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SUBROUTINE SB04QU( N, M, IND, A, LDA, B, LDB, C, LDC, D, IPR,
$ INFO )
C
C SLICOT RELEASE 5.0.
C
C Copyright (c) 2002-2009 NICONET e.V.
C
C This program is free software: you can redistribute it and/or
C modify it under the terms of the GNU General Public License as
C published by the Free Software Foundation, either version 2 of
C the License, or (at your option) any later version.
C
C This program is distributed in the hope that it will be useful,
C but WITHOUT ANY WARRANTY; without even the implied warranty of
C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
C GNU General Public License for more details.
C
C You should have received a copy of the GNU General Public License
C along with this program. If not, see
C <http://www.gnu.org/licenses/>.
C
C PURPOSE
C
C To construct and solve a linear algebraic system of order 2*M
C whose coefficient matrix has zeros below the third subdiagonal,
C and zero elements on the third subdiagonal with even column
C indices. Such systems appear when solving discrete-time Sylvester
C equations using the Hessenberg-Schur method.
C
C ARGUMENTS
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the matrix B. N >= 0.
C
C M (input) INTEGER
C The order of the matrix A. M >= 0.
C
C IND (input) INTEGER
C IND and IND - 1 specify the indices of the columns in C
C to be computed. IND > 1.
C
C A (input) DOUBLE PRECISION array, dimension (LDA,M)
C The leading M-by-M part of this array must contain an
C upper Hessenberg matrix.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,M).
C
C B (input) DOUBLE PRECISION array, dimension (LDB,N)
C The leading N-by-N part of this array must contain a
C matrix in real Schur form.
C
C LDB INTEGER
C The leading dimension of array B. LDB >= MAX(1,N).
C
C C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
C On entry, the leading M-by-N part of this array must
C contain the coefficient matrix C of the equation.
C On exit, the leading M-by-N part of this array contains
C the matrix C with columns IND-1 and IND updated.
C
C LDC INTEGER
C The leading dimension of array C. LDC >= MAX(1,M).
C
C Workspace
C
C D DOUBLE PRECISION array, dimension (2*M*M+8*M)
C
C IPR INTEGER array, dimension (4*M)
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C > 0: if INFO = IND, a singular matrix was encountered.
C
C METHOD
C
C A special linear algebraic system of order 2*M, whose coefficient
C matrix has zeros below the third subdiagonal and zero elements on
C the third subdiagonal with even column indices, is constructed and
C solved. The coefficient matrix is stored compactly, row-wise.
C
C REFERENCES
C
C [1] Golub, G.H., Nash, S. and Van Loan, C.F.
C A Hessenberg-Schur method for the problem AX + XB = C.
C IEEE Trans. Auto. Contr., AC-24, pp. 909-913, 1979.
C
C [2] Sima, V.
C Algorithms for Linear-quadratic Optimization.
C Marcel Dekker, Inc., New York, 1996.
C
C NUMERICAL ASPECTS
C
C None.
C
C CONTRIBUTORS
C
C D. Sima, University of Bucharest, May 2000.
C
C REVISIONS
C
C -
C
C KEYWORDS
C
C Hessenberg form, orthogonal transformation, real Schur form,
C Sylvester equation.
C
C ******************************************************************
C
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
C .. Scalar Arguments ..
INTEGER INFO, IND, LDA, LDB, LDC, M, N
C .. Array Arguments ..
INTEGER IPR(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(*)
C .. Local Scalars ..
INTEGER I, I2, IND1, J, K, K1, K2, M2
DOUBLE PRECISION TEMP
C .. Local Arrays ..
DOUBLE PRECISION DUM(1)
C .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DTRMV, SB04QR
C .. Intrinsic Functions ..
INTRINSIC MAX
C .. Executable Statements ..
C
IND1 = IND - 1
C
IF ( IND.LT.N ) THEN
DUM(1) = ZERO
CALL DCOPY ( M, DUM, 0, D, 1 )
DO 10 I = IND + 1, N
CALL DAXPY ( M, B(IND1,I), C(1,I), 1, D, 1 )
10 CONTINUE
C
DO 20 I = 2, M
C(I,IND1) = C(I,IND1) - A(I,I-1)*D(I-1)
20 CONTINUE
CALL DTRMV ( 'Upper', 'No Transpose', 'Non Unit', M, A, LDA,
$ D, 1 )
DO 30 I = 1, M
C(I,IND1) = C(I,IND1) - D(I)
30 CONTINUE
C
CALL DCOPY ( M, DUM, 0, D, 1 )
DO 40 I = IND + 1, N
CALL DAXPY ( M, B(IND,I), C(1,I), 1, D, 1 )
40 CONTINUE
C
DO 50 I = 2, M
C(I,IND) = C(I,IND) - A(I,I-1)*D(I-1)
50 CONTINUE
CALL DTRMV ( 'Upper', 'No Transpose', 'Non Unit', M, A, LDA,
$ D, 1 )
DO 60 I = 1, M
C(I,IND) = C(I,IND) - D(I)
60 CONTINUE
END IF
C
C Construct the linear algebraic system of order 2*M.
C
K1 = -1
M2 = 2*M
I2 = M2*(M + 3)
K = M2
C
DO 80 I = 1, M
C
DO 70 J = MAX( 1, I - 1 ), M
K1 = K1 + 2
K2 = K1 + K
TEMP = A(I,J)
D(K1) = TEMP * B(IND1,IND1)
D(K1+1) = TEMP * B(IND1,IND)
D(K2) = TEMP * B(IND,IND1)
D(K2+1) = TEMP * B(IND,IND)
IF ( I.EQ.J ) THEN
D(K1) = D(K1) + ONE
D(K2+1) = D(K2+1) + ONE
END IF
70 CONTINUE
C
K1 = K2
IF ( I.GT.1 ) K = K - 2
C
C Store the right hand side.
C
I2 = I2 + 2
D(I2) = C(I,IND)
D(I2-1) = C(I,IND1)
80 CONTINUE
C
C Solve the linear algebraic system and store the solution in C.
C
CALL SB04QR( M2, D, IPR, INFO )
C
IF ( INFO.NE.0 ) THEN
INFO = IND
ELSE
I2 = 0
C
DO 90 I = 1, M
I2 = I2 + 2
C(I,IND1) = D(IPR(I2-1))
C(I,IND) = D(IPR(I2))
90 CONTINUE
C
END IF
C
RETURN
C *** Last line of SB04QU ***
END
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