This subroutine translates from a BLAST-specified integer constant to
 the character string specifying a transposition operation.
 CHLA_TRANSTYPE returns an CHARACTER*1.  If CHLA_TRANSTYPE is 'X',
 then input is not an integer indicating a transposition operator.
 Otherwise CHLA_TRANSTYPE returns the constant value corresponding to
 TRANS.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DBDSDC computes the singular value decomposition (SVD) of a real
 N-by-N (upper or lower) bidiagonal matrix B:  B = U * S * VT,
 using a divide and conquer method, where S is a diagonal matrix
 with non-negative diagonal elements (the singular values of B), and
 U and VT are orthogonal matrices of left and right singular vectors,
 respectively. DBDSDC can be used to compute all singular values,
 and optionally, singular vectors or singular vectors in compact form.
 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 X-MP, Cray Y-MP, Cray C-90, or Cray-2.
 It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.  See DLASD3 for details.
 The code currently calls DLASDQ if singular values only are desired.
 However, it can be slightly modified to compute singular values
 using the divide and conquer method.

UPLO

          UPLO is CHARACTER*1
          = 'U':  B is upper bidiagonal.
          = 'L':  B is lower bidiagonal.

COMPQ

          COMPQ is CHARACTER*1
          Specifies whether singular vectors are to be computed
          as follows:
          = 'N':  Compute singular values only;
          = 'P':  Compute singular values and compute singular
                  vectors in compact form;
          = 'I':  Compute singular values and singular vectors.

N

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

D

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

E

          E is DOUBLE PRECISION array, dimension (N-1)
          On entry, the elements of E contain the offdiagonal
          elements of the bidiagonal matrix whose SVD is desired.
          On exit, E has been destroyed.

U

          U is DOUBLE PRECISION array, dimension (LDU,N)
          If  COMPQ = 'I', then:
             On exit, if INFO = 0, U contains the left singular vectors
             of the bidiagonal matrix.
          For other values of COMPQ, U is not referenced.

LDU

          LDU is INTEGER
          The leading dimension of the array U.  LDU >= 1.
          If singular vectors are desired, then LDU >= max( 1, N ).

VT

          VT is DOUBLE PRECISION array, dimension (LDVT,N)
          If  COMPQ = 'I', then:
             On exit, if INFO = 0, VT**T contains the right singular
             vectors of the bidiagonal matrix.
          For other values of COMPQ, VT is not referenced.

LDVT

          LDVT is INTEGER
          The leading dimension of the array VT.  LDVT >= 1.
          If singular vectors are desired, then LDVT >= max( 1, N ).

Q

          Q is DOUBLE PRECISION array, dimension (LDQ)
          If  COMPQ = 'P', then:
             On exit, if INFO = 0, Q and IQ contain the left
             and right singular vectors in a compact form,
             requiring O(N log N) space instead of 2*N**2.
             In particular, Q contains all the DOUBLE PRECISION data in
             LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1))))
             words of memory, where SMLSIZ is returned by ILAENV and
             is equal to the maximum size of the subproblems at the
             bottom of the computation tree (usually about 25).
          For other values of COMPQ, Q is not referenced.

IQ

          IQ is INTEGER array, dimension (LDIQ)
          If  COMPQ = 'P', then:
             On exit, if INFO = 0, Q and IQ contain the left
             and right singular vectors in a compact form,
             requiring O(N log N) space instead of 2*N**2.
             In particular, IQ contains all INTEGER data in
             LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
             words of memory, where SMLSIZ is returned by ILAENV and
             is equal to the maximum size of the subproblems at the
             bottom of the computation tree (usually about 25).
          For other values of COMPQ, IQ is not referenced.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          If COMPQ = 'N' then LWORK >= (4 * N).
          If COMPQ = 'P' then LWORK >= (6 * N).
          If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).

IWORK

          IWORK is INTEGER array, dimension (8*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.
                The update process of divide and conquer failed.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

 DBDSQR computes the singular values and, optionally, the right and/or
 left singular vectors from the singular value decomposition (SVD) of
 a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
 zero-shift QR algorithm.  The SVD of B has the form
    B = Q * S * P**T
 where S is the diagonal matrix of singular values, Q is an orthogonal
 matrix of left singular vectors, and P is an orthogonal matrix of
 right singular vectors.  If left singular vectors are requested, this
 subroutine actually returns U*Q instead of Q, and, if right singular
 vectors are requested, this subroutine returns P**T*VT instead of
 P**T, for given real input matrices U and VT.  When U and VT are the
 orthogonal matrices that reduce a general matrix A to bidiagonal
 form:  A = U*B*VT, as computed by DGEBRD, then
    A = (U*Q) * S * (P**T*VT)
 is the SVD of A.  Optionally, the subroutine may also compute Q**T*C
 for a given real input matrix C.
 See "Computing  Small Singular Values of Bidiagonal Matrices With
 Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
 LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
 no. 5, pp. 873-912, Sept 1990) and
 "Accurate singular values and differential qd algorithms," by
 B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
 Department, University of California at Berkeley, July 1992
 for a detailed description of the algorithm.

UPLO

          UPLO is CHARACTER*1
          = 'U':  B is upper bidiagonal;
          = 'L':  B is lower bidiagonal.

N

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

NCVT

          NCVT is INTEGER
          The number of columns of the matrix VT. NCVT >= 0.

NRU

          NRU is INTEGER
          The number of rows of the matrix U. NRU >= 0.

NCC

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

D

          D is DOUBLE PRECISION array, dimension (N)
          On entry, the n diagonal elements of the bidiagonal matrix B.
          On exit, if INFO=0, the singular values of B in decreasing
          order.

E

          E is DOUBLE PRECISION array, dimension (N-1)
          On entry, the N-1 offdiagonal elements of the bidiagonal
          matrix B.
          On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
          will contain the diagonal and superdiagonal elements of a
          bidiagonal matrix orthogonally equivalent to the one given
          as input.

VT

          VT is DOUBLE PRECISION array, dimension (LDVT, NCVT)
          On entry, an N-by-NCVT matrix VT.
          On exit, VT is overwritten by P**T * VT.
          Not referenced if NCVT = 0.

LDVT

          LDVT is INTEGER
          The leading dimension of the array VT.
          LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.

U

          U is DOUBLE PRECISION array, dimension (LDU, N)
          On entry, an NRU-by-N matrix U.
          On exit, U is overwritten by U * Q.
          Not referenced if NRU = 0.

LDU

          LDU is INTEGER
          The leading dimension of the array U.  LDU >= max(1,NRU).

C

          C is DOUBLE PRECISION array, dimension (LDC, NCC)
          On entry, an N-by-NCC matrix C.
          On exit, C is overwritten by Q**T * C.
          Not referenced if NCC = 0.

LDC

          LDC is INTEGER
          The leading dimension of the array C.
          LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.

WORK

          WORK is DOUBLE PRECISION array, dimension (4*(N-1))

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  If INFO = -i, the i-th argument had an illegal value
          > 0:
             if NCVT = NRU = NCC = 0,
                = 1, a split was marked by a positive value in E
                = 2, current block of Z not diagonalized after 30*N
                     iterations (in inner while loop)
                = 3, termination criterion of outer while loop not met
                     (program created more than N unreduced blocks)
             else NCVT = NRU = NCC = 0,
                   the algorithm did not converge; D and E contain the
                   elements of a bidiagonal matrix which is orthogonally
                   similar to the input matrix B;  if INFO = i, i
                   elements of E have not converged to zero.

  TOLMUL  DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8)))
          TOLMUL controls the convergence criterion of the QR loop.
          If it is positive, TOLMUL*EPS is the desired relative
             precision in the computed singular values.
          If it is negative, abs(TOLMUL*EPS*sigma_max) is the
             desired absolute accuracy in the computed singular
             values (corresponds to relative accuracy
             abs(TOLMUL*EPS) in the largest singular value.
          abs(TOLMUL) should be between 1 and 1/EPS, and preferably
             between 10 (for fast convergence) and .1/EPS
             (for there to be some accuracy in the results).
          Default is to lose at either one eighth or 2 of the
             available decimal digits in each computed singular value
             (whichever is smaller).
  MAXITR  INTEGER, default = 6
          MAXITR controls the maximum number of passes of the
          algorithm through its inner loop. The algorithms stops
          (and so fails to converge) if the number of passes
          through the inner loop exceeds MAXITR*N**2.

  Bug report from Cezary Dendek.
  On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is
  removed since it can overflow pretty easily (for N larger or equal
  than 18,919). We instead use MAXITDIVN = MAXITR*N.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DDISNA computes the reciprocal condition numbers for the eigenvectors
 of a real symmetric or complex Hermitian matrix or for the left or
 right singular vectors of a general m-by-n matrix. The reciprocal
 condition number is the 'gap' between the corresponding eigenvalue or
 singular value and the nearest other one.
 The bound on the error, measured by angle in radians, in the I-th
 computed vector is given by
        DLAMCH( 'E' ) * ( ANORM / SEP( I ) )
 where ANORM = 2-norm(A) = max( abs( D(j) ) ).  SEP(I) is not allowed
 to be smaller than DLAMCH( 'E' )*ANORM in order to limit the size of
 the error bound.
 DDISNA may also be used to compute error bounds for eigenvectors of
 the generalized symmetric definite eigenproblem.

JOB

          JOB is CHARACTER*1
          Specifies for which problem the reciprocal condition numbers
          should be computed:
          = 'E':  the eigenvectors of a symmetric/Hermitian matrix;
          = 'L':  the left singular vectors of a general matrix;
          = 'R':  the right singular vectors of a general matrix.

M

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

N

          N is INTEGER
          If JOB = 'L' or 'R', the number of columns of the matrix,
          in which case N >= 0. Ignored if JOB = 'E'.

D

          D is DOUBLE PRECISION array, dimension (M) if JOB = 'E'
                              dimension (min(M,N)) if JOB = 'L' or 'R'
          The eigenvalues (if JOB = 'E') or singular values (if JOB =
          'L' or 'R') of the matrix, in either increasing or decreasing
          order. If singular values, they must be non-negative.

SEP

          SEP is DOUBLE PRECISION array, dimension (M) if JOB = 'E'
                               dimension (min(M,N)) if JOB = 'L' or 'R'
          The reciprocal condition numbers of the vectors.

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.

 DLAED0 computes all eigenvalues and corresponding eigenvectors of a
 symmetric tridiagonal matrix using the divide and conquer method.

ICOMPQ

          ICOMPQ is INTEGER
          = 0:  Compute eigenvalues only.
          = 1:  Compute eigenvectors of original dense symmetric matrix
                also.  On entry, Q contains the orthogonal matrix used
                to reduce the original matrix to tridiagonal form.
          = 2:  Compute eigenvalues and eigenvectors of tridiagonal
                matrix.

QSIZ

          QSIZ is INTEGER
         The dimension of the orthogonal matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

N

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

D

          D is DOUBLE PRECISION array, dimension (N)
         On entry, the main diagonal of the tridiagonal matrix.
         On exit, its eigenvalues.

E

          E is DOUBLE PRECISION array, dimension (N-1)
         The off-diagonal elements of the tridiagonal matrix.
         On exit, E has been destroyed.

Q

          Q is DOUBLE PRECISION array, dimension (LDQ, N)
         On entry, Q must contain an N-by-N orthogonal matrix.
         If ICOMPQ = 0    Q is not referenced.
         If ICOMPQ = 1    On entry, Q is a subset of the columns of the
                          orthogonal matrix used to reduce the full
                          matrix to tridiagonal form corresponding to
                          the subset of the full matrix which is being
                          decomposed at this time.
         If ICOMPQ = 2    On entry, Q will be the identity matrix.
                          On exit, Q contains the eigenvectors of the
                          tridiagonal matrix.

LDQ

          LDQ is INTEGER
         The leading dimension of the array Q.  If eigenvectors are
         desired, then  LDQ >= max(1,N).  In any case,  LDQ >= 1.

QSTORE

          QSTORE is DOUBLE PRECISION array, dimension (LDQS, N)
         Referenced only when ICOMPQ = 1.  Used to store parts of
         the eigenvector matrix when the updating matrix multiplies
         take place.

LDQS

          LDQS is INTEGER
         The leading dimension of the array QSTORE.  If ICOMPQ = 1,
         then  LDQS >= max(1,N).  In any case,  LDQS >= 1.

WORK

          WORK is DOUBLE PRECISION array,
         If ICOMPQ = 0 or 1, the dimension of WORK must be at least
                     1 + 3*N + 2*N*lg N + 3*N**2
                     ( lg( N ) = smallest integer k
                                 such that 2^k >= N )
         If ICOMPQ = 2, the dimension of WORK must be at least
                     4*N + N**2.

IWORK

          IWORK is INTEGER array,
         If ICOMPQ = 0 or 1, the dimension of IWORK must be at least
                        6 + 6*N + 5*N*lg N.
                        ( lg( N ) = smallest integer k
                                    such that 2^k >= N )
         If ICOMPQ = 2, the dimension of IWORK must be at least
                        3 + 5*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 an eigenvalue while
                working on the submatrix lying in rows and columns
                INFO/(N+1) through mod(INFO,N+1).

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

 DLAED1 computes the updated eigensystem of a diagonal
 matrix after modification by a rank-one symmetric matrix.  This
 routine is used only for the eigenproblem which requires all
 eigenvalues and eigenvectors of a tridiagonal matrix.  DLAED7 handles
 the case in which eigenvalues only or eigenvalues and eigenvectors
 of a full symmetric matrix (which was reduced to tridiagonal form)
 are desired.
   T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
    where Z = Q**T*u, u is a vector of length N with ones in the
    CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
    The eigenvectors of the original matrix are stored in Q, and the
    eigenvalues are in D.  The algorithm consists of three stages:
       The first stage consists of deflating the size of the problem
       when there are multiple eigenvalues or if there is a zero in
       the Z vector.  For each such occurrence the dimension of the
       secular equation problem is reduced by one.  This stage is
       performed by the routine DLAED2.
       The second stage consists of calculating the updated
       eigenvalues. This is done by finding the roots of the secular
       equation via the routine DLAED4 (as called by DLAED3).
       This routine also calculates the eigenvectors of the current
       problem.
       The final stage consists of computing the updated eigenvectors
       directly using the updated eigenvalues.  The eigenvectors for
       the current problem are multiplied with the eigenvectors from
       the overall problem.

N

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

D

          D is DOUBLE PRECISION array, dimension (N)
         On entry, the eigenvalues of the rank-1-perturbed matrix.
         On exit, the eigenvalues of the repaired matrix.

Q

          Q is DOUBLE PRECISION array, dimension (LDQ,N)
         On entry, the eigenvectors of the rank-1-perturbed matrix.
         On exit, the eigenvectors of the repaired tridiagonal matrix.

LDQ

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

INDXQ

          INDXQ is INTEGER array, dimension (N)
         On entry, the permutation which separately sorts the two
         subproblems in D into ascending order.
         On exit, the permutation which will reintegrate the
         subproblems back into sorted order,
         i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.

RHO

          RHO is DOUBLE PRECISION
         The subdiagonal entry used to create the rank-1 modification.

CUTPNT

          CUTPNT is INTEGER
         The location of the last eigenvalue in the leading sub-matrix.
         min(1,N) <= CUTPNT <= N/2.

WORK

          WORK is DOUBLE PRECISION array, dimension (4*N + N**2)

IWORK

          IWORK is INTEGER array, dimension (4*N)

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA Modified by Francoise Tisseur, University of Tennessee

 DLAED2 merges the two sets of eigenvalues together into a single
 sorted set.  Then it tries to deflate the size of the problem.
 There are two ways in which deflation can occur:  when two or more
 eigenvalues are close together or if there is a tiny entry in the
 Z vector.  For each such occurrence the order of the related secular
 equation problem is reduced by one.

