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746 lines
23 KiB
746 lines
23 KiB
2 years ago
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*> \brief \b SLAQTR solves a real quasi-triangular system of equations, or a complex quasi-triangular system of special form, in real arithmetic.
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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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*> \htmlonly
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*> Download SLAQTR + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slaqtr.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slaqtr.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaqtr.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE SLAQTR( LTRAN, LREAL, N, T, LDT, B, W, SCALE, X, WORK,
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* INFO )
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*
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* .. Scalar Arguments ..
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* LOGICAL LREAL, LTRAN
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* INTEGER INFO, LDT, N
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* REAL SCALE, W
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* ..
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* .. Array Arguments ..
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* REAL B( * ), T( LDT, * ), WORK( * ), X( * )
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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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*> SLAQTR solves the real quasi-triangular system
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*>
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*> op(T)*p = scale*c, if LREAL = .TRUE.
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*>
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*> or the complex quasi-triangular systems
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*>
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*> op(T + iB)*(p+iq) = scale*(c+id), if LREAL = .FALSE.
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*>
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*> in real arithmetic, where T is upper quasi-triangular.
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*> If LREAL = .FALSE., then the first diagonal block of T must be
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*> 1 by 1, B is the specially structured matrix
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*>
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*> B = [ b(1) b(2) ... b(n) ]
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*> [ w ]
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*> [ w ]
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*> [ . ]
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*> [ w ]
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*>
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*> op(A) = A or A**T, A**T denotes the transpose of
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*> matrix A.
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*>
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*> On input, X = [ c ]. On output, X = [ p ].
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*> [ d ] [ q ]
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*>
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*> This subroutine is designed for the condition number estimation
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*> in routine STRSNA.
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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] LTRAN
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*> \verbatim
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*> LTRAN is LOGICAL
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*> On entry, LTRAN specifies the option of conjugate transpose:
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*> = .FALSE., op(T+i*B) = T+i*B,
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*> = .TRUE., op(T+i*B) = (T+i*B)**T.
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*> \endverbatim
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*>
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*> \param[in] LREAL
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*> \verbatim
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*> LREAL is LOGICAL
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*> On entry, LREAL specifies the input matrix structure:
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*> = .FALSE., the input is complex
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*> = .TRUE., the input is real
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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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*> On entry, N specifies the order of T+i*B. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] T
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*> \verbatim
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*> T is REAL array, dimension (LDT,N)
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*> On entry, T contains a matrix in Schur canonical form.
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*> If LREAL = .FALSE., then the first diagonal block of T must
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*> be 1 by 1.
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*> \endverbatim
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*>
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*> \param[in] LDT
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*> \verbatim
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*> LDT is INTEGER
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*> The leading dimension of the matrix T. LDT >= max(1,N).
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*> \endverbatim
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*>
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*> \param[in] B
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*> \verbatim
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*> B is REAL array, dimension (N)
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*> On entry, B contains the elements to form the matrix
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*> B as described above.
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*> If LREAL = .TRUE., B is not referenced.
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*> \endverbatim
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*>
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*> \param[in] W
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*> \verbatim
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*> W is REAL
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*> On entry, W is the diagonal element of the matrix B.
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*> If LREAL = .TRUE., W is not referenced.
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*> \endverbatim
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*>
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*> \param[out] SCALE
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*> \verbatim
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*> SCALE is REAL
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*> On exit, SCALE is the scale factor.
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*> \endverbatim
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*>
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*> \param[in,out] X
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*> \verbatim
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*> X is REAL array, dimension (2*N)
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*> On entry, X contains the right hand side of the system.
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*> On exit, X is overwritten by the solution.
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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 (N)
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> On exit, INFO is set to
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*> 0: successful exit.
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*> 1: the some diagonal 1 by 1 block has been perturbed by
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*> a small number SMIN to keep nonsingularity.
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*> 2: the some diagonal 2 by 2 block has been perturbed by
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*> a small number in SLALN2 to keep nonsingularity.
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*> NOTE: In the interests of speed, this routine does not
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*> check the inputs for errors.
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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 realOTHERauxiliary
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*
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* =====================================================================
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SUBROUTINE SLAQTR( LTRAN, LREAL, N, T, LDT, B, W, SCALE, X, WORK,
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$ INFO )
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*
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* -- LAPACK auxiliary 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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LOGICAL LREAL, LTRAN
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INTEGER INFO, LDT, N
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REAL SCALE, W
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* ..
