sgtsvx(3F)

NAME
     SGTSVX - use the LU factorization to compute the solution to a real
     system of linear equations A * X = B or A**T * X = B,

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

         CHARACTER      FACT, TRANS

         INTEGER        INFO, LDB, LDX, N, NRHS

         REAL           RCOND

         INTEGER        IPIV( * ), IWORK( * )

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

PURPOSE
     SGTSVX 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.


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

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

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

     DLF     (input or output) REAL 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 SGTTRF.

             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) REAL 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) REAL 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) REAL 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 SGTTRF.

             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) REAL 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) REAL 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 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) 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) REAL array, dimension (3*N)

     IWORK   (workspace) INTEGER 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.