CPTSVX (3lapack)

use the factorization A = L*D*L**H to compute the solution to a complex system of linear equations A*X = B, where A is an N-by-N Hermitian positive definite tridiagonal matrix and X and B are N-by-NRHS matrices

SYNOPSIS

  SUBROUTINE CPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND,
                     FERR, BERR, WORK, RWORK, INFO )

      CHARACTER      FACT

      INTEGER        INFO, LDB, LDX, N, NRHS

      REAL           RCOND

      REAL           BERR( * ), D( * ), DF( * ), FERR( * ), RWORK( * )

      COMPLEX        B( LDB, * ), E( * ), EF( * ), WORK( * ), X( LDX, * )

PURPOSE

  CPTSVX uses the factorization A = L*D*L**H to compute the solution to a
  complex system of linear equations A*X = B, where A is an N-by-N Hermitian
  positive definite tridiagonal matrix 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 matrix A is factored as A = L*D*L**H, where L
     is a unit lower bidiagonal matrix and D is diagonal.  The
     factorization can also be regarded as having the form
     A = U**H*D*U.

  2. The factored form of A is used to compute 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 the matrix A is
          supplied on entry.  = 'F':  On entry, DF and EF contain the
          factored form of A.  D, E, DF, and EF will not be modified.  = 'N':
          The matrix A will be copied to DF and EF and factored.

  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
          matrices B and X.  NRHS >= 0.

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

  E       (input) COMPLEX array, dimension (N-1)
          The (n-1) subdiagonal elements of the tridiagonal matrix A.

  DF      (input or output) REAL array, dimension (N)
          If FACT = 'F', then DF is an input argument and on entry contains
          the n diagonal elements of the diagonal matrix D from the L*D*L**H
          factorization of A.  If FACT = 'N', then DF is an output argument
          and on exit contains the n diagonal elements of the diagonal matrix
          D from the L*D*L**H factorization of A.

  EF      (input or output) COMPLEX array, dimension (N-1)
          If FACT = 'F', then EF is an input argument and on entry contains
          the (n-1) subdiagonal elements of the unit bidiagonal factor L from
          the L*D*L**H factorization of A.  If FACT = 'N', then EF is an
          output argument and on exit contains the (n-1) subdiagonal elements
          of the unit bidiagonal factor L from the L*D*L**H factorization of
          A.

  B       (input) COMPLEX 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 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) REAL
          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) REAL array, dimension (NRHS)
          The 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).

  BERR    (output) REAL 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 array, dimension (N)

  RWORK   (workspace) REAL 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  the leading minor of order i of A
          is not positive definite, so the factorization could not be
          completed unless i = N, and 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.