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* .. Array Arguments ..
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REAL B( * ), T( LDT, * ), WORK( * ), X( * )
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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 = 0.0E+0, ONE = 1.0E+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL NOTRAN
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INTEGER I, IERR, J, J1, J2, JNEXT, K, N1, N2
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REAL BIGNUM, EPS, REC, SCALOC, SI, SMIN, SMINW,
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$ SMLNUM, SR, TJJ, TMP, XJ, XMAX, XNORM, Z
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* ..
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* .. Local Arrays ..
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REAL D( 2, 2 ), V( 2, 2 )
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* ..
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* .. External Functions ..
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INTEGER ISAMAX
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REAL SASUM, SDOT, SLAMCH, SLANGE
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EXTERNAL ISAMAX, SASUM, SDOT, SLAMCH, SLANGE
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* ..
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* .. External Subroutines ..
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EXTERNAL SAXPY, SLADIV, SLALN2, SSCAL
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX
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* ..
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* .. Executable Statements ..
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*
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* Do not test the input parameters for errors
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*
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NOTRAN = .NOT.LTRAN
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INFO = 0
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*
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* Quick return if possible
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*
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IF( N.EQ.0 )
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$ RETURN
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*
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* Set constants to control overflow
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*
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EPS = SLAMCH( 'P' )
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SMLNUM = SLAMCH( 'S' ) / EPS
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BIGNUM = ONE / SMLNUM
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*
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XNORM = SLANGE( 'M', N, N, T, LDT, D )
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IF( .NOT.LREAL )
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$ XNORM = MAX( XNORM, ABS( W ), SLANGE( 'M', N, 1, B, N, D ) )
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SMIN = MAX( SMLNUM, EPS*XNORM )
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*
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* Compute 1-norm of each column of strictly upper triangular
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* part of T to control overflow in triangular solver.
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*
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WORK( 1 ) = ZERO
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DO 10 J = 2, N
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WORK( J ) = SASUM( J-1, T( 1, J ), 1 )
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10 CONTINUE
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*
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IF( .NOT.LREAL ) THEN
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DO 20 I = 2, N
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WORK( I ) = WORK( I ) + ABS( B( I ) )
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20 CONTINUE
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END IF
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*
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N2 = 2*N
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N1 = N
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IF( .NOT.LREAL )
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$ N1 = N2
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K = ISAMAX( N1, X, 1 )
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XMAX = ABS( X( K ) )
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SCALE = ONE
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*
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IF( XMAX.GT.BIGNUM ) THEN
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SCALE = BIGNUM / XMAX
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CALL SSCAL( N1, SCALE, X, 1 )
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XMAX = BIGNUM
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END IF
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*
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IF( LREAL ) THEN
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*
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IF( NOTRAN ) THEN
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*
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* Solve T*p = scale*c
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*
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JNEXT = N
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DO 30 J = N, 1, -1
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IF( J.GT.JNEXT )
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$ GO TO 30
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J1 = J
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J2 = J
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JNEXT = J - 1
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IF( J.GT.1 ) THEN
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IF( T( J, J-1 ).NE.ZERO ) THEN
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J1 = J - 1
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JNEXT = J - 2
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END IF
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END IF
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*
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IF( J1.EQ.J2 ) THEN
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*
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* Meet 1 by 1 diagonal block
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*
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* Scale to avoid overflow when computing
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* x(j) = b(j)/T(j,j)
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*
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XJ = ABS( X( J1 ) )
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TJJ = ABS( T( J1, J1 ) )
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TMP = T( J1, J1 )
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IF( TJJ.LT.SMIN ) THEN
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TMP = SMIN
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TJJ = SMIN
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INFO = 1
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END IF
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*
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IF( XJ.EQ.ZERO )
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$ GO TO 30
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*
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IF( TJJ.LT.ONE ) THEN
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IF( XJ.GT.BIGNUM*TJJ ) THEN
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REC = ONE / XJ
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CALL SSCAL( N, REC, X, 1 )
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SCALE = SCALE*REC
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XMAX = XMAX*REC
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END IF
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END IF
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X( J1 ) = X( J1 ) / TMP
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XJ = ABS( X( J1 ) )
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*
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* Scale x if necessary to avoid overflow when adding a
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* multiple of column j1 of T.
