sspsvx(3P) Sun Performance Library sspsvx(3P)NAMEsspsvx - 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
SYNOPSIS
SUBROUTINE SSPSVX(FACT, UPLO, N, NRHS, AP, 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(*)
REAL RCOND
REAL AP(*), AF(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*)
SUBROUTINE SSPSVX_64(FACT, UPLO, N, NRHS, AP, 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(*)
REAL RCOND
REAL AP(*), AF(*), B(LDB,*), X(LDX,*), FERR(*), BERR(*), WORK(*)
F95 INTERFACE
SUBROUTINE SPSVX(FACT, UPLO, [N], [NRHS], AP, 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 :: RCOND
REAL, DIMENSION(:) :: AP, AF, FERR, BERR, WORK
REAL, DIMENSION(:,:) :: B, X
SUBROUTINE SPSVX_64(FACT, UPLO, [N], [NRHS], AP, 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 :: RCOND
REAL, DIMENSION(:) :: AP, AF, FERR, BERR, WORK
REAL, DIMENSION(:,:) :: B, X
C INTERFACE
#include <sunperf.h>
void sspsvx(char fact, char uplo, int n, int nrhs, float *ap, float
*af, int *ipivot, float *b, int ldb, float *x, int ldx, float
*rcond, float *ferr, float *berr, int *info);
void sspsvx_64(char fact, char uplo, long n, long nrhs, float *ap,
float *af, long *ipivot, float *b, long ldb, float *x, long
ldx, float *rcond, float *ferr, float *berr, long *info);
PURPOSE
SSPSVX 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 pro‐
vided.
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.
ARGUMENTS
FACT (input)
Specifies whether or not the factored form of A has been sup‐
plied on entry. = 'F': On entry, AF and IPIVOT contain the
factored form of A. AP, 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.
AP (input)
Real array, dimension (N*(N+1)/2) The upper or lower triangle
of the symmetric matrix A, packed columnwise in a linear
array. The j-th column of A is stored in the array AP as
follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for
1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for
j<=i<=n. See below for further details.
AF (input or output)
Real array, dimension (N*(N+1)/2) 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 com‐
puted by SSPTRF, stored as a packed triangular matrix in the
same storage format as A.
If FACT = 'N', then AF is an output argument and on exit con‐
tains 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 (input or output)
Integer array, dimension (N) If FACT = 'F', then IPIVOT is an
input argument and on entry contains details of the inter‐
changes and the block structure of D, as determined by SSP‐
TRF. 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 SSPTRF.
B (input) Real array, dimension (LDB,NRHS) The N-by-NRHS right hand
side matrix B.
LDB (input)
The leading dimension of the array B. LDB >= max(1,N).
X (output)
Real array, dimension (LDX,NRHS) 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 particu‐
lar, if RCOND = 0), the matrix is singular to working preci‐
sion. This condition is indicated by a return code of INFO >
0.
FERR (output)
Real array, dimension (NRHS) The estimated forward error
bound for each solution vector X(j) (the j-th column of the
solution matrix X). If XTRUE is the true solution corre‐
sponding 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.
BERR (output)
Real array, dimension (NRHS) The componentwise relative back‐
ward 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)
Real array, dimension(3*N)
WORK2 (workspace)
Integer array, dimension(N)
INFO (output)
= 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 solu‐
tion 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.
FURTHER DETAILS
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:
AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
6 Mar 2009 sspsvx(3P)
