spst01.f

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*> \brief \b SPST01
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*  Definition:
*  ===========
*
*       SUBROUTINE SPST01( UPLO, N, A, LDA, AFAC, LDAFAC, PERM, LDPERM,
*                          PIV, RWORK, RESID, RANK )
*
*       .. Scalar Arguments ..
*       REAL               RESID
*       INTEGER            LDA, LDAFAC, LDPERM, N, RANK
*       CHARACTER          UPLO
*       ..
*       .. Array Arguments ..
*       REAL               A( LDA, * ), AFAC( LDAFAC, * ),
*      $                   PERM( LDPERM, * ), RWORK( * )
*       INTEGER            PIV( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SPST01 reconstructs a symmetric positive semidefinite matrix A
*> from its L or U factors and the permutation matrix P and computes
*> the residual
*>    norm( P*L*L'*P' - A ) / ( N * norm(A) * EPS ) or
*>    norm( P*U'*U*P' - A ) / ( N * norm(A) * EPS ),
*> where EPS is the machine epsilon.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is stored:
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of rows and columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is REAL array, dimension (LDA,N)
*>          The original symmetric matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N)
*> \endverbatim
*>
*> \param[in] AFAC
*> \verbatim
*>          AFAC is REAL array, dimension (LDAFAC,N)
*>          The factor L or U from the L*L' or U'*U
*>          factorization of A.
*> \endverbatim
*>
*> \param[in] LDAFAC
*> \verbatim
*>          LDAFAC is INTEGER
*>          The leading dimension of the array AFAC.  LDAFAC >= max(1,N).
*> \endverbatim
*>
*> \param[out] PERM
*> \verbatim
*>          PERM is REAL array, dimension (LDPERM,N)
*>          Overwritten with the reconstructed matrix, and then with the
*>          difference P*L*L'*P' - A (or P*U'*U*P' - A)
*> \endverbatim
*>
*> \param[in] LDPERM
*> \verbatim
*>          LDPERM is INTEGER
*>          The leading dimension of the array PERM.
*>          LDAPERM >= max(1,N).
*> \endverbatim
*>
*> \param[in] PIV
*> \verbatim
*>          PIV is INTEGER array, dimension (N)
*>          PIV is such that the nonzero entries are
*>          P( PIV( K ), K ) = 1.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] RESID
*> \verbatim
*>          RESID is REAL
*>          If UPLO = 'L', norm(L*L' - A) / ( N * norm(A) * EPS )
*>          If UPLO = 'U', norm(U'*U - A) / ( N * norm(A) * EPS )
*> \endverbatim
*>
*> \param[in] RANK
*> \verbatim
*>          RANK is INTEGER
*>          number of nonzero singular values of A.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup single_lin
*
*  =====================================================================
      SUBROUTINE spst01( UPLO, N, A, LDA, AFAC, LDAFAC, PERM, LDPERM,
     $                   PIV, RWORK, RESID, RANK )
*
*  -- LAPACK test routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      REAL               RESID
      INTEGER            LDA, LDAFAC, LDPERM, N, RANK
      CHARACTER          UPLO
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * ), AFAC( LDAFAC, * ),
     $                   perm( ldperm, * ), rwork( * )
      INTEGER            PIV( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter( zero = 0.0e+0, one = 1.0e+0 )
*     ..
*     .. Local Scalars ..
      REAL               ANORM, EPS, T
      INTEGER            I, J, K
*     ..
*     .. External Functions ..
      REAL               SDOT, SLAMCH, SLANSY
      LOGICAL            LSAME
      EXTERNAL           sdot, slamch, slansy, lsame
*     ..
*     .. External Subroutines ..
      EXTERNAL           sscal, ssyr, strmv
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          real
*     ..
*     .. Executable Statements ..
*
*     Quick exit if N = 0.
*
      IF( n.LE.0 ) THEN
         resid = zero
         RETURN
      END IF
*
*     Exit with RESID = 1/EPS if ANORM = 0.
*
      eps = slamch( 'Epsilon' )
      anorm = slansy( '1', uplo, n, a, lda, rwork )
      IF( anorm.LE.zero ) THEN
         resid = one / eps
         RETURN
      END IF
*
*     Compute the product U'*U, overwriting U.
*
      IF( lsame( uplo, 'U' ) ) THEN
*
         IF( rank.LT.n ) THEN
            DO 110 j = rank + 1, n
               DO 100 i = rank + 1, j
                  afac( i, j ) = zero
  100          CONTINUE
  110       CONTINUE
         END IF
*
         DO 120 k = n, 1, -1
*
*           Compute the (K,K) element of the result.
*
            t = sdot( k, afac( 1, k ), 1, afac( 1, k ), 1 )
            afac( k, k ) = t
*
*           Compute the rest of column K.
*
            CALL strmv( 'Upper', 'Transpose', 'Non-unit', k-1, afac,
     $                  ldafac, afac( 1, k ), 1 )
*
  120    CONTINUE
*
*     Compute the product L*L', overwriting L.
*
      ELSE
*
         IF( rank.LT.n ) THEN
            DO 140 j = rank + 1, n
               DO 130 i = j, n
                  afac( i, j ) = zero
  130          CONTINUE
  140       CONTINUE
         END IF
*
         DO 150 k = n, 1, -1
*           Add a multiple of column K of the factor L to each of
*           columns K+1 through N.
*
            IF( k+1.LE.n )
     $         CALL ssyr( 'Lower', n-k, one, afac( k+1, k ), 1,
     $                    afac( k+1, k+1 ), ldafac )
*
*           Scale column K by the diagonal element.
*
            t = afac( k, k )
            CALL sscal( n-k+1, t, afac( k, k ), 1 )
  150    CONTINUE
*
      END IF
*
*        Form P*L*L'*P' or P*U'*U*P'
*
      IF( lsame( uplo, 'U' ) ) THEN
*
         DO 170 j = 1, n
            DO 160 i = 1, n
               IF( piv( i ).LE.piv( j ) ) THEN
                  IF( i.LE.j ) THEN
                     perm( piv( i ), piv( j ) ) = afac( i, j )
                  ELSE
                     perm( piv( i ), piv( j ) ) = afac( j, i )
                  END IF
               END IF
  160       CONTINUE
  170    CONTINUE
*
*
      ELSE
*
         DO 190 j = 1, n
            DO 180 i = 1, n
               IF( piv( i ).GE.piv( j ) ) THEN
                  IF( i.GE.j ) THEN
                     perm( piv( i ), piv( j ) ) = afac( i, j )
                  ELSE
                     perm( piv( i ), piv( j ) ) = afac( j, i )
                  END IF
               END IF
  180       CONTINUE
  190    CONTINUE
*
      END IF
*
*     Compute the difference  P*L*L'*P' - A (or P*U'*U*P' - A).
*
      IF( lsame( uplo, 'U' ) ) THEN
         DO 210 j = 1, n
            DO 200 i = 1, j
               perm( i, j ) = perm( i, j ) - a( i, j )
  200       CONTINUE
  210    CONTINUE
      ELSE
         DO 230 j = 1, n
            DO 220 i = j, n
               perm( i, j ) = perm( i, j ) - a( i, j )
  220       CONTINUE
  230    CONTINUE
      END IF
*
*     Compute norm( P*L*L'P - A ) / ( N * norm(A) * EPS ), or
*     ( P*U'*U*P' - A )/ ( N * norm(A) * EPS ).
*
      resid = slansy( '1', uplo, n, perm, ldafac, rwork )
*
      resid = ( ( resid / real( n ) ) / anorm ) / eps
*
      RETURN
*
*     End of SPST01
*
      END