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*
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IF( XJ.GT.ONE ) THEN
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REC = ONE / XJ
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IF( WORK( J1 ).GT.( BIGNUM-XMAX )*REC ) THEN
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CALL SSCAL( N, REC, X, 1 )
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SCALE = SCALE*REC
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END IF
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END IF
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IF( J1.GT.1 ) THEN
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CALL SAXPY( J1-1, -X( J1 ), T( 1, J1 ), 1, X, 1 )
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K = ISAMAX( J1-1, X, 1 )
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XMAX = ABS( X( K ) )
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END IF
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*
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ELSE
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*
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* Meet 2 by 2 diagonal block
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*
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* Call 2 by 2 linear system solve, to take
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* care of possible overflow by scaling factor.
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*
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D( 1, 1 ) = X( J1 )
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D( 2, 1 ) = X( J2 )
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CALL SLALN2( .FALSE., 2, 1, SMIN, ONE, T( J1, J1 ),
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$ LDT, ONE, ONE, D, 2, ZERO, ZERO, V, 2,
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$ SCALOC, XNORM, IERR )
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IF( IERR.NE.0 )
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$ INFO = 2
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*
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IF( SCALOC.NE.ONE ) THEN
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CALL SSCAL( N, SCALOC, X, 1 )
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SCALE = SCALE*SCALOC
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END IF
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X( J1 ) = V( 1, 1 )
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X( J2 ) = V( 2, 1 )
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*
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* Scale V(1,1) (= X(J1)) and/or V(2,1) (=X(J2))
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* to avoid overflow in updating right-hand side.
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*
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XJ = MAX( ABS( V( 1, 1 ) ), ABS( V( 2, 1 ) ) )
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IF( XJ.GT.ONE ) THEN
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REC = ONE / XJ
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IF( MAX( WORK( J1 ), WORK( J2 ) ).GT.
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$ ( BIGNUM-XMAX )*REC ) THEN
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CALL SSCAL( N, REC, X, 1 )
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SCALE = SCALE*REC
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END IF
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END IF
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*
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* Update right-hand side
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*
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IF( J1.GT.1 ) THEN
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CALL SAXPY( J1-1, -X( J1 ), T( 1, J1 ), 1, X, 1 )
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CALL SAXPY( J1-1, -X( J2 ), T( 1, J2 ), 1, X, 1 )
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K = ISAMAX( J1-1, X, 1 )
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XMAX = ABS( X( K ) )
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END IF
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*
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END IF
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*
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30 CONTINUE
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*
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ELSE
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*
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|
|
* Solve T**T*p = scale*c
|
||
|
|
*
|
||
|
|
JNEXT = 1
|
||
|
|
DO 40 J = 1, N
|
||
|
|
IF( J.LT.JNEXT )
|
||
|
|
$ GO TO 40
|
||
|
|
J1 = J
|
||
|
|
J2 = J
|
||
|
|
JNEXT = J + 1
|
||
|
|
IF( J.LT.N ) THEN
|
||
|
|
IF( T( J+1, J ).NE.ZERO ) THEN
|
||
|
|
J2 = J + 1
|
||
|
|
JNEXT = J + 2
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
IF( J1.EQ.J2 ) THEN
|
||
|
|
*
|
||
|
|
* 1 by 1 diagonal block
|
||
|
|
*
|
||
|
|
* Scale if necessary to avoid overflow in forming the
|
||
|
|
* right-hand side element by inner product.
|
||
|
|
*
|
||
|
|
XJ = ABS( X( J1 ) )
|
||
|
|
IF( XMAX.GT.ONE ) THEN
|
||
|
|
REC = ONE / XMAX
|
||
|
|
IF( WORK( J1 ).GT.( BIGNUM-XJ )*REC ) THEN
|
||
|
|
CALL SSCAL( N, REC, X, 1 )
|
||
|
|
SCALE = SCALE*REC
|
||
|
|
XMAX = XMAX*REC
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
X( J1 ) = X( J1 ) - SDOT( J1-1, T( 1, J1 ), 1, X, 1 )
|
||
|
|
*
|
||
|
|
XJ = ABS( X( J1 ) )
|
||
|
|
TJJ = ABS( T( J1, J1 ) )
|
||
|
|
TMP = T( J1, J1 )
|
||
|
|
IF( TJJ.LT.SMIN ) THEN
|
||
|
|
TMP = SMIN
|
||
|
|
TJJ = SMIN
|
||
|
|
INFO = 1
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
IF( TJJ.LT.ONE ) THEN
|
||
|
|
IF( XJ.GT.BIGNUM*TJJ ) THEN
|
||
|
|
REC = ONE / XJ
|
||
|
|
CALL SSCAL( N, REC, X, 1 )
|
||
|
|
SCALE = SCALE*REC
|
||
|
|
XMAX = XMAX*REC
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
X( J1 ) = X( J1 ) / TMP
|
||
|
|
XMAX = MAX( XMAX, ABS( X( J1 ) ) )
|
||
|
|
*
|
||
|
|
ELSE
|
||
|
|
*
|
||
|
|
* 2 by 2 diagonal block
|
||
|
|
*
|
||
|
|
* Scale if necessary to avoid overflow in forming the
|
||
|
|
* right-hand side elements by inner product.
