zgbsvx - use the LU factorization to compute the solution to a complex system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
SUBROUTINE ZGBSVX( FACT, TRANSA, N, NSUB, NSUPER, NRHS, A, LDA, AF, * LDAF, IPIVOT, EQUED, ROWSC, COLSC, B, LDB, X, LDX, RCOND, FERR, * BERR, WORK, WORK2, INFO) CHARACTER * 1 FACT, TRANSA, EQUED DOUBLE COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*) INTEGER N, NSUB, NSUPER, NRHS, LDA, LDAF, LDB, LDX, INFO INTEGER IPIVOT(*) DOUBLE PRECISION RCOND DOUBLE PRECISION ROWSC(*), COLSC(*), FERR(*), BERR(*), WORK2(*)
SUBROUTINE ZGBSVX_64( FACT, TRANSA, N, NSUB, NSUPER, NRHS, A, LDA, * AF, LDAF, IPIVOT, EQUED, ROWSC, COLSC, B, LDB, X, LDX, RCOND, * FERR, BERR, WORK, WORK2, INFO) CHARACTER * 1 FACT, TRANSA, EQUED DOUBLE COMPLEX A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), WORK(*) INTEGER*8 N, NSUB, NSUPER, NRHS, LDA, LDAF, LDB, LDX, INFO INTEGER*8 IPIVOT(*) DOUBLE PRECISION RCOND DOUBLE PRECISION ROWSC(*), COLSC(*), FERR(*), BERR(*), WORK2(*)
SUBROUTINE GBSVX( FACT, [TRANSA], [N], NSUB, NSUPER, [NRHS], A, [LDA], * AF, [LDAF], IPIVOT, EQUED, ROWSC, COLSC, B, [LDB], X, [LDX], * RCOND, FERR, BERR, [WORK], [WORK2], [INFO]) CHARACTER(LEN=1) :: FACT, TRANSA, EQUED COMPLEX(8), DIMENSION(:) :: WORK COMPLEX(8), DIMENSION(:,:) :: A, AF, B, X INTEGER :: N, NSUB, NSUPER, NRHS, LDA, LDAF, LDB, LDX, INFO INTEGER, DIMENSION(:) :: IPIVOT REAL(8) :: RCOND REAL(8), DIMENSION(:) :: ROWSC, COLSC, FERR, BERR, WORK2
SUBROUTINE GBSVX_64( FACT, [TRANSA], [N], NSUB, NSUPER, [NRHS], A, * [LDA], AF, [LDAF], IPIVOT, EQUED, ROWSC, COLSC, B, [LDB], X, [LDX], * RCOND, FERR, BERR, [WORK], [WORK2], [INFO]) CHARACTER(LEN=1) :: FACT, TRANSA, EQUED COMPLEX(8), DIMENSION(:) :: WORK COMPLEX(8), DIMENSION(:,:) :: A, AF, B, X INTEGER(8) :: N, NSUB, NSUPER, NRHS, LDA, LDAF, LDB, LDX, INFO INTEGER(8), DIMENSION(:) :: IPIVOT REAL(8) :: RCOND REAL(8), DIMENSION(:) :: ROWSC, COLSC, FERR, BERR, WORK2
#include <sunperf.h>
void zgbsvx(char fact, char transa, int n, int nsub, int nsuper, int nrhs, doublecomplex *a, int lda, doublecomplex *af, int ldaf, int *ipivot, char equed, double *rowsc, double *colsc, doublecomplex *b, int ldb, doublecomplex *x, int ldx, double *rcond, double *ferr, double *berr, int *info);
void zgbsvx_64(char fact, char transa, long n, long nsub, long nsuper, long nrhs, doublecomplex *a, long lda, doublecomplex *af, long ldaf, long *ipivot, char equed, double *rowsc, double *colsc, doublecomplex *b, long ldb, doublecomplex *x, long ldx, double *rcond, double *ferr, double *berr, long *info);
zgbsvx uses the LU factorization to compute the solution to a complex system of linear equations A * X = B, A**T * X = B, or A**H * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, 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 by this subroutine:
1. If FACT = 'E', real scaling factors are computed to equilibrate the system:
TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*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(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
or diag(C)*B (if TRANS = 'T' or 'C').
2. If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as
A = L * U,
where L is a product of permutation and unit lower triangular matrices with KL subdiagonals, and U is upper triangular with KL+KU superdiagonals.
3. If some U(i,i)=0, so that U 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.
4. The system of equations is solved for X using the factored 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(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
that it solves the original system before equilibration.
= 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored.
= 'T': A**T * X = B (Transpose)
= 'COLSC': A**H * X = B (Conjugate transpose)
A(NSUPER+1+i-j,j) = A(i,j) for max(1,j-NSUPER) < =i < =min(N,j+kl)
If FACT = 'F' and EQUED is not 'N', then A must have been equilibrated by the scaling factors in ROWSC and/or COLSC. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
On exit, if EQUED .ne. 'N', A is scaled as follows:
EQUED = 'ROWSC': A : = diag(ROWSC) * A
EQUED = 'COLSC': A : = A * diag(COLSC)
EQUED = 'B': A : = diag(ROWSC) * A * diag(COLSC).
If FACT = 'N', then AF is an output argument and on exit returns details of the LU factorization of A.
If FACT = 'E', then AF is an output argument and on exit returns details of the LU factorization of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix).
If FACT = 'N', then IPIVOT is an output argument and on exit contains the pivot indices from the factorization A = L*U of the original matrix A.
If FACT = 'E', then IPIVOT is an output argument and on exit contains the pivot indices from the factorization A = L*U of the equilibrated matrix A.
= 'ROWSC': Row equilibration, i.e., A has been premultiplied by diag(ROWSC). = 'COLSC': Column equilibration, i.e., A has been postmultiplied by diag(COLSC). = 'B': Both row and column equilibration, i.e., A has been replaced by diag(ROWSC) * A * diag(COLSC). EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.
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).
dimension(2*N)
dimension(N)
On exit, WORK2(1) contains the reciprocal pivot growth
factor norm(A)/norm(U). The ``max absolute element'' norm is
used. If WORK2(1) is much less than 1, then the stability
of the LU factorization of the (equilibrated) matrix A
could be poor. This also means that the solution X, condition
estimator RCOND, and forward error bound FERR could be
unreliable. If factorization fails with 0 <INFO < =N, then
WORK2(1) contains the reciprocal pivot growth factor for the
leading INFO columns of A.
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
< = N: U(i,i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be 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.