Contents
zposvx - use the Cholesky factorization A = U**H*U or A =
L*L**H to compute the solution to a complex system of linear
equations A * X = B,
SUBROUTINE ZPOSVX(FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO)
CHARACTER * 1 FACT, UPLO, EQUED
DOUBLE COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*),
WORK(*)
INTEGER N, NRHS, LDA, LDAF, LDB, LDX, INFO
DOUBLE PRECISION RCOND
DOUBLE PRECISION S(*), FERR(*), BERR(*), WORK2(*)
SUBROUTINE ZPOSVX_64(FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO)
CHARACTER * 1 FACT, UPLO, EQUED
DOUBLE COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*),
WORK(*)
INTEGER*8 N, NRHS, LDA, LDAF, LDB, LDX, INFO
DOUBLE PRECISION RCOND
DOUBLE PRECISION S(*), FERR(*), BERR(*), WORK2(*)
F95 INTERFACE
SUBROUTINE POSVX(FACT, UPLO, [N], [NRHS], A, [LDA], AF, [LDAF],
EQUED, S, B, [LDB], X, [LDX], RCOND, FERR, BERR, [WORK],
[WORK2], [INFO])
CHARACTER(LEN=1) :: FACT, UPLO, EQUED
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: A, AF, B, X
INTEGER :: N, NRHS, LDA, LDAF, LDB, LDX, INFO
REAL(8) :: RCOND
REAL(8), DIMENSION(:) :: S, FERR, BERR, WORK2
SUBROUTINE POSVX_64(FACT, UPLO, [N], [NRHS], A, [LDA], AF, [LDAF],
EQUED, S, B, [LDB], X, [LDX], RCOND, FERR, BERR, [WORK],
[WORK2], [INFO])
CHARACTER(LEN=1) :: FACT, UPLO, EQUED
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: A, AF, B, X
INTEGER(8) :: N, NRHS, LDA, LDAF, LDB, LDX, INFO
REAL(8) :: RCOND
REAL(8), DIMENSION(:) :: S, FERR, BERR, WORK2
C INTERFACE
#include <sunperf.h>
void zposvx(char fact, char uplo, int n, int nrhs, doub-
lecomplex *a, int lda, doublecomplex *af, int
ldaf, char equed, double *s, doublecomplex *b, int
ldb, doublecomplex *x, int ldx, double *rcond,
double *ferr, double *berr, int *info);
void zposvx_64(char fact, char uplo, long n, long nrhs,
doublecomplex *a, long lda, doublecomplex *af,
long ldaf, char equed, double *s, doublecomplex
*b, long ldb, doublecomplex *x, long ldx, double
*rcond, double *ferr, double *berr, long *info);
zposvx uses the Cholesky factorization A = U**H*U or A =
L*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 defin-
ite 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 = 'E', real scaling factors are computed to
equilibrate
the system:
diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
Whether or not the system will be equilibrated depends on
the
scaling of the matrix A, but if equilibration is used, A
is
overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
2. If FACT = 'N' or 'E', the Cholesky decomposition is used
to
factor the matrix A (after equilibration if FACT = 'E')
as
A = U**H* U, if UPLO = 'U', or
A = L * L**H, if UPLO = 'L',
where U is an upper triangular matrix and L is a lower
triangular
matrix.
3. 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.
4. The system of equations is solved for X using the fac-
tored form
of A.
5. Iterative refinement is applied to improve the computed
solution
matrix and calculate error bounds and backward error
estimates
for it.
6. If equilibration was used, the matrix X is premultiplied
by
diag(S) so that it solves the original system before
equilibration.
FACT (input)
Specifies whether or not the factored form of the
matrix A is supplied on entry, and if not, whether
the matrix A should be equilibrated before it is
factored. = 'F': On entry, AF contains the fac-
tored form of A. If EQUED = 'Y', the matrix A has
been equilibrated with scaling factors given by S.
A and AF will not be modified. = 'N': The matrix
A will be copied to AF and factored.
= 'E': The matrix A will be equilibrated if
necessary, then copied to AF and factored.
UPLO (input)
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) The number of linear equations, i.e., 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.
A (input/output)
On entry, the Hermitian matrix A, except if FACT =
'F' and EQUED = 'Y', then A must contain the
equilibrated matrix diag(S)*A*diag(S). If UPLO =
'U', the leading N-by-N upper triangular part of A
contains the upper triangular part of the matrix
A, and the strictly lower triangular part of A is
not referenced. If UPLO = 'L', the leading N-by-N
lower triangular part of A contains the lower tri-
angular part of the matrix A, and the strictly
upper triangular part of A is not referenced. A
is not modified if FACT = 'F' or 'N', or if FACT =
'E' and EQUED = 'N' on exit.
On exit, if FACT = 'E' and EQUED = 'Y', A is
overwritten by diag(S)*A*diag(S).
LDA (input)
The leading dimension of the array A. LDA >=
max(1,N).
AF (input or output)
If FACT = 'F', then AF is an input argument and on
entry contains the triangular factor U or L from
the Cholesky factorization A = U**H*U or A =
L*L**H, in the same storage format as A. If EQUED
.ne. 'N', then AF is the factored form of the
equilibrated matrix diag(S)*A*diag(S).
If FACT = 'N', then AF is an output argument and
on exit returns the triangular factor U or L from
the Cholesky factorization A = U**H*U or A =
L*L**H of the original matrix A.
If FACT = 'E', then AF is an output argument and
on exit returns the triangular factor U or L from
the Cholesky factorization A = U**H*U or A =
L*L**H of the equilibrated matrix A (see the
description of A for the form of the equilibrated
matrix).
LDAF (input)
The leading dimension of the array AF. LDAF >=
max(1,N).
EQUED (input or output)
Specifies the form of equilibration that was done.
= 'N': No equilibration (always true if FACT =
'N').
= 'Y': Equilibration was done, i.e., A has been
replaced by diag(S) * A * diag(S). EQUED is an
input argument if FACT = 'F'; otherwise, it is an
output argument.
S (input or output)
The scale factors for A; not accessed if EQUED =
'N'. S is an input argument if FACT = 'F'; other-
wise, S is an output argument. If FACT = 'F' and
EQUED = 'Y', each element of S must be positive.
B (input/output)
On entry, the N-by-NRHS righthand side matrix B.
On exit, if EQUED = 'N', B is not modified; if
EQUED = 'Y', B is overwritten by diag(S) * 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 to the original system of equations.
Note that if EQUED = 'Y', A and B are modified on
exit, and the solution to the equilibrated system
is inv(diag(S))*X.
LDX (input)
The leading dimension of the array X. LDX >=
max(1,N).
RCOND (output)
The estimate of the reciprocal condition number of
the matrix A after equilibration (if done). If
RCOND is less than the machine precision (in par-
ticular, if RCOND = 0), the matrix is singular to
working precision. This condition is indicated by
a return code of INFO > 0.
FERR (output)
The estimated forward error bound for each solu-
tion 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 ele-
ment 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)
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)
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.