468 lines
15 KiB
Fortran
468 lines
15 KiB
Fortran
SUBROUTINE SB03MU( LTRANL, LTRANR, ISGN, N1, N2, TL, LDTL, TR,
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$ LDTR, B, LDB, SCALE, X, LDX, XNORM, INFO )
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C
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C SLICOT RELEASE 5.0.
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C
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C Copyright (c) 2002-2009 NICONET e.V.
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C
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C This program is free software: you can redistribute it and/or
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C modify it under the terms of the GNU General Public License as
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C published by the Free Software Foundation, either version 2 of
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C the License, or (at your option) any later version.
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C
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C This program is distributed in the hope that it will be useful,
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C but WITHOUT ANY WARRANTY; without even the implied warranty of
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C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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C GNU General Public License for more details.
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C
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C You should have received a copy of the GNU General Public License
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C along with this program. If not, see
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C <http://www.gnu.org/licenses/>.
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C
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C PURPOSE
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C
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C To solve for the N1-by-N2 matrix X, 1 <= N1,N2 <= 2, in
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C
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C ISGN*op(TL)*X*op(TR) - X = SCALE*B,
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C
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C where TL is N1-by-N1, TR is N2-by-N2, B is N1-by-N2, and ISGN = 1
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C or -1. op(T) = T or T', where T' denotes the transpose of T.
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C
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C ARGUMENTS
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C
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C Mode Parameters
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C
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C LTRANL LOGICAL
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C Specifies the form of op(TL) to be used, as follows:
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C = .FALSE.: op(TL) = TL,
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C = .TRUE. : op(TL) = TL'.
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C
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C LTRANR LOGICAL
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C Specifies the form of op(TR) to be used, as follows:
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C = .FALSE.: op(TR) = TR,
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C = .TRUE. : op(TR) = TR'.
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C
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C ISGN INTEGER
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C Specifies the sign of the equation as described before.
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C ISGN may only be 1 or -1.
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C
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C Input/Output Parameters
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C
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C N1 (input) INTEGER
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C The order of matrix TL. N1 may only be 0, 1 or 2.
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C
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C N2 (input) INTEGER
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C The order of matrix TR. N2 may only be 0, 1 or 2.
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C
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C TL (input) DOUBLE PRECISION array, dimension (LDTL,2)
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C The leading N1-by-N1 part of this array must contain the
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C matrix TL.
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C
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C LDTL INTEGER
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C The leading dimension of array TL. LDTL >= MAX(1,N1).
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C
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C TR (input) DOUBLE PRECISION array, dimension (LDTR,2)
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C The leading N2-by-N2 part of this array must contain the
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C matrix TR.
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C
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C LDTR INTEGER
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C The leading dimension of array TR. LDTR >= MAX(1,N2).
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C
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C B (input) DOUBLE PRECISION array, dimension (LDB,2)
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C The leading N1-by-N2 part of this array must contain the
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C right-hand side of the equation.
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C
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C LDB INTEGER
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C The leading dimension of array B. LDB >= MAX(1,N1).
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C
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C SCALE (output) DOUBLE PRECISION
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C The scale factor. SCALE is chosen less than or equal to 1
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C to prevent the solution overflowing.
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C
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C X (output) DOUBLE PRECISION array, dimension (LDX,N2)
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C The leading N1-by-N2 part of this array contains the
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C solution of the equation.
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C Note that X may be identified with B in the calling
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C statement.
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C
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C LDX INTEGER
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C The leading dimension of array X. LDX >= MAX(1,N1).
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C
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C XNORM (output) DOUBLE PRECISION
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C The infinity-norm of the solution.
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C
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C Error Indicator
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C
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C INFO INTEGER
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C = 0: successful exit;
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C = 1: if TL and TR have almost reciprocal eigenvalues, so
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C TL or TR is perturbed to get a nonsingular equation.
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C
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C NOTE: In the interests of speed, this routine does not
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C check the inputs for errors.
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C
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C METHOD
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C
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C The equivalent linear algebraic system of equations is formed and
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C solved using Gaussian elimination with complete pivoting.
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C
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C REFERENCES
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C
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C [1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
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C Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
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C Ostrouchov, S., and Sorensen, D.
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C LAPACK Users' Guide: Second Edition.
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C SIAM, Philadelphia, 1995.
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C
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C NUMERICAL ASPECTS
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C
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C The algorithm is stable and reliable, since Gaussian elimination
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C with complete pivoting is used.
