DGTSV  solves the equation
    A*X = B,
 where A is an n by n tridiagonal matrix, by Gaussian elimination with
 partial pivoting.
 Note that the equation  A**T*X = B  may be solved by interchanging the
 order of the arguments DU and DL.

N

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

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

DL

          DL is DOUBLE PRECISION array, dimension (N-1)
          On entry, DL must contain the (n-1) sub-diagonal elements of
          A.
          On exit, DL is overwritten by the (n-2) elements of the
          second super-diagonal of the upper triangular matrix U from
          the LU factorization of A, in DL(1), ..., DL(n-2).

D

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

DU

          DU is DOUBLE PRECISION array, dimension (N-1)
          On entry, DU must contain the (n-1) super-diagonal elements
          of A.
          On exit, DU is overwritten by the (n-1) elements of the first
          super-diagonal of U.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
          On entry, the N by NRHS matrix of right hand side matrix B.
          On exit, if INFO = 0, the N by NRHS solution matrix X.

LDB

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

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          > 0: if INFO = i, U(i,i) is exactly zero, and the solution
               has not been computed.  The factorization has not been
               completed unless i = N.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DGTSVX uses the LU factorization to compute the solution to a real
 system of linear equations A * X = B or A**T * X = B,
 where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
 matrices.
 Error bounds on the solution and a condition estimate are also
 provided.

 The following steps are performed:
 1. If FACT = 'N', the LU decomposition is used to factor the matrix A
    as A = L * U, where L is a product of permutation and unit lower
    bidiagonal matrices and U is upper triangular with nonzeros in
    only the main diagonal and first two superdiagonals.
 2. If some U(i,i)=0, so that U is exactly singular, then the routine
    returns with INFO = i. Otherwise, the factored form of A is used
    to estimate the condition number of the matrix A.  If the
    reciprocal of the condition number is less than machine precision,
    INFO = N+1 is returned as a warning, but the routine still goes on
    to solve for X and compute error bounds as described below.
 3. The system of equations is solved for X using the factored form
    of A.
 4. Iterative refinement is applied to improve the computed solution
    matrix and calculate error bounds and backward error estimates
    for it.

FACT

          FACT is CHARACTER*1
          Specifies whether or not the factored form of A has been
          supplied on entry.
          = 'F':  DLF, DF, DUF, DU2, and IPIV contain the factored
                  form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
                  will not be modified.
          = 'N':  The matrix will be copied to DLF, DF, and DUF
                  and factored.

TRANS

          TRANS is CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B     (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)

N

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

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.

DL

          DL is DOUBLE PRECISION array, dimension (N-1)
          The (n-1) subdiagonal elements of A.

D

          D is DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of A.

DU

          DU is DOUBLE PRECISION array, dimension (N-1)
          The (n-1) superdiagonal elements of A.

DLF

          DLF is DOUBLE PRECISION array, dimension (N-1)
          If FACT = 'F', then DLF is an input argument and on entry
          contains the (n-1) multipliers that define the matrix L from
          the LU factorization of A as computed by DGTTRF.
          If FACT = 'N', then DLF is an output argument and on exit
          contains the (n-1) multipliers that define the matrix L from
          the LU factorization of A.

DF

          DF is DOUBLE PRECISION array, dimension (N)
          If FACT = 'F', then DF is an input argument and on entry
          contains the n diagonal elements of the upper triangular
          matrix U from the LU factorization of A.
          If FACT = 'N', then DF is an output argument and on exit
          contains the n diagonal elements of the upper triangular
          matrix U from the LU factorization of A.

DUF

          DUF is DOUBLE PRECISION array, dimension (N-1)
          If FACT = 'F', then DUF is an input argument and on entry
          contains the (n-1) elements of the first superdiagonal of U.
          If FACT = 'N', then DUF is an output argument and on exit
          contains the (n-1) elements of the first superdiagonal of U.

DU2

          DU2 is DOUBLE PRECISION array, dimension (N-2)
          If FACT = 'F', then DU2 is an input argument and on entry
          contains the (n-2) elements of the second superdiagonal of
          U.
          If FACT = 'N', then DU2 is an output argument and on exit
          contains the (n-2) elements of the second superdiagonal of
          U.

IPIV

          IPIV is INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry
          contains the pivot indices from the LU factorization of A as
          computed by DGTTRF.
          If FACT = 'N', then IPIV is an output argument and on exit
          contains the pivot indices from the LU factorization of A;
          row i of the matrix was interchanged with row IPIV(i).
          IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
          a row interchange was not required.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
          The N-by-NRHS right hand side matrix B.

LDB

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

X

          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

LDX

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

RCOND

          RCOND is DOUBLE PRECISION
          The estimate of the reciprocal condition number of the matrix
          A.  If RCOND is less than the machine precision (in
          particular, if RCOND = 0), the matrix is singular to working
          precision.  This condition is indicated by a return code of
          INFO > 0.

FERR

          FERR is DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)
          is an estimated upper bound for the magnitude of the largest
          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.

BERR

          BERR is DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).

WORK

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

IWORK

          IWORK is INTEGER array, dimension (N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, and i is
                <= N:  U(i,i) is exactly zero.  The factorization
                       has not been completed unless i = N, but the
                       factor U is exactly singular, so the solution
                       and error bounds could not be computed.
                       RCOND = 0 is returned.
                = N+1: U is nonsingular, but RCOND is less than machine
                       precision, meaning that the matrix is singular
                       to working precision.  Nevertheless, the
                       solution and error bounds are computed because
                       there are a number of situations where the
                       computed solution can be more accurate than the
                       value of RCOND would suggest.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.