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480 lines
14 KiB
480 lines
14 KiB
2 years ago
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*> \brief \b SLASY2 solves the Sylvester matrix equation where the matrices are of order 1 or 2.
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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 SLASY2 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slasy2.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/slasy2.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/slasy2.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 SLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
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* LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
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*
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* .. Scalar Arguments ..
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* LOGICAL LTRANL, LTRANR
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* INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
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* REAL SCALE, XNORM
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* ..
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* .. Array Arguments ..
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* REAL B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
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* $ 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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*> SLASY2 solves for the N1 by N2 matrix X, 1 <= N1,N2 <= 2, in
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*>
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*> op(TL)*X + ISGN*X*op(TR) = SCALE*B,
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*>
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*> where TL is N1 by N1, TR is N2 by N2, B is N1 by N2, and ISGN = 1 or
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*> -1. op(T) = T or T**T, where T**T denotes the transpose of T.
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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] LTRANL
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*> \verbatim
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*> LTRANL is LOGICAL
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*> On entry, LTRANL specifies the op(TL):
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*> = .FALSE., op(TL) = TL,
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*> = .TRUE., op(TL) = TL**T.
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*> \endverbatim
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*>
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*> \param[in] LTRANR
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*> \verbatim
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*> LTRANR is LOGICAL
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*> On entry, LTRANR specifies the op(TR):
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*> = .FALSE., op(TR) = TR,
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*> = .TRUE., op(TR) = TR**T.
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*> \endverbatim
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*>
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*> \param[in] ISGN
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*> \verbatim
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*> ISGN is INTEGER
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*> On entry, ISGN specifies the sign of the equation
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*> as described before. ISGN may only be 1 or -1.
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*> \endverbatim
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*>
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*> \param[in] N1
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*> \verbatim
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*> N1 is INTEGER
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*> On entry, N1 specifies the order of matrix TL.
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*> N1 may only be 0, 1 or 2.
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*> \endverbatim
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*>
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*> \param[in] N2
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*> \verbatim
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*> N2 is INTEGER
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*> On entry, N2 specifies the order of matrix TR.
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*> N2 may only be 0, 1 or 2.
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*> \endverbatim
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*>
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*> \param[in] TL
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*> \verbatim
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*> TL is REAL array, dimension (LDTL,2)
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*> On entry, TL contains an N1 by N1 matrix.
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*> \endverbatim
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*>
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*> \param[in] LDTL
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*> \verbatim
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*> LDTL is INTEGER
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*> The leading dimension of the matrix TL. LDTL >= max(1,N1).
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*> \endverbatim
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*>
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*> \param[in] TR
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*> \verbatim
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*> TR is REAL array, dimension (LDTR,2)
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*> On entry, TR contains an N2 by N2 matrix.
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*> \endverbatim
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*>
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*> \param[in] LDTR
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*> \verbatim
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*> LDTR is INTEGER
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*> The leading dimension of the matrix TR. LDTR >= max(1,N2).
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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 (LDB,2)
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*> On entry, the N1 by N2 matrix B contains the right-hand
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*> side of the equation.
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*> \endverbatim
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*>
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*> \param[in] LDB
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*> \verbatim
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*> LDB is INTEGER
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*> The leading dimension of the matrix B. LDB >= max(1,N1).
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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 contains the scale factor. SCALE is chosen
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*> less than or equal to 1 to prevent the solution overflowing.
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*> \endverbatim
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*>
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*> \param[out] X
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*> \verbatim
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*> X is REAL array, dimension (LDX,2)
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*> On exit, X contains the N1 by N2 solution.
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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 matrix X. LDX >= max(1,N1).
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*> \endverbatim
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*>
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*> \param[out] XNORM
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*> \verbatim
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*> XNORM is REAL
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*> On exit, XNORM is the infinity-norm of the solution.
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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: TL and TR have too close eigenvalues, so TL or
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*> TR is perturbed to get a nonsingular equation.
