Contents
sppsvx - 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 SPPSVX(FACT, UPLO, N, NRHS, A, AF, EQUED, S, B, LDB,
X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO)
CHARACTER * 1 FACT, UPLO, EQUED
INTEGER N, NRHS, LDB, LDX, INFO
INTEGER WORK2(*)
REAL RCOND
REAL A(*), AF(*), S(*), B(LDB,*), X(LDX,*), FERR(*),
BERR(*), WORK(*)
SUBROUTINE SPPSVX_64(FACT, UPLO, N, NRHS, A, AF, EQUED, S, B,
LDB, X, LDX, RCOND, FERR, BERR, WORK, WORK2, INFO)
CHARACTER * 1 FACT, UPLO, EQUED
INTEGER*8 N, NRHS, LDB, LDX, INFO
INTEGER*8 WORK2(*)
REAL RCOND
REAL A(*), AF(*), S(*), B(LDB,*), X(LDX,*), FERR(*),
BERR(*), WORK(*)
F95 INTERFACE
SUBROUTINE PPSVX(FACT, UPLO, [N], [NRHS], A, AF, EQUED, S, B,
[LDB], X, [LDX], RCOND, FERR, BERR, [WORK], [WORK2], [INFO])
CHARACTER(LEN=1) :: FACT, UPLO, EQUED
INTEGER :: N, NRHS, LDB, LDX, INFO
INTEGER, DIMENSION(:) :: WORK2
REAL :: RCOND
REAL, DIMENSION(:) :: A, AF, S, FERR, BERR, WORK
REAL, DIMENSION(:,:) :: B, X
SUBROUTINE PPSVX_64(FACT, UPLO, [N], [NRHS], A, AF, EQUED, S, B,
[LDB], X, [LDX], RCOND, FERR, BERR, [WORK], [WORK2], [INFO])
CHARACTER(LEN=1) :: FACT, UPLO, EQUED
INTEGER(8) :: N, NRHS, LDB, LDX, INFO
INTEGER(8), DIMENSION(:) :: WORK2
REAL :: RCOND
REAL, DIMENSION(:) :: A, AF, S, FERR, BERR, WORK
REAL, DIMENSION(:,:) :: B, X
C INTERFACE
#include <sunperf.h>
void sppsvx(char fact, char uplo, int n, int nrhs, float *a,
float *af, char equed, float *s, float *b, int
ldb, float *x, int ldx, float *rcond, float *ferr,
float *berr, int *info);
void sppsvx_64(char fact, char uplo, long n, long nrhs,
float *a, float *af, char equed, float *s, float
*b, long ldb, float *x, long ldx, float *rcond,
float *ferr, float *berr, long *info);
sppsvx 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 defin-
ite 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 = '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 matrix and L is a lower
triangular
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 fac-
tored 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.
FACT (input)
Specifies whether or not the factored form of the
matrix A is supplied on entry, and if not, whether
the matrix A should be equilibrated before it is
factored. = 'F': On entry, AF contains the fac-
tored form of A. If EQUED = 'Y', the matrix A has
been equilibrated with scaling factors given by S.
A and AF will not be modified. = 'N': The matrix
A will be copied to AF and factored.
= 'E': The matrix A will be equilibrated if
necessary, then 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.
A (input/output) REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the sym-
metric matrix A, packed columnwise in a linear
array, except if FACT = 'F' and EQUED = 'Y', then
A must contain the equilibrated matrix
diag(S)*A*diag(S). The j-th column of A is stored
in the array A as follows: if UPLO = 'U', A(i +
(j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L',
A(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. See
below for further details. A is not modified if
FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
'N' on exit.
On exit, if FACT = 'E' and EQUED = 'Y', A is
overwritten by diag(S)*A*diag(S).
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 triangular factor U or L from
the Cholesky factorization A = U'*U or A = L*L',
in the same storage format as A. If EQUED .ne.
'N', then AF is the factored form of the equili-
brated matrix A.
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'*U or A = L*L' of
the original matrix A.
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'*U or A = L*L' of
the equilibrated matrix A (see the description of
A for the form of the equilibrated matrix).
EQUED (input or output)
Specifies the form of equilibration that was done.
= 'N': No equilibration (always true if FACT =
'N').
= 'Y': Equilibration was done, i.e., A has been
replaced by diag(S) * A * diag(S). EQUED is an
input argument if FACT = 'F'; otherwise, it is an
output argument.
S (input or output) REAL array, dimension (N)
The scale factors for A; not accessed if EQUED =
'N'. S is an input argument if FACT = 'F'; other-
wise, S is an output argument. If FACT = 'F' and
EQUED = 'Y', each element of S must be positive.
B (input/output) REAL array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit, if EQUED = 'N', B is not modified; if
EQUED = 'Y', B is overwritten by diag(S) * 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 to the original system of equations.
Note that if EQUED = 'Y', A and B are modified on
exit, and the solution to the equilibrated system
is inv(diag(S))*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 after equilibration (if done). If
RCOND is less than the machine precision (in par-
ticular, 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 solu-
tion 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 ele-
ment 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 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 ille-
gal 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 com-
puted. RCOND = 0 is returned. = N+1: U is non-
singular, but RCOND is less than machine preci-
sion, meaning that the matrix is singular to work-
ing 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 = conjg(aji))
a44
Packed storage of the upper triangle of A:
A = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]