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dposvx


 NAME
      DPOSVX - use the Cholesky factorization A = U**T*U or A =
      L*L**T to compute the solution to a real system of linear
      equations  A * X = B,

 SYNOPSIS
      SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF,
                         EQUED, S, B, LDB, X, LDX, RCOND, FERR,
                         BERR, WORK, IWORK, INFO )

          CHARACTER      EQUED, FACT, UPLO

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

          DOUBLE         PRECISION RCOND

          INTEGER        IWORK( * )

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

 PURPOSE
      DPOSVX uses the Cholesky factorization A = U**T*U or A =
      L*L**T to compute the solution to a real system of linear
      equations
         A * X = B, where A is an N-by-N symmetric positive defin-
      ite 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**T* U,  if UPLO = 'U', or
            A = L * L**T,  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 fac-
      tored 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 fac-
              tored 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 neces-
              sary, 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 righthand sides, i.e., the number of
              columns of the matrices B and X.  NRHS >= 0.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)

              On entry, the symmetric matrix A, except if FACT =
              'F' and EQUED = 'Y', then A must contain the equili-
              brated 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 refer-
              enced.  If UPLO = 'L', the leading N-by-N lower tri-
              angular 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).

 (LDAF,N)
      AF      (input or output) DOUBLE PRECISION array, dimension
              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**T*U or A = L*L**T, 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**T*U or A = L*L**T 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**T*U or A = L*L**T 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/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 variable if FACT = 'F'; otherwise, it is an
              output variable.

      S       (input/output) DOUBLE PRECISION array, dimension (N)
              The scale factors for A; not accessed if EQUED =
              'N'.  S is an input variable if FACT = 'F'; other-
              wise, S is an output variable.  If FACT = 'F' and
              EQUED = 'Y', each element of S must be positive.

 (LDB,NRHS)
      B       (input/output) DOUBLE PRECISION array, dimension
              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) DOUBLE PRECISION 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) 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 partic-
              ular, if RCOND = 0), the matrix is singular to work-
              ing 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 bounds for each solution
              vector X(j) (the j-th column of the solution matrix
              X).  If XTRUE is the true solution, FERR(j) bounds
              the magnitude of the largest entry in (X(j) - XTRUE)
              divided by the magnitude of the largest entry in
              X(j).  The quality of the error bound depends on the
              quality of the estimate of norm(inv(A)) computed in
              the code; if the estimate of norm(inv(A)) is accu-
              rate, the error bound is guaranteed.

      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 entry of A or B that makes X(j) an
              exact solution).

      WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)

      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: if INFO = i, 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 com-
              pleted, but the matrix is singular to working preci-
              sion, and the solution and error bounds have not
              been computed.