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C
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C CONTRIBUTOR
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C
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C Release 3.0: V. Sima, Katholieke Univ. Leuven, Belgium, May 1997.
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C Based on DLASD2 by P. Petkov, Tech. University of Sofia, September
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C 1993.
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C
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C REVISIONS
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C
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C V. Sima, Katholieke Univ. Leuven, Belgium, May 1999.
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C
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C KEYWORDS
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C
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C Discrete-time system, Sylvester equation, matrix algebra.
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C
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C ******************************************************************
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C
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C .. Parameters ..
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DOUBLE PRECISION ZERO, ONE, TWO, HALF, EIGHT
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0,
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$ TWO = 2.0D+0, HALF = 0.5D+0, EIGHT = 8.0D+0 )
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C ..
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C .. 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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DOUBLE PRECISION SCALE, XNORM
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C ..
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C .. Array Arguments ..
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DOUBLE PRECISION B( LDB, * ), TL( LDTL, * ), TR( LDTR, * ),
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$ X( LDX, * )
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C ..
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C .. 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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DOUBLE PRECISION BET, EPS, GAM, L21, SGN, SMIN, SMLNUM, TAU1,
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$ TEMP, U11, U12, U22, XMAX
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C ..
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C .. 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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DOUBLE PRECISION BTMP( 4 ), T16( 4, 4 ), TMP( 4 ), X2( 2 )
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C ..
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C .. External Functions ..
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INTEGER IDAMAX
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DOUBLE PRECISION DLAMCH
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EXTERNAL DLAMCH, IDAMAX
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C ..
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C .. External Subroutines ..
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EXTERNAL DSWAP
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C ..
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C .. Intrinsic Functions ..
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INTRINSIC ABS, MAX
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C ..
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C .. 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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C ..
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C .. Executable Statements ..
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C
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C Do not check the input parameters for errors.
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C
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INFO = 0
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SCALE = ONE
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C
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C Quick return if possible.
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C
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IF( N1.EQ.0 .OR. N2.EQ.0 ) THEN
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XNORM = ZERO
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RETURN
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END IF
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C
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C Set constants to control overflow.
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C
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EPS = DLAMCH( 'P' )
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SMLNUM = DLAMCH( 'S' ) / EPS
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SGN = ISGN
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C
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K = N1 + N1 + N2 - 2
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GO TO ( 10, 20, 30, 50 )K
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C
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C 1-by-1: SGN*TL11*X*TR11 - X = B11.
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C
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10 CONTINUE
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TAU1 = SGN*TL( 1, 1 )*TR( 1, 1 ) - ONE
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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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C
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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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C
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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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C
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C 1-by-2:
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C ISGN*TL11*[X11 X12]*op[TR11 TR12] = [B11 B12].
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C [TR21 TR22]
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C
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20 CONTINUE
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C
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SMIN = MAX( MAX( ABS( TR( 1, 1 ) ), ABS( TR( 1, 2 ) ),
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$ ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) )
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$ *ABS( TL( 1, 1 ) )*EPS,
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$ SMLNUM )
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TMP( 1 ) = SGN*TL( 1, 1 )*TR( 1, 1 ) - ONE
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TMP( 4 ) = SGN*TL( 1, 1 )*TR( 2, 2 ) - ONE
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IF( LTRANR ) THEN
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TMP( 2 ) = SGN*TL( 1, 1 )*TR( 2, 1 )
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TMP( 3 ) = SGN*TL( 1, 1 )*TR( 1, 2 )
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ELSE
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TMP( 2 ) = SGN*TL( 1, 1 )*TR( 1, 2 )
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TMP( 3 ) = SGN*TL( 1, 1 )*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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C
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C 2-by-1:
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C ISGN*op[TL11 TL12]*[X11]*TR11 = [B11].
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C [TL21 TL22] [X21] [B21]
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C
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30 CONTINUE
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SMIN = MAX( MAX( ABS( TL( 1, 1 ) ), ABS( TL( 1, 2 ) ),
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$ ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) )
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$ *ABS( TR( 1, 1 ) )*EPS,
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$ SMLNUM )
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TMP( 1 ) = SGN*TL( 1, 1 )*TR( 1, 1 ) - ONE
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TMP( 4 ) = SGN*TL( 2, 2 )*TR( 1, 1 ) - ONE
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IF( LTRANL ) THEN
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TMP( 2 ) = SGN*TL( 1, 2 )*TR( 1, 1 )
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TMP( 3 ) = SGN*TL( 2, 1 )*TR( 1, 1 )
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ELSE
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TMP( 2 ) = SGN*TL( 2, 1 )*TR( 1, 1 )
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TMP( 3 ) = SGN*TL( 1, 2 )*TR( 1, 1 )
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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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C
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C Solve 2-by-2 system using complete pivoting.
