dspsvx - use the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format and X and B are N-by-NRHS matrices
SUBROUTINE DSPSVX( FACT, UPLO, N, NRHS, A, AF, IPIVOT, B, LDB, X, * LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER * 1 FACT, UPLO INTEGER N, NRHS, LDB, LDX, INFO INTEGER IPIVOT(*), WORK2(*) DOUBLE PRECISION RCOND DOUBLE PRECISION A(*), AF(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*)
SUBROUTINE DSPSVX_64( FACT, UPLO, N, NRHS, A, AF, IPIVOT, B, LDB, X, * LDX, RCOND, FERR, BERR, WORK, WORK2, INFO) CHARACTER * 1 FACT, UPLO INTEGER*8 N, NRHS, LDB, LDX, INFO INTEGER*8 IPIVOT(*), WORK2(*) DOUBLE PRECISION RCOND DOUBLE PRECISION A(*), AF(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*)
SUBROUTINE SPSVX( FACT, UPLO, N, NRHS, A, AF, IPIVOT, B, [LDB], X, * [LDX], RCOND, FERR, BERR, [WORK], [WORK2], [INFO]) CHARACTER(LEN=1) :: FACT, UPLO INTEGER :: N, NRHS, LDB, LDX, INFO INTEGER, DIMENSION(:) :: IPIVOT, WORK2 REAL(8) :: RCOND REAL(8), DIMENSION(:) :: A, AF, FERR, BERR, WORK REAL(8), DIMENSION(:,:) :: B, X
SUBROUTINE SPSVX_64( FACT, UPLO, N, NRHS, A, AF, IPIVOT, B, [LDB], * X, [LDX], RCOND, FERR, BERR, [WORK], [WORK2], [INFO]) CHARACTER(LEN=1) :: FACT, UPLO INTEGER(8) :: N, NRHS, LDB, LDX, INFO INTEGER(8), DIMENSION(:) :: IPIVOT, WORK2 REAL(8) :: RCOND REAL(8), DIMENSION(:) :: A, AF, FERR, BERR, WORK REAL(8), DIMENSION(:,:) :: B, X
#include <sunperf.h>
void dspsvx(char fact, char uplo, int n, int nrhs, double *a, double *af, int *ipivot, double *b, int ldb, double *x, int ldx, double *rcond, double *ferr, double *berr, int *info);
void dspsvx_64(char fact, char uplo, long n, long nrhs, double *a, double *af, long *ipivot, double *b, long ldb, double *x, long ldx, double *rcond, double *ferr, double *berr, long *info);
dspsvx uses the diagonal pivoting factorization A = U*D*U**T or A = L*D*L**T to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix stored in packed format 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 as 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.
A(i,j) for 1 < =i < =j;
if UPLO = 'L', A(i + (j-1)*(2*n-j)/2) = A(i,j) for j < =i < =n.
See below for further details.
If FACT = 'N', then AF is an output argument and on exit 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 SSPTRF, stored as a packed triangular matrix in the same storage format as A.
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 SSPTRF.
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: 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.
The packed storage scheme is illustrated by the following example when N = 4, UPLO = 'U':
Two-dimensional storage of the symmetric matrix A:
a11 a12 a13 a14
a22 a23 a24
a33 a34 (aij = aji)
a44
Packed storage of the upper triangle of A:
A = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]