K

          K is INTEGER
         The number of non-deflated eigenvalues, and the order of the
         related secular equation. 0 <= K <=N.

N

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

N1

          N1 is INTEGER
         The location of the last eigenvalue in the leading sub-matrix.
         min(1,N) <= N1 <= N/2.

D

          D is DOUBLE PRECISION array, dimension (N)
         On entry, D contains the eigenvalues of the two submatrices to
         be combined.
         On exit, D contains the trailing (N-K) updated eigenvalues
         (those which were deflated) sorted into increasing order.

Q

          Q is DOUBLE PRECISION array, dimension (LDQ, N)
         On entry, Q contains the eigenvectors of two submatrices in
         the two square blocks with corners at (1,1), (N1,N1)
         and (N1+1, N1+1), (N,N).
         On exit, Q contains the trailing (N-K) updated eigenvectors
         (those which were deflated) in its last N-K columns.

LDQ

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

INDXQ

          INDXQ is INTEGER array, dimension (N)
         The permutation which separately sorts the two sub-problems
         in D into ascending order.  Note that elements in the second
         half of this permutation must first have N1 added to their
         values. Destroyed on exit.

RHO

          RHO is DOUBLE PRECISION
         On entry, the off-diagonal element associated with the rank-1
         cut which originally split the two submatrices which are now
         being recombined.
         On exit, RHO has been modified to the value required by
         DLAED3.

Z

          Z is DOUBLE PRECISION array, dimension (N)
         On entry, Z contains the updating vector (the last
         row of the first sub-eigenvector matrix and the first row of
         the second sub-eigenvector matrix).
         On exit, the contents of Z have been destroyed by the updating
         process.

DLAMDA

          DLAMDA is DOUBLE PRECISION array, dimension (N)
         A copy of the first K eigenvalues which will be used by
         DLAED3 to form the secular equation.

W

          W is DOUBLE PRECISION array, dimension (N)
         The first k values of the final deflation-altered z-vector
         which will be passed to DLAED3.

Q2

          Q2 is DOUBLE PRECISION array, dimension (N1**2+(N-N1)**2)
         A copy of the first K eigenvectors which will be used by
         DLAED3 in a matrix multiply (DGEMM) to solve for the new
         eigenvectors.

INDX

          INDX is INTEGER array, dimension (N)
         The permutation used to sort the contents of DLAMDA into
         ascending order.

INDXC

          INDXC is INTEGER array, dimension (N)
         The permutation used to arrange the columns of the deflated
         Q matrix into three groups:  the first group contains non-zero
         elements only at and above N1, the second contains
         non-zero elements only below N1, and the third is dense.

INDXP

          INDXP is INTEGER array, dimension (N)
         The permutation used to place deflated values of D at the end
         of the array.  INDXP(1:K) points to the nondeflated D-values
         and INDXP(K+1:N) points to the deflated eigenvalues.

COLTYP

          COLTYP is INTEGER array, dimension (N)
         During execution, a label which will indicate which of the
         following types a column in the Q2 matrix is:
         1 : non-zero in the upper half only;
         2 : dense;
         3 : non-zero in the lower half only;
         4 : deflated.
         On exit, COLTYP(i) is the number of columns of type i,
         for i=1 to 4 only.

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.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA Modified by Francoise Tisseur, University of Tennessee

 DLAED3 finds the roots of the secular equation, as defined by the
 values in D, W, and RHO, between 1 and K.  It makes the
 appropriate calls to DLAED4 and then updates the eigenvectors by
 multiplying the matrix of eigenvectors of the pair of eigensystems
 being combined by the matrix of eigenvectors of the K-by-K system
 which is solved here.
 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 X-MP, Cray Y-MP, Cray C-90, or Cray-2.
 It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.

K

          K is INTEGER
          The number of terms in the rational function to be solved by
          DLAED4.  K >= 0.

N

          N is INTEGER
          The number of rows and columns in the Q matrix.
          N >= K (deflation may result in N>K).

N1

          N1 is INTEGER
          The location of the last eigenvalue in the leading submatrix.
          min(1,N) <= N1 <= N/2.

D

          D is DOUBLE PRECISION array, dimension (N)
          D(I) contains the updated eigenvalues for
          1 <= I <= K.

Q

          Q is DOUBLE PRECISION array, dimension (LDQ,N)
          Initially the first K columns are used as workspace.
          On output the columns 1 to K contain
          the updated eigenvectors.

LDQ

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

RHO

          RHO is DOUBLE PRECISION
          The value of the parameter in the rank one update equation.
          RHO >= 0 required.

DLAMDA

          DLAMDA is DOUBLE PRECISION array, dimension (K)
          The first K elements of this array contain the old roots
          of the deflated updating problem.  These are the poles
          of the secular equation. May be changed on output by
          having lowest order bit set to zero on Cray X-MP, Cray Y-MP,
          Cray-2, or Cray C-90, as described above.

Q2

          Q2 is DOUBLE PRECISION array, dimension (LDQ2*N)
          The first K columns of this matrix contain the non-deflated
          eigenvectors for the split problem.

INDX

          INDX is INTEGER array, dimension (N)
          The permutation used to arrange the columns of the deflated
          Q matrix into three groups (see DLAED2).
          The rows of the eigenvectors found by DLAED4 must be likewise
          permuted before the matrix multiply can take place.

CTOT

          CTOT is INTEGER array, dimension (4)
          A count of the total number of the various types of columns
          in Q, as described in INDX.  The fourth column type is any
          column which has been deflated.

W

          W is DOUBLE PRECISION array, dimension (K)
          The first K elements of this array contain the components
          of the deflation-adjusted updating vector. Destroyed on
          output.

S

          S is DOUBLE PRECISION array, dimension (N1 + 1)*K
          Will contain the eigenvectors of the repaired matrix which
          will be multiplied by the previously accumulated eigenvectors
          to update the system.

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA Modified by Francoise Tisseur, University of Tennessee

 This subroutine computes the I-th updated eigenvalue of a symmetric
 rank-one modification to a diagonal matrix whose elements are
 given in the array d, and that
            D(i) < D(j)  for  i < j
 and that RHO > 0.  This is arranged by the calling routine, and is
 no loss in generality.  The rank-one modified system is thus
            diag( D )  +  RHO * Z * Z_transpose.
 where we assume the Euclidean norm of Z is 1.
 The method consists of approximating the rational functions in the
 secular equation by simpler interpolating rational functions.

N

          N is INTEGER
         The length of all arrays.

I

          I is INTEGER
         The index of the eigenvalue to be computed.  1 <= I <= N.

D

          D is DOUBLE PRECISION array, dimension (N)
         The original eigenvalues.  It is assumed that they are in
         order, D(I) < D(J)  for I < J.

Z

          Z is DOUBLE PRECISION array, dimension (N)
         The components of the updating vector.

DELTA

          DELTA is DOUBLE PRECISION array, dimension (N)
         If N > 2, DELTA contains (D(j) - lambda_I) in its  j-th
         component.  If N = 1, then DELTA(1) = 1. If N = 2, see DLAED5
         for detail. The vector DELTA contains the information necessary
         to construct the eigenvectors by DLAED3 and DLAED9.

RHO

          RHO is DOUBLE PRECISION
         The scalar in the symmetric updating formula.

DLAM

          DLAM is DOUBLE PRECISION
         The computed lambda_I, the I-th updated eigenvalue.

INFO

          INFO is INTEGER
         = 0:  successful exit
         > 0:  if INFO = 1, the updating process failed.

  Logical variable ORGATI (origin-at-i?) is used for distinguishing
  whether D(i) or D(i+1) is treated as the origin.
            ORGATI = .true.    origin at i
            ORGATI = .false.   origin at i+1
   Logical variable SWTCH3 (switch-for-3-poles?) is for noting
   if we are working with THREE poles!
   MAXIT is the maximum number of iterations allowed for each
   eigenvalue.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

 This subroutine computes the I-th eigenvalue of a symmetric rank-one
 modification of a 2-by-2 diagonal matrix
            diag( D )  +  RHO * Z * transpose(Z) .
 The diagonal elements in the array D are assumed to satisfy
            D(i) < D(j)  for  i < j .
 We also assume RHO > 0 and that the Euclidean norm of the vector
 Z is one.

I

          I is INTEGER
         The index of the eigenvalue to be computed.  I = 1 or I = 2.

D

          D is DOUBLE PRECISION array, dimension (2)
         The original eigenvalues.  We assume D(1) < D(2).

Z

          Z is DOUBLE PRECISION array, dimension (2)
         The components of the updating vector.

DELTA

          DELTA is DOUBLE PRECISION array, dimension (2)
         The vector DELTA contains the information necessary
         to construct the eigenvectors.

RHO

          RHO is DOUBLE PRECISION
         The scalar in the symmetric updating formula.

DLAM

          DLAM is DOUBLE PRECISION
         The computed lambda_I, the I-th updated eigenvalue.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

 DLAED6 computes the positive or negative root (closest to the origin)
 of
                  z(1)        z(2)        z(3)
 f(x) =   rho + --------- + ---------- + ---------
                 d(1)-x      d(2)-x      d(3)-x
 It is assumed that
       if ORGATI = .true. the root is between d(2) and d(3);
       otherwise it is between d(1) and d(2)
 This routine will be called by DLAED4 when necessary. In most cases,
 the root sought is the smallest in magnitude, though it might not be
 in some extremely rare situations.

KNITER

          KNITER is INTEGER
               Refer to DLAED4 for its significance.

ORGATI

          ORGATI is LOGICAL
               If ORGATI is true, the needed root is between d(2) and
               d(3); otherwise it is between d(1) and d(2).  See
               DLAED4 for further details.

RHO

          RHO is DOUBLE PRECISION
               Refer to the equation f(x) above.

D

          D is DOUBLE PRECISION array, dimension (3)
               D satisfies d(1) < d(2) < d(3).

Z

          Z is DOUBLE PRECISION array, dimension (3)
               Each of the elements in z must be positive.

FINIT

          FINIT is DOUBLE PRECISION
               The value of f at 0. It is more accurate than the one
               evaluated inside this routine (if someone wants to do
               so).

TAU

          TAU is DOUBLE PRECISION
               The root of the equation f(x).

INFO

          INFO is INTEGER
               = 0: successful exit
               > 0: if INFO = 1, failure to converge

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  10/02/03: This version has a few statements commented out for thread
  safety (machine parameters are computed on each entry). SJH.
  05/10/06: Modified from a new version of Ren-Cang Li, use
     Gragg-Thornton-Warner cubic convergent scheme for better stability.

Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

 DLAED7 computes the updated eigensystem of a diagonal
 matrix after modification by a rank-one symmetric matrix. This
 routine is used only for the eigenproblem which requires all
 eigenvalues and optionally eigenvectors of a dense symmetric matrix
 that has been reduced to tridiagonal form.  DLAED1 handles
 the case in which all eigenvalues and eigenvectors of a symmetric
 tridiagonal matrix are desired.
   T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
    where Z = Q**Tu, u is a vector of length N with ones in the
    CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
    The eigenvectors of the original matrix are stored in Q, and the
    eigenvalues are in D.  The algorithm consists of three stages:
       The first stage consists of deflating the size of the problem
       when there are multiple eigenvalues or if there is a zero in
       the Z vector.  For each such occurrence the dimension of the
       secular equation problem is reduced by one.  This stage is
       performed by the routine DLAED8.
       The second stage consists of calculating the updated
       eigenvalues. This is done by finding the roots of the secular
       equation via the routine DLAED4 (as called by DLAED9).
       This routine also calculates the eigenvectors of the current
       problem.
       The final stage consists of computing the updated eigenvectors
       directly using the updated eigenvalues.  The eigenvectors for
       the current problem are multiplied with the eigenvectors from
       the overall problem.

ICOMPQ

          ICOMPQ is INTEGER
          = 0:  Compute eigenvalues only.
          = 1:  Compute eigenvectors of original dense symmetric matrix
                also.  On entry, Q contains the orthogonal matrix used
                to reduce the original matrix to tridiagonal form.

N

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

QSIZ

          QSIZ is INTEGER
         The dimension of the orthogonal matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

TLVLS

          TLVLS is INTEGER
         The total number of merging levels in the overall divide and
         conquer tree.

CURLVL

          CURLVL is INTEGER
         The current level in the overall merge routine,
         0 <= CURLVL <= TLVLS.

CURPBM

          CURPBM is INTEGER
         The current problem in the current level in the overall
         merge routine (counting from upper left to lower right).

D

          D is DOUBLE PRECISION array, dimension (N)
         On entry, the eigenvalues of the rank-1-perturbed matrix.
         On exit, the eigenvalues of the repaired matrix.

Q

          Q is DOUBLE PRECISION array, dimension (LDQ, N)
         On entry, the eigenvectors of the rank-1-perturbed matrix.
         On exit, the eigenvectors of the repaired tridiagonal matrix.

LDQ

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

INDXQ

          INDXQ is INTEGER array, dimension (N)
         The permutation which will reintegrate the subproblem just
         solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
         will be in ascending order.

RHO

          RHO is DOUBLE PRECISION
         The subdiagonal element used to create the rank-1
         modification.

CUTPNT

          CUTPNT is INTEGER
         Contains the location of the last eigenvalue in the leading
         sub-matrix.  min(1,N) <= CUTPNT <= N.

QSTORE

          QSTORE is DOUBLE PRECISION array, dimension (N**2+1)
         Stores eigenvectors of submatrices encountered during
         divide and conquer, packed together. QPTR points to
         beginning of the submatrices.

QPTR

          QPTR is INTEGER array, dimension (N+2)
         List of indices pointing to beginning of submatrices stored
         in QSTORE. The submatrices are numbered starting at the
         bottom left of the divide and conquer tree, from left to
         right and bottom to top.

PRMPTR

          PRMPTR is INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in PERM a
         level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
         indicates the size of the permutation and also the size of
         the full, non-deflated problem.

PERM

          PERM is INTEGER array, dimension (N lg N)
         Contains the permutations (from deflation and sorting) to be
         applied to each eigenblock.

GIVPTR

          GIVPTR is INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in GIVCOL a
         level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
         indicates the number of Givens rotations.

GIVCOL

          GIVCOL is INTEGER array, dimension (2, N lg N)
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation.

GIVNUM

          GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
         Each number indicates the S value to be used in the
         corresponding Givens rotation.

WORK

          WORK is DOUBLE PRECISION array, dimension (3*N+2*QSIZ*N)

IWORK

          IWORK is INTEGER array, dimension (4*N)

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

 DLAED8 merges the two sets of eigenvalues together into a single
 sorted set.  Then it tries to deflate the size of the problem.
 There are two ways in which deflation can occur:  when two or more
 eigenvalues are close together or if there is a tiny element in the
 Z vector.  For each such occurrence the order of the related secular
 equation problem is reduced by one.

ICOMPQ

          ICOMPQ is INTEGER
          = 0:  Compute eigenvalues only.
          = 1:  Compute eigenvectors of original dense symmetric matrix
                also.  On entry, Q contains the orthogonal matrix used
                to reduce the original matrix to tridiagonal form.

K

          K is INTEGER
         The number of non-deflated eigenvalues, and the order of the
         related secular equation.

