# man csysvx (Fonctions bibliothèques) - use the diagonal pivoting factorization to compute the solution to a complex system of linear equations A * X = B,

## NAME

CSYSVX - use the diagonal pivoting factorization to compute the solution to a complex system of linear equations A * X = B,

## SYNOPSIS

- SUBROUTINE CSYSVX(
- FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK, RWORK, INFO )
- CHARACTER FACT, UPLO
- INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
- REAL RCOND
- INTEGER IPIV( * )
- REAL BERR( * ), FERR( * ), RWORK( * )
- COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ), WORK( * ), X( LDX, * )

## PURPOSE

CSYSVX uses the diagonal pivoting factorization to compute the solution to a complex system of linear equations A * X = B, where A is an N-by-N symmetric 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 = 'N', the diagonal pivoting method is used to factor A.
The form of the factorization is

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 factored form
of A.

4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.

## ARGUMENTS

- FACT (input) CHARACTER*1
- Specifies whether or not the factored form of A has been supplied on entry. = 'F': On entry, AF and IPIV contain the factored form of A. A, AF and IPIV will not be modified. = 'N': The matrix A will be 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) COMPLEX array, dimension (LDA,N)
- The symmetric matrix A. 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.
- 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 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 CSYTRF.

If FACT = 'N', then AF is an output argument and on exit returns 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.

- LDAF (input) INTEGER
- The leading dimension of the array AF. LDAF >= max(1,N).
- IPIV (input or output) INTEGER array, dimension (N)
- If FACT = 'F', then IPIV is an input argument and on entry contains details of the interchanges and the block structure of D, as determined by CSYTRF. If IPIV(k) > 0, then rows and columns k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

If FACT = 'N', then IPIV is an output argument and on exit contains details of the interchanges and the block structure of D, as determined by CSYTRF.

- B (input) COMPLEX array, dimension (LDB,NRHS)
- The N-by-NRHS right hand side matrix 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.
- 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. 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 solution).
- WORK (workspace/output) COMPLEX array, dimension (LWORK)
- On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
- LWORK (input) INTEGER
- The length of WORK. LWORK >= 2*N, and for best performance LWORK >= N*NB, where NB is the optimal blocksize for CSYTRF.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

- 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: 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.