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C Set pivots less than SMIN to SMIN.
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C
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IPIV = IDAMAX( 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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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( X2( 1 ) ) + ABS( X2( 2 ) )
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ELSE
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X( 2, 1 ) = X2( 2 )
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XNORM = MAX( ABS( X2( 1 ) ), ABS( X2( 2 ) ) )
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END IF
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RETURN
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C
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C 2-by-2:
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C ISGN*op[TL11 TL12]*[X11 X12]*op[TR11 TR12]-[X11 X12] = [B11 B12].
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C [TL21 TL22] [X21 X22] [TR21 TR22] [X21 X22] [B21 B22]
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C
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C Solve equivalent 4-by-4 system using complete pivoting.
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C Set pivots less than SMIN to SMIN.
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C
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50 CONTINUE
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SMIN = MAX( ABS( TR( 1, 1 ) ), ABS( TR( 1, 2 ) ),
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$ ABS( TR( 2, 1 ) ), ABS( TR( 2, 2 ) ) )
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SMIN = MAX( ABS( TL( 1, 1 ) ), ABS( TL( 1, 2 ) ),
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$ ABS( TL( 2, 1 ) ), ABS( TL( 2, 2 ) ) )*SMIN
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SMIN = MAX( EPS*SMIN, SMLNUM )
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T16( 1, 1 ) = SGN*TL( 1, 1 )*TR( 1, 1 ) - ONE
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T16( 2, 2 ) = SGN*TL( 2, 2 )*TR( 1, 1 ) - ONE
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T16( 3, 3 ) = SGN*TL( 1, 1 )*TR( 2, 2 ) - ONE
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T16( 4, 4 ) = SGN*TL( 2, 2 )*TR( 2, 2 ) - ONE
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IF( LTRANL ) THEN
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T16( 1, 2 ) = SGN*TL( 2, 1 )*TR( 1, 1 )
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T16( 2, 1 ) = SGN*TL( 1, 2 )*TR( 1, 1 )
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T16( 3, 4 ) = SGN*TL( 2, 1 )*TR( 2, 2 )
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T16( 4, 3 ) = SGN*TL( 1, 2 )*TR( 2, 2 )
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ELSE
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T16( 1, 2 ) = SGN*TL( 1, 2 )*TR( 1, 1 )
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T16( 2, 1 ) = SGN*TL( 2, 1 )*TR( 1, 1 )
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T16( 3, 4 ) = SGN*TL( 1, 2 )*TR( 2, 2 )
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T16( 4, 3 ) = SGN*TL( 2, 1 )*TR( 2, 2 )
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END IF
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IF( LTRANR ) THEN
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T16( 1, 3 ) = SGN*TL( 1, 1 )*TR( 1, 2 )
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T16( 2, 4 ) = SGN*TL( 2, 2 )*TR( 1, 2 )
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T16( 3, 1 ) = SGN*TL( 1, 1 )*TR( 2, 1 )
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T16( 4, 2 ) = SGN*TL( 2, 2 )*TR( 2, 1 )
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ELSE
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T16( 1, 3 ) = SGN*TL( 1, 1 )*TR( 2, 1 )
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T16( 2, 4 ) = SGN*TL( 2, 2 )*TR( 2, 1 )
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T16( 3, 1 ) = SGN*TL( 1, 1 )*TR( 1, 2 )
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T16( 4, 2 ) = SGN*TL( 2, 2 )*TR( 1, 2 )
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END IF
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IF( LTRANL .AND. LTRANR ) THEN
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T16( 1, 4 ) = SGN*TL( 2, 1 )*TR( 1, 2 )
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T16( 2, 3 ) = SGN*TL( 1, 2 )*TR( 1, 2 )
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T16( 3, 2 ) = SGN*TL( 2, 1 )*TR( 2, 1 )
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T16( 4, 1 ) = SGN*TL( 1, 2 )*TR( 2, 1 )
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ELSE IF( LTRANL .AND. .NOT.LTRANR ) THEN
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T16( 1, 4 ) = SGN*TL( 2, 1 )*TR( 2, 1 )
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T16( 2, 3 ) = SGN*TL( 1, 2 )*TR( 2, 1 )
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T16( 3, 2 ) = SGN*TL( 2, 1 )*TR( 1, 2 )
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T16( 4, 1 ) = SGN*TL( 1, 2 )*TR( 1, 2 )
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ELSE IF( .NOT.LTRANL .AND. LTRANR ) THEN
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T16( 1, 4 ) = SGN*TL( 1, 2 )*TR( 1, 2 )
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T16( 2, 3 ) = SGN*TL( 2, 1 )*TR( 1, 2 )
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T16( 3, 2 ) = SGN*TL( 1, 2 )*TR( 2, 1 )
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T16( 4, 1 ) = SGN*TL( 2, 1 )*TR( 2, 1 )
