399 lines
13 KiB
Fortran
399 lines
13 KiB
Fortran
SUBROUTINE SB03SX( TRANA, UPLO, LYAPUN, N, XANORM, T, LDT, U, LDU,
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$ R, LDR, FERR, IWORK, DWORK, LDWORK, 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 estimate a forward error bound for the solution X of a real
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C discrete-time Lyapunov matrix equation,
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C
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C op(A)'*X*op(A) - X = C,
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C
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C where op(A) = A or A' (A**T) and C is symmetric (C = C**T). The
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C matrix A, the right hand side C, and the solution X are N-by-N.
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C An absolute residual matrix, which takes into account the rounding
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C errors in forming it, is given in the array R.
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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 TRANA CHARACTER*1
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C Specifies the form of op(A) to be used, as follows:
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C = 'N': op(A) = A (No transpose);
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C = 'T': op(A) = A**T (Transpose);
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C = 'C': op(A) = A**T (Conjugate transpose = Transpose).
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C
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C UPLO CHARACTER*1
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C Specifies which part of the symmetric matrix R is to be
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C used, as follows:
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C = 'U': Upper triangular part;
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C = 'L': Lower triangular part.
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C
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C LYAPUN CHARACTER*1
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C Specifies whether or not the original Lyapunov equations
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C should be solved, as follows:
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C = 'O': Solve the original Lyapunov equations, updating
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C the right-hand sides and solutions with the
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C matrix U, e.g., X <-- U'*X*U;
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C = 'R': Solve reduced Lyapunov equations only, without
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C updating the right-hand sides and solutions.
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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 matrices A and R. N >= 0.
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C
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C XANORM (input) DOUBLE PRECISION
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C The absolute (maximal) norm of the symmetric solution
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C matrix X of the Lyapunov equation. XANORM >= 0.
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C
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C T (input) DOUBLE PRECISION array, dimension (LDT,N)
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C The leading N-by-N upper Hessenberg part of this array
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C must contain the upper quasi-triangular matrix T in Schur
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C canonical form from a Schur factorization of A.
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C
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C LDT INTEGER
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C The leading dimension of array T. LDT >= MAX(1,N).
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C
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C U (input) DOUBLE PRECISION array, dimension (LDU,N)
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C The leading N-by-N part of this array must contain the
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C orthogonal matrix U from a real Schur factorization of A.
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C If LYAPUN = 'R', the array U is not referenced.
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C
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C LDU INTEGER
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C The leading dimension of array U.
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C LDU >= 1, if LYAPUN = 'R';
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C LDU >= MAX(1,N), if LYAPUN = 'O'.
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C
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C R (input/output) DOUBLE PRECISION array, dimension (LDR,N)
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C On entry, if UPLO = 'U', the leading N-by-N upper
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C triangular part of this array must contain the upper
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C triangular part of the absolute residual matrix R, with
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C bounds on rounding errors added.
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C On entry, if UPLO = 'L', the leading N-by-N lower
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C triangular part of this array must contain the lower
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C triangular part of the absolute residual matrix R, with
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C bounds on rounding errors added.
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C On exit, the leading N-by-N part of this array contains
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C the symmetric absolute residual matrix R (with bounds on
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C rounding errors added), fully stored.
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C
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C LDR INTEGER
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C The leading dimension of array R. LDR >= MAX(1,N).
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C
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C FERR (output) DOUBLE PRECISION
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C An estimated forward error bound for the solution X.
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C If XTRUE is the true solution, FERR bounds the magnitude
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C of the largest entry in (X - XTRUE) divided by the
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C magnitude of the largest entry in X.
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C If N = 0 or XANORM = 0, FERR is set to 0, without any
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C calculations.
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C
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C Workspace
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C
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C IWORK INTEGER array, dimension (N*N)
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C
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C DWORK DOUBLE PRECISION array, dimension (LDWORK)
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C
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C LDWORK INTEGER
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C The length of the array DWORK.
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C LDWORK >= 0, if N = 0;
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C LDWORK >= MAX(3,2*N*N), if N > 0.
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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 = -i, the i-th argument had an illegal
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C value;
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C = N+1: if T has almost reciprocal eigenvalues; perturbed
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C values were used to solve Lyapunov equations (but
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C the matrix T is unchanged).
