*> \brief \b SQRT02
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at 
*            http://www.netlib.org/lapack/explore-html/ 
*
*  Definition:
*  ===========
*
*       SUBROUTINE SQRT02( 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
*>
*> SQRT02 tests SORGQR, which generates an m-by-n matrix Q with
*> orthonornmal columns that is defined as the product of k elementary
*> reflectors.
*>
*> Given the QR factorization of an m-by-n matrix A, SQRT02 generates
*> the orthogonal matrix Q defined by the factorization of the first k
*> columns of A; it compares R(1:n,1:k) with Q(1:m,1:n)'*A(1:m,1:k),
*> and checks that the columns 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.
*>          M >= N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of elementary reflectors whose product defines the
*>          matrix Q. N >= K >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is REAL array, dimension (LDA,N)
*>          The m-by-n matrix A which was factorized by SQRT01.
*> \endverbatim
*>
*> \param[in] 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,N)
*> \endverbatim
*>
*> \param[out] R
*> \verbatim
*>          R is REAL array, dimension (LDA,N)
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the arrays A, AF, Q and R. LDA >= M.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is REAL array, dimension (N)
*>          The scalar factors of the elementary reflectors corresponding
*>          to the QR 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 - 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 November 2011
*
*> \ingroup single_lin
*
*  =====================================================================
      SUBROUTINE SQRT02( M, N, K, A, AF, Q, R, LDA, TAU, WORK, LWORK,
     $                   RWORK, RESULT )
*
*  -- LAPACK test routine (version 3.4.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2011
*
*     .. 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, SORGQR, SSYRK
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, REAL
*     ..
*     .. Scalars in Common ..
      CHARACTER*32       SRNAMT
*     ..
*     .. Common blocks ..
      COMMON             / SRNAMC / SRNAMT
*     ..
*     .. Executable Statements ..
*
      EPS = SLAMCH( 'Epsilon' )
*
*     Copy the first k columns of the factorization to the array Q
*
      CALL SLASET( 'Full', M, N, ROGUE, ROGUE, Q, LDA )
      CALL SLACPY( 'Lower', M-1, K, AF( 2, 1 ), LDA, Q( 2, 1 ), LDA )
*
*     Generate the first n columns of the matrix Q
*
      SRNAMT = 'SORGQR'
      CALL SORGQR( M, N, K, Q, LDA, TAU, WORK, LWORK, INFO )
*
*     Copy R(1:n,1:k)
*
      CALL SLASET( 'Full', N, K, ZERO, ZERO, R, LDA )
      CALL SLACPY( 'Upper', N, K, AF, LDA, R, LDA )
*
*     Compute R(1:n,1:k) - Q(1:m,1:n)' * A(1:m,1:k)
*
      CALL SGEMM( 'Transpose', 'No transpose', N, K, M, -ONE, Q, LDA, A,
     $            LDA, ONE, R, LDA )
*
*     Compute norm( R - Q'*A ) / ( M * norm(A) * EPS ) .
*
      ANORM = SLANGE( '1', M, K, A, LDA, RWORK )
      RESID = SLANGE( '1', N, K, 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', N, N, ZERO, ONE, R, LDA )
      CALL SSYRK( 'Upper', 'Transpose', N, M, -ONE, Q, LDA, ONE, R,
     $            LDA )
*
*     Compute norm( I - Q'*Q ) / ( M * EPS ) .
*
      RESID = SLANSY( '1', 'Upper', N, R, LDA, RWORK )
*
      RESULT( 2 ) = ( RESID / REAL( MAX( 1, M ) ) ) / EPS
*
      RETURN
*
*     End of SQRT02
*
      END
