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NAME
cppsvx - use the Cholesky factorization A = U**H*U or A =
L*L**H to compute the solution to a complex system of linear
equations A * X = B,
SYNOPSIS
SUBROUTINE CPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S,
B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK,
INFO )
CHARACTER EQUED, FACT, UPLO
INTEGER INFO, LDB, LDX, N, NRHS
REAL RCOND
REAL BERR( * ), FERR( * ), RWORK( * ), S( * )
COMPLEX AFP( * ), AP( * ), B( LDB, * ), WORK( * ), X( LDX, *
)
#include <sunperf.h>
void cppsvx(char fact, char uplo, int n, int nrhs, complex
*cap, complex *afp, char *equed, float *s, complex
*cb, int ldb, complex *cx, int ldx, float *srcond,
float *ferr, float *berr, int *info) ;
PURPOSE
CPPSVX uses the Cholesky factorization A = U**H*U or A =
L*L**H to compute the solution to a complex system of linear
equations A * X = B, where A is an N-by-N Hermitian positive
definite 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.
DESCRIPTION
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'* U , if UPLO = 'U', or
A = L * L', if UPLO = 'L',
where U is an upper triangular matrix, L is a lower triangu-
lar matrix, and ' indicates conjugate transpose.
3. 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, steps 4-6 are
skipped.
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.
ARGUMENTS
FACT (input) CHARACTER*1
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, AFP contains the fac-
tored form of A. If EQUED = 'Y', the matrix A has
been equilibrated with scaling factors given by S.
AP and AFP will not be modified. = 'N': The
matrix A will be copied to AFP and factored.
= 'E': The matrix A will be equilibrated if
necessary, then copied to AFP and factored.
UPLO (input) CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The number of linear equations, i.e., the order of
the matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number
of columns of the matrices B and X. NRHS >= 0.
AP (input/output) COMPLEX array, dimension
(N*(N+1)/2)
On entry, the upper or lower triangle of the
Hermitian 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 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)*(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).
AFP (input or output) COMPLEX array, dimension
(N*(N+1)/2)
If FACT = 'F', then AFP is an input argument and
on entry contains the triangular factor U or L
from the Cholesky factorization A = U**H*U or A =
L*L**H, in the same storage format as A. If EQUED
.ne. 'N', then AFP is the factored form of the
equilibrated matrix A.
If FACT = 'N', then AFP is an output argument and
on exit returns the triangular factor U or L from
the Cholesky factorization A = U**H*U or A =
L*L**H of the original matrix A.
If FACT = 'E', then AFP is an output argument and
on exit returns the triangular factor U or L from
the Cholesky factorization A = U**H*U or A =
L*L**H of the equilibrated matrix A (see the
description of AP for the form of the equilibrated
matrix).
EQUED (input or output) CHARACTER*1
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) COMPLEX 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) INTEGER
The leading dimension of the array B. LDB >=
max(1,N).
X (output) COMPLEX array, dimension (LDX,NRHS)
If INFO = 0, 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) INTEGER
The leading dimension of the array X. LDX >=
max(1,N).
RCOND (output) REAL
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, and the solution and
error bounds are not computed.
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) COMPLEX array, dimension (2*N)
RWORK (workspace) REAL array, dimension (N)
INFO (output) INTEGER
= 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 and error bounds
could not be computed. = N+1: RCOND is less than
machine precision. The factorization has been
completed, but the matrix is singular to working
precision, and the solution and error bounds have
not been computed.
FURTHER DETAILS
The packed storage scheme is illustrated by the following
example when N = 4, UPLO = 'U':
Two-dimensional storage of the Hermitian matrix A:
a11 a12 a13 a14
a22 a23 a24
a33 a34 (aij = conjg(aji))
a44
Packed storage of the upper triangle of A:
AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
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