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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 realSYauxiliary
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*
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* =====================================================================
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SUBROUTINE SLASY2( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
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$ LDTR, B, LDB, SCALE, X, LDX, XNORM, 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 LTRANL, LTRANR
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INTEGER INFO, ISGN, LDB, LDTL, LDTR, LDX, N1, N2
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REAL SCALE, XNORM
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* ..
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* .. Array Arguments ..
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REAL B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
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$ 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 = 0.0E+0, ONE = 1.0E+0 )
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REAL TWO, HALF, EIGHT
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PARAMETER ( TWO = 2.0E+0, HALF = 0.5E+0, EIGHT = 8.0E+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL BSWAP, XSWAP
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INTEGER I, IP, IPIV, IPSV, J, JP, JPSV, K
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REAL BET, EPS, GAM, L21, SGN, SMIN, SMLNUM, TAU1,
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$ TEMP, U11, U12, U22, XMAX
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* ..
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* .. Local Arrays ..
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LOGICAL BSWPIV( 4 ), XSWPIV( 4 )
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INTEGER JPIV( 4 ), LOCL21( 4 ), LOCU12( 4 ),
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$ LOCU22( 4 )
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REAL BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 )
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* ..
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* .. External Functions ..
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INTEGER ISAMAX
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REAL SLAMCH
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EXTERNAL ISAMAX, SLAMCH
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* ..
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* .. External Subroutines ..
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EXTERNAL SCOPY, SSWAP
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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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* .. Data statements ..
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DATA LOCU12 / 3, 4, 1, 2 / , LOCL21 / 2, 1, 4, 3 / ,
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$ LOCU22 / 4, 3, 2, 1 /
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DATA XSWPIV / .FALSE., .FALSE., .TRUE., .TRUE. /
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DATA BSWPIV / .FALSE., .TRUE., .FALSE., .TRUE. /
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* ..
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* .. Executable Statements ..
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*
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* Do not check the input parameters for errors
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*
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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( N1.EQ.0 .OR. N2.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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SGN = ISGN
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*
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K = N1 + N1 + N2 - 2
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GO TO ( 10, 20, 30, 50 )K
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*
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* 1 by 1: TL11*X + SGN*X*TR11 = B11
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*
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10 CONTINUE
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TAU1 = TL( 1, 1 ) + SGN*TR( 1, 1 )
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BET = ABS( TAU1 )
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IF( BET.LE.SMLNUM ) THEN
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TAU1 = SMLNUM
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BET = SMLNUM
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INFO = 1
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END IF
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*
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SCALE = ONE
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GAM = ABS( B( 1, 1 ) )
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IF( SMLNUM*GAM.GT.BET )
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$ SCALE = ONE / GAM
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*
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X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / TAU1
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XNORM = ABS( X( 1, 1 ) )
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RETURN
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*
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* 1 by 2:
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* TL11*[X11 X12] + ISGN*[X11 X12]*op[TR11 TR12] = [B11 B12]
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* [TR21 TR22]
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*
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20 CONTINUE
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*
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SMIN = MAX( EPS*MAX( ABS( TL( 1, 1 ) ), ABS( TR( 1, 1 ) ),
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$ ABS( TR( 1, 2 ) ), ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) ),
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$ SMLNUM )
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TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
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TMP( 4 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
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IF( LTRANR ) THEN
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TMP( 2 ) = SGN*TR( 2, 1 )
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TMP( 3 ) = SGN*TR( 1, 2 )
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ELSE
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TMP( 2 ) = SGN*TR( 1, 2 )
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TMP( 3 ) = SGN*TR( 2, 1 )
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END IF
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BTMP( 1 ) = B( 1, 1 )
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BTMP( 2 ) = B( 1, 2 )
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GO TO 40
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*
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* 2 by 1:
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* op[TL11 TL12]*[X11] + ISGN* [X11]*TR11 = [B11]
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* [TL21 TL22] [X21] [X21] [B21]
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*
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30 CONTINUE
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SMIN = MAX( EPS*MAX( ABS( TR( 1, 1 ) ), ABS( TL( 1, 1 ) ),
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$ ABS( TL( 1, 2 ) ), ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) ),
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$ SMLNUM )
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TMP( 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
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TMP( 4 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
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IF( LTRANL ) THEN
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TMP( 2 ) = TL( 1, 2 )
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TMP( 3 ) = TL( 2, 1 )
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ELSE
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TMP( 2 ) = TL( 2, 1 )
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TMP( 3 ) = TL( 1, 2 )
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END IF
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BTMP( 1 ) = B( 1, 1 )
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BTMP( 2 ) = B( 2, 1 )
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40 CONTINUE
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*
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* Solve 2 by 2 system using complete pivoting.
