Contents
zptsvx - 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
SUBROUTINE ZPTSVX(FACT, N, NRHS, DIAG, SUB, DIAGF, SUBF, B, LDB, X,
LDX, RCOND, FERR, BERR, WORK, WORK2, INFO)
CHARACTER * 1 FACT
DOUBLE COMPLEX SUB(*), SUBF(*), B(LDB,*), X(LDX,*), WORK(*)
INTEGER N, NRHS, LDB, LDX, INFO
DOUBLE PRECISION RCOND
DOUBLE PRECISION DIAG(*), DIAGF(*), FERR(*), BERR(*),
WORK2(*)
SUBROUTINE ZPTSVX_64(FACT, N, NRHS, DIAG, SUB, DIAGF, SUBF, B, LDB,
X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO)
CHARACTER * 1 FACT
DOUBLE COMPLEX SUB(*), SUBF(*), B(LDB,*), X(LDX,*), WORK(*)
INTEGER*8 N, NRHS, LDB, LDX, INFO
DOUBLE PRECISION RCOND
DOUBLE PRECISION 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(8), DIMENSION(:) :: SUB, SUBF, WORK
COMPLEX(8), DIMENSION(:,:) :: B, X
INTEGER :: N, NRHS, LDB, LDX, INFO
REAL(8) :: RCOND
REAL(8), 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(8), DIMENSION(:) :: SUB, SUBF, WORK
COMPLEX(8), DIMENSION(:,:) :: B, X
INTEGER(8) :: N, NRHS, LDB, LDX, INFO
REAL(8) :: RCOND
REAL(8), DIMENSION(:) :: DIAG, DIAGF, FERR, BERR, WORK2
C INTERFACE
#include <sunperf.h>
void zptsvx(char fact, int n, int nrhs, double *diag, doub-
lecomplex *sub, double *diagf, doublecomplex
*subf, doublecomplex *b, int ldb, doublecomplex
*x, int ldx, double *rcond, double *ferr, double
*berr, int *info);
void zptsvx_64(char fact, long n, long nrhs, double *diag,
doublecomplex *sub, double *diagf, doublecomplex
*subf, doublecomplex *b, long ldb, doublecomplex
*x, long ldx, double *rcond, double *ferr, double
*berr, long *info);
zptsvx 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 fac-
tored form
of A.
4. Iterative refinement is applied to improve the computed
solution
matrix and calculate error bounds and backward error
estimates
for it.
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 factori-
zation of A. If FACT = 'N', then DIAGF is an out-
put 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 con-
tains the (n-1) subdiagonal elements of the unit
bidiagonal factor L from the L*DIAG*L**H
factorization of A.
B (input) On entry, the N-by-NRHS right hand side matrix B.
Unchanged on exit.
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 ille-
gal 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 com-
puted. RCOND = 0 is returned. = N+1: U is non-
singular, but RCOND is less than machine preci-
sion, meaning that the matrix is singular to work-
ing 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.