|
||
|
|
*
|
||
|
|
XJ = MAX( ABS( X( J1 ) ), ABS( X( J2 ) ) )
|
||
|
|
IF( XMAX.GT.ONE ) THEN
|
||
|
|
REC = ONE / XMAX
|
||
|
|
IF( MAX( WORK( J2 ), WORK( J1 ) ).GT.( BIGNUM-XJ )*
|
||
|
|
$ REC ) THEN
|
||
|
|
CALL SSCAL( N, REC, X, 1 )
|
||
|
|
SCALE = SCALE*REC
|
||
|
|
XMAX = XMAX*REC
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
D( 1, 1 ) = X( J1 ) - SDOT( J1-1, T( 1, J1 ), 1, X,
|
||
|
|
$ 1 )
|
||
|
|
D( 2, 1 ) = X( J2 ) - SDOT( J1-1, T( 1, J2 ), 1, X,
|
||
|
|
$ 1 )
|
||
|
|
*
|
||
|
|
CALL SLALN2( .TRUE., 2, 1, SMIN, ONE, T( J1, J1 ),
|
||
|
|
$ LDT, ONE, ONE, D, 2, ZERO, ZERO, V, 2,
|
||
|
|
$ SCALOC, XNORM, IERR )
|
||
|
|
IF( IERR.NE.0 )
|
||
|
|
$ INFO = 2
|
||
|
|
*
|
||
|
|
IF( SCALOC.NE.ONE ) THEN
|
||
|
|
CALL SSCAL( N, SCALOC, X, 1 )
|
||
|
|
SCALE = SCALE*SCALOC
|
||
|
|
END IF
|
||
|
|
X( J1 ) = V( 1, 1 )
|
||
|
|
X( J2 ) = V( 2, 1 )
|
||
|
|
XMAX = MAX( ABS( X( J1 ) ), ABS( X( J2 ) ), XMAX )
|
||
|
|
*
|
||
|
|
END IF
|
||
|
|
40 CONTINUE
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
ELSE
|
||
|
|
*
|
||
|
|
SMINW = MAX( EPS*ABS( W ), SMIN )
|
||
|
|
IF( NOTRAN ) THEN
|
||
|
|
*
|
||
|
|
* Solve (T + iB)*(p+iq) = c+id
|
||
|
|
*
|
||
|
|
JNEXT = N
|
||
|
|
DO 70 J = N, 1, -1
|
||
|
|
IF( J.GT.JNEXT )
|
||
|
|
$ GO TO 70
|
||
|
|
J1 = J
|
||
|
|
J2 = J
|
||
|
|
JNEXT = J - 1
|
||
|
|
IF( J.GT.1 ) THEN
|
||
|
|
IF( T( J, J-1 ).NE.ZERO ) THEN
|
||
|
|
J1 = J - 1
|
||
|
|
JNEXT = J - 2
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
IF( J1.EQ.J2 ) THEN
|
||
|
|
*
|
||
|
|
* 1 by 1 diagonal block
|
||
|
|
*
|
||
|
|
* Scale if necessary to avoid overflow in division
|
||
|
|
*
|
||
|
|
Z = W
|
||
|
|
IF( J1.EQ.1 )
|
||
|
|
$ Z = B( 1 )
|
||
|
|
XJ = ABS( X( J1 ) ) + ABS( X( N+J1 ) )
|
||
|
|
TJJ = ABS( T( J1, J1 ) ) + ABS( Z )
|
||
|
|
TMP = T( J1, J1 )
|
||
|
|
IF( TJJ.LT.SMINW ) THEN
|
||
|
|
TMP = SMINW
|
||
|
|
TJJ = SMINW
|
||
|
|
INFO = 1
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
IF( XJ.EQ.ZERO )
|
||
|
|
$ GO TO 70
|
||
|
|
*
|
||
|
|
IF( TJJ.LT.ONE ) THEN
|
||
|
|
IF( XJ.GT.BIGNUM*TJJ ) THEN
|
||
|
|
REC = ONE / XJ
|
||
|
|
CALL SSCAL( N2, REC, X, 1 )
|
||
|
|
SCALE = SCALE*REC
|
||
|
|
XMAX = XMAX*REC
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
CALL SLADIV( X( J1 ), X( N+J1 ), TMP, Z, SR, SI )
|
||
|
|
X( J1 ) = SR
|
||
|
|
X( N+J1 ) = SI
|
||
|
|
XJ = ABS( X( J1 ) ) + ABS( X( N+J1 ) )
|
||
|
|
*
|
||
|
|
* Scale x if necessary to avoid overflow when adding a
|
||
|
|
* multiple of column j1 of T.
