sqrt14.f

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*> \brief \b SQRT14
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at 
*            http://www.netlib.org/lapack/explore-html/ 
*
*  Definition:
*  ===========
*
*       REAL             FUNCTION SQRT14( TRANS, M, N, NRHS, A, LDA, X,
*                        LDX, WORK, LWORK )
* 
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       INTEGER            LDA, LDX, LWORK, M, N, NRHS
*       ..
*       .. Array Arguments ..
*       REAL               A( LDA, * ), WORK( LWORK ), X( LDX, * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SQRT14 checks whether X is in the row space of A or A'.  It does so
*> by scaling both X and A such that their norms are in the range
*> [sqrt(eps), 1/sqrt(eps)], then computing a QR factorization of [A,X]
*> (if TRANS = 'T') or an LQ factorization of [A',X]' (if TRANS = 'N'),
*> and returning the norm of the trailing triangle, scaled by
*> MAX(M,N,NRHS)*eps.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          = 'N':  No transpose, check for X in the row space of A
*>          = 'T':  Transpose, check for X in the row space of A'.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of columns
*>          of X.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is REAL array, dimension (LDA,N)
*>          The M-by-N matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*>          X is REAL array, dimension (LDX,NRHS)
*>          If TRANS = 'N', the N-by-NRHS matrix X.
*>          IF TRANS = 'T', the M-by-NRHS matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          length of workspace array required
*>          If TRANS = 'N', LWORK >= (M+NRHS)*(N+2);
*>          if TRANS = 'T', LWORK >= (N+NRHS)*(M+2).
*> \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
*
*  =====================================================================
      REAL             FUNCTION sqrt14( TRANS, M, N, NRHS, A, LDA, X,
     $                 ldx, work, lwork )
*
*  -- 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 ..
      CHARACTER          TRANS
      INTEGER            LDA, LDX, LWORK, M, N, NRHS
*     ..
*     .. Array Arguments ..
      REAL               A( lda, * ), WORK( lwork ), X( ldx, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter                ( zero = 0.0e0, one = 1.0e0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            TPSD
      INTEGER            I, INFO, J, LDWORK
      REAL               ANRM, ERR, XNRM
*     ..
*     .. Local Arrays ..
      REAL               RWORK( 1 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      REAL               SLAMCH, SLANGE
      EXTERNAL           lsame, slamch, slange
*     ..
*     .. External Subroutines ..
      EXTERNAL           sgelq2, sgeqr2, slacpy, slascl, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min, real
*     ..
*     .. Executable Statements ..
*
      sqrt14 = zero
      IF( lsame( trans, 'N' ) ) THEN
         ldwork = m + nrhs
         tpsd = .false.
         IF( lwork.LT.( m+nrhs )*( n+2 ) ) THEN
            CALL xerbla( 'SQRT14', 10 )
            RETURN
         ELSE IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
            RETURN
         END IF
      ELSE IF( lsame( trans, 'T' ) ) THEN
         ldwork = m
         tpsd = .true.
         IF( lwork.LT.( n+nrhs )*( m+2 ) ) THEN
            CALL xerbla( 'SQRT14', 10 )
            RETURN
         ELSE IF( m.LE.0 .OR. nrhs.LE.0 ) THEN
            RETURN
         END IF
      ELSE
         CALL xerbla( 'SQRT14', 1 )
         RETURN
      END IF
*
*     Copy and scale A
*
      CALL slacpy( 'All', m, n, a, lda, work, ldwork )
      anrm = slange( 'M', m, n, work, ldwork, rwork )
      IF( anrm.NE.zero )
     $   CALL slascl( 'G', 0, 0, anrm, one, m, n, work, ldwork, info )
*
*     Copy X or X' into the right place and scale it
*
      IF( tpsd ) THEN
*
*        Copy X into columns n+1:n+nrhs of work
*
         CALL slacpy( 'All', m, nrhs, x, ldx, work( n*ldwork+1 ),
     $                ldwork )
         xnrm = slange( 'M', m, nrhs, work( n*ldwork+1 ), ldwork,
     $          rwork )
         IF( xnrm.NE.zero )
     $      CALL slascl( 'G', 0, 0, xnrm, one, m, nrhs,
     $                   work( n*ldwork+1 ), ldwork, info )
         anrm = slange( 'One-norm', m, n+nrhs, work, ldwork, rwork )
*
*        Compute QR factorization of X
*
         CALL sgeqr2( m, n+nrhs, work, ldwork,
     $                work( ldwork*( n+nrhs )+1 ),
     $                work( ldwork*( n+nrhs )+min( m, n+nrhs )+1 ),
     $                info )
*
*        Compute largest entry in upper triangle of
*        work(n+1:m,n+1:n+nrhs)
*
         err = zero
         DO 20 j = n + 1, n + nrhs
            DO 10 i = n + 1, min( m, j )
               err = max( err, abs( work( i+( j-1 )*m ) ) )
   10       CONTINUE
   20    CONTINUE
*
      ELSE
*
*        Copy X' into rows m+1:m+nrhs of work
*
         DO 40 i = 1, n
            DO 30 j = 1, nrhs
               work( m+j+( i-1 )*ldwork ) = x( i, j )
   30       CONTINUE
   40    CONTINUE
*
         xnrm = slange( 'M', nrhs, n, work( m+1 ), ldwork, rwork )
         IF( xnrm.NE.zero )
     $      CALL slascl( 'G', 0, 0, xnrm, one, nrhs, n, work( m+1 ),
     $                   ldwork, info )
*
*        Compute LQ factorization of work
*
         CALL sgelq2( ldwork, n, work, ldwork, work( ldwork*n+1 ),
     $                work( ldwork*( n+1 )+1 ), info )
*
*        Compute largest entry in lower triangle in
*        work(m+1:m+nrhs,m+1:n)
*
         err = zero
         DO 60 j = m + 1, n
            DO 50 i = j, ldwork
               err = max( err, abs( work( i+( j-1 )*ldwork ) ) )
   50       CONTINUE
   60    CONTINUE
*
      END IF
*
      sqrt14 = err / ( REAL( MAX( M, N, NRHS ) )*slamch( 'Epsilon' ) )
*
      RETURN
*
*     End of SQRT14
*
      END