed storage using the Cholesky
 factorization A = U**T*U or A = L*L**T computed by DPPTRF.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The triangular factor U or L from the Cholesky factorization
          A = U**T*U or A = L*L**T, packed columnwise in a linear
          array.  The j-th column of U or L is stored in the array AP
          as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = U(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = L(i,j) for j<=i<=n.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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 DPSTF2 computes the Cholesky factorization with complete
 pivoting of a real symmetric positive semidefinite matrix A.
 The factorization has the form
    P**T * A * P = U**T * U ,  if UPLO = 'U',
    P**T * A * P = L  * L**T,  if UPLO = 'L',
 where U is an upper triangular matrix and L is lower triangular, and
 P is stored as vector PIV.
 This algorithm does not attempt to check that A is positive
 semidefinite. This version of the algorithm calls level 2 BLAS.

UPLO

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n by n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n by n lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit, if INFO = 0, the factor U or L from the Cholesky
          factorization as above.

PIV

          PIV is INTEGER array, dimension (N)
          PIV is such that the nonzero entries are P( PIV(K), K ) = 1.

RANK

          RANK is INTEGER
          The rank of A given by the number of steps the algorithm
          completed.

TOL

          TOL is DOUBLE PRECISION
          User defined tolerance. If TOL < 0, then N*U*MAX( A( K,K ) )
          will be used. The algorithm terminates at the (K-1)st step
          if the pivot <= TOL.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

WORK

          WORK is DOUBLE PRECISION array, dimension (2*N)
          Work space.

INFO

          INFO is INTEGER
          < 0: If INFO = -K, the K-th argument had an illegal value,
          = 0: algorithm completed successfully, and
          > 0: the matrix A is either rank deficient with computed rank
               as returned in RANK, or is not positive semidefinite. See
               Section 7 of LAPACK Working Note #161 for further
               information.

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 DPSTRF computes the Cholesky factorization with complete
 pivoting of a real symmetric positive semidefinite matrix A.
 The factorization has the form
    P**T * A * P = U**T * U ,  if UPLO = 'U',
    P**T * A * P = L  * L**T,  if UPLO = 'L',
 where U is an upper triangular matrix and L is lower triangular, and
 P is stored as vector PIV.
 This algorithm does not attempt to check that A is positive
 semidefinite. This version of the algorithm calls level 3 BLAS.

UPLO

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
          n by n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n by n lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit, if INFO = 0, the factor U or L from the Cholesky
          factorization as above.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

PIV

          PIV is INTEGER array, dimension (N)
          PIV is such that the nonzero entries are P( PIV(K), K ) = 1.

RANK

          RANK is INTEGER
          The rank of A given by the number of steps the algorithm
          completed.

TOL

          TOL is DOUBLE PRECISION
          User defined tolerance. If TOL < 0, then N*U*MAX( A(K,K) )
          will be used. The algorithm terminates at the (K-1)st step
          if the pivot <= TOL.

WORK

          WORK is DOUBLE PRECISION array, dimension (2*N)
          Work space.

INFO

          INFO is INTEGER
          < 0: If INFO = -K, the K-th argument had an illegal value,
          = 0: algorithm completed successfully, and
          > 0: the matrix A is either rank deficient with computed rank
               as returned in RANK, or is not positive semidefinite. See
               Section 7 of LAPACK Working Note #161 for further
               information.

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 DSBGST reduces a real symmetric-definite banded generalized
 eigenproblem  A*x = lambda*B*x  to standard form  C*y = lambda*y,
 such that C has the same bandwidth as A.
 B must have been previously factorized as S**T*S by DPBSTF, using a
 split Cholesky factorization. A is overwritten by C = X**T*A*X, where
 X = S**(-1)*Q and Q is an orthogonal matrix chosen to preserve the
 bandwidth of A.

VECT

          VECT is CHARACTER*1
          = 'N':  do not form the transformation matrix X;
          = 'V':  form X.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrices A and B.  N >= 0.

KA

          KA is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KA >= 0.

KB

          KB is INTEGER
          The number of superdiagonals of the matrix B if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KA >= KB >= 0.

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first ka+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+ka).
          On exit, the transformed matrix X**T*A*X, stored in the same
          format as A.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KA+1.

BB

          BB is DOUBLE PRECISION array, dimension (LDBB,N)
          The banded factor S from the split Cholesky factorization of
          B, as returned by DPBSTF, stored in the first KB+1 rows of
          the array.

LDBB

          LDBB is INTEGER
          The leading dimension of the array BB.  LDBB >= KB+1.

X

          X is DOUBLE PRECISION array, dimension (LDX,N)
          If VECT = 'V', the n-by-n matrix X.
          If VECT = 'N', the array X is not referenced.

LDX

          LDX is INTEGER
          The leading dimension of the array X.
          LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise.

WORK

          WORK is DOUBLE PRECISION array, dimension (2*N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.

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 DSBTRD reduces a real symmetric band matrix A to symmetric
 tridiagonal form T by an orthogonal similarity transformation:
 Q**T * A * Q = T.

VECT

          VECT is CHARACTER*1
          = 'N':  do not form Q;
          = 'V':  form Q;
          = 'U':  update a matrix X, by forming X*Q.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U',
          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          On exit, the diagonal elements of AB are overwritten by the
          diagonal elements of the tridiagonal matrix T; if KD > 0, the
          elements on the first superdiagonal (if UPLO = 'U') or the
          first subdiagonal (if UPLO = 'L') are overwritten by the
          off-diagonal elements of T; the rest of AB is overwritten by
          values generated during the reduction.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

D

          D is DOUBLE PRECISION array, dimension (N)
          The diagonal elements of the tridiagonal matrix T.

E

          E is DOUBLE PRECISION array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.

Q

          Q is DOUBLE PRECISION array, dimension (LDQ,N)
          On entry, if VECT = 'U', then Q must contain an N-by-N
          matrix X; if VECT = 'N' or 'V', then Q need not be set.
          On exit:
          if VECT = 'V', Q contains the N-by-N orthogonal matrix Q;
          if VECT = 'U', Q contains the product X*Q;
          if VECT = 'N', the array Q is not referenced.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.
          LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'.

WORK

          WORK is DOUBLE PRECISION array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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  Modified by Linda Kaufman, Bell Labs.

 Level 3 BLAS like routine for C in RFP Format.
 DSFRK performs one of the symmetric rank--k operations
    C := alpha*A*A**T + beta*C,
 or
    C := alpha*A**T*A + beta*C,
 where alpha and beta are real scalars, C is an n--by--n symmetric
 matrix and A is an n--by--k matrix in the first case and a k--by--n
 matrix in the second case.

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  The Normal Form of RFP A is stored;
          = 'T':  The Transpose Form of RFP A is stored.

UPLO

          UPLO is CHARACTER*1
           On  entry, UPLO specifies whether the upper or lower
           triangular part of the array C is to be referenced as
           follows:
              UPLO = 'U' or 'u'   Only the upper triangular part of C
                                  is to be referenced.
              UPLO = 'L' or 'l'   Only the lower triangular part of C
                                  is to be referenced.
           Unchanged on exit.