|
||
|
|
*
|
||
|
|
IF( XJ.GT.ONE ) THEN
|
||
|
|
REC = ONE / XJ
|
||
|
|
IF( WORK( J1 ).GT.( BIGNUM-XMAX )*REC ) THEN
|
||
|
|
CALL SSCAL( N2, REC, X, 1 )
|
||
|
|
SCALE = SCALE*REC
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
IF( J1.GT.1 ) THEN
|
||
|
|
CALL SAXPY( J1-1, -X( J1 ), T( 1, J1 ), 1, X, 1 )
|
||
|
|
CALL SAXPY( J1-1, -X( N+J1 ), T( 1, J1 ), 1,
|
||
|
|
$ X( N+1 ), 1 )
|
||
|
|
*
|
||
|
|
X( 1 ) = X( 1 ) + B( J1 )*X( N+J1 )
|
||
|
|
X( N+1 ) = X( N+1 ) - B( J1 )*X( J1 )
|
||
|
|
*
|
||
|
|
XMAX = ZERO
|
||
|
|
DO 50 K = 1, J1 - 1
|
||
|
|
XMAX = MAX( XMAX, ABS( X( K ) )+
|
||
|
|
$ ABS( X( K+N ) ) )
|
||
|
|
50 CONTINUE
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
ELSE
|
||
|
|
*
|
||
|
|
* Meet 2 by 2 diagonal block
|
||
|
|
*
|
||
|
|
D( 1, 1 ) = X( J1 )
|
||
|
|
D( 2, 1 ) = X( J2 )
|
||
|
|
D( 1, 2 ) = X( N+J1 )
|
||
|
|
D( 2, 2 ) = X( N+J2 )
|
||
|
|
CALL SLALN2( .FALSE., 2, 2, SMINW, ONE, T( J1, J1 ),
|
||
|
|
$ LDT, ONE, ONE, D, 2, ZERO, -W, V, 2,
|
||
|
|
$ SCALOC, XNORM, IERR )
|
||
|
|
IF( IERR.NE.0 )
|
||
|
|
$ INFO = 2
|
||
|
|
*
|
||
|
|
IF( SCALOC.NE.ONE ) THEN
|
||
|
|
CALL SSCAL( 2*N, SCALOC, X, 1 )
|
||
|
|
SCALE = SCALOC*SCALE
|
||
|
|
END IF
|
||
|
|
X( J1 ) = V( 1, 1 )
|
||
|
|
X( J2 ) = V( 2, 1 )
|
||
|
|
X( N+J1 ) = V( 1, 2 )
|
||
|
|
X( N+J2 ) = V( 2, 2 )
|
||
|
|
*
|
||
|
|
* Scale X(J1), .... to avoid overflow in
|
||
|
|
* updating right hand side.
|
||
|
|
*
|
||
|
|
XJ = MAX( ABS( V( 1, 1 ) )+ABS( V( 1, 2 ) ),
|
||
|
|
$ ABS( V( 2, 1 ) )+ABS( V( 2, 2 ) ) )
|
||
|
|
IF( XJ.GT.ONE ) THEN
|
||
|
|
REC = ONE / XJ
|
||
|
|
IF( MAX( WORK( J1 ), WORK( J2 ) ).GT.
