Contents
sspsvx - 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 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);
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 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 fac-
tored form
of A.
4. Iterative refinement is applied to improve the computed
solution
matrix and calculate error bounds and backward error
estimates
for it.
FACT (input)
Specifies whether or not the factored form of A
has been supplied 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 factori-
zation 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.
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 (input or output)
Integer array, dimension (N) If FACT = 'F', then
IPIVOT is an input argument and on entry contains
details of the interchanges and the block struc-
ture of D, as determined by SSPTRF. 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 diago-
nal 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 particular, if RCOND = 0), the
matrix is singular to working precision. 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 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.
BERR (output)
Real array, dimension (NRHS) The componentwise
relative backward 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 solu-
tion).
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 ille-
gal 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:
AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]