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*> \brief \b SQRT12
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* REAL FUNCTION SQRT12( M, N, A, LDA, S, WORK, LWORK )
*
* .. Scalar Arguments ..
* INTEGER LDA, LWORK, M, N
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), S( * ), WORK( LWORK )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SQRT12 computes the singular values `svlues' of the upper trapezoid
*> of A(1:M,1:N) and returns the ratio
*>
*> || s - svlues||/(||svlues||*eps*max(M,N))
*> \endverbatim
*
* Arguments:
* ==========
*
*> \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] A
*> \verbatim
*> A is REAL array, dimension (LDA,N)
*> The M-by-N matrix A. Only the upper trapezoid is referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is REAL array, dimension (min(M,N))
*> The singular values of the matrix A.
*> \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 >= max(M*N + 4*min(M,N) +
*> max(M,N), M*N+2*MIN( M, N )+4*N).
*> \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 SQRT12( M, N, A, LDA, S, 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 ..
INTEGER LDA, LWORK, M, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), S( * ), WORK( LWORK )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
INTEGER I, INFO, ISCL, J, MN
REAL ANRM, BIGNUM, NRMSVL, SMLNUM
* ..
* .. External Functions ..
REAL SASUM, SLAMCH, SLANGE, SNRM2
EXTERNAL SASUM, SLAMCH, SLANGE, SNRM2
* ..
* .. External Subroutines ..
EXTERNAL SAXPY, SBDSQR, SGEBD2, SLABAD, SLASCL, SLASET,
$ XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, REAL
* ..
* .. Local Arrays ..
REAL DUMMY( 1 )
* ..
* .. Executable Statements ..
*
SQRT12 = ZERO
*
* Test that enough workspace is supplied
*
IF( LWORK.LT.MAX( M*N+4*MIN( M, N )+MAX( M, N ),
$ M*N+2*MIN( M, N )+4*N) ) THEN
CALL XERBLA( 'SQRT12', 7 )
RETURN
END IF
*
* Quick return if possible
*
MN = MIN( M, N )
IF( MN.LE.ZERO )
$ RETURN
*
NRMSVL = SNRM2( MN, S, 1 )
*
* Copy upper triangle of A into work
*
CALL SLASET( 'Full', M, N, ZERO, ZERO, WORK, M )
DO 20 J = 1, N
DO 10 I = 1, MIN( J, M )
WORK( ( J-1 )*M+I ) = A( I, J )
10 CONTINUE
20 CONTINUE
*
* Get machine parameters
*
SMLNUM = SLAMCH( 'S' ) / SLAMCH( 'P' )
BIGNUM = ONE / SMLNUM
CALL SLABAD( SMLNUM, BIGNUM )
*
* Scale work if max entry outside range [SMLNUM,BIGNUM]
*
ANRM = SLANGE( 'M', M, N, WORK, M, DUMMY )
ISCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM
*
CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, WORK, M, INFO )
ISCL = 1
ELSE IF( ANRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM
*
CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, WORK, M, INFO )
ISCL = 1
END IF
*
IF( ANRM.NE.ZERO ) THEN
*
* Compute SVD of work
*
CALL SGEBD2( M, N, WORK, M, WORK( M*N+1 ), WORK( M*N+MN+1 ),
$ WORK( M*N+2*MN+1 ), WORK( M*N+3*MN+1 ),
$ WORK( M*N+4*MN+1 ), INFO )
CALL SBDSQR( 'Upper', MN, 0, 0, 0, WORK( M*N+1 ),
$ WORK( M*N+MN+1 ), DUMMY, MN, DUMMY, 1, DUMMY, MN,
$ WORK( M*N+2*MN+1 ), INFO )
*
IF( ISCL.EQ.1 ) THEN
IF( ANRM.GT.BIGNUM ) THEN
CALL SLASCL( 'G', 0, 0, BIGNUM, ANRM, MN, 1,
$ WORK( M*N+1 ), MN, INFO )
END IF
IF( ANRM.LT.SMLNUM ) THEN
CALL SLASCL( 'G', 0, 0, SMLNUM, ANRM, MN, 1,
$ WORK( M*N+1 ), MN, INFO )
END IF
END IF
*
ELSE
*
DO 30 I = 1, MN
WORK( M*N+I ) = ZERO
30 CONTINUE
END IF
*
* Compare s and singular values of work
*
CALL SAXPY( MN, -ONE, S, 1, WORK( M*N+1 ), 1 )
SQRT12 = SASUM( MN, WORK( M*N+1 ), 1 ) /
$ ( SLAMCH( 'Epsilon' )*REAL( MAX( M, N ) ) )
IF( NRMSVL.NE.ZERO )
$ SQRT12 = SQRT12 / NRMSVL
*
RETURN
*
* End of SQRT12
*
END
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