sqrt01.f

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*> \brief \b SQRT01
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*  Definition:
*  ===========
*
*       SUBROUTINE SQRT01( M, N, A, AF, Q, R, LDA, TAU, WORK, LWORK,
*                          RWORK, RESULT )
*
*       .. Scalar Arguments ..
*       INTEGER            LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       REAL               A( LDA, * ), AF( LDA, * ), Q( LDA, * ),
*      $                   R( LDA, * ), RESULT( * ), RWORK( * ), TAU( * ),
*      $                   WORK( LWORK )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SQRT01 tests SGEQRF, which computes the QR factorization of an m-by-n
*> matrix A, and partially tests SORGQR which forms the m-by-m
*> orthogonal matrix Q.
*>
*> SQRT01 compares R with Q'*A, and checks that Q is orthogonal.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is REAL array, dimension (LDA,N)
*>          The m-by-n matrix A.
*> \endverbatim
*>
*> \param[out] AF
*> \verbatim
*>          AF is REAL array, dimension (LDA,N)
*>          Details of the QR factorization of A, as returned by SGEQRF.
*>          See SGEQRF for further details.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is REAL array, dimension (LDA,M)
*>          The m-by-m orthogonal matrix Q.
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*>          R is REAL array, dimension (LDA,max(M,N))
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the arrays A, AF, Q and R.
*>          LDA >= max(M,N).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*>          TAU is REAL array, dimension (min(M,N))
*>          The scalar factors of the elementary reflectors, as returned
*>          by SGEQRF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is REAL array, dimension (M)
*> \endverbatim
*>
*> \param[out] RESULT
*> \verbatim
*>          RESULT is REAL array, dimension (2)
*>          The test ratios:
*>          RESULT(1) = norm( R - Q'*A ) / ( M * norm(A) * EPS )
*>          RESULT(2) = norm( I - Q'*Q ) / ( M * EPS )
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup single_lin
*
*  =====================================================================
      SUBROUTINE sqrt01( M, N, A, AF, Q, R, LDA, TAU, WORK, LWORK,
     $                   RWORK, RESULT )
*
*  -- LAPACK test routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
*     .. Scalar Arguments ..
      INTEGER            LDA, LWORK, M, N
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * ), AF( LDA, * ), Q( LDA, * ),
     $                   r( lda, * ), result( * ), rwork( * ), tau( * ),
     $                   work( lwork )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter( zero = 0.0e+0, one = 1.0e+0 )
      REAL               ROGUE
      parameter( rogue = -1.0e+10 )
*     ..
*     .. Local Scalars ..
      INTEGER            INFO, MINMN
      REAL               ANORM, EPS, RESID
*     ..
*     .. External Functions ..
      REAL               SLAMCH, SLANGE, SLANSY
      EXTERNAL           slamch, slange, slansy
*     ..
*     .. External Subroutines ..
      EXTERNAL           sgemm, sgeqrf, slacpy, slaset, sorgqr, ssyrk
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min, real
*     ..
*     .. Scalars in Common ..
      CHARACTER*32       SRNAMT
*     ..
*     .. Common blocks ..
      COMMON             / srnamc / srnamt
*     ..
*     .. Executable Statements ..
*
      minmn = min( m, n )
      eps = slamch( 'Epsilon' )
*
*     Copy the matrix A to the array AF.
*
      CALL slacpy( 'Full', m, n, a, lda, af, lda )
*
*     Factorize the matrix A in the array AF.
*
      srnamt = 'SGEQRF'
      CALL sgeqrf( m, n, af, lda, tau, work, lwork, info )
*
*     Copy details of Q
*
      CALL slaset( 'Full', m, m, rogue, rogue, q, lda )
      CALL slacpy( 'Lower', m-1, n, af( 2, 1 ), lda, q( 2, 1 ), lda )
*
*     Generate the m-by-m matrix Q
*
      srnamt = 'SORGQR'
      CALL sorgqr( m, m, minmn, q, lda, tau, work, lwork, info )
*
*     Copy R
*
      CALL slaset( 'Full', m, n, zero, zero, r, lda )
      CALL slacpy( 'Upper', m, n, af, lda, r, lda )
*
*     Compute R - Q'*A
*
      CALL sgemm( 'Transpose', 'No transpose', m, n, m, -one, q, lda, a,
     $            lda, one, r, lda )
*
*     Compute norm( R - Q'*A ) / ( M * norm(A) * EPS ) .
*
      anorm = slange( '1', m, n, a, lda, rwork )
      resid = slange( '1', m, n, r, lda, rwork )
      IF( anorm.GT.zero ) THEN
         result( 1 ) = ( ( resid / real( max( 1, m ) ) ) / anorm ) / eps
      ELSE
         result( 1 ) = zero
      END IF
*
*     Compute I - Q'*Q
*
      CALL slaset( 'Full', m, m, zero, one, r, lda )
      CALL ssyrk( 'Upper', 'Transpose', m, m, -one, q, lda, one, r,
     $            lda )
*
*     Compute norm( I - Q'*Q ) / ( M * EPS ) .
*
      resid = slansy( '1', 'Upper', m, r, lda, rwork )
*
      result( 2 ) = ( resid / real( max( 1, m ) ) ) / eps
*
      RETURN
*
*     End of SQRT01
*
      END