TRANS

          TRANS is CHARACTER*1
           On entry, TRANS specifies the operation to be performed as
           follows:
              TRANS = 'N' or 'n'   C := alpha*A*A**T + beta*C.
              TRANS = 'T' or 't'   C := alpha*A**T*A + beta*C.
           Unchanged on exit.

N

          N is INTEGER
           On entry, N specifies the order of the matrix C. N must be
           at least zero.
           Unchanged on exit.

K

          K is INTEGER
           On entry with TRANS = 'N' or 'n', K specifies the number
           of  columns of the matrix A, and on entry with TRANS = 'T'
           or 't', K specifies the number of rows of the matrix A. K
           must be at least zero.
           Unchanged on exit.

ALPHA

          ALPHA is DOUBLE PRECISION
           On entry, ALPHA specifies the scalar alpha.
           Unchanged on exit.

A

          A is DOUBLE PRECISION array, dimension (LDA,ka)
           where KA
           is K  when TRANS = 'N' or 'n', and is N otherwise. Before
           entry with TRANS = 'N' or 'n', the leading N--by--K part of
           the array A must contain the matrix A, otherwise the leading
           K--by--N part of the array A must contain the matrix A.
           Unchanged on exit.

LDA

          LDA is INTEGER
           On entry, LDA specifies the first dimension of A as declared
           in  the  calling  (sub)  program.   When  TRANS = 'N' or 'n'
           then  LDA must be at least  max( 1, n ), otherwise  LDA must
           be at least  max( 1, k ).
           Unchanged on exit.

BETA

          BETA is DOUBLE PRECISION
           On entry, BETA specifies the scalar beta.
           Unchanged on exit.

C

          C is DOUBLE PRECISION array, dimension (NT)
           NT = N*(N+1)/2. On entry, the symmetric matrix C in RFP
           Format. RFP Format is described by TRANSR, UPLO and N.

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 DSPCON estimates the reciprocal of the condition number (in the
 1-norm) of a real symmetric packed matrix A using the factorization
 A = U*D*U**T or A = L*D*L**T computed by DSPTRF.
 An estimate is obtained for norm(inv(A)), and the reciprocal of the
 condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).

UPLO

          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by DSPTRF, stored as a
          packed triangular matrix.

IPIV

          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by DSPTRF.

ANORM

          ANORM is DOUBLE PRECISION
          The 1-norm of the original matrix A.

RCOND

          RCOND is DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
          estimate of the 1-norm of inv(A) computed in this routine.

WORK

          WORK is DOUBLE PRECISION array, dimension (2*N)

IWORK

          IWORK is INTEGER array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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 DSPGST reduces a real symmetric-definite generalized eigenproblem
 to standard form, using packed storage.
 If ITYPE = 1, the problem is A*x = lambda*B*x,
 and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
 If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
 B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
 B must have been previously factorized as U**T*U or L*L**T by DPPTRF.

ITYPE

          ITYPE is INTEGER
          = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
          = 2 or 3: compute U*A*U**T or L**T*A*L.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored and B is factored as
                  U**T*U;
          = 'L':  Lower triangle of A is stored and B is factored as
                  L*L**T.

N

          N is INTEGER
          The order of the matrices A and B.  N >= 0.

AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          On exit, if INFO = 0, the transformed matrix, stored in the
          same format as A.

BP

          BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The triangular factor from the Cholesky factorization of B,
          stored in the same format as A, as returned by DPPTRF.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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 DSPRFS improves the computed solution to a system of linear
 equations when the coefficient matrix is symmetric indefinite
 and packed, and provides error bounds and backward error estimates
 for the solution.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The upper or lower triangle of the symmetric matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.

AFP

          AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The factored form of the matrix A.  AFP contains the block
          diagonal matrix D and the multipliers used to obtain the
          factor U or L from the factorization A = U*D*U**T or
          A = L*D*L**T as computed by DSPTRF, stored as a packed
          triangular matrix.

IPIV

          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by DSPTRF.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by DSPTRS.
          On exit, the improved solution matrix X.

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

FERR

          FERR is DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)
          is an estimated upper bound for the magnitude of the largest
          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

BERR

          BERR is DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

WORK

          WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

          IWORK is INTEGER array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

  ITMAX is the maximum number of steps of iterative refinement.

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 DSPTRD reduces a real symmetric matrix A stored in packed form to
 symmetric tridiagonal form T by an orthogonal similarity
 transformation: Q**T * A * Q = T.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
          On exit, if UPLO = 'U', the diagonal and first superdiagonal
          of A are overwritten by the corresponding elements of the
          tridiagonal matrix T, and the elements above the first
          superdiagonal, with the array TAU, represent the orthogonal
          matrix Q as a product of elementary reflectors; if UPLO
          = 'L', the diagonal and first subdiagonal of A are over-
          written by the corresponding elements of the tridiagonal
          matrix T, and the elements below the first subdiagonal, with
          the array TAU, represent the orthogonal matrix Q as a product
          of elementary reflectors. See Further Details.

D

          D is DOUBLE PRECISION array, dimension (N)
          The diagonal elements of the tridiagonal matrix T:
          D(i) = A(i,i).

E

          E is DOUBLE PRECISION array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T:
          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.

TAU

          TAU is DOUBLE PRECISION array, dimension (N-1)
          The scalar factors of the elementary reflectors (see Further
          Details).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors
     Q = H(n-1) . . . H(2) H(1).
  Each H(i) has the form
     H(i) = I - tau * v * v**T
  where tau is a real scalar, and v is a real vector with
  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
  overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors
     Q = H(1) H(2) . . . H(n-1).
  Each H(i) has the form
     H(i) = I - tau * v * v**T
  where tau is a real scalar, and v is a real vector with
  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
  overwriting A(i+2:n,i), and tau is stored in TAU(i).

 DSPTRF computes the factorization of a real symmetric matrix A stored
 in packed format using the Bunch-Kaufman diagonal pivoting method:
    A = U*D*U**T  or  A = L*D*L**T
 where U (or L) is a product of permutation and unit upper (lower)
 triangular matrices, and D is symmetric and block diagonal with
 1-by-1 and 2-by-2 diagonal blocks.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L, stored as a packed triangular
          matrix overwriting A (see below for further details).

IPIV

          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D.
          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization
               has been completed, but the block diagonal matrix D is
               exactly singular, and division by zero will occur if it
               is used to solve a system of equations.

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  If UPLO = 'U', then A = U*D*U**T, where
     U = P(n)*U(n)* ... *P(k)U(k)* ...,
  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then
             (   I    v    0   )   k-s
     U(k) =  (   0    I    0   )   s
             (   0    0    I   )   n-k
                k-s   s   n-k
  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  and A(k,k), and v overwrites A(1:k-2,k-1:k).
  If UPLO = 'L', then A = L*D*L**T, where
     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then
             (   I    0     0   )  k-1
     L(k) =  (   0    I     0   )  s
             (   0    v     I   )  n-k-s+1
                k-1   s  n-k-s+1
  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

J. Lewis, Boeing Computer Services Company

 DSPTRI computes the inverse of a real symmetric indefinite matrix
 A in packed storage using the factorization A = U*D*U**T or
 A = L*D*L**T computed by DSPTRF.

