219 lines
6.3 KiB
Fortran
219 lines
6.3 KiB
Fortran
SUBROUTINE SB04QU( N, M, IND, A, LDA, B, LDB, C, LDC, D, IPR,
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$ 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 construct and solve a linear algebraic system of order 2*M
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C whose coefficient matrix has zeros below the third subdiagonal,
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C and zero elements on the third subdiagonal with even column
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C indices. Such systems appear when solving discrete-time Sylvester
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C equations using the Hessenberg-Schur method.
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C
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C ARGUMENTS
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C
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C Input/Output Parameters
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C
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C N (input) INTEGER
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C The order of the matrix B. N >= 0.
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C
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C M (input) INTEGER
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C The order of the matrix A. M >= 0.
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C
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C IND (input) INTEGER
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C IND and IND - 1 specify the indices of the columns in C
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C to be computed. IND > 1.
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C
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C A (input) DOUBLE PRECISION array, dimension (LDA,M)
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C The leading M-by-M part of this array must contain an
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C upper Hessenberg matrix.
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C
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C LDA INTEGER
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C The leading dimension of array A. LDA >= MAX(1,M).
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C
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C B (input) DOUBLE PRECISION array, dimension (LDB,N)
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C The leading N-by-N part of this array must contain a
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C matrix in real Schur form.
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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,N).
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C
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C C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
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C On entry, the leading M-by-N part of this array must
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C contain the coefficient matrix C of the equation.
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C On exit, the leading M-by-N part of this array contains
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C the matrix C with columns IND-1 and IND updated.
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C
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C LDC INTEGER
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C The leading dimension of array C. LDC >= MAX(1,M).
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C
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C Workspace
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C
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C D DOUBLE PRECISION array, dimension (2*M*M+8*M)
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C
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C IPR INTEGER array, dimension (4*M)
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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 > 0: if INFO = IND, a singular matrix was encountered.
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C
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C METHOD
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C
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C A special linear algebraic system of order 2*M, whose coefficient
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C matrix has zeros below the third subdiagonal and zero elements on
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C the third subdiagonal with even column indices, is constructed and
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C solved. The coefficient matrix is stored compactly, row-wise.
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C
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C REFERENCES
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C
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C [1] Golub, G.H., Nash, S. and Van Loan, C.F.
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C A Hessenberg-Schur method for the problem AX + XB = C.
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C IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
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C
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C [2] Sima, V.
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C Algorithms for Linear-quadratic Optimization.
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C Marcel Dekker, Inc., New York, 1996.
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C
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C NUMERICAL ASPECTS
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C
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C None.
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C
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C CONTRIBUTORS
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C
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C D. Sima, University of Bucharest, May 2000.
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C
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C REVISIONS
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C
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C -
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C
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C KEYWORDS
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C
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C Hessenberg form, orthogonal transformation, real Schur form,
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C Sylvester equation.
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C
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C ******************************************************************
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C
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DOUBLE PRECISION ONE, ZERO
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PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
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C .. Scalar Arguments ..
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INTEGER INFO, IND, LDA, LDB, LDC, M, N
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C .. Array Arguments ..
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INTEGER IPR(*)
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DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(*)
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C .. Local Scalars ..
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INTEGER I, I2, IND1, J, K, K1, K2, M2
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DOUBLE PRECISION TEMP
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C .. Local Arrays ..
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DOUBLE PRECISION DUM(1)
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C .. External Subroutines ..
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EXTERNAL DAXPY, DCOPY, DTRMV, SB04QR
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C .. Intrinsic Functions ..
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INTRINSIC MAX
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C .. Executable Statements ..
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C
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IND1 = IND - 1
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C
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IF ( IND.LT.N ) THEN
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DUM(1) = ZERO
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CALL DCOPY ( M, DUM, 0, D, 1 )
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DO 10 I = IND + 1, N
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CALL DAXPY ( M, B(IND1,I), C(1,I), 1, D, 1 )
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10 CONTINUE
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C
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DO 20 I = 2, M
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C(I,IND1) = C(I,IND1) - A(I,I-1)*D(I-1)
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20 CONTINUE
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CALL DTRMV ( 'Upper', 'No Transpose', 'Non Unit', M, A, LDA,
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$ D, 1 )
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DO 30 I = 1, M
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C(I,IND1) = C(I,IND1) - D(I)
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30 CONTINUE
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C
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CALL DCOPY ( M, DUM, 0, D, 1 )
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DO 40 I = IND + 1, N
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CALL DAXPY ( M, B(IND,I), C(1,I), 1, D, 1 )
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40 CONTINUE
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C
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DO 50 I = 2, M
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C(I,IND) = C(I,IND) - A(I,I-1)*D(I-1)
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50 CONTINUE
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CALL DTRMV ( 'Upper', 'No Transpose', 'Non Unit', M, A, LDA,
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$ D, 1 )
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DO 60 I = 1, M
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C(I,IND) = C(I,IND) - D(I)
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60 CONTINUE
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END IF
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C
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C Construct the linear algebraic system of order 2*M.
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C
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K1 = -1
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M2 = 2*M
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I2 = M2*(M + 3)
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K = M2
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C
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DO 80 I = 1, M
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C
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DO 70 J = MAX( 1, I - 1 ), M
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K1 = K1 + 2
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K2 = K1 + K
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TEMP = A(I,J)
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D(K1) = TEMP * B(IND1,IND1)
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D(K1+1) = TEMP * B(IND1,IND)
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D(K2) = TEMP * B(IND,IND1)
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D(K2+1) = TEMP * B(IND,IND)
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IF ( I.EQ.J ) THEN
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D(K1) = D(K1) + ONE
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D(K2+1) = D(K2+1) + ONE
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END IF
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70 CONTINUE
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C
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K1 = K2
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IF ( I.GT.1 ) K = K - 2
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C
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C Store the right hand side.
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C
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I2 = I2 + 2
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D(I2) = C(I,IND)
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D(I2-1) = C(I,IND1)
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80 CONTINUE
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C
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C Solve the linear algebraic system and store the solution in C.
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C
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CALL SB04QR( M2, D, IPR, INFO )
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C
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IF ( INFO.NE.0 ) THEN
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INFO = IND
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ELSE
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I2 = 0
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C
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DO 90 I = 1, M
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I2 = I2 + 2
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C(I,IND1) = D(IPR(I2-1))
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C(I,IND) = D(IPR(I2))
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90 CONTINUE
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C
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END IF
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C
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RETURN
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C *** Last line of SB04QU ***
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END
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