N

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

QSIZ

          QSIZ is INTEGER
         The dimension of the orthogonal matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

D

          D is DOUBLE PRECISION array, dimension (N)
         On entry, the eigenvalues of the two submatrices to be
         combined.  On exit, the trailing (N-K) updated eigenvalues
         (those which were deflated) sorted into increasing order.

Q

          Q is DOUBLE PRECISION array, dimension (LDQ,N)
         If ICOMPQ = 0, Q is not referenced.  Otherwise,
         on entry, Q contains the eigenvectors of the partially solved
         system which has been previously updated in matrix
         multiplies with other partially solved eigensystems.
         On exit, Q contains the trailing (N-K) updated eigenvectors
         (those which were deflated) in its last N-K columns.

LDQ

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

INDXQ

          INDXQ is INTEGER array, dimension (N)
         The permutation which separately sorts the two sub-problems
         in D into ascending order.  Note that elements in the second
         half of this permutation must first have CUTPNT added to
         their values in order to be accurate.

RHO

          RHO is DOUBLE PRECISION
         On entry, the off-diagonal element associated with the rank-1
         cut which originally split the two submatrices which are now
         being recombined.
         On exit, RHO has been modified to the value required by
         DLAED3.

CUTPNT

          CUTPNT is INTEGER
         The location of the last eigenvalue in the leading
         sub-matrix.  min(1,N) <= CUTPNT <= N.

Z

          Z is DOUBLE PRECISION array, dimension (N)
         On entry, Z contains the updating vector (the last row of
         the first sub-eigenvector matrix and the first row of the
         second sub-eigenvector matrix).
         On exit, the contents of Z are destroyed by the updating
         process.

DLAMDA

          DLAMDA is DOUBLE PRECISION array, dimension (N)
         A copy of the first K eigenvalues which will be used by
         DLAED3 to form the secular equation.

Q2

          Q2 is DOUBLE PRECISION array, dimension (LDQ2,N)
         If ICOMPQ = 0, Q2 is not referenced.  Otherwise,
         a copy of the first K eigenvectors which will be used by
         DLAED7 in a matrix multiply (DGEMM) to update the new
         eigenvectors.

LDQ2

          LDQ2 is INTEGER
         The leading dimension of the array Q2.  LDQ2 >= max(1,N).

W

          W is DOUBLE PRECISION array, dimension (N)
         The first k values of the final deflation-altered z-vector and
         will be passed to DLAED3.

PERM

          PERM is INTEGER array, dimension (N)
         The permutations (from deflation and sorting) to be applied
         to each eigenblock.

GIVPTR

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

GIVCOL

          GIVCOL is INTEGER array, dimension (2, N)
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation.

GIVNUM

          GIVNUM is DOUBLE PRECISION array, dimension (2, N)
         Each number indicates the S value to be used in the
         corresponding Givens rotation.

INDXP

          INDXP is INTEGER array, dimension (N)
         The permutation used to place deflated values of D at the end
         of the array.  INDXP(1:K) points to the nondeflated D-values
         and INDXP(K+1:N) points to the deflated eigenvalues.

INDX

          INDX is INTEGER array, dimension (N)
         The permutation used to sort the contents of D into ascending
         order.

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.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

 DLAED9 finds the roots of the secular equation, as defined by the
 values in D, Z, and RHO, between KSTART and KSTOP.  It makes the
 appropriate calls to DLAED4 and then stores the new matrix of
 eigenvectors for use in calculating the next level of Z vectors.

K

          K is INTEGER
          The number of terms in the rational function to be solved by
          DLAED4.  K >= 0.

KSTART

          KSTART is INTEGER

KSTOP

          KSTOP is INTEGER
          The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
          are to be computed.  1 <= KSTART <= KSTOP <= K.

N

          N is INTEGER
          The number of rows and columns in the Q matrix.
          N >= K (delation may result in N > K).

D

          D is DOUBLE PRECISION array, dimension (N)
          D(I) contains the updated eigenvalues
          for KSTART <= I <= KSTOP.

Q

          Q is DOUBLE PRECISION array, dimension (LDQ,N)

LDQ

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

RHO

          RHO is DOUBLE PRECISION
          The value of the parameter in the rank one update equation.
          RHO >= 0 required.

DLAMDA

          DLAMDA is DOUBLE PRECISION array, dimension (K)
          The first K elements of this array contain the old roots
          of the deflated updating problem.  These are the poles
          of the secular equation.

W

          W is DOUBLE PRECISION array, dimension (K)
          The first K elements of this array contain the components
          of the deflation-adjusted updating vector.

S

          S is DOUBLE PRECISION array, dimension (LDS, K)
          Will contain the eigenvectors of the repaired matrix which
          will be stored for subsequent Z vector calculation and
          multiplied by the previously accumulated eigenvectors
          to update the system.

LDS

          LDS is INTEGER
          The leading dimension of S.  LDS >= max( 1, K ).

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

 DLAEDA computes the Z vector corresponding to the merge step in the
 CURLVLth step of the merge process with TLVLS steps for the CURPBMth
 problem.

N

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

TLVLS

          TLVLS is INTEGER
         The total number of merging levels in the overall divide and
         conquer tree.

CURLVL

          CURLVL is INTEGER
         The current level in the overall merge routine,
         0 <= curlvl <= tlvls.

CURPBM

          CURPBM is INTEGER
         The current problem in the current level in the overall
         merge routine (counting from upper left to lower right).

PRMPTR

          PRMPTR is INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in PERM a
         level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
         indicates the size of the permutation and incidentally the
         size of the full, non-deflated problem.

PERM

          PERM is INTEGER array, dimension (N lg N)
         Contains the permutations (from deflation and sorting) to be
         applied to each eigenblock.

GIVPTR

          GIVPTR is INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in GIVCOL a
         level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
         indicates the number of Givens rotations.

GIVCOL

          GIVCOL is INTEGER array, dimension (2, N lg N)
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation.

GIVNUM

          GIVNUM is DOUBLE PRECISION array, dimension (2, N lg N)
         Each number indicates the S value to be used in the
         corresponding Givens rotation.

Q

          Q is DOUBLE PRECISION array, dimension (N**2)
         Contains the square eigenblocks from previous levels, the
         starting positions for blocks are given by QPTR.

QPTR

          QPTR is INTEGER array, dimension (N+2)
         Contains a list of pointers which indicate where in Q an
         eigenblock is stored.  SQRT( QPTR(i+1) - QPTR(i) ) indicates
         the size of the block.

Z

          Z is DOUBLE PRECISION array, dimension (N)
         On output this vector contains the updating vector (the last
         row of the first sub-eigenvector matrix and the first row of
         the second sub-eigenvector matrix).

ZTEMP

          ZTEMP 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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

 DLAGTF factorizes the matrix (T - lambda*I), where T is an n by n
 tridiagonal matrix and lambda is a scalar, as
    T - lambda*I = PLU,
 where P is a permutation matrix, L is a unit lower tridiagonal matrix
 with at most one non-zero sub-diagonal elements per column and U is
 an upper triangular matrix with at most two non-zero super-diagonal
 elements per column.
 The factorization is obtained by Gaussian elimination with partial
 pivoting and implicit row scaling.
 The parameter LAMBDA is included in the routine so that DLAGTF may
 be used, in conjunction with DLAGTS, to obtain eigenvectors of T by
 inverse iteration.

N

          N is INTEGER
          The order of the matrix T.

A

          A is DOUBLE PRECISION array, dimension (N)
          On entry, A must contain the diagonal elements of T.
          On exit, A is overwritten by the n diagonal elements of the
          upper triangular matrix U of the factorization of T.

LAMBDA

          LAMBDA is DOUBLE PRECISION
          On entry, the scalar lambda.

B

          B is DOUBLE PRECISION array, dimension (N-1)
          On entry, B must contain the (n-1) super-diagonal elements of
          T.
          On exit, B is overwritten by the (n-1) super-diagonal
          elements of the matrix U of the factorization of T.

C

          C is DOUBLE PRECISION array, dimension (N-1)
          On entry, C must contain the (n-1) sub-diagonal elements of
          T.
          On exit, C is overwritten by the (n-1) sub-diagonal elements
          of the matrix L of the factorization of T.

TOL

          TOL is DOUBLE PRECISION
          On entry, a relative tolerance used to indicate whether or
          not the matrix (T - lambda*I) is nearly singular. TOL should
          normally be chose as approximately the largest relative error
          in the elements of T. For example, if the elements of T are
          correct to about 4 significant figures, then TOL should be
          set to about 5*10**(-4). If TOL is supplied as less than eps,
          where eps is the relative machine precision, then the value
          eps is used in place of TOL.

D

          D is DOUBLE PRECISION array, dimension (N-2)
          On exit, D is overwritten by the (n-2) second super-diagonal
          elements of the matrix U of the factorization of T.

IN

          IN is INTEGER array, dimension (N)
          On exit, IN contains details of the permutation matrix P. If
          an interchange occurred at the kth step of the elimination,
          then IN(k) = 1, otherwise IN(k) = 0. The element IN(n)
          returns the smallest positive integer j such that
             abs( u(j,j) ) <= norm( (T - lambda*I)(j) )*TOL,
          where norm( A(j) ) denotes the sum of the absolute values of
          the jth row of the matrix A. If no such j exists then IN(n)
          is returned as zero. If IN(n) is returned as positive, then a
          diagonal element of U is small, indicating that
          (T - lambda*I) is singular or nearly singular,

INFO

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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DLAMRG will create a permutation list which will merge the elements
 of A (which is composed of two independently sorted sets) into a
 single set which is sorted in ascending order.

N1

          N1 is INTEGER

N2

          N2 is INTEGER
         These arguments contain the respective lengths of the two
         sorted lists to be merged.

A

          A is DOUBLE PRECISION array, dimension (N1+N2)
         The first N1 elements of A contain a list of numbers which
         are sorted in either ascending or descending order.  Likewise
         for the final N2 elements.

DTRD1

          DTRD1 is INTEGER

DTRD2

          DTRD2 is INTEGER
         These are the strides to be taken through the array A.
         Allowable strides are 1 and -1.  They indicate whether a
         subset of A is sorted in ascending (DTRDx = 1) or descending
         (DTRDx = -1) order.

INDEX

          INDEX is INTEGER array, dimension (N1+N2)
         On exit this array will contain a permutation such that
         if B( I ) = A( INDEX( I ) ) for I=1,N1+N2, then B will be
         sorted in ascending order.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DLARTGS generates a plane rotation designed to introduce a bulge in
 Golub-Reinsch-style implicit QR iteration for the bidiagonal SVD
 problem. X and Y are the top-row entries, and SIGMA is the shift.
 The computed CS and SN define a plane rotation satisfying
    [  CS  SN  ]  .  [ X^2 - SIGMA ]  =  [ R ],
    [ -SN  CS  ]     [    X * Y    ]     [ 0 ]
 with R nonnegative.  If X^2 - SIGMA and X * Y are 0, then the
 rotation is by PI/2.

X

          X is DOUBLE PRECISION
          The (1,1) entry of an upper bidiagonal matrix.

Y

          Y is DOUBLE PRECISION
          The (1,2) entry of an upper bidiagonal matrix.

SIGMA

          SIGMA is DOUBLE PRECISION
          The shift.

CS

          CS is DOUBLE PRECISION
          The cosine of the rotation.

SN

          SN is DOUBLE PRECISION
          The sine of the rotation.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DLASQ1 computes the singular values of a real N-by-N bidiagonal
 matrix with diagonal D and off-diagonal E. The singular values
 are computed to high relative accuracy, in the absence of
 denormalization, underflow and overflow. The algorithm was first
 presented in
 "Accurate singular values and differential qd algorithms" by K. V.
 Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
 1994,
 and the present implementation is described in "An implementation of
 the dqds Algorithm (Positive Case)", LAPACK Working Note.

N

          N is INTEGER
        The number of rows and columns in the matrix. N >= 0.

D

          D is DOUBLE PRECISION array, dimension (N)
        On entry, D contains the diagonal elements of the
        bidiagonal matrix whose SVD is desired. On normal exit,
        D contains the singular values in decreasing order.

E

          E is DOUBLE PRECISION array, dimension (N)
        On entry, elements E(1:N-1) contain the off-diagonal elements
        of the bidiagonal matrix whose SVD is desired.
        On exit, E is overwritten.

WORK

          WORK is DOUBLE PRECISION array, dimension (4*N)

INFO

          INFO is INTEGER
        = 0: successful exit
        < 0: if INFO = -i, the i-th argument had an illegal value
        > 0: the algorithm failed
             = 1, a split was marked by a positive value in E
             = 2, current block of Z not diagonalized after 100*N
                  iterations (in inner while loop)  On exit D and E
                  represent a matrix with the same singular values
                  which the calling subroutine could use to finish the
                  computation, or even feed back into DLASQ1
             = 3, termination criterion of outer while loop not met
                  (program created more than N unreduced blocks)

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DLASQ2 computes all the eigenvalues of the symmetric positive
 definite tridiagonal matrix associated with the qd array Z to high
 relative accuracy are computed to high relative accuracy, in the
 absence of denormalization, underflow and overflow.
 To see the relation of Z to the tridiagonal matrix, let L be a
 unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
 let U be an upper bidiagonal matrix with 1's above and diagonal
 Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
 symmetric tridiagonal to which it is similar.
 Note : DLASQ2 defines a logical variable, IEEE, which is true
 on machines which follow ieee-754 floating-point standard in their
 handling of infinities and NaNs, and false otherwise. This variable
 is passed to DLASQ3.

N

          N is INTEGER
        The number of rows and columns in the matrix. N >= 0.

Z

          Z is DOUBLE PRECISION array, dimension ( 4*N )
        On entry Z holds the qd array. On exit, entries 1 to N hold
        the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
        trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
        N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
        holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
        shifts that failed.

INFO

          INFO is INTEGER
        = 0: successful exit
        < 0: if the i-th argument is a scalar and had an illegal
             value, then INFO = -i, if the i-th argument is an
             array and the j-entry had an illegal value, then
             INFO = -(i*100+j)
        > 0: the algorithm failed
              = 1, a split was marked by a positive value in E
              = 2, current block of Z not diagonalized after 100*N
                   iterations (in inner while loop).  On exit Z holds
                   a qd array with the same eigenvalues as the given Z.
              = 3, termination criterion of outer while loop not met
                   (program created more than N unreduced blocks)

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  Local Variables: I0:N0 defines a current unreduced segment of Z.
  The shifts are accumulated in SIGMA. Iteration count is in ITER.
  Ping-pong is controlled by PP (alternates between 0 and 1).

 DLASQ3 checks for deflation, computes a shift (TAU) and calls dqds.
 In case of failure it changes shifts, and tries again until output
 is positive.

I0

          I0 is INTEGER
         First index.

N0

          N0 is INTEGER
         Last index.

Z

          Z is DOUBLE PRECISION array, dimension ( 4*N0 )
         Z holds the qd array.

PP

          PP is INTEGER
         PP=0 for ping, PP=1 for pong.
         PP=2 indicates that flipping was applied to the Z array
         and that the initial tests for deflation should not be
         performed.

DMIN

          DMIN is DOUBLE PRECISION
         Minimum value of d.

SIGMA

          SIGMA is DOUBLE PRECISION
         Sum of shifts used in current segment.

DESIG

          DESIG is DOUBLE PRECISION
         Lower order part of SIGMA

QMAX

          QMAX is DOUBLE PRECISION
         Maximum value of q.

NFAIL

          NFAIL is INTEGER
         Increment NFAIL by 1 each time the shift was too big.

ITER

          ITER is INTEGER
         Increment ITER by 1 for each iteration.