UPLO

          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          On entry, the block diagonal matrix D and the multipliers
          used to obtain the factor U or L as computed by DSPTRF,
          stored as a packed triangular matrix.
          On exit, if INFO = 0, the (symmetric) inverse of the original
          matrix, stored as a packed triangular matrix. The j-th column
          of inv(A) is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = inv(A)(i,j) for 1<=i<=j;
          if UPLO = 'L',
             AP(i + (j-1)*(2n-j)/2) = inv(A)(i,j) for j<=i<=n.

IPIV

          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by DSPTRF.

WORK

          WORK is DOUBLE PRECISION array, dimension (N)

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
               inverse could not be computed.

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 DSPTRS solves a system of linear equations A*X = B with a real
 symmetric matrix A stored in packed format using the factorization
 A = U*D*U**T or A = L*D*L**T computed by DSPTRF.

UPLO

          UPLO is CHARACTER*1
          Specifies whether the details of the factorization are stored
          as an upper or lower triangular matrix.
          = 'U':  Upper triangular, form is A = U*D*U**T;
          = 'L':  Lower triangular, form is A = L*D*L**T.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

AP

          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The block diagonal matrix D and the multipliers used to
          obtain the factor U or L as computed by DSPTRF, stored as a
          packed triangular matrix.

IPIV

          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D
          as determined by DSPTRF.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value

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 DSTEGR computes selected eigenvalues and, optionally, eigenvectors
 of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
 a well defined set of pairwise different real eigenvalues, the corresponding
 real eigenvectors are pairwise orthogonal.
 The spectrum may be computed either completely or partially by specifying
 either an interval (VL,VU] or a range of indices IL:IU for the desired
 eigenvalues.
 DSTEGR is a compatibility wrapper around the improved DSTEMR routine.
 See DSTEMR for further details.
 One important change is that the ABSTOL parameter no longer provides any
 benefit and hence is no longer used.
 Note : DSTEGR and DSTEMR work only on machines which follow
 IEEE-754 floating-point standard in their handling of infinities and
 NaNs.  Normal execution may create these exceptiona values and hence
 may abort due to a floating point exception in environments which
 do not conform to the IEEE-754 standard.

JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

RANGE

          RANGE is CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

N

          N is INTEGER
          The order of the matrix.  N >= 0.

D

          D is DOUBLE PRECISION array, dimension (N)
          On entry, the N diagonal elements of the tridiagonal matrix
          T. On exit, D is overwritten.

E

          E is DOUBLE PRECISION array, dimension (N)
          On entry, the (N-1) subdiagonal elements of the tridiagonal
          matrix T in elements 1 to N-1 of E. E(N) need not be set on
          input, but is used internally as workspace.
          On exit, E is overwritten.

VL

          VL is DOUBLE PRECISION
          If RANGE='V', the lower bound of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

VU

          VU is DOUBLE PRECISION
          If RANGE='V', the upper bound of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

IL

          IL is INTEGER
          If RANGE='I', the index of the
          smallest eigenvalue to be returned.
          1 <= IL <= IU <= N, if N > 0.
          Not referenced if RANGE = 'A' or 'V'.

IU

          IU is INTEGER
          If RANGE='I', the index of the
          largest eigenvalue to be returned.
          1 <= IL <= IU <= N, if N > 0.
          Not referenced if RANGE = 'A' or 'V'.

ABSTOL

          ABSTOL is DOUBLE PRECISION
          Unused.  Was the absolute error tolerance for the
          eigenvalues/eigenvectors in previous versions.

M

          M is INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

W

          W is DOUBLE PRECISION array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
          contain the orthonormal eigenvectors of the matrix T
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.
          Supplying N columns is always safe.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', then LDZ >= max(1,N).

ISUPPZ

          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
          The support of the eigenvectors in Z, i.e., the indices
          indicating the nonzero elements in Z. The i-th computed eigenvector
          is nonzero only in elements ISUPPZ( 2*i-1 ) through
          ISUPPZ( 2*i ). This is relevant in the case when the matrix
          is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal
          (and minimal) LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,18*N)
          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (LIWORK)
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK

          LIWORK is INTEGER
          The dimension of the array IWORK.  LIWORK >= max(1,10*N)
          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
          if only the eigenvalues are to be computed.
          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,
          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          On exit, INFO
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = 1X, internal error in DLARRE,
                if INFO = 2X, internal error in DLARRV.
                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                the nonzero error code returned by DLARRE or
                DLARRV, respectively.

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 DSTEIN computes the eigenvectors of a real symmetric tridiagonal
 matrix T corresponding to specified eigenvalues, using inverse
 iteration.
 The maximum number of iterations allowed for each eigenvector is
 specified by an internal parameter MAXITS (currently set to 5).

N

          N is INTEGER
          The order of the matrix.  N >= 0.

D

          D is DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the tridiagonal matrix T.

E

          E is DOUBLE PRECISION array, dimension (N-1)
          The (n-1) subdiagonal elements of the tridiagonal matrix
          T, in elements 1 to N-1.

M

          M is INTEGER
          The number of eigenvectors to be found.  0 <= M <= N.

W

          W is DOUBLE PRECISION array, dimension (N)
          The first M elements of W contain the eigenvalues for
          which eigenvectors are to be computed.  The eigenvalues
          should be grouped by split-off block and ordered from
          smallest to largest within the block.  ( The output array
          W from DSTEBZ with ORDER = 'B' is expected here. )

IBLOCK

          IBLOCK is INTEGER array, dimension (N)
          The submatrix indices associated with the corresponding
          eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
          the first submatrix from the top, =2 if W(i) belongs to
          the second submatrix, etc.  ( The output array IBLOCK
          from DSTEBZ is expected here. )

ISPLIT

          ISPLIT is INTEGER array, dimension (N)
          The splitting points, at which T breaks up into submatrices.
          The first submatrix consists of rows/columns 1 to
          ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
          through ISPLIT( 2 ), etc.
          ( The output array ISPLIT from DSTEBZ is expected here. )

Z

          Z is DOUBLE PRECISION array, dimension (LDZ, M)
          The computed eigenvectors.  The eigenvector associated
          with the eigenvalue W(i) is stored in the i-th column of
          Z.  Any vector which fails to converge is set to its current
          iterate after MAXITS iterations.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= max(1,N).

WORK

          WORK is DOUBLE PRECISION array, dimension (5*N)

IWORK

          IWORK is INTEGER array, dimension (N)

IFAIL

          IFAIL is INTEGER array, dimension (M)
          On normal exit, all elements of IFAIL are zero.
          If one or more eigenvectors fail to converge after
          MAXITS iterations, then their indices are stored in
          array IFAIL.

INFO

          INFO is INTEGER
          = 0: successful exit.
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, then i eigenvectors failed to converge
               in MAXITS iterations.  Their indices are stored in
               array IFAIL.

  MAXITS  INTEGER, default = 5
          The maximum number of iterations performed.
  EXTRA   INTEGER, default = 2
          The number of iterations performed after norm growth
          criterion is satisfied, should be at least 1.