|
||
|
|
$ ( BIGNUM-XMAX )*REC ) THEN
|
||
|
|
CALL SSCAL( N2, REC, X, 1 )
|
||
|
|
SCALE = SCALE*REC
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
* Update the right-hand side.
|
||
|
|
*
|
||
|
|
IF( J1.GT.1 ) THEN
|
||
|
|
CALL SAXPY( J1-1, -X( J1 ), T( 1, J1 ), 1, X, 1 )
|
||
|
|
CALL SAXPY( J1-1, -X( J2 ), T( 1, J2 ), 1, X, 1 )
|
||
|
|
*
|
||
|
|
CALL SAXPY( J1-1, -X( N+J1 ), T( 1, J1 ), 1,
|
||
|
|
$ X( N+1 ), 1 )
|
||
|
|
CALL SAXPY( J1-1, -X( N+J2 ), T( 1, J2 ), 1,
|
||
|
|
$ X( N+1 ), 1 )
|
||
|
|
*
|
||
|
|
X( 1 ) = X( 1 ) + B( J1 )*X( N+J1 ) +
|
||
|
|
$ B( J2 )*X( N+J2 )
|
||
|
|
X( N+1 ) = X( N+1 ) - B( J1 )*X( J1 ) -
|
||
|
|
$ B( J2 )*X( J2 )
|
||
|
|
*
|
||
|
|
XMAX = ZERO
|
||
|
|
DO 60 K = 1, J1 - 1
|
||
|
|
XMAX = MAX( ABS( X( K ) )+ABS( X( K+N ) ),
|
||
|
|
$ XMAX )
|
||
|
|
60 CONTINUE
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
END IF
|
||
|
|
70 CONTINUE
|
||
|
|
*
|
||
|
|
ELSE
|
||
|
|
*
|
||
|
|
* Solve (T + iB)**T*(p+iq) = c+id
|
||
|
|
*
|
||
|
|
JNEXT = 1
|
||
|
|
DO 80 J = 1, N
|
||
|
|
IF( J.LT.JNEXT )
|
||
|
|
$ GO TO 80
|
||
|
|
J1 = J
|
||
|
|
J2 = J
|
||
|
|
JNEXT = J + 1
|
||
|
|
IF( J.LT.N ) THEN
|
||
|
|
IF( T( J+1, J ).NE.ZERO ) THEN
|
||
|
|
J2 = J + 1
|
||
|
|
JNEXT = J + 2
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
IF( J1.EQ.J2 ) THEN
|
||
|
|
*
|
||
|
|
* 1 by 1 diagonal block
|
||
|
|
*
|
||
|
|
* Scale if necessary to avoid overflow in forming the
|
||
|
|
* right-hand side element by inner product.
|
||
|
|
*
|
||
|
|
XJ = ABS( X( J1 ) ) + ABS( X( J1+N ) )
|
||
|
|
IF( XMAX.GT.ONE ) THEN
|
||
|
|
REC = ONE / XMAX
|
||
|
|
IF( WORK( J1 ).GT.( BIGNUM-XJ )*REC ) THEN
|
||
|
|
CALL SSCAL( N2, REC, X, 1 )
|
||
|
|
SCALE = SCALE*REC
|
||
|
|
XMAX = XMAX*REC
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
X( J1 ) = X( J1 ) - SDOT( J1-1, T( 1, J1 ), 1, X, 1 )
|
||
|
|
X( N+J1 ) = X( N+J1 ) - SDOT( J1-1, T( 1, J1 ), 1,
|
||
|
|
$ X( N+1 ), 1 )
|
||
|
|
IF( J1.GT.1 ) THEN
|
||
|
|
X( J1 ) = X( J1 ) - B( J1 )*X( N+1 )
|
||
|
|
X( N+J1 ) = X( N+J1 ) + B( J1 )*X( 1 )
|
||
|
|
END IF
|
||
|
|
XJ = ABS( X( J1 ) ) + ABS( X( J1+N ) )
|
||
|
|
*
|
||
|
|
Z = W
|
||
|
|
IF( J1.EQ.1 )
|
||
|
|
$ Z = B( 1 )
|
||
|
|
*
|
||
|
|
* Scale if necessary to avoid overflow in
|
||
|
|
* complex division
|
||
|
|
*
|
||
|
|
TJJ = ABS( T( J1, J1 ) ) + ABS( Z )
|
||
|
|
TMP = T( J1, J1 )
|
||
|
|
IF( TJJ.LT.SMINW ) THEN
|
||
|
|
TMP = SMINW
|
||
|
|
TJJ = SMINW
|
||
|
|
INFO = 1
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
IF( TJJ.LT.ONE ) THEN
|
||
|
|
IF( XJ.GT.BIGNUM*TJJ ) THEN
|
||
|
|
REC = ONE / XJ
|
||
|
|
CALL SSCAL( N2, REC, X, 1 )
|
||
|
|
SCALE = SCALE*REC
|
||
|
|
XMAX = XMAX*REC
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
CALL SLADIV( X( J1 ), X( N+J1 ), TMP, -Z, SR, SI )
|
||
|
|
X( J1 ) = SR
|
||
|
|
X( J1+N ) = SI
|
||
|
|
XMAX = MAX( ABS( X( J1 ) )+ABS( X( J1+N ) ), XMAX )
|
||
|
|
*
|
||
|
|
ELSE
|
||
|
|
*
|
||
|
|
* 2 by 2 diagonal block
|
||
|
|
*
|
||
|
|
* Scale if necessary to avoid overflow in forming the
|
||
|
|
* right-hand side element by inner product.
