• NAME
• SYNOPSIS
• F95 INTERFACE
• C INTERFACE
• PURPOSE
• ARGUMENTS

NAME

sptsvx - 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 SPTSVX( FACT, N, NRHS, DIAG, SUB, DIAGF, SUBF, B, LDB, X, 
 *      LDX, RCOND, FERR, BERR, WORK, INFO)
  CHARACTER * 1 FACT
  INTEGER N, NRHS, LDB, LDX, INFO
  REAL RCOND
  REAL DIAG(*), SUB(*), DIAGF(*), SUBF(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*)

  SUBROUTINE SPTSVX_64( FACT, N, NRHS, DIAG, SUB, DIAGF, SUBF, B, LDB, 
 *      X, LDX, RCOND, FERR, BERR, WORK, INFO)
  CHARACTER * 1 FACT
  INTEGER*8 N, NRHS, LDB, LDX, INFO
  REAL RCOND
  REAL DIAG(*), SUB(*), DIAGF(*), SUBF(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*)

F95 INTERFACE

  SUBROUTINE PTSVX( FACT, [N], [NRHS], DIAG, SUB, DIAGF, SUBF, B, [LDB], 
 *       X, [LDX], RCOND, FERR, BERR, [WORK], [INFO])
  CHARACTER(LEN=1) :: FACT
  INTEGER :: N, NRHS, LDB, LDX, INFO
  REAL :: RCOND
  REAL, DIMENSION(:) :: DIAG, SUB, DIAGF, SUBF, FERR, BERR, WORK
  REAL, DIMENSION(:,:) :: B, X

  SUBROUTINE PTSVX_64( FACT, [N], [NRHS], DIAG, SUB, DIAGF, SUBF, B, 
 *       [LDB], X, [LDX], RCOND, FERR, BERR, [WORK], [INFO])
  CHARACTER(LEN=1) :: FACT
  INTEGER(8) :: N, NRHS, LDB, LDX, INFO
  REAL :: RCOND
  REAL, DIMENSION(:) :: DIAG, SUB, DIAGF, SUBF, FERR, BERR, WORK
  REAL, DIMENSION(:,:) :: B, X

C INTERFACE

#include <sunperf.h>

void sptsvx(char fact, int n, int nrhs, float *diag, float *sub, float *diagf, float *subf, float *b, int ldb, float *x, int ldx, float *rcond, float *ferr, float *berr, int *info);

void sptsvx_64(char fact, long n, long nrhs, float *diag, float *sub, float *diagf, float *subf, float *b, long ldb, float *x, long ldx, float *rcond, float *ferr, float *berr, long *info);

PURPOSE

sptsvx 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.

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)
  Specifies whether or not the factored form of A has been supplied on entry. = 'F': On entry, DIAGF and SUBF contain the factored form of A. DIAG, SUB, DIAGF, and SUBF will not be modified. = 'N': The matrix A will be copied to DIAGF and SUBF and factored.

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

• NRHS (input)
  The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS > = 0.

• DIAG (input)
  The n diagonal elements of the tridiagonal matrix A.

• SUB (input)
  The (n-1) subdiagonal elements of the tridiagonal matrix A.

• DIAGF (input/output)
  If FACT = 'F', then DIAGF is an input argument and on entry contains the n diagonal elements of the diagonal matrix DIAG from the L*DIAG*L**T factorization of A. If FACT = 'N', then DIAGF is an output argument and on exit contains the n diagonal elements of the diagonal matrix DIAG from the L*DIAG*L**T factorization of A.

• SUBF (input/output)
  If FACT = 'F', then SUBF is an input argument and on entry contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*DIAG*L**T factorization of A. If FACT = 'N', then SUBF is an output argument and on exit contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*DIAG*L**T factorization of A.

• B (input)
  The N-by-NRHS right hand side matrix B.

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

• X (output)
  If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X.

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

• RCOND (output)
  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)
  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)
  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)
  dimension(2*N)

• INFO (output)
  

   = 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.