srqt02.f

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*> \brief \b SRQT02
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*  Definition:
*  ===========
*
*       SUBROUTINE SRQT02( M, N, K, A, AF, Q, R, LDA, TAU, WORK, LWORK,
*                          RWORK, RESULT )
*
*       .. Scalar Arguments ..
*       INTEGER            K, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       REAL               A( LDA, * ), AF( LDA, * ), Q( LDA, * ),
*      $                   R( LDA, * ), RESULT( * ), RWORK( * ), TAU( * ),
*      $                   WORK( LWORK )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SRQT02 tests SORGRQ, which generates an m-by-n matrix Q with
*> orthonornmal rows that is defined as the product of k elementary
*> reflectors.
*>
*> Given the RQ factorization of an m-by-n matrix A, SRQT02 generates
*> the orthogonal matrix Q defined by the factorization of the last k
*> rows of A; it compares R(m-k+1:m,n-m+1:n) with
*> A(m-k+1:m,1:n)*Q(n-m+1:n,1:n)', and checks that the rows of Q are
*> orthonormal.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix Q to be generated.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix Q to be generated.
*>          N >= M >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines the
*>          matrix Q. M >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is REAL array, dimension (LDA,N)
*>          The m-by-n matrix A which was factorized by SRQT01.
*> \endverbatim
*>
*> \param[in] AF
*> \verbatim
*>          AF is REAL array, dimension (LDA,N)
*>          Details of the RQ factorization of A, as returned by SGERQF.
*>          See SGERQF for further details.
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is REAL array, dimension (LDA,N)
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*>          R is REAL array, dimension (LDA,M)
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the arrays A, AF, Q and L. LDA >= N.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is REAL array, dimension (M)
*>          The scalar factors of the elementary reflectors corresponding
*>          to the RQ factorization in AF.
*> \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 - A*Q' ) / ( N * norm(A) * EPS )
*>          RESULT(2) = norm( I - Q*Q' ) / ( N * 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 srqt02( M, N, K, 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            K, 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
      REAL               ANORM, EPS, RESID
*     ..
*     .. External Functions ..
      REAL               SLAMCH, SLANGE, SLANSY
      EXTERNAL           slamch, slange, slansy
*     ..
*     .. External Subroutines ..
      EXTERNAL           sgemm, slacpy, slaset, sorgrq, ssyrk
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, real
*     ..
*     .. Scalars in Common ..
      CHARACTER*32       SRNAMT
*     ..
*     .. Common blocks ..
      COMMON             / srnamc / srnamt
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( m.EQ.0 .OR. n.EQ.0 .OR. k.EQ.0 ) THEN
         result( 1 ) = zero
         result( 2 ) = zero
         RETURN
      END IF
*
      eps = slamch( 'Epsilon' )
*
*     Copy the last k rows of the factorization to the array Q
*
      CALL slaset( 'Full', m, n, rogue, rogue, q, lda )
      IF( k.LT.n )
     $   CALL slacpy( 'Full', k, n-k, af( m-k+1, 1 ), lda,
     $                q( m-k+1, 1 ), lda )
      IF( k.GT.1 )
     $   CALL slacpy( 'Lower', k-1, k-1, af( m-k+2, n-k+1 ), lda,
     $                q( m-k+2, n-k+1 ), lda )
*
*     Generate the last n rows of the matrix Q
*
      srnamt = 'SORGRQ'
      CALL sorgrq( m, n, k, q, lda, tau( m-k+1 ), work, lwork, info )
*
*     Copy R(m-k+1:m,n-m+1:n)
*
      CALL slaset( 'Full', k, m, zero, zero, r( m-k+1, n-m+1 ), lda )
      CALL slacpy( 'Upper', k, k, af( m-k+1, n-k+1 ), lda,
     $             r( m-k+1, n-k+1 ), lda )
*
*     Compute R(m-k+1:m,n-m+1:n) - A(m-k+1:m,1:n) * Q(n-m+1:n,1:n)'
*
      CALL sgemm( 'No transpose', 'Transpose', k, m, n, -one,
     $            a( m-k+1, 1 ), lda, q, lda, one, r( m-k+1, n-m+1 ),
     $            lda )
*
*     Compute norm( R - A*Q' ) / ( N * norm(A) * EPS ) .
*
      anorm = slange( '1', k, n, a( m-k+1, 1 ), lda, rwork )
      resid = slange( '1', k, m, r( m-k+1, n-m+1 ), lda, rwork )
      IF( anorm.GT.zero ) THEN
         result( 1 ) = ( ( resid / real( max( 1, n ) ) ) / anorm ) / eps
      ELSE
         result( 1 ) = zero
      END IF
*
*     Compute I - Q*Q'
*
      CALL slaset( 'Full', m, m, zero, one, r, lda )
      CALL ssyrk( 'Upper', 'No transpose', m, n, -one, q, lda, one, r,
     $            lda )
*
*     Compute norm( I - Q*Q' ) / ( N * EPS ) .
*
      resid = slansy( '1', 'Upper', m, r, lda, rwork )
*
      result( 2 ) = ( resid / real( max( 1, n ) ) ) / eps
*
      RETURN
*
*     End of SRQT02
*
      END