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* Set pivots less than SMIN to SMIN.
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*
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IPIV = ISAMAX( 4, TMP, 1 )
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U11 = TMP( IPIV )
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IF( ABS( U11 ).LE.SMIN ) THEN
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INFO = 1
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U11 = SMIN
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END IF
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U12 = TMP( LOCU12( IPIV ) )
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L21 = TMP( LOCL21( IPIV ) ) / U11
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U22 = TMP( LOCU22( IPIV ) ) - U12*L21
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XSWAP = XSWPIV( IPIV )
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BSWAP = BSWPIV( IPIV )
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IF( ABS( U22 ).LE.SMIN ) THEN
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INFO = 1
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U22 = SMIN
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END IF
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IF( BSWAP ) THEN
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TEMP = BTMP( 2 )
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BTMP( 2 ) = BTMP( 1 ) - L21*TEMP
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BTMP( 1 ) = TEMP
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ELSE
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BTMP( 2 ) = BTMP( 2 ) - L21*BTMP( 1 )
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END IF
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SCALE = ONE
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IF( ( TWO*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( U22 ) .OR.
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$ ( TWO*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( U11 ) ) THEN
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SCALE = HALF / MAX( ABS( BTMP( 1 ) ), ABS( BTMP( 2 ) ) )
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BTMP( 1 ) = BTMP( 1 )*SCALE
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BTMP( 2 ) = BTMP( 2 )*SCALE
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END IF
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X2( 2 ) = BTMP( 2 ) / U22
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X2( 1 ) = BTMP( 1 ) / U11 - ( U12 / U11 )*X2( 2 )
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IF( XSWAP ) THEN
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TEMP = X2( 2 )
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X2( 2 ) = X2( 1 )
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X2( 1 ) = TEMP
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END IF
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X( 1, 1 ) = X2( 1 )
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IF( N1.EQ.1 ) THEN
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X( 1, 2 ) = X2( 2 )
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XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) )
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ELSE
|
||
|
|
X( 2, 1 ) = X2( 2 )
|
||
|
|
XNORM = MAX( ABS( X( 1, 1 ) ), ABS( X( 2, 1 ) ) )
|
||
|
|
END IF
|
||
|
|
RETURN
|
||
|
|
*
|
||
|
|
* 2 by 2:
|
||
|
|
* op[TL11 TL12]*[X11 X12] +ISGN* [X11 X12]*op[TR11 TR12] = [B11 B12]
|
||
|
|
* [TL21 TL22] [X21 X22] [X21 X22] [TR21 TR22] [B21 B22]
|
||
|
|
*
|
||
|
|
* Solve equivalent 4 by 4 system using complete pivoting.
|
||
|
|
* Set pivots less than SMIN to SMIN.