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 DSTEMR computes selected eigenvalues and, optionally, eigenvectors
 of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
 a well defined set of pairwise different real eigenvalues, the corresponding
 real eigenvectors are pairwise orthogonal.
 The spectrum may be computed either completely or partially by specifying
 either an interval (VL,VU] or a range of indices IL:IU for the desired
 eigenvalues.
 Depending on the number of desired eigenvalues, these are computed either
 by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
 computed by the use of various suitable L D L^T factorizations near clusters
 of close eigenvalues (referred to as RRRs, Relatively Robust
 Representations). An informal sketch of the algorithm follows.
 For each unreduced block (submatrix) of T,
    (a) Compute T - sigma I  = L D L^T, so that L and D
        define all the wanted eigenvalues to high relative accuracy.
        This means that small relative changes in the entries of D and L
        cause only small relative changes in the eigenvalues and
        eigenvectors. The standard (unfactored) representation of the
        tridiagonal matrix T does not have this property in general.
    (b) Compute the eigenvalues to suitable accuracy.
        If the eigenvectors are desired, the algorithm attains full
        accuracy of the computed eigenvalues only right before
        the corresponding vectors have to be computed, see steps c) and d).
    (c) For each cluster of close eigenvalues, select a new
        shift close to the cluster, find a new factorization, and refine
        the shifted eigenvalues to suitable accuracy.
    (d) For each eigenvalue with a large enough relative separation compute
        the corresponding eigenvector by forming a rank revealing twisted
        factorization. Go back to (c) for any clusters that remain.
 For more details, see:
 - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
   to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
   Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
 - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
   Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
   2004.  Also LAPACK Working Note 154.
 - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
   tridiagonal eigenvalue/eigenvector problem",
   Computer Science Division Technical Report No. UCB/CSD-97-971,
   UC Berkeley, May 1997.
 Further Details
 1.DSTEMR works only on machines which follow IEEE-754
 floating-point standard in their handling of infinities and NaNs.
 This permits the use of efficient inner loops avoiding a check for
 zero divisors.

JOBZ

          JOBZ is CHARACTER*1
          = 'N':  Compute eigenvalues only;
          = 'V':  Compute eigenvalues and eigenvectors.

RANGE

          RANGE is CHARACTER*1
          = 'A': all eigenvalues will be found.
          = 'V': all eigenvalues in the half-open interval (VL,VU]
                 will be found.
          = 'I': the IL-th through IU-th eigenvalues will be found.

N

          N is INTEGER
          The order of the matrix.  N >= 0.

D

          D is DOUBLE PRECISION array, dimension (N)
          On entry, the N diagonal elements of the tridiagonal matrix
          T. On exit, D is overwritten.

E

          E is DOUBLE PRECISION array, dimension (N)
          On entry, the (N-1) subdiagonal elements of the tridiagonal
          matrix T in elements 1 to N-1 of E. E(N) need not be set on
          input, but is used internally as workspace.
          On exit, E is overwritten.

VL

          VL is DOUBLE PRECISION
          If RANGE='V', the lower bound of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

VU

          VU is DOUBLE PRECISION
          If RANGE='V', the upper bound of the interval to
          be searched for eigenvalues. VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

IL

          IL is INTEGER
          If RANGE='I', the index of the
          smallest eigenvalue to be returned.
          1 <= IL <= IU <= N, if N > 0.
          Not referenced if RANGE = 'A' or 'V'.

IU

          IU is INTEGER
          If RANGE='I', the index of the
          largest eigenvalue to be returned.
          1 <= IL <= IU <= N, if N > 0.
          Not referenced if RANGE = 'A' or 'V'.

M

          M is INTEGER
          The total number of eigenvalues found.  0 <= M <= N.
          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

W

          W is DOUBLE PRECISION array, dimension (N)
          The first M elements contain the selected eigenvalues in
          ascending order.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
          If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
          contain the orthonormal eigenvectors of the matrix T
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          If JOBZ = 'N', then Z is not referenced.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and can be computed with a workspace
          query by setting NZC = -1, see below.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', then LDZ >= max(1,N).

NZC

          NZC is INTEGER
          The number of eigenvectors to be held in the array Z.
          If RANGE = 'A', then NZC >= max(1,N).
          If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
          If RANGE = 'I', then NZC >= IU-IL+1.
          If NZC = -1, then a workspace query is assumed; the
          routine calculates the number of columns of the array Z that
          are needed to hold the eigenvectors.
          This value is returned as the first entry of the Z array, and
          no error message related to NZC is issued by XERBLA.

ISUPPZ

          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
          The support of the eigenvectors in Z, i.e., the indices
          indicating the nonzero elements in Z. The i-th computed eigenvector
          is nonzero only in elements ISUPPZ( 2*i-1 ) through
          ISUPPZ( 2*i ). This is relevant in the case when the matrix
          is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.

TRYRAC

          TRYRAC is LOGICAL
          If TRYRAC = .TRUE., indicates that the code should check whether
          the tridiagonal matrix defines its eigenvalues to high relative
          accuracy.  If so, the code uses relative-accuracy preserving
          algorithms that might be (a bit) slower depending on the matrix.
          If the matrix does not define its eigenvalues to high relative
          accuracy, the code can uses possibly faster algorithms.
          If TRYRAC = .FALSE., the code is not required to guarantee
          relatively accurate eigenvalues and can use the fastest possible
          techniques.
          On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
          does not define its eigenvalues to high relative accuracy.

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal
          (and minimal) LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,18*N)
          if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (LIWORK)
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK

          LIWORK is INTEGER
          The dimension of the array IWORK.  LIWORK >= max(1,10*N)
          if the eigenvectors are desired, and LIWORK >= max(1,8*N)
          if only the eigenvalues are to be computed.
          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,
          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

INFO

          INFO is INTEGER
          On exit, INFO
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = 1X, internal error in DLARRE,
                if INFO = 2X, internal error in DLARRV.
                Here, the digit X = ABS( IINFO ) < 10, where IINFO is
                the nonzero error code returned by DLARRE or
                DLARRV, respectively.

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Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USA

 DTBCON estimates the reciprocal of the condition number of a
 triangular band matrix A, in either the 1-norm or the infinity-norm.
 The norm of A is computed and an estimate is obtained for
 norm(inv(A)), then the reciprocal of the condition number is
 computed as
    RCOND = 1 / ( norm(A) * norm(inv(A)) ).

NORM

          NORM is CHARACTER*1
          Specifies whether the 1-norm condition number or the
          infinity-norm condition number is required:
          = '1' or 'O':  1-norm;
          = 'I':         Infinity-norm.

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

DIAG

          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals or subdiagonals of the
          triangular band matrix A.  KD >= 0.

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first kd+1 rows of the array. The j-th column of A is stored
          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          If DIAG = 'U', the diagonal elements of A are not referenced
          and are assumed to be 1.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

RCOND

          RCOND is DOUBLE PRECISION
          The reciprocal of the condition number of the matrix A,
          computed as RCOND = 1/(norm(A) * norm(inv(A))).