NDIV

          NDIV is INTEGER
         Increment NDIV by 1 for each division.

IEEE

          IEEE is LOGICAL
         Flag for IEEE or non IEEE arithmetic (passed to DLASQ5).

TTYPE

          TTYPE is INTEGER
         Shift type.

DMIN1

          DMIN1 is DOUBLE PRECISION

DMIN2

          DMIN2 is DOUBLE PRECISION

DN

          DN is DOUBLE PRECISION

DN1

          DN1 is DOUBLE PRECISION

DN2

          DN2 is DOUBLE PRECISION

G

          G is DOUBLE PRECISION

TAU

          TAU is DOUBLE PRECISION
         These are passed as arguments in order to save their values
         between calls to DLASQ3.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DLASQ4 computes an approximation TAU to the smallest eigenvalue
 using values of d from the previous transform.

I0

          I0 is INTEGER
        First index.

N0

          N0 is INTEGER
        Last index.

Z

          Z is DOUBLE PRECISION array, dimension ( 4*N0 )
        Z holds the qd array.

PP

          PP is INTEGER
        PP=0 for ping, PP=1 for pong.

N0IN

          N0IN is INTEGER
        The value of N0 at start of EIGTEST.

DMIN

          DMIN is DOUBLE PRECISION
        Minimum value of d.

DMIN1

          DMIN1 is DOUBLE PRECISION
        Minimum value of d, excluding D( N0 ).

DMIN2

          DMIN2 is DOUBLE PRECISION
        Minimum value of d, excluding D( N0 ) and D( N0-1 ).

DN

          DN is DOUBLE PRECISION
        d(N)

DN1

          DN1 is DOUBLE PRECISION
        d(N-1)

DN2

          DN2 is DOUBLE PRECISION
        d(N-2)

TAU

          TAU is DOUBLE PRECISION
        This is the shift.

TTYPE

          TTYPE is INTEGER
        Shift type.

G

          G is DOUBLE PRECISION
        G is passed as an argument in order to save its value between
        calls to DLASQ4.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  CNST1 = 9/16

 DLASQ5 computes one dqds transform in ping-pong form, one
 version for IEEE machines another for non IEEE machines.

I0

          I0 is INTEGER
        First index.

N0

          N0 is INTEGER
        Last index.

Z

          Z is DOUBLE PRECISION array, dimension ( 4*N )
        Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
        an extra argument.

PP

          PP is INTEGER
        PP=0 for ping, PP=1 for pong.

TAU

          TAU is DOUBLE PRECISION
        This is the shift.

SIGMA

          SIGMA is DOUBLE PRECISION
        This is the accumulated shift up to this step.

DMIN

          DMIN is DOUBLE PRECISION
        Minimum value of d.

DMIN1

          DMIN1 is DOUBLE PRECISION
        Minimum value of d, excluding D( N0 ).

DMIN2

          DMIN2 is DOUBLE PRECISION
        Minimum value of d, excluding D( N0 ) and D( N0-1 ).

DN

          DN is DOUBLE PRECISION
        d(N0), the last value of d.

DNM1

          DNM1 is DOUBLE PRECISION
        d(N0-1).

DNM2

          DNM2 is DOUBLE PRECISION
        d(N0-2).

IEEE

          IEEE is LOGICAL
        Flag for IEEE or non IEEE arithmetic.

EPS

          EPS is DOUBLE PRECISION
        This is the value of epsilon used.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DLASQ6 computes one dqd (shift equal to zero) transform in
 ping-pong form, with protection against underflow and overflow.

I0

          I0 is INTEGER
        First index.

N0

          N0 is INTEGER
        Last index.

Z

          Z is DOUBLE PRECISION array, dimension ( 4*N )
        Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
        an extra argument.

PP

          PP is INTEGER
        PP=0 for ping, PP=1 for pong.

DMIN

          DMIN is DOUBLE PRECISION
        Minimum value of d.

DMIN1

          DMIN1 is DOUBLE PRECISION
        Minimum value of d, excluding D( N0 ).

DMIN2

          DMIN2 is DOUBLE PRECISION
        Minimum value of d, excluding D( N0 ) and D( N0-1 ).

DN

          DN is DOUBLE PRECISION
        d(N0), the last value of d.

DNM1

          DNM1 is DOUBLE PRECISION
        d(N0-1).

DNM2

          DNM2 is DOUBLE PRECISION
        d(N0-2).

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 Sort the numbers in D in increasing order (if ID = 'I') or
 in decreasing order (if ID = 'D' ).
 Use Quick Sort, reverting to Insertion sort on arrays of
 size <= 20. Dimension of STACK limits N to about 2**32.

ID

          ID is CHARACTER*1
          = 'I': sort D in increasing order;
          = 'D': sort D in decreasing order.

N

          N is INTEGER
          The length of the array D.

D

          D is DOUBLE PRECISION array, dimension (N)
          On entry, the array to be sorted.
          On exit, D has been sorted into increasing order
          (D(1) <= ... <= D(N) ) or into decreasing order
          (D(1) >= ... >= D(N) ), depending on ID.

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.

 DSTEBZ computes the eigenvalues of a symmetric tridiagonal
 matrix T.  The user may ask for all eigenvalues, all eigenvalues
 in the half-open interval (VL, VU], or the IL-th through IU-th
 eigenvalues.
 To avoid overflow, the matrix must be scaled so that its
 largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
 accuracy, it should not be much smaller than that.
 See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
 Matrix", Report CS41, Computer Science Dept., Stanford
 University, July 21, 1966.

RANGE

          RANGE is CHARACTER*1
          = 'A': ("All")   all eigenvalues will be found.
          = 'V': ("Value") all eigenvalues in the half-open interval
                           (VL, VU] will be found.
          = 'I': ("Index") the IL-th through IU-th eigenvalues (of the
                           entire matrix) will be found.

ORDER

          ORDER is CHARACTER*1
          = 'B': ("By Block") the eigenvalues will be grouped by
                              split-off block (see IBLOCK, ISPLIT) and
                              ordered from smallest to largest within
                              the block.
          = 'E': ("Entire matrix")
                              the eigenvalues for the entire matrix
                              will be ordered from smallest to
                              largest.

N

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

VL

          VL is DOUBLE PRECISION
          If RANGE='V', the lower bound of the interval to
          be searched for eigenvalues.  Eigenvalues less than or equal
          to VL, or greater than VU, will not be returned.  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.  Eigenvalues less than or equal
          to VL, or greater than VU, will not be returned.  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; IL = 1 and IU = 0 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; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

ABSTOL

          ABSTOL is DOUBLE PRECISION
          The absolute tolerance for the eigenvalues.  An eigenvalue
          (or cluster) is considered to be located if it has been
          determined to lie in an interval whose width is ABSTOL or
          less.  If ABSTOL is less than or equal to zero, then ULP*|T|
          will be used, where |T| means the 1-norm of T.
          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*DLAMCH('S'), not zero.

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) off-diagonal elements of the tridiagonal matrix T.

M

          M is INTEGER
          The actual number of eigenvalues found. 0 <= M <= N.
          (See also the description of INFO=2,3.)

NSPLIT

          NSPLIT is INTEGER
          The number of diagonal blocks in the matrix T.
          1 <= NSPLIT <= N.

W

          W is DOUBLE PRECISION array, dimension (N)
          On exit, the first M elements of W will contain the
          eigenvalues.  (DSTEBZ may use the remaining N-M elements as
          workspace.)

IBLOCK

          IBLOCK is INTEGER array, dimension (N)
          At each row/column j where E(j) is zero or small, the
          matrix T is considered to split into a block diagonal
          matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which
          block (from 1 to the number of blocks) the eigenvalue W(i)
          belongs.  (DSTEBZ may use the remaining N-M elements as
          workspace.)

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., and the NSPLIT-th consists of rows/columns
          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
          (Only the first NSPLIT elements will actually be used, but
          since the user cannot know a priori what value NSPLIT will
          have, N words must be reserved for ISPLIT.)

WORK

          WORK is DOUBLE PRECISION array, dimension (4*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
          > 0:  some or all of the eigenvalues failed to converge or
                were not computed:
                =1 or 3: Bisection failed to converge for some
                        eigenvalues; these eigenvalues are flagged by a
                        negative block number.  The effect is that the
                        eigenvalues may not be as accurate as the
                        absolute and relative tolerances.  This is
                        generally caused by unexpectedly inaccurate
                        arithmetic.
                =2 or 3: RANGE='I' only: Not all of the eigenvalues
                        IL:IU were found.
                        Effect: M < IU+1-IL
                        Cause:  non-monotonic arithmetic, causing the
                                Sturm sequence to be non-monotonic.
                        Cure:   recalculate, using RANGE='A', and pick
                                out eigenvalues IL:IU.  In some cases,
                                increasing the PARAMETER "FUDGE" may
                                make things work.
                = 4:    RANGE='I', and the Gershgorin interval
                        initially used was too small.  No eigenvalues
                        were computed.
                        Probable cause: your machine has sloppy
                                        floating-point arithmetic.
                        Cure: Increase the PARAMETER "FUDGE",
                              recompile, and try again.

  RELFAC  DOUBLE PRECISION, default = 2.0e0
          The relative tolerance.  An interval (a,b] lies within
          "relative tolerance" if  b-a < RELFAC*ulp*max(|a|,|b|),
          where "ulp" is the machine precision (distance from 1 to
          the next larger floating point number.)
  FUDGE   DOUBLE PRECISION, default = 2
          A "fudge factor" to widen the Gershgorin intervals.  Ideally,
          a value of 1 should work, but on machines with sloppy
          arithmetic, this needs to be larger.  The default for
          publicly released versions should be large enough to handle
          the worst machine around.  Note that this has no effect
          on accuracy of the solution.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DSTEDC computes all eigenvalues and, optionally, eigenvectors of a
 symmetric tridiagonal matrix using the divide and conquer method.
 The eigenvectors of a full or band real symmetric matrix can also be
 found if DSYTRD or DSPTRD or DSBTRD has been used to reduce this
 matrix to tridiagonal form.
 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 X-MP, Cray Y-MP, Cray C-90, or Cray-2.
 It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.  See DLAED3 for details.

COMPZ

          COMPZ is CHARACTER*1
          = 'N':  Compute eigenvalues only.
          = 'I':  Compute eigenvectors of tridiagonal matrix also.
          = 'V':  Compute eigenvectors of original dense symmetric
                  matrix also.  On entry, Z contains the orthogonal
                  matrix used to reduce the original matrix to
                  tridiagonal form.

N

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

D

          D is DOUBLE PRECISION array, dimension (N)
          On entry, the diagonal elements of the tridiagonal matrix.
          On exit, if INFO = 0, the eigenvalues in ascending order.

E

          E is DOUBLE PRECISION array, dimension (N-1)
          On entry, the subdiagonal elements of the tridiagonal matrix.
          On exit, E has been destroyed.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ,N)
          On entry, if COMPZ = 'V', then Z contains the orthogonal
          matrix used in the reduction to tridiagonal form.
          On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
          orthonormal eigenvectors of the original symmetric matrix,
          and if COMPZ = 'I', Z contains the orthonormal eigenvectors
          of the symmetric tridiagonal matrix.
          If  COMPZ = 'N', then Z is not referenced.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1.
          If eigenvectors are desired, then LDZ >= max(1,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 COMPZ = 'N' or N <= 1 then LWORK must be at least 1.
          If COMPZ = 'V' and N > 1 then LWORK must be at least
                         ( 1 + 3*N + 2*N*lg N + 4*N**2 ),
                         where lg( N ) = smallest integer k such
                         that 2**k >= N.
          If COMPZ = 'I' and N > 1 then LWORK must be at least
                         ( 1 + 4*N + N**2 ).
          Note that for COMPZ = 'I' or 'V', then if N is less than or
          equal to the minimum divide size, usually 25, then LWORK need
          only be max(1,2*(N-1)).
          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.
          If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1.
          If COMPZ = 'V' and N > 1 then LIWORK must be at least
                         ( 6 + 6*N + 5*N*lg N ).
          If COMPZ = 'I' and N > 1 then LIWORK must be at least
                         ( 3 + 5*N ).
          Note that for COMPZ = 'I' or 'V', then if N is less than or
          equal to the minimum divide size, usually 25, then LIWORK
          need only be 1.
          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.
          > 0:  The algorithm failed to compute an eigenvalue while
                working on the submatrix lying in rows and columns
                INFO/(N+1) through mod(INFO,N+1).

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA Modified by Francoise Tisseur, University of Tennessee

 DSTEQR computes all eigenvalues and, optionally, eigenvectors of a
 symmetric tridiagonal matrix using the implicit QL or QR method.
 The eigenvectors of a full or band symmetric matrix can also be found
 if DSYTRD or DSPTRD or DSBTRD has been used to reduce this matrix to
 tridiagonal form.

COMPZ

          COMPZ is CHARACTER*1
          = 'N':  Compute eigenvalues only.
          = 'V':  Compute eigenvalues and eigenvectors of the original
                  symmetric matrix.  On entry, Z must contain the
                  orthogonal matrix used to reduce the original matrix
                  to tridiagonal form.
          = 'I':  Compute eigenvalues and eigenvectors of the
                  tridiagonal matrix.  Z is initialized to the identity
                  matrix.

N

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

D

          D is DOUBLE PRECISION array, dimension (N)
          On entry, the diagonal elements of the tridiagonal matrix.
          On exit, if INFO = 0, the eigenvalues in ascending order.

E

          E is DOUBLE PRECISION array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix.
          On exit, E has been destroyed.

Z

          Z is DOUBLE PRECISION array, dimension (LDZ, N)
          On entry, if  COMPZ = 'V', then Z contains the orthogonal
          matrix used in the reduction to tridiagonal form.
          On exit, if INFO = 0, then if  COMPZ = 'V', Z contains the
          orthonormal eigenvectors of the original symmetric matrix,
          and if COMPZ = 'I', Z contains the orthonormal eigenvectors
          of the symmetric tridiagonal matrix.
          If COMPZ = 'N', then Z is not referenced.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          eigenvectors are desired, then  LDZ >= max(1,N).

WORK

          WORK is DOUBLE PRECISION array, dimension (max(1,2*N-2))
          If COMPZ = 'N', then WORK is not referenced.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  the algorithm has failed to find all the eigenvalues in
                a total of 30*N iterations; if INFO = i, then i
                elements of E have not converged to zero; on exit, D
                and E contain the elements of a symmetric tridiagonal
                matrix which is orthogonally similar to the original
                matrix.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DSTERF computes all eigenvalues of a symmetric tridiagonal matrix
 using the Pal-Walker-Kahan variant of the QL or QR algorithm.

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.
          On exit, if INFO = 0, the eigenvalues in ascending order.

