Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  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 ZPOSVX( 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

           DOUBLE         PRECISION RCOND

           DOUBLE         PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )

           COMPLEX*16     A( LDA, * ), AF( LDAF, * ), B( LDB, * ), WORK( * ), X( LDX, * )

PURPOSE

       ZPOSVX 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. 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 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*16 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*16 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) DOUBLE PRECISION 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*16 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*16 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
                equilibrated system is inv(diag(S))*X.

        LDX     (input) INTEGER
                The leading dimension of the array X.  LDX >= max(1,N).

        RCOND   (output) DOUBLE PRECISION
                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) DOUBLE PRECISION 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) DOUBLE PRECISION 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*16 array, dimension (2*N)

        RWORK   (workspace) DOUBLE PRECISION 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 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.  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.

 LAPACK driver routine (version 3.3.1)      April 2011                            ZPOSVX(3lapack)