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ELSE
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T16( 1, 4 ) = SGN*TL( 1, 2 )*TR( 2, 1 )
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T16( 2, 3 ) = SGN*TL( 2, 1 )*TR( 2, 1 )
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T16( 3, 2 ) = SGN*TL( 1, 2 )*TR( 1, 2 )
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T16( 4, 1 ) = SGN*TL( 2, 1 )*TR( 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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BTMP( 3 ) = B( 1, 2 )
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BTMP( 4 ) = B( 2, 2 )
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C
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C Perform elimination
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C
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DO 100 I = 1, 3
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XMAX = ZERO
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C
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DO 70 IP = I, 4
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C
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DO 60 JP = I, 4
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IF( ABS( T16( IP, JP ) ).GE.XMAX ) THEN
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XMAX = ABS( T16( IP, JP ) )
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IPSV = IP
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JPSV = JP
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END IF
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60 CONTINUE
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C
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70 CONTINUE
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C
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IF( IPSV.NE.I ) THEN
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CALL DSWAP( 4, T16( IPSV, 1 ), 4, T16( I, 1 ), 4 )
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TEMP = BTMP( I )
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BTMP( I ) = BTMP( IPSV )
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BTMP( IPSV ) = TEMP
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END IF
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IF( JPSV.NE.I )
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$ CALL DSWAP( 4, T16( 1, JPSV ), 1, T16( 1, I ), 1 )
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JPIV( I ) = JPSV
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IF( ABS( T16( I, I ) ).LT.SMIN ) THEN
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INFO = 1
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T16( I, I ) = SMIN
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END IF
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C
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DO 90 J = I + 1, 4
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T16( J, I ) = T16( J, I ) / T16( I, I )
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BTMP( J ) = BTMP( J ) - T16( J, I )*BTMP( I )
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C
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DO 80 K = I + 1, 4
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T16( J, K ) = T16( J, K ) - T16( J, I )*T16( I, K )
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80 CONTINUE
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C
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90 CONTINUE
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C
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100 CONTINUE
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C
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IF( ABS( T16( 4, 4 ) ).LT.SMIN )
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$ T16( 4, 4 ) = SMIN
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IF( ( EIGHT*SMLNUM )*ABS( BTMP( 1 ) ).GT.ABS( T16( 1, 1 ) ) .OR.
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$ ( EIGHT*SMLNUM )*ABS( BTMP( 2 ) ).GT.ABS( T16( 2, 2 ) ) .OR.
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$ ( EIGHT*SMLNUM )*ABS( BTMP( 3 ) ).GT.ABS( T16( 3, 3 ) ) .OR.
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$ ( EIGHT*SMLNUM )*ABS( BTMP( 4 ) ).GT.ABS( T16( 4, 4 ) ) ) THEN
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SCALE = ( ONE / EIGHT ) / MAX( ABS( BTMP( 1 ) ),
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$ ABS( BTMP( 2 ) ), ABS( BTMP( 3 ) ),
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$ ABS( BTMP( 4 ) ) )
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|
BTMP( 1 ) = BTMP( 1 )*SCALE
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|
BTMP( 2 ) = BTMP( 2 )*SCALE
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|
BTMP( 3 ) = BTMP( 3 )*SCALE
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BTMP( 4 ) = BTMP( 4 )*SCALE
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|
END IF
|
|
C
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DO 120 I = 1, 4
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|
K = 5 - I
|
|
TEMP = ONE / T16( K, K )
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|
TMP( K ) = BTMP( K )*TEMP
|
|
C
|
|
DO 110 J = K + 1, 4
|
|
TMP( K ) = TMP( K ) - ( TEMP*T16( K, J ) )*TMP( J )
|
|
110 CONTINUE
|
|
C
|
|
120 CONTINUE
|
|
C
|
|
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
|
|
C
|
|
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 ) ) )
|
|
C
|
|
RETURN
|
|
C *** Last line of SB03MU ***
|
|
END
|