E

          E is DOUBLE PRECISION array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix.
          On exit, E has been destroyed.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  the algorithm failed to find all of the eigenvalues in
                a total of 30*N iterations; if INFO = i, then i
                elements of E have not converged to zero.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 This subroutine translated from a character string specifying if a
 matrix has unit diagonal or not to the relevant BLAST-specified
 integer constant.
 ILADIAG returns an INTEGER.  If ILADIAG < 0, then the input is not a
 character indicating a unit or non-unit diagonal.  Otherwise ILADIAG
 returns the constant value corresponding to DIAG.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 This subroutine translated from a character string specifying an
 intermediate precision to the relevant BLAST-specified integer
 constant.
 ILAPREC returns an INTEGER.  If ILAPREC < 0, then the input is not a
 character indicating a supported intermediate precision.  Otherwise
 ILAPREC returns the constant value corresponding to PREC.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 This subroutine translates from a character string specifying a
 transposition operation to the relevant BLAST-specified integer
 constant.
 ILATRANS returns an INTEGER.  If ILATRANS < 0, then the input is not
 a character indicating a transposition operator.  Otherwise ILATRANS
 returns the constant value corresponding to TRANS.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 This subroutine translated from a character string specifying a
 upper- or lower-triangular matrix to the relevant BLAST-specified
 integer constant.
 ILAUPLO returns an INTEGER.  If ILAUPLO < 0, then the input is not
 a character indicating an upper- or lower-triangular matrix.
 Otherwise ILAUPLO returns the constant value corresponding to UPLO.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 SBDSDC computes the singular value decomposition (SVD) of a real
 N-by-N (upper or lower) bidiagonal matrix B:  B = U * S * VT,
 using a divide and conquer method, where S is a diagonal matrix
 with non-negative diagonal elements (the singular values of B), and
 U and VT are orthogonal matrices of left and right singular vectors,
 respectively. SBDSDC can be used to compute all singular values,
 and optionally, singular vectors or singular vectors in compact form.
 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 X-MP, Cray Y-MP, Cray C-90, or Cray-2.
 It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.  See SLASD3 for details.
 The code currently calls SLASDQ if singular values only are desired.
 However, it can be slightly modified to compute singular values
 using the divide and conquer method.

UPLO

          UPLO is CHARACTER*1
          = 'U':  B is upper bidiagonal.
          = 'L':  B is lower bidiagonal.

COMPQ

          COMPQ is CHARACTER*1
          Specifies whether singular vectors are to be computed
          as follows:
          = 'N':  Compute singular values only;
          = 'P':  Compute singular values and compute singular
                  vectors in compact form;
          = 'I':  Compute singular values and singular vectors.

N

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

D

          D is REAL array, dimension (N)
          On entry, the n diagonal elements of the bidiagonal matrix B.
          On exit, if INFO=0, the singular values of B.

E

          E is REAL array, dimension (N-1)
          On entry, the elements of E contain the offdiagonal
          elements of the bidiagonal matrix whose SVD is desired.
          On exit, E has been destroyed.

U

          U is REAL array, dimension (LDU,N)
          If  COMPQ = 'I', then:
             On exit, if INFO = 0, U contains the left singular vectors
             of the bidiagonal matrix.
          For other values of COMPQ, U is not referenced.

LDU

          LDU is INTEGER
          The leading dimension of the array U.  LDU >= 1.
          If singular vectors are desired, then LDU >= max( 1, N ).

VT

          VT is REAL array, dimension (LDVT,N)
          If  COMPQ = 'I', then:
             On exit, if INFO = 0, VT**T contains the right singular
             vectors of the bidiagonal matrix.
          For other values of COMPQ, VT is not referenced.

LDVT

          LDVT is INTEGER
          The leading dimension of the array VT.  LDVT >= 1.
          If singular vectors are desired, then LDVT >= max( 1, N ).

Q

          Q is REAL array, dimension (LDQ)
          If  COMPQ = 'P', then:
             On exit, if INFO = 0, Q and IQ contain the left
             and right singular vectors in a compact form,
             requiring O(N log N) space instead of 2*N**2.
             In particular, Q contains all the REAL data in
             LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1))))
             words of memory, where SMLSIZ is returned by ILAENV and
             is equal to the maximum size of the subproblems at the
             bottom of the computation tree (usually about 25).
          For other values of COMPQ, Q is not referenced.

IQ

          IQ is INTEGER array, dimension (LDIQ)
          If  COMPQ = 'P', then:
             On exit, if INFO = 0, Q and IQ contain the left
             and right singular vectors in a compact form,
             requiring O(N log N) space instead of 2*N**2.
             In particular, IQ contains all INTEGER data in
             LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
             words of memory, where SMLSIZ is returned by ILAENV and
             is equal to the maximum size of the subproblems at the
             bottom of the computation tree (usually about 25).
          For other values of COMPQ, IQ is not referenced.

WORK

          WORK is REAL array, dimension (MAX(1,LWORK))
          If COMPQ = 'N' then LWORK >= (4 * N).
          If COMPQ = 'P' then LWORK >= (6 * N).
          If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).

IWORK

          IWORK is INTEGER array, dimension (8*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.
                The update process of divide and conquer failed.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

 SBDSQR computes the singular values and, optionally, the right and/or
 left singular vectors from the singular value decomposition (SVD) of
 a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
 zero-shift QR algorithm.  The SVD of B has the form
    B = Q * S * P**T
 where S is the diagonal matrix of singular values, Q is an orthogonal
 matrix of left singular vectors, and P is an orthogonal matrix of
 right singular vectors.  If left singular vectors are requested, this
 subroutine actually returns U*Q instead of Q, and, if right singular
 vectors are requested, this subroutine returns P**T*VT instead of
 P**T, for given real input matrices U and VT.  When U and VT are the
 orthogonal matrices that reduce a general matrix A to bidiagonal
 form:  A = U*B*VT, as computed by SGEBRD, then
    A = (U*Q) * S * (P**T*VT)
 is the SVD of A.  Optionally, the subroutine may also compute Q**T*C
 for a given real input matrix C.
 See "Computing  Small Singular Values of Bidiagonal Matrices With
 Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
 LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
 no. 5, pp. 873-912, Sept 1990) and
 "Accurate singular values and differential qd algorithms," by
 B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
 Department, University of California at Berkeley, July 1992
 for a detailed description of the algorithm.

UPLO

          UPLO is CHARACTER*1
          = 'U':  B is upper bidiagonal;
          = 'L':  B is lower bidiagonal.

N

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

NCVT

          NCVT is INTEGER
          The number of columns of the matrix VT. NCVT >= 0.

NRU

          NRU is INTEGER
          The number of rows of the matrix U. NRU >= 0.

NCC

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

D

          D is REAL array, dimension (N)
          On entry, the n diagonal elements of the bidiagonal matrix B.
          On exit, if INFO=0, the singular values of B in decreasing
          order.

E

          E is REAL array, dimension (N-1)
          On entry, the N-1 offdiagonal elements of the bidiagonal
          matrix B.
          On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
          will contain the diagonal and superdiagonal elements of a
          bidiagonal matrix orthogonally equivalent to the one given
          as input.

VT

          VT is REAL array, dimension (LDVT, NCVT)
          On entry, an N-by-NCVT matrix VT.
          On exit, VT is overwritten by P**T * VT.
          Not referenced if NCVT = 0.

LDVT

          LDVT is INTEGER
          The leading dimension of the array VT.
          LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.

U

          U is REAL array, dimension (LDU, N)
          On entry, an NRU-by-N matrix U.
          On exit, U is overwritten by U * Q.
          Not referenced if NRU = 0.

LDU

          LDU is INTEGER
          The leading dimension of the array U.  LDU >= max(1,NRU).

C

          C is REAL array, dimension (LDC, NCC)
          On entry, an N-by-NCC matrix C.
          On exit, C is overwritten by Q**T * C.
          Not referenced if NCC = 0.

LDC

          LDC is INTEGER
          The leading dimension of the array C.
          LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.

WORK

          WORK is REAL array, dimension (4*N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  If INFO = -i, the i-th argument had an illegal value
          > 0:
             if NCVT = NRU = NCC = 0,
                = 1, a split was marked by a positive value in E
                = 2, current block of Z not diagonalized after 30*N
                     iterations (in inner while loop)
                = 3, termination criterion of outer while loop not met
                     (program created more than N unreduced blocks)
             else NCVT = NRU = NCC = 0,
                   the algorithm did not converge; D and E contain the
                   elements of a bidiagonal matrix which is orthogonally
                   similar to the input matrix B;  if INFO = i, i
                   elements of E have not converged to zero.

  TOLMUL  REAL, default = max(10,min(100,EPS**(-1/8)))
          TOLMUL controls the convergence criterion of the QR loop.
          If it is positive, TOLMUL*EPS is the desired relative
             precision in the computed singular values.
          If it is negative, abs(TOLMUL*EPS*sigma_max) is the
             desired absolute accuracy in the computed singular
             values (corresponds to relative accuracy
             abs(TOLMUL*EPS) in the largest singular value.
          abs(TOLMUL) should be between 1 and 1/EPS, and preferably
             between 10 (for fast convergence) and .1/EPS
             (for there to be some accuracy in the results).
          Default is to lose at either one eighth or 2 of the
             available decimal digits in each computed singular value
             (whichever is smaller).
  MAXITR  INTEGER, default = 6
          MAXITR controls the maximum number of passes of the
          algorithm through its inner loop. The algorithms stops
          (and so fails to converge) if the number of passes
          through the inner loop exceeds MAXITR*N**2.

  Bug report from Cezary Dendek.
  On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is
  removed since it can overflow pretty easily (for N larger or equal
  than 18,919). We instead use MAXITDIVN = MAXITR*N.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 SDISNA computes the reciprocal condition numbers for the eigenvectors
 of a real symmetric or complex Hermitian matrix or for the left or
 right singular vectors of a general m-by-n matrix. The reciprocal
 condition number is the 'gap' between the corresponding eigenvalue or
 singular value and the nearest other one.
 The bound on the error, measured by angle in radians, in the I-th
 computed vector is given by
        SLAMCH( 'E' ) * ( ANORM / SEP( I ) )
 where ANORM = 2-norm(A) = max( abs( D(j) ) ).  SEP(I) is not allowed
 to be smaller than SLAMCH( 'E' )*ANORM in order to limit the size of
 the error bound.
 SDISNA may also be used to compute error bounds for eigenvectors of
 the generalized symmetric definite eigenproblem.

JOB

          JOB is CHARACTER*1
          Specifies for which problem the reciprocal condition numbers
          should be computed:
          = 'E':  the eigenvectors of a symmetric/Hermitian matrix;
          = 'L':  the left singular vectors of a general matrix;
          = 'R':  the right singular vectors of a general matrix.

M

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

N

          N is INTEGER
          If JOB = 'L' or 'R', the number of columns of the matrix,
          in which case N >= 0. Ignored if JOB = 'E'.

D

          D is REAL array, dimension (M) if JOB = 'E'
                              dimension (min(M,N)) if JOB = 'L' or 'R'
          The eigenvalues (if JOB = 'E') or singular values (if JOB =
          'L' or 'R') of the matrix, in either increasing or decreasing
          order. If singular values, they must be non-negative.

SEP

          SEP is REAL array, dimension (M) if JOB = 'E'
                               dimension (min(M,N)) if JOB = 'L' or 'R'
          The reciprocal condition numbers of the vectors.

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.

 SLAED0 computes all eigenvalues and corresponding eigenvectors of a
 symmetric tridiagonal matrix using the divide and conquer method.

ICOMPQ

          ICOMPQ is INTEGER
          = 0:  Compute eigenvalues only.
          = 1:  Compute eigenvectors of original dense symmetric matrix
                also.  On entry, Q contains the orthogonal matrix used
                to reduce the original matrix to tridiagonal form.
          = 2:  Compute eigenvalues and eigenvectors of tridiagonal
                matrix.

QSIZ

          QSIZ is INTEGER
         The dimension of the orthogonal matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

N

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

D

          D is REAL array, dimension (N)
         On entry, the main diagonal of the tridiagonal matrix.
         On exit, its eigenvalues.

E

          E is REAL array, dimension (N-1)
         The off-diagonal elements of the tridiagonal matrix.
         On exit, E has been destroyed.

Q

          Q is REAL array, dimension (LDQ, N)
         On entry, Q must contain an N-by-N orthogonal matrix.
         If ICOMPQ = 0    Q is not referenced.
         If ICOMPQ = 1    On entry, Q is a subset of the columns of the
                          orthogonal matrix used to reduce the full
                          matrix to tridiagonal form corresponding to
                          the subset of the full matrix which is being
                          decomposed at this time.
         If ICOMPQ = 2    On entry, Q will be the identity matrix.
                          On exit, Q contains the eigenvectors of the
                          tridiagonal matrix.

LDQ

          LDQ is INTEGER
         The leading dimension of the array Q.  If eigenvectors are
         desired, then  LDQ >= max(1,N).  In any case,  LDQ >= 1.

QSTORE

          QSTORE is REAL array, dimension (LDQS, N)
         Referenced only when ICOMPQ = 1.  Used to store parts of
         the eigenvector matrix when the updating matrix multiplies
         take place.

LDQS

          LDQS is INTEGER
         The leading dimension of the array QSTORE.  If ICOMPQ = 1,
         then  LDQS >= max(1,N).  In any case,  LDQS >= 1.

WORK

          WORK is REAL array,
         If ICOMPQ = 0 or 1, the dimension of WORK must be at least
                     1 + 3*N + 2*N*lg N + 3*N**2
                     ( lg( N ) = smallest integer k
                                 such that 2^k >= N )
         If ICOMPQ = 2, the dimension of WORK must be at least
                     4*N + N**2.

IWORK

          IWORK is INTEGER array,
         If ICOMPQ = 0 or 1, the dimension of IWORK must be at least
                        6 + 6*N + 5*N*lg N.
                        ( lg( N ) = smallest integer k
                                    such that 2^k >= N )
         If ICOMPQ = 2, the dimension of IWORK must be at least
                        3 + 5*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 an eigenvalue while
                working on the submatrix lying in rows and columns
                INFO/(N+1) through mod(INFO,N+1).

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

 SLAED1 computes the updated eigensystem of a diagonal
 matrix after modification by a rank-one symmetric matrix.  This
 routine is used only for the eigenproblem which requires all
 eigenvalues and eigenvectors of a tridiagonal matrix.  SLAED7 handles
 the case in which eigenvalues only or eigenvalues and eigenvectors
 of a full symmetric matrix (which was reduced to tridiagonal form)
 are desired.
   T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
    where Z = Q**T*u, u is a vector of length N with ones in the
    CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
    The eigenvectors of the original matrix are stored in Q, and the
    eigenvalues are in D.  The algorithm consists of three stages:
       The first stage consists of deflating the size of the problem
       when there are multiple eigenvalues or if there is a zero in
       the Z vector.  For each such occurrence the dimension of the
       secular equation problem is reduced by one.  This stage is
       performed by the routine SLAED2.
       The second stage consists of calculating the updated
       eigenvalues. This is done by finding the roots of the secular
       equation via the routine SLAED4 (as called by SLAED3).
       This routine also calculates the eigenvectors of the current
       problem.
       The final stage consists of computing the updated eigenvectors
       directly using the updated eigenvalues.  The eigenvectors for
       the current problem are multiplied with the eigenvectors from
       the overall problem.

N

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

D

          D is REAL array, dimension (N)
         On entry, the eigenvalues of the rank-1-perturbed matrix.
         On exit, the eigenvalues of the repaired matrix.

Q

          Q is REAL array, dimension (LDQ,N)
         On entry, the eigenvectors of the rank-1-perturbed matrix.
         On exit, the eigenvectors of the repaired tridiagonal matrix.

LDQ

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

INDXQ

          INDXQ is INTEGER array, dimension (N)
         On entry, the permutation which separately sorts the two
         subproblems in D into ascending order.
         On exit, the permutation which will reintegrate the
         subproblems back into sorted order,
         i.e. D( INDXQ( I = 1, N ) ) will be in ascending order.

RHO

          RHO is REAL
         The subdiagonal entry used to create the rank-1 modification.

CUTPNT

          CUTPNT is INTEGER
         The location of the last eigenvalue in the leading sub-matrix.
         min(1,N) <= CUTPNT <= N/2.

