chesvx(3F)

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

SYNOPSIS
     SUBROUTINE CHESVX( 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
     CHESVX 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
     Hermitian 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**H,  if UPLO = 'U', or
           A = L * D * L**H,  if UPLO = 'L',
        where U (or L) is a product of permutation and unit upper (lower)
        triangular matrices, and D is Hermitian and block diagonal with
        1-by-1 and 2-by-2 diagonal blocks.

     2. 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 3 and 4 are skipped.

     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 Hermitian 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**H or A =
             L*D*L**H as computed by CHETRF.

             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**H or A =
             L*D*L**H.

     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 CHETRF.  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 CHETRF.

     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, 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, 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/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 CHETRF.

     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 has been
             completed, but the block diagonal matrix D is exactly singular,
             so the solution and error bounds could not be computed.  = N+1:
             the block diagonal matrix D is nonsingular, but RCOND is less
             than machine precision.  The factorization has been completed,
             but the matrix is singular to working precision, so the solution
             and error bounds have not been computed.