WORK

          WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

          IWORK is INTEGER array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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 DTBRFS provides error bounds and backward error estimates for the
 solution to a system of linear equations with a triangular band
 coefficient matrix.
 The solution matrix X must be computed by DTBTRS or some other
 means before entering this routine.  DTBRFS does not do iterative
 refinement because doing so cannot improve the backward error.

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

TRANS

          TRANS is CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

DIAG

          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals or subdiagonals of the
          triangular band matrix A.  KD >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first kd+1 rows of the array. The j-th column of A is stored
          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          If DIAG = 'U', the diagonal elements of A are not referenced
          and are assumed to be 1.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
          The solution matrix X.

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).

FERR

          FERR is DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)
          is an estimated upper bound for the magnitude of the largest
          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

BERR

          BERR is DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

WORK

          WORK is DOUBLE PRECISION array, dimension (3*N)

IWORK

          IWORK is INTEGER array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

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 DTBTRS solves a triangular system of the form
    A * X = B  or  A**T * X = B,
 where A is a triangular band matrix of order N, and B is an
 N-by NRHS matrix.  A check is made to verify that A is nonsingular.

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

TRANS

          TRANS is CHARACTER*1
          Specifies the form the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

DIAG

          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of superdiagonals or subdiagonals of the
          triangular band matrix A.  KD >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

AB

          AB is DOUBLE PRECISION array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first kd+1 rows of AB.  The j-th column of A is stored
          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
          If DIAG = 'U', the diagonal elements of A are not referenced
          and are assumed to be 1.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, if INFO = 0, the solution matrix X.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the i-th diagonal element of A is zero,
                indicating that the matrix is singular and the
                solutions X have not been computed.

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 Level 3 BLAS like routine for A in RFP Format.
 DTFSM  solves the matrix equation
    op( A )*X = alpha*B  or  X*op( A ) = alpha*B
 where alpha is a scalar, X and B are m by n matrices, A is a unit, or
 non-unit,  upper or lower triangular matrix  and  op( A )  is one  of
    op( A ) = A   or   op( A ) = A**T.
 A is in Rectangular Full Packed (RFP) Format.
 The matrix X is overwritten on B.

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  The Normal Form of RFP A is stored;
          = 'T':  The Transpose Form of RFP A is stored.

SIDE

          SIDE is CHARACTER*1
           On entry, SIDE specifies whether op( A ) appears on the left
           or right of X as follows:
              SIDE = 'L' or 'l'   op( A )*X = alpha*B.
              SIDE = 'R' or 'r'   X*op( A ) = alpha*B.
           Unchanged on exit.

UPLO

          UPLO is CHARACTER*1
           On entry, UPLO specifies whether the RFP matrix A came from
           an upper or lower triangular matrix as follows:
           UPLO = 'U' or 'u' RFP A came from an upper triangular matrix
           UPLO = 'L' or 'l' RFP A came from a  lower triangular matrix
           Unchanged on exit.

TRANS

          TRANS is CHARACTER*1
           On entry, TRANS  specifies the form of op( A ) to be used
           in the matrix multiplication as follows:
              TRANS  = 'N' or 'n'   op( A ) = A.
              TRANS  = 'T' or 't'   op( A ) = A'.
           Unchanged on exit.

DIAG

          DIAG is CHARACTER*1
           On entry, DIAG specifies whether or not RFP A is unit
           triangular as follows:
              DIAG = 'U' or 'u'   A is assumed to be unit triangular.
              DIAG = 'N' or 'n'   A is not assumed to be unit
                                  triangular.
           Unchanged on exit.

M

          M is INTEGER
           On entry, M specifies the number of rows of B. M must be at
           least zero.
           Unchanged on exit.

N

          N is INTEGER
           On entry, N specifies the number of columns of B.  N must be
           at least zero.
           Unchanged on exit.

ALPHA

          ALPHA is DOUBLE PRECISION
           On entry,  ALPHA specifies the scalar  alpha. When  alpha is
           zero then  A is not referenced and  B need not be set before
           entry.
           Unchanged on exit.

A

          A is DOUBLE PRECISION array, dimension (NT)
           NT = N*(N+1)/2. On entry, the matrix A in RFP Format.
           RFP Format is described by TRANSR, UPLO and N as follows:
           If TRANSR='N' then RFP A is (0:N,0:K-1) when N is even;
           K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If
           TRANSR = 'T' then RFP is the transpose of RFP A as
           defined when TRANSR = 'N'. The contents of RFP A are defined
           by UPLO as follows: If UPLO = 'U' the RFP A contains the NT
           elements of upper packed A either in normal or
           transpose Format. If UPLO = 'L' the RFP A contains
           the NT elements of lower packed A either in normal or
           transpose Format. The LDA of RFP A is (N+1)/2 when
           TRANSR = 'T'. When TRANSR is 'N' the LDA is N+1 when N is
           even and is N when is odd.
           See the Note below for more details. Unchanged on exit.

B

          B is DOUBLE PRECISION array, dimension (LDB,N)
           Before entry,  the leading  m by n part of the array  B must
           contain  the  right-hand  side  matrix  B,  and  on exit  is
           overwritten by the solution matrix  X.

LDB

          LDB is INTEGER
           On entry, LDB specifies the first dimension of B as declared
           in  the  calling  (sub)  program.   LDB  must  be  at  least
           max( 1, m ).
           Unchanged on exit.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  We first consider Rectangular Full Packed (RFP) Format when N is
  even. We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  the transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  the transpose of the last three columns of AP lower.
  This covers the case N even and TRANSR = 'N'.
         RFP A                   RFP A
        03 04 05                33 43 53
        13 14 15                00 44 54
        23 24 25                10 11 55
        33 34 35                20 21 22
        00 44 45                30 31 32
        01 11 55                40 41 42
        02 12 22                50 51 52
  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We then consider Rectangular Full Packed (RFP) Format when N is
  odd. We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  the transpose of the first two columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  the transpose of the last two columns of AP lower.
  This covers the case N odd and TRANSR = 'N'.
         RFP A                   RFP A
        02 03 04                00 33 43
        12 13 14                10 11 44
        22 23 24                20 21 22
        00 33 34                30 31 32
        01 11 44                40 41 42
  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     02 12 22 00 01             00 10 20 30 40 50
     03 13 23 33 11             33 11 21 31 41 51
     04 14 24 34 44             43 44 22 32 42 52

 DTFTRI computes the inverse of a triangular matrix A stored in RFP
 format.
 This is a Level 3 BLAS version of the algorithm.

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  The Normal TRANSR of RFP A is stored;
          = 'T':  The Transpose TRANSR of RFP A is stored.

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

DIAG

          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is DOUBLE PRECISION array, dimension (0:nt-1);
          nt=N*(N+1)/2. On entry, the triangular factor of a Hermitian
          Positive Definite matrix A in RFP format. RFP format is
          described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
          then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
          (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is
          the transpose of RFP A as defined when
          TRANSR = 'N'. The contents of RFP A are defined by UPLO as
          follows: If UPLO = 'U' the RFP A contains the nt elements of
          upper packed A; If UPLO = 'L' the RFP A contains the nt
          elements of lower packed A. The LDA of RFP A is (N+1)/2 when
          TRANSR = 'T'. When TRANSR is 'N' the LDA is N+1 when N is
          even and N is odd. See the Note below for more details.
          On exit, the (triangular) inverse of the original matrix, in
          the same storage format.