WORK

          WORK is REAL array, dimension (4*N + N**2)

IWORK

          IWORK is INTEGER array, dimension (4*N)

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA Modified by Francoise Tisseur, University of Tennessee

 SLAED2 merges the two sets of eigenvalues together into a single
 sorted set.  Then it tries to deflate the size of the problem.
 There are two ways in which deflation can occur:  when two or more
 eigenvalues are close together or if there is a tiny entry in the
 Z vector.  For each such occurrence the order of the related secular
 equation problem is reduced by one.

K

          K is INTEGER
         The number of non-deflated eigenvalues, and the order of the
         related secular equation. 0 <= K <=N.

N

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

N1

          N1 is INTEGER
         The location of the last eigenvalue in the leading sub-matrix.
         min(1,N) <= N1 <= N/2.

D

          D is REAL array, dimension (N)
         On entry, D contains the eigenvalues of the two submatrices to
         be combined.
         On exit, D contains the trailing (N-K) updated eigenvalues
         (those which were deflated) sorted into increasing order.

Q

          Q is REAL array, dimension (LDQ, N)
         On entry, Q contains the eigenvectors of two submatrices in
         the two square blocks with corners at (1,1), (N1,N1)
         and (N1+1, N1+1), (N,N).
         On exit, Q contains the trailing (N-K) updated eigenvectors
         (those which were deflated) in its last N-K columns.

LDQ

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

INDXQ

          INDXQ is INTEGER array, dimension (N)
         The permutation which separately sorts the two sub-problems
         in D into ascending order.  Note that elements in the second
         half of this permutation must first have N1 added to their
         values. Destroyed on exit.

RHO

          RHO is REAL
         On entry, the off-diagonal element associated with the rank-1
         cut which originally split the two submatrices which are now
         being recombined.
         On exit, RHO has been modified to the value required by
         SLAED3.

Z

          Z is REAL array, dimension (N)
         On entry, Z contains the updating vector (the last
         row of the first sub-eigenvector matrix and the first row of
         the second sub-eigenvector matrix).
         On exit, the contents of Z have been destroyed by the updating
         process.

DLAMDA

          DLAMDA is REAL array, dimension (N)
         A copy of the first K eigenvalues which will be used by
         SLAED3 to form the secular equation.

W

          W is REAL array, dimension (N)
         The first k values of the final deflation-altered z-vector
         which will be passed to SLAED3.

Q2

          Q2 is REAL array, dimension (N1**2+(N-N1)**2)
         A copy of the first K eigenvectors which will be used by
         SLAED3 in a matrix multiply (SGEMM) to solve for the new
         eigenvectors.

INDX

          INDX is INTEGER array, dimension (N)
         The permutation used to sort the contents of DLAMDA into
         ascending order.

INDXC

          INDXC is INTEGER array, dimension (N)
         The permutation used to arrange the columns of the deflated
         Q matrix into three groups:  the first group contains non-zero
         elements only at and above N1, the second contains
         non-zero elements only below N1, and the third is dense.

INDXP

          INDXP is INTEGER array, dimension (N)
         The permutation used to place deflated values of D at the end
         of the array.  INDXP(1:K) points to the nondeflated D-values
         and INDXP(K+1:N) points to the deflated eigenvalues.

COLTYP

          COLTYP is INTEGER array, dimension (N)
         During execution, a label which will indicate which of the
         following types a column in the Q2 matrix is:
         1 : non-zero in the upper half only;
         2 : dense;
         3 : non-zero in the lower half only;
         4 : deflated.
         On exit, COLTYP(i) is the number of columns of type i,
         for i=1 to 4 only.

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.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA Modified by Francoise Tisseur, University of Tennessee

 SLAED3 finds the roots of the secular equation, as defined by the
 values in D, W, and RHO, between 1 and K.  It makes the
 appropriate calls to SLAED4 and then updates the eigenvectors by
 multiplying the matrix of eigenvectors of the pair of eigensystems
 being combined by the matrix of eigenvectors of the K-by-K system
 which is solved here.
 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 X-MP, Cray Y-MP, Cray C-90, or Cray-2.
 It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.

K

          K is INTEGER
          The number of terms in the rational function to be solved by
          SLAED4.  K >= 0.

N

          N is INTEGER
          The number of rows and columns in the Q matrix.
          N >= K (deflation may result in N>K).

N1

          N1 is INTEGER
          The location of the last eigenvalue in the leading submatrix.
          min(1,N) <= N1 <= N/2.

D

          D is REAL array, dimension (N)
          D(I) contains the updated eigenvalues for
          1 <= I <= K.

Q

          Q is REAL array, dimension (LDQ,N)
          Initially the first K columns are used as workspace.
          On output the columns 1 to K contain
          the updated eigenvectors.

LDQ

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

RHO

          RHO is REAL
          The value of the parameter in the rank one update equation.
          RHO >= 0 required.

DLAMDA

          DLAMDA is REAL array, dimension (K)
          The first K elements of this array contain the old roots
          of the deflated updating problem.  These are the poles
          of the secular equation. May be changed on output by
          having lowest order bit set to zero on Cray X-MP, Cray Y-MP,
          Cray-2, or Cray C-90, as described above.

Q2

          Q2 is REAL array, dimension (LDQ2*N)
          The first K columns of this matrix contain the non-deflated
          eigenvectors for the split problem.

INDX

          INDX is INTEGER array, dimension (N)
          The permutation used to arrange the columns of the deflated
          Q matrix into three groups (see SLAED2).
          The rows of the eigenvectors found by SLAED4 must be likewise
          permuted before the matrix multiply can take place.

CTOT

          CTOT is INTEGER array, dimension (4)
          A count of the total number of the various types of columns
          in Q, as described in INDX.  The fourth column type is any
          column which has been deflated.

W

          W is REAL array, dimension (K)
          The first K elements of this array contain the components
          of the deflation-adjusted updating vector. Destroyed on
          output.

S

          S is REAL array, dimension (N1 + 1)*K
          Will contain the eigenvectors of the repaired matrix which
          will be multiplied by the previously accumulated eigenvectors
          to update the system.

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA Modified by Francoise Tisseur, University of Tennessee

 This subroutine computes the I-th updated eigenvalue of a symmetric
 rank-one modification to a diagonal matrix whose elements are
 given in the array d, and that
            D(i) < D(j)  for  i < j
 and that RHO > 0.  This is arranged by the calling routine, and is
 no loss in generality.  The rank-one modified system is thus
            diag( D )  +  RHO * Z * Z_transpose.
 where we assume the Euclidean norm of Z is 1.
 The method consists of approximating the rational functions in the
 secular equation by simpler interpolating rational functions.

N

          N is INTEGER
         The length of all arrays.

I

          I is INTEGER
         The index of the eigenvalue to be computed.  1 <= I <= N.

D

          D is REAL array, dimension (N)
         The original eigenvalues.  It is assumed that they are in
         order, D(I) < D(J)  for I < J.

Z

          Z is REAL array, dimension (N)
         The components of the updating vector.

DELTA

          DELTA is REAL array, dimension (N)
         If N > 2, DELTA contains (D(j) - lambda_I) in its  j-th
         component.  If N = 1, then DELTA(1) = 1. If N = 2, see SLAED5
         for detail. The vector DELTA contains the information necessary
         to construct the eigenvectors by SLAED3 and SLAED9.

RHO

          RHO is REAL
         The scalar in the symmetric updating formula.

DLAM

          DLAM is REAL
         The computed lambda_I, the I-th updated eigenvalue.

INFO

          INFO is INTEGER
         = 0:  successful exit
         > 0:  if INFO = 1, the updating process failed.

  Logical variable ORGATI (origin-at-i?) is used for distinguishing
  whether D(i) or D(i+1) is treated as the origin.
            ORGATI = .true.    origin at i
            ORGATI = .false.   origin at i+1
   Logical variable SWTCH3 (switch-for-3-poles?) is for noting
   if we are working with THREE poles!
   MAXIT is the maximum number of iterations allowed for each
   eigenvalue.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

 This subroutine computes the I-th eigenvalue of a symmetric rank-one
 modification of a 2-by-2 diagonal matrix
            diag( D )  +  RHO * Z * transpose(Z) .
 The diagonal elements in the array D are assumed to satisfy
            D(i) < D(j)  for  i < j .
 We also assume RHO > 0 and that the Euclidean norm of the vector
 Z is one.

I

          I is INTEGER
         The index of the eigenvalue to be computed.  I = 1 or I = 2.

D

          D is REAL array, dimension (2)
         The original eigenvalues.  We assume D(1) < D(2).

Z

          Z is REAL array, dimension (2)
         The components of the updating vector.

DELTA

          DELTA is REAL array, dimension (2)
         The vector DELTA contains the information necessary
         to construct the eigenvectors.

RHO

          RHO is REAL
         The scalar in the symmetric updating formula.

DLAM

          DLAM is REAL
         The computed lambda_I, the I-th updated eigenvalue.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

 SLAED6 computes the positive or negative root (closest to the origin)
 of
                  z(1)        z(2)        z(3)
 f(x) =   rho + --------- + ---------- + ---------
                 d(1)-x      d(2)-x      d(3)-x
 It is assumed that
       if ORGATI = .true. the root is between d(2) and d(3);
       otherwise it is between d(1) and d(2)
 This routine will be called by SLAED4 when necessary. In most cases,
 the root sought is the smallest in magnitude, though it might not be
 in some extremely rare situations.

KNITER

          KNITER is INTEGER
               Refer to SLAED4 for its significance.

ORGATI

          ORGATI is LOGICAL
               If ORGATI is true, the needed root is between d(2) and
               d(3); otherwise it is between d(1) and d(2).  See
               SLAED4 for further details.

RHO

          RHO is REAL
               Refer to the equation f(x) above.

D

          D is REAL array, dimension (3)
               D satisfies d(1) < d(2) < d(3).

Z

          Z is REAL array, dimension (3)
               Each of the elements in z must be positive.

FINIT

          FINIT is REAL
               The value of f at 0. It is more accurate than the one
               evaluated inside this routine (if someone wants to do
               so).

TAU

          TAU is REAL
               The root of the equation f(x).

INFO

          INFO is INTEGER
               = 0: successful exit
               > 0: if INFO = 1, failure to converge

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  10/02/03: This version has a few statements commented out for thread
  safety (machine parameters are computed on each entry). SJH.
  05/10/06: Modified from a new version of Ren-Cang Li, use
     Gragg-Thornton-Warner cubic convergent scheme for better stability.

Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA

 SLAED7 computes the updated eigensystem of a diagonal
 matrix after modification by a rank-one symmetric matrix. This
 routine is used only for the eigenproblem which requires all
 eigenvalues and optionally eigenvectors of a dense symmetric matrix
 that has been reduced to tridiagonal form.  SLAED1 handles
 the case in which all eigenvalues and eigenvectors of a symmetric
 tridiagonal matrix are desired.
   T = Q(in) ( D(in) + RHO * Z*Z**T ) Q**T(in) = Q(out) * D(out) * Q**T(out)
    where Z = Q**Tu, u is a vector of length N with ones in the
    CUTPNT and CUTPNT + 1 th elements and zeros elsewhere.
    The eigenvectors of the original matrix are stored in Q, and the
    eigenvalues are in D.  The algorithm consists of three stages:
       The first stage consists of deflating the size of the problem
       when there are multiple eigenvalues or if there is a zero in
       the Z vector.  For each such occurrence the dimension of the
       secular equation problem is reduced by one.  This stage is
       performed by the routine SLAED8.
       The second stage consists of calculating the updated
       eigenvalues. This is done by finding the roots of the secular
       equation via the routine SLAED4 (as called by SLAED9).
       This routine also calculates the eigenvectors of the current
       problem.
       The final stage consists of computing the updated eigenvectors
       directly using the updated eigenvalues.  The eigenvectors for
       the current problem are multiplied with the eigenvectors from
       the overall problem.

ICOMPQ

          ICOMPQ is INTEGER
          = 0:  Compute eigenvalues only.
          = 1:  Compute eigenvectors of original dense symmetric matrix
                also.  On entry, Q contains the orthogonal matrix used
                to reduce the original matrix to tridiagonal form.

N

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

QSIZ

          QSIZ is INTEGER
         The dimension of the orthogonal matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

TLVLS

          TLVLS is INTEGER
         The total number of merging levels in the overall divide and
         conquer tree.

CURLVL

          CURLVL is INTEGER
         The current level in the overall merge routine,
         0 <= CURLVL <= TLVLS.

CURPBM

          CURPBM is INTEGER
         The current problem in the current level in the overall
         merge routine (counting from upper left to lower right).

D

          D is REAL array, dimension (N)
         On entry, the eigenvalues of the rank-1-perturbed matrix.
         On exit, the eigenvalues of the repaired matrix.

Q

          Q is REAL array, dimension (LDQ, N)
         On entry, the eigenvectors of the rank-1-perturbed matrix.
         On exit, the eigenvectors of the repaired tridiagonal matrix.

LDQ

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

INDXQ

          INDXQ is INTEGER array, dimension (N)
         The permutation which will reintegrate the subproblem just
         solved back into sorted order, i.e., D( INDXQ( I = 1, N ) )
         will be in ascending order.

RHO

          RHO is REAL
         The subdiagonal element used to create the rank-1
         modification.

CUTPNT

          CUTPNT is INTEGER
         Contains the location of the last eigenvalue in the leading
         sub-matrix.  min(1,N) <= CUTPNT <= N.

QSTORE

          QSTORE is REAL array, dimension (N**2+1)
         Stores eigenvectors of submatrices encountered during
         divide and conquer, packed together. QPTR points to
         beginning of the submatrices.

QPTR

          QPTR is INTEGER array, dimension (N+2)
         List of indices pointing to beginning of submatrices stored
         in QSTORE. The submatrices are numbered starting at the
         bottom left of the divide and conquer tree, from left to
         right and bottom to top.

PRMPTR

          PRMPTR is INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in PERM a
         level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
         indicates the size of the permutation and also the size of
         the full, non-deflated problem.

PERM

          PERM is INTEGER array, dimension (N lg N)
         Contains the permutations (from deflation and sorting) to be
         applied to each eigenblock.

GIVPTR

          GIVPTR is INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in GIVCOL a
         level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
         indicates the number of Givens rotations.

GIVCOL

          GIVCOL is INTEGER array, dimension (2, N lg N)
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation.

GIVNUM

          GIVNUM is REAL array, dimension (2, N lg N)
         Each number indicates the S value to be used in the
         corresponding Givens rotation.

WORK

          WORK is REAL array, dimension (3*N+2*QSIZ*N)

IWORK

          IWORK is INTEGER array, dimension (4*N)

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

 SLAED8 merges the two sets of eigenvalues together into a single
 sorted set.  Then it tries to deflate the size of the problem.
 There are two ways in which deflation can occur:  when two or more
 eigenvalues are close together or if there is a tiny element in the
 Z vector.  For each such occurrence the order of the related secular
 equation problem is reduced by one.

ICOMPQ

          ICOMPQ is INTEGER
          = 0:  Compute eigenvalues only.
          = 1:  Compute eigenvectors of original dense symmetric matrix
                also.  On entry, Q contains the orthogonal matrix used
                to reduce the original matrix to tridiagonal form.

K

          K is INTEGER
         The number of non-deflated eigenvalues, and the order of the
         related secular equation.

N

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

QSIZ

          QSIZ is INTEGER
         The dimension of the orthogonal matrix used to reduce
         the full matrix to tridiagonal form.  QSIZ >= N if ICOMPQ = 1.

