dpbsvx - use the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations A * X = B,
SUBROUTINE DPBSVX( FACT, UPLO, N, NDIAG, NRHS, A, LDA, AF, LDAF, * EQUED, SCALE, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, * INFO) CHARACTER * 1 FACT, UPLO, EQUED INTEGER N, NDIAG, NRHS, LDA, LDAF, LDB, LDX, INFO INTEGER WORK2(*) DOUBLE PRECISION RCOND DOUBLE PRECISION A(LDA,*), AF(LDAF,*), SCALE(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*)
SUBROUTINE DPBSVX_64( FACT, UPLO, N, NDIAG, NRHS, A, LDA, AF, LDAF, * EQUED, SCALE, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, * INFO) CHARACTER * 1 FACT, UPLO, EQUED INTEGER*8 N, NDIAG, NRHS, LDA, LDAF, LDB, LDX, INFO INTEGER*8 WORK2(*) DOUBLE PRECISION RCOND DOUBLE PRECISION A(LDA,*), AF(LDAF,*), SCALE(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*)
SUBROUTINE PBSVX( FACT, UPLO, [N], NDIAG, [NRHS], A, [LDA], AF, * [LDAF], EQUED, SCALE, B, [LDB], X, [LDX], RCOND, FERR, BERR, * [WORK], [WORK2], [INFO]) CHARACTER(LEN=1) :: FACT, UPLO, EQUED INTEGER :: N, NDIAG, NRHS, LDA, LDAF, LDB, LDX, INFO INTEGER, DIMENSION(:) :: WORK2 REAL(8) :: RCOND REAL(8), DIMENSION(:) :: SCALE, FERR, BERR, WORK REAL(8), DIMENSION(:,:) :: A, AF, B, X
SUBROUTINE PBSVX_64( FACT, UPLO, [N], NDIAG, [NRHS], A, [LDA], AF, * [LDAF], EQUED, SCALE, B, [LDB], X, [LDX], RCOND, FERR, BERR, * [WORK], [WORK2], [INFO]) CHARACTER(LEN=1) :: FACT, UPLO, EQUED INTEGER(8) :: N, NDIAG, NRHS, LDA, LDAF, LDB, LDX, INFO INTEGER(8), DIMENSION(:) :: WORK2 REAL(8) :: RCOND REAL(8), DIMENSION(:) :: SCALE, FERR, BERR, WORK REAL(8), DIMENSION(:,:) :: A, AF, B, X
#include <sunperf.h>
void dpbsvx(char fact, char uplo, int n, int ndiag, int nrhs, double *a, int lda, double *af, int ldaf, char equed, double *scale, double *b, int ldb, double *x, int ldx, double *rcond, double *ferr, double *berr, int *info);
void dpbsvx_64(char fact, char uplo, long n, long ndiag, long nrhs, double *a, long lda, double *af, long ldaf, char equed, double *scale, double *b, long ldb, double *x, long ldx, double *rcond, double *ferr, double *berr, long *info);
dpbsvx uses the Cholesky factorization A = U**T*U or A = L*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric positive definite band 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**T * U, if UPLO = 'U', or
A = L * L**T, if UPLO = 'L',
where U is an upper triangular band matrix, and L is a lower triangular band 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 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(S) so that it solves the original system before
equilibration.
= 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored.
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
A(NDIAG+1+i-j,j) = A(i,j) for max(1,j-NDIAG) < =i < =j;
if UPLO = 'L', A(1+i-j,j) = A(i,j) for j < =i < =min(N,j+NDIAG).
See below for further details.
On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(SCALE)*A*diag(SCALE).
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**T*U or A = L*L**T.
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**T*U or A = L*L**T of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix).
= 'Y': Equilibration was done, i.e., A has been replaced by diag(SCALE) * A * diag(SCALE). EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.
diag(SCALE) * B.
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(3*N)
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: 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 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.
The band storage scheme is illustrated by the following example, when N = 6, NDIAG = 2, and UPLO = 'U':
Two-dimensional storage of the symmetric matrix A:
a11 a12 a13
a22 a23 a24
a33 a34 a35
a44 a45 a46
a55 a56
(aij =conjg(aji)) a66
Band storage of the upper triangle of A:
* * a13 a24 a35 a46
* a12 a23 a34 a45 a56
a11 a22 a33 a44 a55 a66
Similarly, if UPLO = 'L' the format of A is as follows:
a11 a22 a33 a44 a55 a66
a21 a32 a43 a54 a65 *
a31 a42 a53 a64 * *
Array elements marked * are not used by the routine.