CPOSVX(l) — LAPACK driver routine (version 2.0)

NAME

CPOSVX - 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 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 compute 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 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∗∗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. 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 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 contains 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 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 AF is the factored form of the equilibrated 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 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 righthand 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 particular, 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 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 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 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.