|
||
|
|
*
|
||
|
|
50 CONTINUE
|
||
|
|
SMIN = MAX( ABS( TR( 1, 1 ) ), ABS( TR( 1, 2 ) ),
|
||
|
|
$ ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) )
|
||
|
|
SMIN = MAX( SMIN, ABS( TL( 1, 1 ) ), ABS( TL( 1, 2 ) ),
|
||
|
|
$ ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) )
|
||
|
|
SMIN = MAX( EPS*SMIN, SMLNUM )
|
||
|
|
BTMP( 1 ) = ZERO
|
||
|
|
CALL SCOPY( 16, BTMP, 0, T16, 1 )
|
||
|
|
T16( 1, 1 ) = TL( 1, 1 ) + SGN*TR( 1, 1 )
|
||
|
|
T16( 2, 2 ) = TL( 2, 2 ) + SGN*TR( 1, 1 )
|
||
|
|
T16( 3, 3 ) = TL( 1, 1 ) + SGN*TR( 2, 2 )
|
||
|
|
T16( 4, 4 ) = TL( 2, 2 ) + SGN*TR( 2, 2 )
|
||
|
|
IF( LTRANL ) THEN
|
||
|
|
T16( 1, 2 ) = TL( 2, 1 )
|
||
|
|
T16( 2, 1 ) = TL( 1, 2 )
|
||
|
|
T16( 3, 4 ) = TL( 2, 1 )
|
||
|
|
T16( 4, 3 ) = TL( 1, 2 )
|
||
|
|
ELSE
|
||
|
|
T16( 1, 2 ) = TL( 1, 2 )
|
||
|
|
T16( 2, 1 ) = TL( 2, 1 )
|
||
|
|
T16( 3, 4 ) = TL( 1, 2 )
|
||
|
|
T16( 4, 3 ) = TL( 2, 1 )
|
||
|
|
END IF
|
||
|
|
IF( LTRANR ) THEN
|
||
|
|
T16( 1, 3 ) = SGN*TR( 1, 2 )
|
||
|
|
T16( 2, 4 ) = SGN*TR( 1, 2 )
|
||
|
|
T16( 3, 1 ) = SGN*TR( 2, 1 )
|
||
|
|
T16( 4, 2 ) = SGN*TR( 2, 1 )
|
||
|
|
ELSE
|
||
|
|
T16( 1, 3 ) = SGN*TR( 2, 1 )
|
||
|
|
T16( 2, 4 ) = SGN*TR( 2, 1 )
|
||
|
|
T16( 3, 1 ) = SGN*TR( 1, 2 )
|
||
|
|
T16( 4, 2 ) = SGN*TR( 1, 2 )
|
||
|
|
END IF
|
||
|
|
BTMP( 1 ) = B( 1, 1 )
|
||
|
|
BTMP( 2 ) = B( 2, 1 )
|
||
|
|
BTMP( 3 ) = B( 1, 2 )
|
||
|
|
BTMP( 4 ) = B( 2, 2 )
|
||
|
|
*
|
||
|
|
* Perform elimination
|
||
|
|
*
|
||
|
|
DO 100 I = 1, 3
|
||
|
|
XMAX = ZERO
|
||
|
|
DO 70 IP = I, 4
|
||
|
|
DO 60 JP = I, 4
|
||
|
|
IF( ABS( T16( IP, JP ) ).GE.XMAX ) THEN
|
||
|
|
XMAX = ABS( T16( IP, JP ) )
|
||
|
|
IPSV = IP
|
||
|
|
JPSV = JP
|
||
|
|
END IF
|
||
|
|
60 CONTINUE
|
||
|
|
70 CONTINUE
|
||
|
|
IF( IPSV.NE.I ) THEN
|
||
|
|
CALL SSWAP( 4, T16( IPSV, 1 ), 4, T16( I, 1 ), 4 )
|
||
|
|
TEMP = BTMP( I )
|
||
|
|
BTMP( I ) = BTMP( IPSV )
|
||
|
|
BTMP( IPSV ) = TEMP
|
||
|
|
END IF
|
||
|
|
IF( JPSV.NE.I )
|
||