D

          D is REAL array, dimension (N)
         On entry, the eigenvalues of the two submatrices to be
         combined.  On exit, the trailing (N-K) updated eigenvalues
         (those which were deflated) sorted into increasing order.

Q

          Q is REAL array, dimension (LDQ,N)
         If ICOMPQ = 0, Q is not referenced.  Otherwise,
         on entry, Q contains the eigenvectors of the partially solved
         system which has been previously updated in matrix
         multiplies with other partially solved eigensystems.
         On exit, Q contains the trailing (N-K) updated eigenvectors
         (those which were deflated) in its last N-K columns.

LDQ

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

INDXQ

          INDXQ is INTEGER array, dimension (N)
         The permutation which separately sorts the two sub-problems
         in D into ascending order.  Note that elements in the second
         half of this permutation must first have CUTPNT added to
         their values in order to be accurate.

RHO

          RHO is REAL
         On entry, the off-diagonal element associated with the rank-1
         cut which originally split the two submatrices which are now
         being recombined.
         On exit, RHO has been modified to the value required by
         SLAED3.

CUTPNT

          CUTPNT is INTEGER
         The location of the last eigenvalue in the leading
         sub-matrix.  min(1,N) <= CUTPNT <= N.

Z

          Z is REAL array, dimension (N)
         On entry, Z contains the updating vector (the last row of
         the first sub-eigenvector matrix and the first row of the
         second sub-eigenvector matrix).
         On exit, the contents of Z are destroyed by the updating
         process.

DLAMDA

          DLAMDA is REAL array, dimension (N)
         A copy of the first K eigenvalues which will be used by
         SLAED3 to form the secular equation.

Q2

          Q2 is REAL array, dimension (LDQ2,N)
         If ICOMPQ = 0, Q2 is not referenced.  Otherwise,
         a copy of the first K eigenvectors which will be used by
         SLAED7 in a matrix multiply (SGEMM) to update the new
         eigenvectors.

LDQ2

          LDQ2 is INTEGER
         The leading dimension of the array Q2.  LDQ2 >= max(1,N).

W

          W is REAL array, dimension (N)
         The first k values of the final deflation-altered z-vector and
         will be passed to SLAED3.

PERM

          PERM is INTEGER array, dimension (N)
         The permutations (from deflation and sorting) to be applied
         to each eigenblock.

GIVPTR

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

GIVCOL

          GIVCOL is INTEGER array, dimension (2, N)
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation.

GIVNUM

          GIVNUM is REAL array, dimension (2, N)
         Each number indicates the S value to be used in the
         corresponding Givens rotation.

INDXP

          INDXP is INTEGER array, dimension (N)
         The permutation used to place deflated values of D at the end
         of the array.  INDXP(1:K) points to the nondeflated D-values
         and INDXP(K+1:N) points to the deflated eigenvalues.

INDX

          INDX is INTEGER array, dimension (N)
         The permutation used to sort the contents of D into ascending
         order.

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.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

 SLAED9 finds the roots of the secular equation, as defined by the
 values in D, Z, and RHO, between KSTART and KSTOP.  It makes the
 appropriate calls to SLAED4 and then stores the new matrix of
 eigenvectors for use in calculating the next level of Z vectors.

K

          K is INTEGER
          The number of terms in the rational function to be solved by
          SLAED4.  K >= 0.

KSTART

          KSTART is INTEGER

KSTOP

          KSTOP is INTEGER
          The updated eigenvalues Lambda(I), KSTART <= I <= KSTOP
          are to be computed.  1 <= KSTART <= KSTOP <= K.

N

          N is INTEGER
          The number of rows and columns in the Q matrix.
          N >= K (delation may result in N > K).

D

          D is REAL array, dimension (N)
          D(I) contains the updated eigenvalues
          for KSTART <= I <= KSTOP.

Q

          Q is REAL array, dimension (LDQ,N)

LDQ

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

RHO

          RHO is REAL
          The value of the parameter in the rank one update equation.
          RHO >= 0 required.

DLAMDA

          DLAMDA is REAL array, dimension (K)
          The first K elements of this array contain the old roots
          of the deflated updating problem.  These are the poles
          of the secular equation.

W

          W is REAL array, dimension (K)
          The first K elements of this array contain the components
          of the deflation-adjusted updating vector.

S

          S is REAL array, dimension (LDS, K)
          Will contain the eigenvectors of the repaired matrix which
          will be stored for subsequent Z vector calculation and
          multiplied by the previously accumulated eigenvectors
          to update the system.

LDS

          LDS is INTEGER
          The leading dimension of S.  LDS >= max( 1, K ).

INFO

          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, an eigenvalue did not converge

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

 SLAEDA computes the Z vector corresponding to the merge step in the
 CURLVLth step of the merge process with TLVLS steps for the CURPBMth
 problem.

N

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

TLVLS

          TLVLS is INTEGER
         The total number of merging levels in the overall divide and
         conquer tree.

CURLVL

          CURLVL is INTEGER
         The current level in the overall merge routine,
         0 <= curlvl <= tlvls.

CURPBM

          CURPBM is INTEGER
         The current problem in the current level in the overall
         merge routine (counting from upper left to lower right).

PRMPTR

          PRMPTR is INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in PERM a
         level's permutation is stored.  PRMPTR(i+1) - PRMPTR(i)
         indicates the size of the permutation and incidentally the
         size of the full, non-deflated problem.

PERM

          PERM is INTEGER array, dimension (N lg N)
         Contains the permutations (from deflation and sorting) to be
         applied to each eigenblock.

GIVPTR

          GIVPTR is INTEGER array, dimension (N lg N)
         Contains a list of pointers which indicate where in GIVCOL a
         level's Givens rotations are stored.  GIVPTR(i+1) - GIVPTR(i)
         indicates the number of Givens rotations.

GIVCOL

          GIVCOL is INTEGER array, dimension (2, N lg N)
         Each pair of numbers indicates a pair of columns to take place
         in a Givens rotation.

GIVNUM

          GIVNUM is REAL array, dimension (2, N lg N)
         Each number indicates the S value to be used in the
         corresponding Givens rotation.

Q

          Q is REAL array, dimension (N**2)
         Contains the square eigenblocks from previous levels, the
         starting positions for blocks are given by QPTR.

QPTR

          QPTR is INTEGER array, dimension (N+2)
         Contains a list of pointers which indicate where in Q an
         eigenblock is stored.  SQRT( QPTR(i+1) - QPTR(i) ) indicates
         the size of the block.

Z

          Z is REAL array, dimension (N)
         On output this vector contains the updating vector (the last
         row of the first sub-eigenvector matrix and the first row of
         the second sub-eigenvector matrix).

ZTEMP

          ZTEMP is REAL array, dimension (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.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA

 SLAGTF factorizes the matrix (T - lambda*I), where T is an n by n
 tridiagonal matrix and lambda is a scalar, as
    T - lambda*I = PLU,
 where P is a permutation matrix, L is a unit lower tridiagonal matrix
 with at most one non-zero sub-diagonal elements per column and U is
 an upper triangular matrix with at most two non-zero super-diagonal
 elements per column.
 The factorization is obtained by Gaussian elimination with partial
 pivoting and implicit row scaling.
 The parameter LAMBDA is included in the routine so that SLAGTF may
 be used, in conjunction with SLAGTS, to obtain eigenvectors of T by
 inverse iteration.

N

          N is INTEGER
          The order of the matrix T.

A

          A is REAL array, dimension (N)
          On entry, A must contain the diagonal elements of T.
          On exit, A is overwritten by the n diagonal elements of the
          upper triangular matrix U of the factorization of T.

LAMBDA

          LAMBDA is REAL
          On entry, the scalar lambda.

B

          B is REAL array, dimension (N-1)
          On entry, B must contain the (n-1) super-diagonal elements of
          T.
          On exit, B is overwritten by the (n-1) super-diagonal
          elements of the matrix U of the factorization of T.

C

          C is REAL array, dimension (N-1)
          On entry, C must contain the (n-1) sub-diagonal elements of
          T.
          On exit, C is overwritten by the (n-1) sub-diagonal elements
          of the matrix L of the factorization of T.

TOL

          TOL is REAL
          On entry, a relative tolerance used to indicate whether or
          not the matrix (T - lambda*I) is nearly singular. TOL should
          normally be chose as approximately the largest relative error
          in the elements of T. For example, if the elements of T are
          correct to about 4 significant figures, then TOL should be
          set to about 5*10**(-4). If TOL is supplied as less than eps,
          where eps is the relative machine precision, then the value
          eps is used in place of TOL.

D

          D is REAL array, dimension (N-2)
          On exit, D is overwritten by the (n-2) second super-diagonal
          elements of the matrix U of the factorization of T.

IN

          IN is INTEGER array, dimension (N)
          On exit, IN contains details of the permutation matrix P. If
          an interchange occurred at the kth step of the elimination,
          then IN(k) = 1, otherwise IN(k) = 0. The element IN(n)
          returns the smallest positive integer j such that
             abs( u(j,j) ) <= norm( (T - lambda*I)(j) )*TOL,
          where norm( A(j) ) denotes the sum of the absolute values of
          the jth row of the matrix A. If no such j exists then IN(n)
          is returned as zero. If IN(n) is returned as positive, then a
          diagonal element of U is small, indicating that
          (T - lambda*I) is singular or nearly singular,

INFO

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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 SLAMRG will create a permutation list which will merge the elements
 of A (which is composed of two independently sorted sets) into a
 single set which is sorted in ascending order.

N1

          N1 is INTEGER

N2

          N2 is INTEGER
         These arguments contain the respective lengths of the two
         sorted lists to be merged.

A

          A is REAL array, dimension (N1+N2)
         The first N1 elements of A contain a list of numbers which
         are sorted in either ascending or descending order.  Likewise
         for the final N2 elements.

STRD1

          STRD1 is INTEGER

STRD2

          STRD2 is INTEGER
         These are the strides to be taken through the array A.
         Allowable strides are 1 and -1.  They indicate whether a
         subset of A is sorted in ascending (STRDx = 1) or descending
         (STRDx = -1) order.

INDEX

          INDEX is INTEGER array, dimension (N1+N2)
         On exit this array will contain a permutation such that
         if B( I ) = A( INDEX( I ) ) for I=1,N1+N2, then B will be
         sorted in ascending order.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 SLARTGS generates a plane rotation designed to introduce a bulge in
 Golub-Reinsch-style implicit QR iteration for the bidiagonal SVD
 problem. X and Y are the top-row entries, and SIGMA is the shift.
 The computed CS and SN define a plane rotation satisfying
    [  CS  SN  ]  .  [ X^2 - SIGMA ]  =  [ R ],
    [ -SN  CS  ]     [    X * Y    ]     [ 0 ]
 with R nonnegative.  If X^2 - SIGMA and X * Y are 0, then the
 rotation is by PI/2.

X

          X is REAL
          The (1,1) entry of an upper bidiagonal matrix.

Y

          Y is REAL
          The (1,2) entry of an upper bidiagonal matrix.

SIGMA

          SIGMA is REAL
          The shift.

CS

          CS is REAL
          The cosine of the rotation.

SN

          SN is REAL
          The sine of the rotation.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 SLASQ1 computes the singular values of a real N-by-N bidiagonal
 matrix with diagonal D and off-diagonal E. The singular values
 are computed to high relative accuracy, in the absence of
 denormalization, underflow and overflow. The algorithm was first
 presented in
 "Accurate singular values and differential qd algorithms" by K. V.
 Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230,
 1994,
 and the present implementation is described in "An implementation of
 the dqds Algorithm (Positive Case)", LAPACK Working Note.

N

          N is INTEGER
        The number of rows and columns in the matrix. N >= 0.

D

          D is REAL array, dimension (N)
        On entry, D contains the diagonal elements of the
        bidiagonal matrix whose SVD is desired. On normal exit,
        D contains the singular values in decreasing order.

E

          E is REAL array, dimension (N)
        On entry, elements E(1:N-1) contain the off-diagonal elements
        of the bidiagonal matrix whose SVD is desired.
        On exit, E is overwritten.

WORK

          WORK is REAL array, dimension (4*N)

INFO

          INFO is INTEGER
        = 0: successful exit
        < 0: if INFO = -i, the i-th argument had an illegal value
        > 0: the algorithm failed
             = 1, a split was marked by a positive value in E
             = 2, current block of Z not diagonalized after 100*N
                  iterations (in inner while loop)  On exit D and E
                  represent a matrix with the same singular values
                  which the calling subroutine could use to finish the
                  computation, or even feed back into SLASQ1
             = 3, termination criterion of outer while loop not met
                  (program created more than N unreduced blocks)

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 SLASQ2 computes all the eigenvalues of the symmetric positive
 definite tridiagonal matrix associated with the qd array Z to high
 relative accuracy are computed to high relative accuracy, in the
 absence of denormalization, underflow and overflow.
 To see the relation of Z to the tridiagonal matrix, let L be a
 unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and
 let U be an upper bidiagonal matrix with 1's above and diagonal
 Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
 symmetric tridiagonal to which it is similar.
 Note : SLASQ2 defines a logical variable, IEEE, which is true
 on machines which follow ieee-754 floating-point standard in their
 handling of infinities and NaNs, and false otherwise. This variable
 is passed to SLASQ3.

N

          N is INTEGER
        The number of rows and columns in the matrix. N >= 0.

Z

          Z is REAL array, dimension ( 4*N )
        On entry Z holds the qd array. On exit, entries 1 to N hold
        the eigenvalues in decreasing order, Z( 2*N+1 ) holds the
        trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If
        N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 )
        holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of
        shifts that failed.

INFO

          INFO is INTEGER
        = 0: successful exit
        < 0: if the i-th argument is a scalar and had an illegal
             value, then INFO = -i, if the i-th argument is an
             array and the j-entry had an illegal value, then
             INFO = -(i*100+j)
        > 0: the algorithm failed
              = 1, a split was marked by a positive value in E
              = 2, current block of Z not diagonalized after 100*N
                   iterations (in inner while loop).  On exit Z holds
                   a qd array with the same eigenvalues as the given Z.
              = 3, termination criterion of outer while loop not met
                   (program created more than N unreduced blocks)

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  Local Variables: I0:N0 defines a current unreduced segment of Z.
  The shifts are accumulated in SIGMA. Iteration count is in ITER.
  Ping-pong is controlled by PP (alternates between 0 and 1).

 SLASQ3 checks for deflation, computes a shift (TAU) and calls dqds.
 In case of failure it changes shifts, and tries again until output
 is positive.

I0

          I0 is INTEGER
         First index.

N0

          N0 is INTEGER
         Last index.

Z

          Z is REAL array, dimension ( 4*N0 )
         Z holds the qd array.

PP

          PP is INTEGER
         PP=0 for ping, PP=1 for pong.
         PP=2 indicates that flipping was applied to the Z array
         and that the initial tests for deflation should not be
         performed.

DMIN

          DMIN is REAL
         Minimum value of d.

SIGMA

          SIGMA is REAL
         Sum of shifts used in current segment.

DESIG

          DESIG is REAL
         Lower order part of SIGMA

QMAX

          QMAX is REAL
         Maximum value of q.

NFAIL

          NFAIL is INTEGER
         Increment NFAIL by 1 each time the shift was too big.

ITER

          ITER is INTEGER
         Increment ITER by 1 for each iteration.

NDIV

          NDIV is INTEGER
         Increment NDIV by 1 for each division.

IEEE

          IEEE is LOGICAL
         Flag for IEEE or non IEEE arithmetic (passed to SLASQ5).

TTYPE

          TTYPE is INTEGER
         Shift type.

