cgtsvx(3S)

NAME
     CGTSVX - 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 CGTSVX( 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

         REAL           RCOND

         INTEGER        IPIV( * )

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

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

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     CGTSVX 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. 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.


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 array, dimension (N-1)
             The (n-1) subdiagonal elements of A.

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

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

     DLF     (input or output) COMPLEX 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 CGTTRF.

             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 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 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 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 CGTTRF.

             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 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 or INFO = N+1, 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 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    (output) REAL 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) 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 (2*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:  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.

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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