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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
|
