NAME

cptsvx - 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, DIAG, SUB, DIAGF, SUBF, B, LDB, X, 
 *      LDX, RCOND, FERR, BERR, WORK, WORK2, INFO)
  CHARACTER * 1 FACT
  COMPLEX SUB(*), SUBF(*), B(LDB,*), X(LDX,*), WORK(*)
  INTEGER N, NRHS, LDB, LDX, INFO
  REAL RCOND
  REAL DIAG(*), DIAGF(*), FERR(*), BERR(*), WORK2(*)

  SUBROUTINE CPTSVX_64( FACT, N, NRHS, DIAG, SUB, DIAGF, SUBF, B, LDB, 
 *      X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO)
  CHARACTER * 1 FACT
  COMPLEX SUB(*), SUBF(*), B(LDB,*), X(LDX,*), WORK(*)
  INTEGER*8 N, NRHS, LDB, LDX, INFO
  REAL RCOND
  REAL DIAG(*), DIAGF(*), FERR(*), BERR(*), WORK2(*)

F95 INTERFACE

  SUBROUTINE PTSVX( FACT, [N], [NRHS], DIAG, SUB, DIAGF, SUBF, B, [LDB], 
 *       X, [LDX], RCOND, FERR, BERR, [WORK], [WORK2], [INFO])
  CHARACTER(LEN=1) :: FACT
  COMPLEX, DIMENSION(:) :: SUB, SUBF, WORK
  COMPLEX, DIMENSION(:,:) :: B, X
  INTEGER :: N, NRHS, LDB, LDX, INFO
  REAL :: RCOND
  REAL, DIMENSION(:) :: DIAG, DIAGF, FERR, BERR, WORK2

  SUBROUTINE PTSVX_64( FACT, [N], [NRHS], DIAG, SUB, DIAGF, SUBF, B, 
 *       [LDB], X, [LDX], RCOND, FERR, BERR, [WORK], [WORK2], [INFO])
  CHARACTER(LEN=1) :: FACT
  COMPLEX, DIMENSION(:) :: SUB, SUBF, WORK
  COMPLEX, DIMENSION(:,:) :: B, X
  INTEGER(8) :: N, NRHS, LDB, LDX, INFO
  REAL :: RCOND
  REAL, DIMENSION(:) :: DIAG, DIAGF, FERR, BERR, WORK2

C INTERFACE

#include <sunperf.h>

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

void cptsvx_64(char fact, long n, long nrhs, float *diag, complex *sub, float *diagf, complex *subf, complex *b, long ldb, complex *x, long ldx, float *rcond, float *ferr, float *berr, long *info);

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.

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. 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 the matrix A is 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**H 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**H 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**H 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**H 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 or 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(N)
WORK2 (workspace)
dimension(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.