DMIN1

          DMIN1 is REAL

DMIN2

          DMIN2 is REAL

DN

          DN is REAL

DN1

          DN1 is REAL

DN2

          DN2 is REAL

G

          G is REAL

TAU

          TAU is REAL
         These are passed as arguments in order to save their values
         between calls to SLASQ3.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 SLASQ4 computes an approximation TAU to the smallest eigenvalue
 using values of d from the previous transform.

I0

          I0 is INTEGER
        First index.

N0

          N0 is INTEGER
        Last index.

Z

          Z is REAL array, dimension ( 4*N0 )
        Z holds the qd array.

PP

          PP is INTEGER
        PP=0 for ping, PP=1 for pong.

N0IN

          N0IN is INTEGER
        The value of N0 at start of EIGTEST.

DMIN

          DMIN is REAL
        Minimum value of d.

DMIN1

          DMIN1 is REAL
        Minimum value of d, excluding D( N0 ).

DMIN2

          DMIN2 is REAL
        Minimum value of d, excluding D( N0 ) and D( N0-1 ).

DN

          DN is REAL
        d(N)

DN1

          DN1 is REAL
        d(N-1)

DN2

          DN2 is REAL
        d(N-2)

TAU

          TAU is REAL
        This is the shift.

TTYPE

          TTYPE is INTEGER
        Shift type.

G

          G is REAL
        G is passed as an argument in order to save its value between
        calls to SLASQ4.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  CNST1 = 9/16

 SLASQ5 computes one dqds transform in ping-pong form, one
 version for IEEE machines another for non IEEE machines.

I0

          I0 is INTEGER
        First index.

N0

          N0 is INTEGER
        Last index.

Z

          Z is REAL array, dimension ( 4*N )
        Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
        an extra argument.

PP

          PP is INTEGER
        PP=0 for ping, PP=1 for pong.

TAU

          TAU is REAL
        This is the shift.

SIGMA

          SIGMA is REAL
        This is the accumulated shift up to this step.

DMIN

          DMIN is REAL
        Minimum value of d.

DMIN1

          DMIN1 is REAL
        Minimum value of d, excluding D( N0 ).

DMIN2

          DMIN2 is REAL
        Minimum value of d, excluding D( N0 ) and D( N0-1 ).

DN

          DN is REAL
        d(N0), the last value of d.

DNM1

          DNM1 is REAL
        d(N0-1).

DNM2

          DNM2 is REAL
        d(N0-2).

IEEE

          IEEE is LOGICAL
        Flag for IEEE or non IEEE arithmetic.

EPS

         EPS is REAL
        This is the value of epsilon used.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 SLASQ6 computes one dqd (shift equal to zero) transform in
 ping-pong form, with protection against underflow and overflow.

I0

          I0 is INTEGER
        First index.

N0

          N0 is INTEGER
        Last index.

Z

          Z is REAL array, dimension ( 4*N )
        Z holds the qd array. EMIN is stored in Z(4*N0) to avoid
        an extra argument.

PP

          PP is INTEGER
        PP=0 for ping, PP=1 for pong.

DMIN

          DMIN is REAL
        Minimum value of d.

DMIN1

          DMIN1 is REAL
        Minimum value of d, excluding D( N0 ).

DMIN2

          DMIN2 is REAL
        Minimum value of d, excluding D( N0 ) and D( N0-1 ).

DN

          DN is REAL
        d(N0), the last value of d.

DNM1

          DNM1 is REAL
        d(N0-1).

DNM2

          DNM2 is REAL
        d(N0-2).

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 Sort the numbers in D in increasing order (if ID = 'I') or
 in decreasing order (if ID = 'D' ).
 Use Quick Sort, reverting to Insertion sort on arrays of
 size <= 20. Dimension of STACK limits N to about 2**32.

ID

          ID is CHARACTER*1
          = 'I': sort D in increasing order;
          = 'D': sort D in decreasing order.

N

          N is INTEGER
          The length of the array D.

D

          D is REAL array, dimension (N)
          On entry, the array to be sorted.
          On exit, D has been sorted into increasing order
          (D(1) <= ... <= D(N) ) or into decreasing order
          (D(1) >= ... >= D(N) ), depending on ID.

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.

 SPTTRF computes the L*D*L**T factorization of a real symmetric
 positive definite tridiagonal matrix A.  The factorization may also
 be regarded as having the form A = U**T*D*U.

N

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

D

          D is REAL array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix
          A.  On exit, the n diagonal elements of the diagonal matrix
          D from the L*D*L**T factorization of A.

E

          E is REAL array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix A.  On exit, the (n-1) subdiagonal elements of the
          unit bidiagonal factor L from the L*D*L**T factorization of A.
          E can also be regarded as the superdiagonal of the unit
          bidiagonal factor U from the U**T*D*U factorization of A.

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; if k < N, the factorization could not
               be completed, while if k = N, the factorization was
               completed, but D(N) <= 0.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 SSTEBZ computes the eigenvalues of a symmetric tridiagonal
 matrix T.  The user may ask for all eigenvalues, all eigenvalues
 in the half-open interval (VL, VU], or the IL-th through IU-th
 eigenvalues.
 To avoid overflow, the matrix must be scaled so that its
 largest element is no greater than overflow**(1/2) * underflow**(1/4) in absolute value, and for greatest
 accuracy, it should not be much smaller than that.
 See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
 Matrix", Report CS41, Computer Science Dept., Stanford
 University, July 21, 1966.

RANGE

          RANGE is CHARACTER*1
          = 'A': ("All")   all eigenvalues will be found.
          = 'V': ("Value") all eigenvalues in the half-open interval
                           (VL, VU] will be found.
          = 'I': ("Index") the IL-th through IU-th eigenvalues (of the
                           entire matrix) will be found.

ORDER

          ORDER is CHARACTER*1
          = 'B': ("By Block") the eigenvalues will be grouped by
                              split-off block (see IBLOCK, ISPLIT) and
                              ordered from smallest to largest within
                              the block.
          = 'E': ("Entire matrix")
                              the eigenvalues for the entire matrix
                              will be ordered from smallest to
                              largest.

N

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

VL

          VL is REAL
          If RANGE='V', the lower bound of the interval to
          be searched for eigenvalues.  Eigenvalues less than or equal
          to VL, or greater than VU, will not be returned.  VL < VU.
          Not referenced if RANGE = 'A' or 'I'.

VU

          VU is REAL
          If RANGE='V', the upper bound of the interval to
          be searched for eigenvalues.  Eigenvalues less than or equal
          to VL, or greater than VU, will not be returned.  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; IL = 1 and IU = 0 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; IL = 1 and IU = 0 if N = 0.
          Not referenced if RANGE = 'A' or 'V'.

ABSTOL

          ABSTOL is REAL
          The absolute tolerance for the eigenvalues.  An eigenvalue
          (or cluster) is considered to be located if it has been
          determined to lie in an interval whose width is ABSTOL or
          less.  If ABSTOL is less than or equal to zero, then ULP*|T|
          will be used, where |T| means the 1-norm of T.
          Eigenvalues will be computed most accurately when ABSTOL is
          set to twice the underflow threshold 2*SLAMCH('S'), not zero.

D

          D is REAL array, dimension (N)
          The n diagonal elements of the tridiagonal matrix T.

E

          E is REAL array, dimension (N-1)
          The (n-1) off-diagonal elements of the tridiagonal matrix T.

M

          M is INTEGER
          The actual number of eigenvalues found. 0 <= M <= N.
          (See also the description of INFO=2,3.)

NSPLIT

          NSPLIT is INTEGER
          The number of diagonal blocks in the matrix T.
          1 <= NSPLIT <= N.

W

          W is REAL array, dimension (N)
          On exit, the first M elements of W will contain the
          eigenvalues.  (SSTEBZ may use the remaining N-M elements as
          workspace.)

IBLOCK

          IBLOCK is INTEGER array, dimension (N)
          At each row/column j where E(j) is zero or small, the
          matrix T is considered to split into a block diagonal
          matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which
          block (from 1 to the number of blocks) the eigenvalue W(i)
          belongs.  (SSTEBZ may use the remaining N-M elements as
          workspace.)

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., and the NSPLIT-th consists of rows/columns
          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
          (Only the first NSPLIT elements will actually be used, but
          since the user cannot know a priori what value NSPLIT will
          have, N words must be reserved for ISPLIT.)

WORK

          WORK is REAL array, dimension (4*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
          > 0:  some or all of the eigenvalues failed to converge or
                were not computed:
                =1 or 3: Bisection failed to converge for some
                        eigenvalues; these eigenvalues are flagged by a
                        negative block number.  The effect is that the
                        eigenvalues may not be as accurate as the
                        absolute and relative tolerances.  This is
                        generally caused by unexpectedly inaccurate
                        arithmetic.
                =2 or 3: RANGE='I' only: Not all of the eigenvalues
                        IL:IU were found.
                        Effect: M < IU+1-IL
                        Cause:  non-monotonic arithmetic, causing the
                                Sturm sequence to be non-monotonic.
                        Cure:   recalculate, using RANGE='A', and pick
                                out eigenvalues IL:IU.  In some cases,
                                increasing the PARAMETER "FUDGE" may
                                make things work.
                = 4:    RANGE='I', and the Gershgorin interval
                        initially used was too small.  No eigenvalues
                        were computed.
                        Probable cause: your machine has sloppy
                                        floating-point arithmetic.
                        Cure: Increase the PARAMETER "FUDGE",
                              recompile, and try again.

  RELFAC  REAL, default = 2.0e0
          The relative tolerance.  An interval (a,b] lies within
          "relative tolerance" if  b-a < RELFAC*ulp*max(|a|,|b|),
          where "ulp" is the machine precision (distance from 1 to
          the next larger floating point number.)
  FUDGE   REAL, default = 2
          A "fudge factor" to widen the Gershgorin intervals.  Ideally,
          a value of 1 should work, but on machines with sloppy
          arithmetic, this needs to be larger.  The default for
          publicly released versions should be large enough to handle
          the worst machine around.  Note that this has no effect
          on accuracy of the solution.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 SSTEDC computes all eigenvalues and, optionally, eigenvectors of a
 symmetric tridiagonal matrix using the divide and conquer method.
 The eigenvectors of a full or band real symmetric matrix can also be
 found if SSYTRD or SSPTRD or SSBTRD has been used to reduce this
 matrix to tridiagonal form.
 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 X-MP, Cray Y-MP, Cray C-90, or Cray-2.
 It could conceivably fail on hexadecimal or decimal machines
 without guard digits, but we know of none.  See SLAED3 for details.

COMPZ

          COMPZ is CHARACTER*1
          = 'N':  Compute eigenvalues only.
          = 'I':  Compute eigenvectors of tridiagonal matrix also.
          = 'V':  Compute eigenvectors of original dense symmetric
                  matrix also.  On entry, Z contains the orthogonal
                  matrix used to reduce the original matrix to
                  tridiagonal form.

N

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

D

          D is REAL array, dimension (N)
          On entry, the diagonal elements of the tridiagonal matrix.
          On exit, if INFO = 0, the eigenvalues in ascending order.

E

          E is REAL array, dimension (N-1)
          On entry, the subdiagonal elements of the tridiagonal matrix.
          On exit, E has been destroyed.

Z

          Z is REAL array, dimension (LDZ,N)
          On entry, if COMPZ = 'V', then Z contains the orthogonal
          matrix used in the reduction to tridiagonal form.
          On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
          orthonormal eigenvectors of the original symmetric matrix,
          and if COMPZ = 'I', Z contains the orthonormal eigenvectors
          of the symmetric tridiagonal matrix.
          If  COMPZ = 'N', then Z is not referenced.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1.
          If eigenvectors are desired, then LDZ >= max(1,N).

WORK

          WORK is REAL 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 COMPZ = 'N' or N <= 1 then LWORK must be at least 1.
          If COMPZ = 'V' and N > 1 then LWORK must be at least
                         ( 1 + 3*N + 2*N*lg N + 4*N**2 ),
                         where lg( N ) = smallest integer k such
                         that 2**k >= N.
          If COMPZ = 'I' and N > 1 then LWORK must be at least
                         ( 1 + 4*N + N**2 ).
          Note that for COMPZ = 'I' or 'V', then if N is less than or
          equal to the minimum divide size, usually 25, then LWORK need
          only be max(1,2*(N-1)).
          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.
          If COMPZ = 'N' or N <= 1 then LIWORK must be at least 1.
          If COMPZ = 'V' and N > 1 then LIWORK must be at least
                         ( 6 + 6*N + 5*N*lg N ).
          If COMPZ = 'I' and N > 1 then LIWORK must be at least
                         ( 3 + 5*N ).
          Note that for COMPZ = 'I' or 'V', then if N is less than or
          equal to the minimum divide size, usually 25, then LIWORK
          need only be 1.
          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.
          > 0:  The algorithm failed to compute an eigenvalue while
                working on the submatrix lying in rows and columns
                INFO/(N+1) through mod(INFO,N+1).

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Jeff Rutter, Computer Science Division, University of California at Berkeley, USA Modified by Francoise Tisseur, University of Tennessee

 SSTEQR computes all eigenvalues and, optionally, eigenvectors of a
 symmetric tridiagonal matrix using the implicit QL or QR method.
 The eigenvectors of a full or band symmetric matrix can also be found
 if SSYTRD or SSPTRD or SSBTRD has been used to reduce this matrix to
 tridiagonal form.

COMPZ

          COMPZ is CHARACTER*1
          = 'N':  Compute eigenvalues only.
          = 'V':  Compute eigenvalues and eigenvectors of the original
                  symmetric matrix.  On entry, Z must contain the
                  orthogonal matrix used to reduce the original matrix
                  to tridiagonal form.
          = 'I':  Compute eigenvalues and eigenvectors of the
                  tridiagonal matrix.  Z is initialized to the identity
                  matrix.

N

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

D

          D is REAL array, dimension (N)
          On entry, the diagonal elements of the tridiagonal matrix.
          On exit, if INFO = 0, the eigenvalues in ascending order.

E

          E is REAL array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix.
          On exit, E has been destroyed.

Z

          Z is REAL array, dimension (LDZ, N)
          On entry, if  COMPZ = 'V', then Z contains the orthogonal
          matrix used in the reduction to tridiagonal form.
          On exit, if INFO = 0, then if  COMPZ = 'V', Z contains the
          orthonormal eigenvectors of the original symmetric matrix,
          and if COMPZ = 'I', Z contains the orthonormal eigenvectors
          of the symmetric tridiagonal matrix.
          If COMPZ = 'N', then Z is not referenced.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          eigenvectors are desired, then  LDZ >= max(1,N).

WORK

          WORK is REAL array, dimension (max(1,2*N-2))
          If COMPZ = 'N', then WORK is not referenced.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  the algorithm has failed to find all the eigenvalues in
                a total of 30*N iterations; if INFO = i, then i
                elements of E have not converged to zero; on exit, D
                and E contain the elements of a symmetric tridiagonal
                matrix which is orthogonally similar to the original
                matrix.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 SSTERF computes all eigenvalues of a symmetric tridiagonal matrix
 using the Pal-Walker-Kahan variant of the QL or QR algorithm.

N

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

D

          D is REAL array, dimension (N)
          On entry, the n diagonal elements of the tridiagonal matrix.
          On exit, if INFO = 0, the eigenvalues in ascending order.

E

          E is REAL array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal
          matrix.
          On exit, E has been destroyed.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  the algorithm failed to find all of the eigenvalues in
                a total of 30*N iterations; if INFO = i, then i
                elements of E have not converged to zero.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.