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(*)
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
#include <sunperf.h>
void zsysvx(char fact, char uplo, int n, int nrhs, doublecomplex *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 diagonal 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 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.
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
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.
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 CSYTRF.
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). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
WORK(1) returns the optimal LDWORK.
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.
dimension(N)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal 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.