|
|
$ CALL SSWAP( 4, T16( 1, JPSV ), 1, T16( 1, I ), 1 )
|
||
|
|
JPIV( I ) = JPSV
|
||
|
|
IF( ABS( T16( I, I ) ).LT.SMIN ) THEN
|
||
|
|
INFO = 1
|
||
|
|
T16( I, I ) = SMIN
|
||
|
|
END IF
|
||
|
|
DO 90 J = I + 1, 4
|
||
|
|
T16( J, I ) = T16( J, I ) / T16( I, I )
|
||
|
|
BTMP( J ) = BTMP( J ) - T16( J, I )*BTMP( I )
|
||
|
|
DO 80 K = I + 1, 4
|
||
|
|
T16( J, K ) = T16( J, K ) - T16( J, I )*T16( I, K )
|
||
|
|
80 CONTINUE
|
||
|
|
90 CONTINUE
|
||
|
|
100 CONTINUE
|
||
|
|
IF( ABS( T16( 4, 4 ) ).LT.SMIN ) THEN
|
||
|
|
INFO = 1
|
||
|
|
T16( 4, 4 ) = SMIN
|
||
|
|
END IF
|
||
|
|
SCALE = ONE
|
||
|
|
IF( ( EIGHT*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T16( 1, 1 ) ) .OR.
|
||
|
|
$ ( EIGHT*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( T16( 2, 2 ) ) .OR.
|
||
|
|
$ ( EIGHT*SMLNUM )*ABS( BTMP( 3 ) ).GT.ABS( T16( 3, 3 ) ) .OR.
|
||
|
|
$ ( EIGHT*SMLNUM )*ABS( BTMP( 4 ) ).GT.ABS( T16( 4, 4 ) ) ) THEN
|
||
|
|
SCALE = ( ONE / EIGHT ) / MAX( ABS( BTMP( 1 ) ),
|
||
|
|
$ ABS( BTMP( 2 ) ), ABS( BTMP( 3 ) ), ABS( BTMP( 4 ) ) )
|
||
|
|
BTMP( 1 ) = BTMP( 1 )*SCALE
|
||
|
|
BTMP( 2 ) = BTMP( 2 )*SCALE
|
||
|
|
BTMP( 3 ) = BTMP( 3 )*SCALE
|
||
|
|
BTMP( 4 ) = BTMP( 4 )*SCALE
|
||
|
|
END IF
|
||
|
|
DO 120 I = 1, 4
|
||
|
|
K = 5 - I
|
||
|
|
TEMP = ONE / T16( K, K )
|
||
|
|
TMP( K ) = BTMP( K )*TEMP
|
||
|
|
DO 110 J = K + 1, 4
|
||
|
|
TMP( K ) = TMP( K ) - ( TEMP*T16( K, J ) )*TMP( J )
|
||
|
|
110 CONTINUE
|
||
|
|
120 CONTINUE
|
||
|
|
DO 130 I = 1, 3
|
||
|
|
IF( JPIV( 4-I ).NE.4-I ) THEN
|
||
|
|
TEMP = TMP( 4-I )
|
||
|
|
TMP( 4-I ) = TMP( JPIV( 4-I ) )
|
||
|
|
TMP( JPIV( 4-I ) ) = TEMP
|
||
|
|
END IF
|
||
|
|
130 CONTINUE
|
||
|
|
X( 1, 1 ) = TMP( 1 )
|
||
|
|
X( 2, 1 ) = TMP( 2 )
|
||
|
|
X( 1, 2 ) = TMP( 3 )
|
||
|
|
X( 2, 2 ) = TMP( 4 )
|
||
|
|
XNORM = MAX( ABS( TMP( 1 ) )+ABS( TMP( 3 ) ),
|
||
|
|
$ ABS( TMP( 2 ) )+ABS( TMP( 4 ) ) )
|
||
|
|
RETURN
|
||
|
|
*
|
||
|
|
* End of SLASY2
|
||
|
|
*
|
||
|
|
END
|