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, A(i,i) is exactly zero.  The triangular
               matrix is singular and its inverse can not be computed.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  We first consider Rectangular Full Packed (RFP) Format when N is
  even. We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  the transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  the transpose of the last three columns of AP lower.
  This covers the case N even and TRANSR = 'N'.
         RFP A                   RFP A
        03 04 05                33 43 53
        13 14 15                00 44 54
        23 24 25                10 11 55
        33 34 35                20 21 22
        00 44 45                30 31 32
        01 11 55                40 41 42
        02 12 22                50 51 52
  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We then consider Rectangular Full Packed (RFP) Format when N is
  odd. We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  the transpose of the first two columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  the transpose of the last two columns of AP lower.
  This covers the case N odd and TRANSR = 'N'.
         RFP A                   RFP A
        02 03 04                00 33 43
        12 13 14                10 11 44
        22 23 24                20 21 22
        00 33 34                30 31 32
        01 11 44                40 41 42
  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     02 12 22 00 01             00 10 20 30 40 50
     03 13 23 33 11             33 11 21 31 41 51
     04 14 24 34 44             43 44 22 32 42 52

 DTFTTP copies a triangular matrix A from rectangular full packed
 format (TF) to standard packed format (TP).

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  ARF is in Normal format;
          = 'T':  ARF is in Transpose format;

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

N

          N is INTEGER
          The order of the matrix A. N >= 0.

ARF

          ARF is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
          On entry, the upper or lower triangular matrix A stored in
          RFP format. For a further discussion see Notes below.

AP

          AP is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ),
          On exit, the upper or lower triangular matrix A, packed
          columnwise in a linear array. The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  We first consider Rectangular Full Packed (RFP) Format when N is
  even. We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  the transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  the transpose of the last three columns of AP lower.
  This covers the case N even and TRANSR = 'N'.
         RFP A                   RFP A
        03 04 05                33 43 53
        13 14 15                00 44 54
        23 24 25                10 11 55
        33 34 35                20 21 22
        00 44 45                30 31 32
        01 11 55                40 41 42
        02 12 22                50 51 52
  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We then consider Rectangular Full Packed (RFP) Format when N is
  odd. We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  the transpose of the first two columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  the transpose of the last two columns of AP lower.
  This covers the case N odd and TRANSR = 'N'.
         RFP A                   RFP A
        02 03 04                00 33 43
        12 13 14                10 11 44
        22 23 24                20 21 22
        00 33 34                30 31 32
        01 11 44                40 41 42
  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     02 12 22 00 01             00 10 20 30 40 50
     03 13 23 33 11             33 11 21 31 41 51
     04 14 24 34 44             43 44 22 32 42 52

 DTFTTR copies a triangular matrix A from rectangular full packed
 format (TF) to standard full format (TR).

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  ARF is in Normal format;
          = 'T':  ARF is in Transpose format.

UPLO

          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.

N

          N is INTEGER
          The order of the matrices ARF and A. N >= 0.

ARF

          ARF is DOUBLE PRECISION array, dimension (N*(N+1)/2).
          On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
          matrix A in RFP format. See the "Notes" below for more
          details.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On exit, the triangular matrix A.  If UPLO = 'U', the
          leading N-by-N upper triangular part of the array A contains
          the upper triangular matrix, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading N-by-N lower triangular part of the array A contains
          the lower triangular matrix, and the strictly upper
          triangular part of A is not referenced.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  We first consider Rectangular Full Packed (RFP) Format when N is
  even. We give an example where N = 6.
      AP is Upper             AP is Lower
   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  the transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  the transpose of the last three columns of AP lower.
  This covers the case N even and TRANSR = 'N'.
         RFP A                   RFP A
        03 04 05                33 43 53
        13 14 15                00 44 54
        23 24 25                10 11 55
        33 34 35                20 21 22
        00 44 45                30 31 32
        01 11 55                40 41 42
        02 12 22                50 51 52
  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     05 15 25 35 45 55 22    53 54 55 22 32 42 52
  We then consider Rectangular Full Packed (RFP) Format when N is
  odd. We give an example where N = 5.
     AP is Upper                 AP is Lower
   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44
  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  the transpose of the first two columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  the transpose of the last two columns of AP lower.
  This covers the case N odd and TRANSR = 'N'.
         RFP A                   RFP A
        02 03 04                00 33 43
        12 13 14                10 11 44
        22 23 24                20 21 22
        00 33 34                30 31 32
        01 11 44                40 41 42
  Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
  transpose of RFP A above. One therefore gets:
           RFP A                   RFP A
     02 12 22 00 01             00 10 20 30 40 50
     03 13 23 33 11             33 11 21 31 41 51
     04 14 24 34 44             43 44 22 32 42 52

 DTGSEN reorders the generalized real Schur decomposition of a real
 matrix pair (A, B) (in terms of an orthonormal equivalence trans-
 formation Q**T * (A, B) * Z), so that a selected cluster of eigenvalues
 appears in the leading diagonal blocks of the upper quasi-triangular
 matrix A and the upper triangular B. The leading columns of Q and
 Z form orthonormal bases of the corresponding left and right eigen-
 spaces (deflating subspaces). (A, B) must be in generalized real
 Schur canonical form (as returned by DGGES), i.e. A is block upper
 triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper
 triangular.
 DTGSEN also computes the generalized eigenvalues
             w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)
 of the reordered matrix pair (A, B).
 Optionally, DTGSEN computes the estimates of reciprocal condition
 numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
 (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
 between the matrix pairs (A11, B11) and (A22,B22) that correspond to
 the selected cluster and the eigenvalues outside the cluster, resp.,
 and norms of "projections" onto left and right eigenspaces w.r.t.
 the selected cluster in the (1,1)-block.

IJOB

          IJOB is INTEGER
          Specifies whether condition numbers are required for the
          cluster of eigenvalues (PL and PR) or the deflating subspaces
          (Difu and Difl):
           =0: Only reorder w.r.t. SELECT. No extras.
           =1: Reciprocal of norms of "projections" onto left and right
               eigenspaces w.r.t. the selected cluster (PL and PR).
           =2: Upper bounds on Difu and Difl. F-norm-based estimate
               (DIF(1:2)).
           =3: Estimate of Difu and Difl. 1-norm-based estimate
               (DIF(1:2)).
               About 5 times as expensive as IJOB = 2.
           =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
               version to get it all.
           =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)

WANTQ

          WANTQ is LOGICAL
          .TRUE. : update the left transformation matrix Q;
          .FALSE.: do not update Q.

WANTZ

          WANTZ is LOGICAL
          .TRUE. : update the right transformation matrix Z;
          .FALSE.: do not update Z.