|
||
|
|
*
|
||
|
|
XJ = MAX( ABS( X( J1 ) )+ABS( X( N+J1 ) ),
|
||
|
|
$ ABS( X( J2 ) )+ABS( X( N+J2 ) ) )
|
||
|
|
IF( XMAX.GT.ONE ) THEN
|
||
|
|
REC = ONE / XMAX
|
||
|
|
IF( MAX( WORK( J1 ), WORK( J2 ) ).GT.
|
||
|
|
$ ( BIGNUM-XJ ) / XMAX ) THEN
|
||
|
|
CALL SSCAL( N2, REC, X, 1 )
|
||
|
|
SCALE = SCALE*REC
|
||
|
|
XMAX = XMAX*REC
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
D( 1, 1 ) = X( J1 ) - SDOT( J1-1, T( 1, J1 ), 1, X,
|
||
|
|
$ 1 )
|
||
|
|
D( 2, 1 ) = X( J2 ) - SDOT( J1-1, T( 1, J2 ), 1, X,
|
||
|
|
$ 1 )
|
||
|
|
D( 1, 2 ) = X( N+J1 ) - SDOT( J1-1, T( 1, J1 ), 1,
|
||
|
|
$ X( N+1 ), 1 )
|
||
|
|
D( 2, 2 ) = X( N+J2 ) - SDOT( J1-1, T( 1, J2 ), 1,
|
||
|
|
$ X( N+1 ), 1 )
|
||
|
|
D( 1, 1 ) = D( 1, 1 ) - B( J1 )*X( N+1 )
|
||
|
|
D( 2, 1 ) = D( 2, 1 ) - B( J2 )*X( N+1 )
|
||
|
|
D( 1, 2 ) = D( 1, 2 ) + B( J1 )*X( 1 )
|
||
|
|
D( 2, 2 ) = D( 2, 2 ) + B( J2 )*X( 1 )
|
||
|
|
*
|
||
|
|
CALL SLALN2( .TRUE., 2, 2, SMINW, ONE, T( J1, J1 ),
|
||
|
|
$ LDT, ONE, ONE, D, 2, ZERO, W, V, 2,
|
||
|
|
$ SCALOC, XNORM, IERR )
|
||
|
|
IF( IERR.NE.0 )
|
||
|
|
$ INFO = 2
|
||
|
|
*
|
||
|
|
IF( SCALOC.NE.ONE ) THEN
|
||
|
|
CALL SSCAL( N2, SCALOC, X, 1 )
|
||
|
|
SCALE = SCALOC*SCALE
|
||
|
|
END IF
|
||
|
|
X( J1 ) = V( 1, 1 )
|
||
|
|
X( J2 ) = V( 2, 1 )
|
||
|
|
X( N+J1 ) = V( 1, 2 )
|
||
|
|
X( N+J2 ) = V( 2, 2 )
|
||
|
|
XMAX = MAX( ABS( X( J1 ) )+ABS( X( N+J1 ) ),
|
||
|
|
$ ABS( X( J2 ) )+ABS( X( N+J2 ) ), XMAX )
|
||
|
|
*
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
80 CONTINUE
|
||
|
|
*
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
RETURN
|
||
|
|
*
|
||
|
|
* End of SLAQTR
|
||
|
|
*
|
||
|
|
END
|