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257 lines
6.8 KiB
257 lines
6.8 KiB
2 years ago
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*> \brief \b SQRT14
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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* Definition:
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* ===========
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*
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* REAL FUNCTION SQRT14( TRANS, M, N, NRHS, A, LDA, X,
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* LDX, WORK, LWORK )
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*
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* .. Scalar Arguments ..
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* CHARACTER TRANS
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* INTEGER LDA, LDX, LWORK, M, N, NRHS
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* ..
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* .. Array Arguments ..
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* REAL A( LDA, * ), WORK( LWORK ), X( LDX, * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> SQRT14 checks whether X is in the row space of A or A'. It does so
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*> by scaling both X and A such that their norms are in the range
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*> [sqrt(eps), 1/sqrt(eps)], then computing a QR factorization of [A,X]
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*> (if TRANS = 'T') or an LQ factorization of [A',X]' (if TRANS = 'N'),
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*> and returning the norm of the trailing triangle, scaled by
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*> MAX(M,N,NRHS)*eps.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] TRANS
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*> \verbatim
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*> TRANS is CHARACTER*1
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*> = 'N': No transpose, check for X in the row space of A
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*> = 'T': Transpose, check for X in the row space of A'.
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*> \endverbatim
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*>
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*> \param[in] M
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*> \verbatim
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*> M is INTEGER
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*> The number of rows of the matrix A.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The number of columns of the matrix A.
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*> \endverbatim
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*>
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*> \param[in] NRHS
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*> \verbatim
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*> NRHS is INTEGER
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*> The number of right hand sides, i.e., the number of columns
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*> of X.
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*> \endverbatim
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*>
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*> \param[in] A
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*> \verbatim
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*> A is REAL array, dimension (LDA,N)
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*> The M-by-N matrix A.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the array A.
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*> \endverbatim
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*>
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*> \param[in] X
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*> \verbatim
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*> X is REAL array, dimension (LDX,NRHS)
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*> If TRANS = 'N', the N-by-NRHS matrix X.
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*> IF TRANS = 'T', the M-by-NRHS matrix X.
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*> \endverbatim
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*>
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*> \param[in] LDX
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*> \verbatim
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*> LDX is INTEGER
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*> The leading dimension of the array X.
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is REAL array dimension (LWORK)
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*> \endverbatim
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> length of workspace array required
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*> If TRANS = 'N', LWORK >= (M+NRHS)*(N+2);
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*> if TRANS = 'T', LWORK >= (N+NRHS)*(M+2).
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \ingroup single_lin
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*
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* =====================================================================
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REAL FUNCTION SQRT14( TRANS, M, N, NRHS, A, LDA, X,
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$ LDX, WORK, LWORK )
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*
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* -- LAPACK test routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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CHARACTER TRANS
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INTEGER LDA, LDX, LWORK, M, N, NRHS
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* ..
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* .. Array Arguments ..
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REAL A( LDA, * ), WORK( LWORK ), X( LDX, * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ZERO, ONE
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PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
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* ..
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* .. Local Scalars ..
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LOGICAL TPSD
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INTEGER I, INFO, J, LDWORK
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REAL ANRM, ERR, XNRM
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* ..
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* .. Local Arrays ..
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REAL RWORK( 1 )
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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REAL SLAMCH, SLANGE
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EXTERNAL LSAME, SLAMCH, SLANGE
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* ..
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* .. External Subroutines ..
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EXTERNAL SGELQ2, SGEQR2, SLACPY, SLASCL, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX, MIN, REAL
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* ..
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* .. Executable Statements ..
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*
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SQRT14 = ZERO
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IF( LSAME( TRANS, 'N' ) ) THEN
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LDWORK = M + NRHS
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TPSD = .FALSE.
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IF( LWORK.LT.( M+NRHS )*( N+2 ) ) THEN
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CALL XERBLA( 'SQRT14', 10 )
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RETURN
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ELSE IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
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RETURN
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END IF
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ELSE IF( LSAME( TRANS, 'T' ) ) THEN
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LDWORK = M
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TPSD = .TRUE.
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IF( LWORK.LT.( N+NRHS )*( M+2 ) ) THEN
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CALL XERBLA( 'SQRT14', 10 )
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RETURN
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ELSE IF( M.LE.0 .OR. NRHS.LE.0 ) THEN
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RETURN
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END IF
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ELSE
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CALL XERBLA( 'SQRT14', 1 )
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RETURN
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END IF
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*
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* Copy and scale A
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*
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CALL SLACPY( 'All', M, N, A, LDA, WORK, LDWORK )
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ANRM = SLANGE( 'M', M, N, WORK, LDWORK, RWORK )
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IF( ANRM.NE.ZERO )
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$ CALL SLASCL( 'G', 0, 0, ANRM, ONE, M, N, WORK, LDWORK, INFO )
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*
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* Copy X or X' into the right place and scale it
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*
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IF( TPSD ) THEN
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*
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* Copy X into columns n+1:n+nrhs of work
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*
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CALL SLACPY( 'All', M, NRHS, X, LDX, WORK( N*LDWORK+1 ),
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$ LDWORK )
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XNRM = SLANGE( 'M', M, NRHS, WORK( N*LDWORK+1 ), LDWORK,
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$ RWORK )
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IF( XNRM.NE.ZERO )
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$ CALL SLASCL( 'G', 0, 0, XNRM, ONE, M, NRHS,
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$ WORK( N*LDWORK+1 ), LDWORK, INFO )
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*
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* Compute QR factorization of X
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*
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CALL SGEQR2( M, N+NRHS, WORK, LDWORK,
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$ WORK( LDWORK*( N+NRHS )+1 ),
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$ WORK( LDWORK*( N+NRHS )+MIN( M, N+NRHS )+1 ),
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$ INFO )
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*
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* Compute largest entry in upper triangle of
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* work(n+1:m,n+1:n+nrhs)
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*
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ERR = ZERO
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DO 20 J = N + 1, N + NRHS
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DO 10 I = N + 1, MIN( M, J )
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ERR = MAX( ERR, ABS( WORK( I+( J-1 )*M ) ) )
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10 CONTINUE
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20 CONTINUE
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*
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ELSE
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*
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* Copy X' into rows m+1:m+nrhs of work
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*
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DO 40 I = 1, N
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DO 30 J = 1, NRHS
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WORK( M+J+( I-1 )*LDWORK ) = X( I, J )
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30 CONTINUE
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40 CONTINUE
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*
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XNRM = SLANGE( 'M', NRHS, N, WORK( M+1 ), LDWORK, RWORK )
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IF( XNRM.NE.ZERO )
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$ CALL SLASCL( 'G', 0, 0, XNRM, ONE, NRHS, N, WORK( M+1 ),
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$ LDWORK, INFO )
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*
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* Compute LQ factorization of work
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*
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CALL SGELQ2( LDWORK, N, WORK, LDWORK, WORK( LDWORK*N+1 ),
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$ WORK( LDWORK*( N+1 )+1 ), INFO )
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*
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* Compute largest entry in lower triangle in
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* work(m+1:m+nrhs,m+1:n)
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*
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ERR = ZERO
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DO 60 J = M + 1, N
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DO 50 I = J, LDWORK
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ERR = MAX( ERR, ABS( WORK( I+( J-1 )*LDWORK ) ) )
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50 CONTINUE
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60 CONTINUE
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*
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END IF
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*
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SQRT14 = ERR / ( REAL( MAX( M, N, NRHS ) )*SLAMCH( 'Epsilon' ) )
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*
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RETURN
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*
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* End of SQRT14
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*
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END
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