SELECT

          SELECT is LOGICAL array, dimension (N)
          SELECT specifies the eigenvalues in the selected cluster.
          To select a real eigenvalue w(j), SELECT(j) must be set to
          .TRUE.. To select a complex conjugate pair of eigenvalues
          w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
          either SELECT(j) or SELECT(j+1) or both must be set to
          .TRUE.; a complex conjugate pair of eigenvalues must be
          either both included in the cluster or both excluded.

N

          N is INTEGER
          The order of the matrices A and B. N >= 0.

A

          A is DOUBLE PRECISION array, dimension(LDA,N)
          On entry, the upper quasi-triangular matrix A, with (A, B) in
          generalized real Schur canonical form.
          On exit, A is overwritten by the reordered matrix A.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,N).

B

          B is DOUBLE PRECISION array, dimension(LDB,N)
          On entry, the upper triangular matrix B, with (A, B) in
          generalized real Schur canonical form.
          On exit, B is overwritten by the reordered matrix B.

LDB

          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,N).

ALPHAR

          ALPHAR is DOUBLE PRECISION array, dimension (N)

ALPHAI

          ALPHAI is DOUBLE PRECISION array, dimension (N)

BETA

          BETA is DOUBLE PRECISION array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i
          and BETA(j),j=1,...,N  are the diagonals of the complex Schur
          form (S,T) that would result if the 2-by-2 diagonal blocks of
          the real generalized Schur form of (A,B) were further reduced
          to triangular form using complex unitary transformations.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
          positive, then the j-th and (j+1)-st eigenvalues are a
          complex conjugate pair, with ALPHAI(j+1) negative.

Q

          Q is DOUBLE PRECISION array, dimension (LDQ,N)
          On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
          On exit, Q has been postmultiplied by the left orthogonal
          transformation matrix which reorder (A, B); The leading M
          columns of Q form orthonormal bases for the specified pair of
          left eigenspaces (deflating subspaces).
          If WANTQ = .FALSE., Q is not referenced.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q.  LDQ >= 1;
          and if WANTQ = .TRUE., LDQ >= N.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ,N)
          On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
          On exit, Z has been postmultiplied by the left orthogonal
          transformation matrix which reorder (A, B); The leading M
          columns of Z form orthonormal bases for the specified pair of
          left eigenspaces (deflating subspaces).
          If WANTZ = .FALSE., Z is not referenced.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z. LDZ >= 1;
          If WANTZ = .TRUE., LDZ >= N.

M

          M is INTEGER
          The dimension of the specified pair of left and right eigen-
          spaces (deflating subspaces). 0 <= M <= N.

PL

          PL is DOUBLE PRECISION

PR

          PR is DOUBLE PRECISION
          If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
          reciprocal of the norm of "projections" onto left and right
          eigenspaces with respect to the selected cluster.
          0 < PL, PR <= 1.
          If M = 0 or M = N, PL = PR  = 1.
          If IJOB = 0, 2 or 3, PL and PR are not referenced.

DIF

          DIF is DOUBLE PRECISION array, dimension (2).
          If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
          If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
          Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
          estimates of Difu and Difl.
          If M = 0 or N, DIF(1:2) = F-norm([A, B]).
          If IJOB = 0 or 1, DIF is not referenced.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >=  4*N+16.
          If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)).
          If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)).
          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK

          LIWORK is INTEGER
          The dimension of the array IWORK. LIWORK >= 1.
          If IJOB = 1, 2 or 4, LIWORK >=  N+6.
          If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6).
          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,
          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.

INFO

          INFO is INTEGER
            =0: Successful exit.
            <0: If INFO = -i, the i-th argument had an illegal value.
            =1: Reordering of (A, B) failed because the transformed
                matrix pair (A, B) would be too far from generalized
                Schur form; the problem is very ill-conditioned.
                (A, B) may have been partially reordered.
                If requested, 0 is returned in DIF(*), PL and PR.

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  DTGSEN first collects the selected eigenvalues by computing
  orthogonal U and W that move them to the top left corner of (A, B).
  In other words, the selected eigenvalues are the eigenvalues of
  (A11, B11) in:
              U**T*(A, B)*W = (A11 A12) (B11 B12) n1
                              ( 0  A22),( 0  B22) n2
                                n1  n2    n1  n2
  where N = n1+n2 and U**T means the transpose of U. The first n1 columns
  of U and W span the specified pair of left and right eigenspaces
  (deflating subspaces) of (A, B).
  If (A, B) has been obtained from the generalized real Schur
  decomposition of a matrix pair (C, D) = Q*(A, B)*Z**T, then the
  reordered generalized real Schur form of (C, D) is given by
           (C, D) = (Q*U)*(U**T*(A, B)*W)*(Z*W)**T,
  and the first n1 columns of Q*U and Z*W span the corresponding
  deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).
  Note that if the selected eigenvalue is sufficiently ill-conditioned,
  then its value may differ significantly from its value before
  reordering.
  The reciprocal condition numbers of the left and right eigenspaces
  spanned by the first n1 columns of U and W (or Q*U and Z*W) may
  be returned in DIF(1:2), corresponding to Difu and Difl, resp.
  The Difu and Difl are defined as:
       Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
  and
       Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],
  where sigma-min(Zu) is the smallest singular value of the
  (2*n1*n2)-by-(2*n1*n2) matrix
       Zu = [ kron(In2, A11)  -kron(A22**T, In1) ]
            [ kron(In2, B11)  -kron(B22**T, In1) ].
  Here, Inx is the identity matrix of size nx and A22**T is the
  transpose of A22. kron(X, Y) is the Kronecker product between
  the matrices X and Y.
  When DIF(2) is small, small changes in (A, B) can cause large changes
  in the deflating subspace. An approximate (asymptotic) bound on the
  maximum angular error in the computed deflating subspaces is
       EPS * norm((A, B)) / DIF(2),
  where EPS is the machine precision.
  The reciprocal norm of the projectors on the left and right
  eigenspaces associated with (A11, B11) may be returned in PL and PR.
  They are computed as follows. First we compute L and R so that
  P*(A, B)*Q is block diagonal, where
       P = ( I -L ) n1           Q = ( I R ) n1
           ( 0  I ) n2    and        ( 0 I ) n2
             n1 n2                    n1 n2
  and (L, R) is the solution to the generalized Sylvester equation
       A11*R - L*A22 = -A12
       B11*R - L*B22 = -B12
  Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
  An approximate (asymptotic) bound on the average absolute error of
  the selected eigenvalues is
       EPS * norm((A, B)) / PL.
  There are also global error bounds which valid for perturbations up
  to a certain restriction:  A lower bound (x) on the smallest
  F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
  coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
  (i.e. (A + E, B + F), is
   x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
  An approximate bound on x can be computed from DIF(1:2), PL and PR.
  If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
  (L', R') and unperturbed (L, R) left and right deflating subspaces
  associated with the selected cluster in the (1,1)-blocks can be
  bounded as
   max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
   max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))
  See LAPACK User's Guide section 4.11 or the following references
  for more information.
  Note that if the default method for computing the Frobenius-norm-
  based estimate DIF is not wanted (see DLATDF), then the parameter
  IDIFJB (see below) should be changed from 3 to 4 (routine DLATDF
  (IJOB = 2 will be used)). See DTGSYL for more details.

