DSYSVX (3lapack)

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

SYNOPSIS

  SUBROUTINE DSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
                     LDX, RCOND, FERR, BERR, WORK, LWORK, IWORK, INFO )

      CHARACTER      FACT, UPLO

      INTEGER        INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS

      DOUBLE         PRECISION RCOND

      INTEGER        IPIV( * ), IWORK( * )

      DOUBLE         PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), BERR(
                     * ), FERR( * ), WORK( * ), X( LDX, * )

PURPOSE

  DSYSVX uses the diagonal pivoting factorization to compute the solution to
  a real 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. 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.  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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 DSYTRF.

          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 DSYTRF.  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 DSYTRF.

  B       (input) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION
          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) 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/output) DOUBLE PRECISION array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The length of WORK.  LWORK >= 3*N, and for best performance LWORK
          >= N*NB, where NB is the optimal blocksize for DSYTRF.

  IWORK   (workspace) INTEGER 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.