SPPSVX (3lapack)

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 SPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB, X, LDX,
                     RCOND, FERR, BERR, WORK, IWORK, INFO )

      CHARACTER      EQUED, FACT, UPLO

      INTEGER        INFO, LDB, LDX, N, NRHS

      REAL           RCOND

      INTEGER        IWORK( * )

      REAL           AFP( * ), AP( * ), B( LDB, * ), BERR( * ), FERR( * ), S(
                     * ), WORK( * ), X( LDX, * )

PURPOSE

  SPPSVX 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 definite matrix
  stored in packed format 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 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, AFP contains
          the factored form of A.  If EQUED = 'Y', the matrix A has been
          equilibrated with scaling factors given by S.  AP and AFP will not
          be modified.  = 'N':  The matrix A will be copied to AFP and
          factored.
          = 'E':  The matrix A will be equilibrated if necessary, then copied
          to AFP 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.

  AP      (input/output) REAL array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix A,
          packed columnwise in a linear array, except if FACT = 'F' and EQUED
          = 'Y', then A must contain the equilibrated matrix
          diag(S)*A*diag(S).  The j-th column of A is stored in the array AP
          as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.  See
          below for further details.  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).

  AFP     (input or output) REAL array, dimension
          (N*(N+1)/2) If FACT = 'F', then AFP is an input argument and on
          entry contains the triangular factor U or L from the Cholesky
          factorization A = U'*U or A = L*L', in the same storage format as
          A.  If EQUED .ne. 'N', then AFP is the factored form of the
          equilibrated matrix A.

          If FACT = 'N', then AFP is an output argument and on exit returns
          the triangular factor U or L from the Cholesky factorization A =
          U'*U or A = L*L' of the original matrix A.

          If FACT = 'E', then AFP is an output argument and on exit returns
          the triangular factor U or L from the Cholesky factorization A =
          U'*U or A = L*L' of the equilibrated matrix A (see the description
          of AP for the form of the equilibrated matrix).

  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) REAL 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) REAL array, dimension (LDB,NRHS)
          On entry, the N-by-NRHS right hand 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) REAL 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) REAL
          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, 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) REAL 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: 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 completed, but the
          matrix is singular to working precision, and the solution and error
          bounds have not been computed.

FURTHER DETAILS

  The packed storage scheme is illustrated by the following example when N =
  4, UPLO = 'U':

  Two-dimensional storage of the symmetric matrix A:

     a11 a12 a13 a14
         a22 a23 a24
             a33 a34     (aij = conjg(aji))
                 a44

  Packed storage of the upper triangle of A:

  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]