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
      Estimation: Theory, Algorithms and Software,
      Report UMINF - 94.04, Department of Computing Science, Umea
      University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
      Note 87. To appear in Numerical Algorithms, 1996.
  [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
      for Solving the Generalized Sylvester Equation and Estimating the
      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
      Department of Computing Science, Umea University, S-901 87 Umea,
      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
      Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
      1996.

 DTGSJA computes the generalized singular value decomposition (GSVD)
 of two real upper triangular (or trapezoidal) matrices A and B.
 On entry, it is assumed that matrices A and B have the following
 forms, which may be obtained by the preprocessing subroutine DGGSVP
 from a general M-by-N matrix A and P-by-N matrix B:
              N-K-L  K    L
    A =    K ( 0    A12  A13 ) if M-K-L >= 0;
           L ( 0     0   A23 )
       M-K-L ( 0     0    0  )
            N-K-L  K    L
    A =  K ( 0    A12  A13 ) if M-K-L < 0;
       M-K ( 0     0   A23 )
            N-K-L  K    L
    B =  L ( 0     0   B13 )
       P-L ( 0     0    0  )
 where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
 upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
 otherwise A23 is (M-K)-by-L upper trapezoidal.
 On exit,
        U**T *A*Q = D1*( 0 R ),    V**T *B*Q = D2*( 0 R ),
 where U, V and Q are orthogonal matrices.
 R is a nonsingular upper triangular matrix, and D1 and D2 are
 ``diagonal'' matrices, which are of the following structures:
 If M-K-L >= 0,
                     K  L
        D1 =     K ( I  0 )
                 L ( 0  C )
             M-K-L ( 0  0 )
                   K  L
        D2 = L   ( 0  S )
             P-L ( 0  0 )
                N-K-L  K    L
   ( 0 R ) = K (  0   R11  R12 ) K
             L (  0    0   R22 ) L
 where
   C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
   S = diag( BETA(K+1),  ... , BETA(K+L) ),
   C**2 + S**2 = I.
   R is stored in A(1:K+L,N-K-L+1:N) on exit.
 If M-K-L < 0,
                K M-K K+L-M
     D1 =   K ( I  0    0   )
          M-K ( 0  C    0   )
                  K M-K K+L-M
     D2 =   M-K ( 0  S    0   )
          K+L-M ( 0  0    I   )
            P-L ( 0  0    0   )
                N-K-L  K   M-K  K+L-M
 ( 0 R ) =    K ( 0    R11  R12  R13  )
           M-K ( 0     0   R22  R23  )
         K+L-M ( 0     0    0   R33  )
 where
 C = diag( ALPHA(K+1), ... , ALPHA(M) ),
 S = diag( BETA(K+1),  ... , BETA(M) ),
 C**2 + S**2 = I.
 R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
     (  0  R22 R23 )
 in B(M-K+1:L,N+M-K-L+1:N) on exit.
 The computation of the orthogonal transformation matrices U, V or Q
 is optional.  These matrices may either be formed explicitly, or they
 may be postmultiplied into input matrices U1, V1, or Q1.

JOBU

          JOBU is CHARACTER*1
          = 'U':  U must contain an orthogonal matrix U1 on entry, and
                  the product U1*U is returned;
          = 'I':  U is initialized to the unit matrix, and the
                  orthogonal matrix U is returned;
          = 'N':  U is not computed.

JOBV

          JOBV is CHARACTER*1
          = 'V':  V must contain an orthogonal matrix V1 on entry, and
                  the product V1*V is returned;
          = 'I':  V is initialized to the unit matrix, and the
                  orthogonal matrix V is returned;
          = 'N':  V is not computed.

JOBQ

          JOBQ is CHARACTER*1
          = 'Q':  Q must contain an orthogonal matrix Q1 on entry, and
                  the product Q1*Q is returned;
          = 'I':  Q is initialized to the unit matrix, and the
                  orthogonal matrix Q is returned;
          = 'N':  Q is not computed.

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

P

          P is INTEGER
          The number of rows of the matrix B.  P >= 0.

N

          N is INTEGER
          The number of columns of the matrices A and B.  N >= 0.

K

          K is INTEGER

L

          L is INTEGER
          K and L specify the subblocks in the input matrices A and B:
          A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
          of A and B, whose GSVD is going to be computed by DTGSJA.
          See Further Details.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
          matrix R or part of R.  See Purpose for details.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

B

          B is DOUBLE PRECISION array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
          a part of R.  See Purpose for details.

LDB

          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,P).

TOLA

          TOLA is DOUBLE PRECISION

TOLB

          TOLB is DOUBLE PRECISION
          TOLA and TOLB are the convergence criteria for the Jacobi-
          Kogbetliantz iteration procedure. Generally, they are the
          same as used in the preprocessing step, say
              TOLA = max(M,N)*norm(A)*MAZHEPS,
              TOLB = max(P,N)*norm(B)*MAZHEPS.

ALPHA

          ALPHA is DOUBLE PRECISION array, dimension (N)

BETA

          BETA is DOUBLE PRECISION array, dimension (N)
          On exit, ALPHA and BETA contain the generalized singular
          value pairs of A and B;
            ALPHA(1:K) = 1,
            BETA(1:K)  = 0,
          and if M-K-L >= 0,
            ALPHA(K+1:K+L) = diag(C),
            BETA(K+1:K+L)  = diag(S),
          or if M-K-L < 0,
            ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
            BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
          Furthermore, if K+L < N,
            ALPHA(K+L+1:N) = 0 and
            BETA(K+L+1:N)  = 0.

U

          U is DOUBLE PRECISION array, dimension (LDU,M)
          On entry, if JOBU = 'U', U must contain a matrix U1 (usually
          the orthogonal matrix returned by DGGSVP).
          On exit,
          if JOBU = 'I', U contains the orthogonal matrix U;
          if JOBU = 'U', U contains the product U1*U.
          If JOBU = 'N', U is not referenced.

LDU

          LDU is INTEGER
          The leading dimension of the array U. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise.

V

          V is DOUBLE PRECISION array, dimension (LDV,P)
          On entry, if JOBV = 'V', V must contain a matrix V1 (usually
          the orthogonal matrix returned by DGGSVP).
          On exit,
          if JOBV = 'I', V contains the orthogonal matrix V;
          if JOBV = 'V', V contains the product V1*V.
          If JOBV = 'N', V is not referenced.

LDV

          LDV is INTEGER
          The leading dimension of the array V. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise.

Q

          Q is DOUBLE PRECISION array, dimension (LDQ,N)
          On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
          the orthogonal matrix returned by DGGSVP).
          On exit,
          if JOBQ = 'I', Q contains the orthogonal matrix Q;
          if JOBQ = 'Q', Q contains the product Q1*Q.
          If JOBQ = 'N', Q is not referenced.

LDQ

          LDQ is INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise.

WORK

          WORK is DOUBLE PRECISION array, dimension (2*N)

NCYCLE

          NCYCLE is INTEGER
          The number of cycles required for convergence.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1:  the procedure does not converge after MAXIT cycles.