sqpt01.f

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*> \brief \b SQPT01
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*  Definition:
*  ===========
*
*       REAL             FUNCTION SQPT01( M, N, K, A, AF, LDA, TAU, JPVT,
*                        WORK, LWORK )
*
*       .. Scalar Arguments ..
*       INTEGER            K, LDA, LWORK, M, N
*       ..
*       .. Array Arguments ..
*       INTEGER            JPVT( * )
*       REAL               A( LDA, * ), AF( LDA, * ), TAU( * ),
*      $                   WORK( LWORK )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SQPT01 tests the QR-factorization with pivoting of a matrix A.  The
*> array AF contains the (possibly partial) QR-factorization of A, where
*> the upper triangle of AF(1:k,1:k) is a partial triangular factor,
*> the entries below the diagonal in the first k columns are the
*> Householder vectors, and the rest of AF contains a partially updated
*> matrix.
*>
*> This function returns ||A*P - Q*R|| / ( ||norm(A)||*eps*max(M,N) )
*> where || . || is matrix one norm.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrices A and AF.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrices A and AF.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*>          K is INTEGER
*>          The number of columns of AF that have been reduced
*>          to upper triangular form.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is REAL array, dimension (LDA, N)
*>          The original matrix A.
*> \endverbatim
*>
*> \param[in] AF
*> \verbatim
*>          AF is REAL array, dimension (LDA,N)
*>          The (possibly partial) output of SGEQPF.  The upper triangle
*>          of AF(1:k,1:k) is a partial triangular factor, the entries
*>          below the diagonal in the first k columns are the Householder
*>          vectors, and the rest of AF contains a partially updated
*>          matrix.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the arrays A and AF.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*>          TAU is REAL array, dimension (K)
*>          Details of the Householder transformations as returned by
*>          SGEQPF.
*> \endverbatim
*>
*> \param[in] JPVT
*> \verbatim
*>          JPVT is INTEGER array, dimension (N)
*>          Pivot information as returned by SGEQPF.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The length of the array WORK.  LWORK >= M*N+N.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup single_lin
*
*  =====================================================================
      REAL             function sqpt01( m, n, k, a, af, lda, tau, jpvt,
     $                 work, lwork )
*
*  -- 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 ..
      INTEGER            k, lda, lwork, m, n
*     ..
*     .. Array Arguments ..
      INTEGER            jpvt( * )
      REAL               a( lda, * ), af( lda, * ), tau( * ),
     $                   work( lwork )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               zero, one
      parameter( zero = 0.0e0, one = 1.0e0 )
*     ..
*     .. Local Scalars ..
      INTEGER            i, info, j
      REAL               norma
*     ..
*     .. Local Arrays ..
      REAL               rwork( 1 )
*     ..
*     .. External Functions ..
      REAL               slamch, slange
      EXTERNAL           slamch, slange
*     ..
*     .. External Subroutines ..
      EXTERNAL           saxpy, scopy, sormqr, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min, real
*     ..
*     .. Executable Statements ..
*
      sqpt01 = zero
*
*     Test if there is enough workspace
*
      IF( lwork.LT.m*n+n ) THEN
         CALL xerbla( 'SQPT01', 10 )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( m.LE.0 .OR. n.LE.0 )
     $   RETURN
*
      norma = slange( 'One-norm', m, n, a, lda, rwork )
*
      DO j = 1, k
         DO i = 1, min( j, m )
            work( ( j-1 )*m+i ) = af( i, j )
         END DO
         DO i = j + 1, m
            work( ( j-1 )*m+i ) = zero
         END DO
      END DO
      DO j = k + 1, n
         CALL scopy( m, af( 1, j ), 1, work( ( j-1 )*m+1 ), 1 )
      END DO
*
      CALL sormqr( 'Left', 'No transpose', m, n, k, af, lda, tau, work,
     $             m, work( m*n+1 ), lwork-m*n, info )
*
      DO j = 1, n
*
*        Compare i-th column of QR and jpvt(i)-th column of A
*
         CALL saxpy( m, -one, a( 1, jpvt( j ) ), 1, work( ( j-1 )*m+1 ),
     $               1 )
      END DO
*
      sqpt01 = slange( 'One-norm', m, n, work, m, rwork ) /
     $         ( real( max( m, n ) )*slamch( 'Epsilon' ) )
      IF( norma.NE.zero )
     $   sqpt01 = sqpt01 / norma
*
      RETURN
*
*     End of SQPT01
*
      END