ZGTSVX (3lapack)

use the LU factorization to compute the solution to a complex system of linear equations A * X = B, A**T * X = B, or A**H * X = B,

SYNOPSIS

  SUBROUTINE ZGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
                     IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK,
                     INFO )

      CHARACTER      FACT, TRANS

      INTEGER        INFO, LDB, LDX, N, NRHS

      DOUBLE         PRECISION RCOND

      INTEGER        IPIV( * )

      DOUBLE         PRECISION BERR( * ), FERR( * ), RWORK( * )

      COMPLEX*16     B( LDB, * ), D( * ), DF( * ), DL( * ), DLF( * ), DU( *
                     ), DU2( * ), DUF( * ), WORK( * ), X( LDX, * )

PURPOSE

  ZGTSVX uses the LU factorization to compute the solution to a complex
  system of linear equations A * X = B, A**T * X = B, or A**H * 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.

DESCRIPTION

  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. 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, steps 3 and 4 are skipped.

  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.

ARGUMENTS

  FACT    (input) 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   (input) 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)

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

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

  DL      (input) COMPLEX*16 array, dimension (N-1)
          The (n-1) subdiagonal elements of A.

  D       (input) COMPLEX*16 array, dimension (N)
          The n diagonal elements of A.

  DU      (input) COMPLEX*16 array, dimension (N-1)
          The (n-1) superdiagonal elements of A.

  DLF     (input or output) COMPLEX*16 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 ZGTTRF.

          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      (input or output) COMPLEX*16 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     (input or output) COMPLEX*16 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     (input or output) COMPLEX*16 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    (input or output) 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
          ZGTTRF.

          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       (input) COMPLEX*16 array, dimension (LDB,NRHS)
          The N-by-NRHS right hand side matrix B.

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

  X       (output) COMPLEX*16 array, dimension (LDX,NRHS)
          If INFO = 0, the N-by-NRHS solution matrix X.

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

  RCOND   (output) 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, and the
          solution and error bounds are not computed.

  FERR    (output) 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    (output) 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    (workspace) COMPLEX*16 array, dimension (2*N)

  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)

  INFO    (output) 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.  = N+1:  RCOND
          is less than machine precision.  The factorization has been
          completed, but the matrix is singular to working precision, and the
          solution and error bounds have not been computed.