Contents
zsysvx - use the diagonal pivoting factorization to compute
the solution to a complex system of linear equations A * X =
B,
SUBROUTINE ZSYSVX(FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIVOT, B,
LDB, X, LDX, RCOND, FERR, BERR, WORK, LDWORK, WORK2, INFO)
CHARACTER * 1 FACT, UPLO
DOUBLE COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*),
WORK(*)
INTEGER N, NRHS, LDA, LDAF, LDB, LDX, LDWORK, INFO
INTEGER IPIVOT(*)
DOUBLE PRECISION RCOND
DOUBLE PRECISION FERR(*), BERR(*), WORK2(*)
SUBROUTINE ZSYSVX_64(FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIVOT,
B, LDB, X, LDX, RCOND, FERR, BERR, WORK, LDWORK, WORK2, INFO)
CHARACTER * 1 FACT, UPLO
DOUBLE COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*),
WORK(*)
INTEGER*8 N, NRHS, LDA, LDAF, LDB, LDX, LDWORK, INFO
INTEGER*8 IPIVOT(*)
DOUBLE PRECISION RCOND
DOUBLE PRECISION FERR(*), BERR(*), WORK2(*)
F95 INTERFACE
SUBROUTINE SYSVX(FACT, UPLO, N, NRHS, A, [LDA], AF, [LDAF], IPIVOT,
B, [LDB], X, [LDX], RCOND, FERR, BERR, [WORK], [LDWORK], [WORK2],
[INFO])
CHARACTER(LEN=1) :: FACT, UPLO
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: A, AF, B, X
INTEGER :: N, NRHS, LDA, LDAF, LDB, LDX, LDWORK, INFO
INTEGER, DIMENSION(:) :: IPIVOT
REAL(8) :: RCOND
REAL(8), DIMENSION(:) :: FERR, BERR, WORK2
SUBROUTINE SYSVX_64(FACT, UPLO, N, NRHS, A, [LDA], AF, [LDAF],
IPIVOT, B, [LDB], X, [LDX], RCOND, FERR, BERR, [WORK], [LDWORK],
[WORK2], [INFO])
CHARACTER(LEN=1) :: FACT, UPLO
COMPLEX(8), DIMENSION(:) :: WORK
COMPLEX(8), DIMENSION(:,:) :: A, AF, B, X
INTEGER(8) :: N, NRHS, LDA, LDAF, LDB, LDX, LDWORK, INFO
INTEGER(8), DIMENSION(:) :: IPIVOT
REAL(8) :: RCOND
REAL(8), DIMENSION(:) :: FERR, BERR, WORK2
C INTERFACE
#include <sunperf.h>
void zsysvx(char fact, char uplo, int n, int nrhs, doub-
lecomplex *a, int lda, doublecomplex *af, int
ldaf, int *ipivot, doublecomplex *b, int ldb,
doublecomplex *x, int ldx, double *rcond, double
*ferr, double *berr, int *info);
void zsysvx_64(char fact, char uplo, long n, long nrhs,
doublecomplex *a, long lda, doublecomplex *af,
long ldaf, long *ipivot, doublecomplex *b, long
ldb, doublecomplex *x, long ldx, double *rcond,
double *ferr, double *berr, long *info);
zsysvx uses the diagonal pivoting factorization to compute
the solution to a complex system of linear equations A * X =
B, where A is an N-by-N symmetric 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 diagonal pivoting method is used to
factor A.
The form of the factorization is
A = U * D * U**T, if UPLO = 'U', or
A = L * D * L**T, if UPLO = 'L',
where U (or L) is a product of permutation and unit upper
(lower)
triangular matrices, and D is symmetric and block diago-
nal with
1-by-1 and 2-by-2 diagonal blocks.
2. If some D(i,i)=0, so that D is exactly singular, 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 A
has been supplied on entry. = 'F': On entry, AF
and IPIVOT contain the factored form of A. A, AF
and IPIVOT will not be modified. = 'N': The
matrix A will be 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) The symmetric matrix A. If UPLO = 'U', the lead-
ing 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 refer-
enced. If UPLO = 'L', the leading N-by-N lower
triangular part of A contains the lower triangular
part of the matrix A, and the strictly upper tri-
angular part of A is not referenced.
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 block diagonal matrix D and the
multipliers used to obtain the factor U or L from
the factorization A = U*D*U**T or A = L*D*L**T as
computed by ZSYTRF.
If FACT = 'N', then AF is an output argument and
on exit returns the block diagonal matrix D and
the multipliers used to obtain the factor U or L
from the factorization A = U*D*U**T or A =
L*D*L**T.
LDAF (input)
The leading dimension of the array AF. LDAF >=
max(1,N).
IPIVOT (input or output)
If FACT = 'F', then IPIVOT is an input argument
and on entry contains details of the interchanges
and the block structure of D, as determined by
ZSYTRF. If IPIVOT(k) > 0, then rows and columns k
and IPIVOT(k) were interchanged and D(k,k) is a
1-by-1 diagonal block. If UPLO = 'U' and
IPIVOT(k) = IPIVOT(k-1) < 0, then rows and columns
k-1 and -IPIVOT(k) were interchanged and D(k-
1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO =
'L' and IPIVOT(k) = IPIVOT(k+1) < 0, then rows and
columns k+1 and -IPIVOT(k) were interchanged and
D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
If FACT = 'N', then IPIVOT is an output argument
and on exit contains details of the interchanges
and the block structure of D, as determined by
ZSYTRF.
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 estimate of 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 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)
On exit, if INFO = 0, WORK(1) returns the optimal
LDWORK.
LDWORK (input)
The length of WORK. LDWORK >= 2*N, and for best
performance LDWORK >= N*NB, where NB is the
optimal blocksize for ZSYTRF.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of
the WORK array, returns this value as the first
entry of the WORK array, and no error message
related to LDWORK is issued by XERBLA.
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: D(i,i) is exactly zero. The factorization
has been completed but the factor D is exactly
singular, so the solution and error bounds could
not be computed. RCOND = 0 is returned. = N+1: D
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.