CPOSVX(1) LAPACK driver routine (version 3.2) CPOSVX(1)NAME
CPOSVX - 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,
SYNOPSIS
SUBROUTINE CPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B,
LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
CHARACTER EQUED, FACT, UPLO
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
REAL RCOND
REAL BERR( * ), FERR( * ), RWORK( * ), S( * )
COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ), WORK( * ),
X( LDX, * )
PURPOSE
CPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to com‐
pute the solution to a complex system of linear equations
A * X = B, where A is an N-by-N Hermitian positive definite matrix
and X and B are N-by-NRHS matrices.
Error bounds on the solution and a condition estimate are also pro‐
vided.
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**H* U, if UPLO = 'U', or
A = L * L**H, 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 factored 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, AF con‐
tains the factored 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) 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.
A (input/output) COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A, except if FACT = 'F' and
EQUED = 'Y', then A must contain the equilibrated matrix
diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper
triangular part of A contains the upper triangular part of the
matrix A, and the strictly lower triangular part of A is not
referenced. If UPLO = 'L', the leading N-by-N lower triangular
part of A contains the lower triangular part of the matrix A,
and the strictly upper triangular part of A is not referenced.
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).
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF (input or output) COMPLEX array, dimension (LDAF,N)
If FACT = 'F', then AF is an input argument and on entry con‐
tains the triangular factor U or L from the Cholesky factoriza‐
tion A = U**H*U or A = L*L**H, in the same storage format as A.
If EQUED .ne. 'N', then AF is the factored form of the equili‐
brated matrix diag(S)*A*diag(S). 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**H*U or A = L*L**H 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**H*U or A = L*L**H of the
equilibrated matrix A (see the description of A for the form of
the equilibrated matrix).
LDAF (input) INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
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'; otherwise, S is an output argu‐
ment. 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 righthand side matrix B. On exit, if
EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwrit‐
ten 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 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 equili‐
brated 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 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 vec‐
tor 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 illegal value
> 0: if INFO = i, and i is
<= N: the leading minor of order i of A is not positive defi‐
nite, so the factorization could not be completed, and the
solution has not been computed. RCOND = 0 is returned. = N+1:
U is nonsingular, but RCOND is less than machine precision,
meaning that the matrix is singular to working precision. Nev‐
ertheless, 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.
LAPACK driver routine (version 3November 2008 CPOSVX(1)
