dptsvx(3S)

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

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

         CHARACTER      FACT

         INTEGER        INFO, LDB, LDX, N, NRHS

         DOUBLE         PRECISION RCOND

         DOUBLE         PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ), E(
                        * ), EF( * ), FERR( * ), 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
     DPTSVX uses the factorization A = L*D*L**T to compute the solution to a
     real system of linear equations A*X = B, where A is an N-by-N symmetric
     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**T, where L
        is a unit lower bidiagonal matrix and D is diagonal.  The
        factorization can also be regarded as having the form
        A = U**T*D*U.

     2. If the leading i-by-i principal minor is not positive definite,
        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':  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) DOUBLE PRECISION array, dimension (N)
             The n diagonal elements of the tridiagonal matrix A.

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

     DF      (input or output) DOUBLE PRECISION 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**T 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**T factorization of A.

     EF      (input or output) DOUBLE PRECISION 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**T 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**T
             factorization of A.

     B       (input) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDX,NRHS)
             If INFO = 0 of 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) DOUBLE PRECISION
             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) DOUBLE PRECISION 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) 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) DOUBLE PRECISION array, dimension (2*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, and the
             solution has not been 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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