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C
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C METHOD
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C
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C The forward error bound is estimated using a practical error bound
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C similar to the one proposed in [1], based on the 1-norm estimator
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C in [2].
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C
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C REFERENCES
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C
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C [1] Higham, N.J.
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C Perturbation theory and backward error for AX-XB=C.
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C BIT, vol. 33, pp. 124-136, 1993.
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C
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C [2] Higham, N.J.
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C FORTRAN codes for estimating the one-norm of a real or
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C complex matrix, with applications to condition estimation.
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C ACM Trans. Math. Softw., 14, pp. 381-396, 1988.
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C
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C NUMERICAL ASPECTS
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C 3
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C The algorithm requires 0(N ) operations.
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C
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C FURTHER COMMENTS
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C
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C The option LYAPUN = 'R' may occasionally produce slightly worse
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C or better estimates, and it is much faster than the option 'O'.
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C The routine can be also used as a final step in estimating a
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C forward error bound for the solution of a discrete-time algebraic
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C matrix Riccati equation.
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C
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C CONTRIBUTOR
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C
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C V. Sima, Research Institute for Informatics, Bucharest, Romania,
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C Oct. 1998. Partly based on DDLSVX (and then SB03SD) by P. Petkov,
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C Tech. University of Sofia, March 1998 (and December 1998).
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C
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C REVISIONS
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C
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C February 6, 1999, V. Sima, Katholieke Univ. Leuven, Belgium.
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C V. Sima, Research Institute for Informatics, Bucharest, Oct. 2004.
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C
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C KEYWORDS
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C
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C Lyapunov equation, orthogonal transformation, real Schur form.
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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, HALF
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, HALF = 0.5D+0 )
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C ..
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C .. Scalar Arguments ..
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CHARACTER LYAPUN, TRANA, UPLO
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INTEGER INFO, LDR, LDT, LDU, LDWORK, N
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DOUBLE PRECISION FERR, XANORM
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C ..
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C .. Array Arguments ..
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INTEGER IWORK( * )
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DOUBLE PRECISION DWORK( * ), R( LDR, * ), T( LDT, * ),
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$ U( LDU, * )
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C ..
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C .. Local Scalars ..
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LOGICAL LOWER, NOTRNA, UPDATE
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CHARACTER TRANAT, UPLOW
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INTEGER I, IJ, INFO2, ITMP, J, KASE, NN
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DOUBLE PRECISION EST, SCALE, TEMP
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C ..
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C .. External Functions ..
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LOGICAL LSAME
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DOUBLE PRECISION DLANSY
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EXTERNAL DLANSY, LSAME
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C ..
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C .. External Subroutines ..
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EXTERNAL DLACON, DSCAL, MA02ED, MB01RU, SB03MX, XERBLA
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C ..
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C .. Intrinsic Functions ..
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INTRINSIC MAX
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C ..
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C .. Executable Statements ..
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C
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C Decode and Test input parameters.
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C
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NOTRNA = LSAME( TRANA, 'N' )
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UPDATE = LSAME( LYAPUN, 'O' )
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C
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NN = N*N
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INFO = 0
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IF( .NOT.( NOTRNA .OR. LSAME( TRANA, 'T' ) .OR.
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$ LSAME( TRANA, 'C' ) ) ) THEN
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INFO = -1
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ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR. LSAME( UPLO, 'U' ) ) )
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$ THEN
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INFO = -2
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ELSE IF( .NOT.( UPDATE .OR. LSAME( LYAPUN, 'R' ) ) ) THEN
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INFO = -3
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ELSE IF( N.LT.0 ) THEN
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INFO = -4
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ELSE IF( XANORM.LT.ZERO ) THEN
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INFO = -5
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ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
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INFO = -7
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ELSE IF( LDU.LT.1 .OR. ( UPDATE .AND. LDU.LT.N ) ) THEN
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INFO = -9
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ELSE IF( LDR.LT.MAX( 1, N ) ) THEN
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INFO = -11
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ELSE IF( LDWORK.LT.0 .OR.
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$ ( LDWORK.LT.MAX( 3, 2*NN ) .AND. N.GT.0 ) ) THEN
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INFO = -15
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END IF
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C
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'SB03SX', -INFO )
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RETURN
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END IF
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C
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C Quick return if possible.
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C
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FERR = ZERO
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IF( N.EQ.0 .OR. XANORM.EQ.ZERO )
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$ RETURN
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C
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ITMP = NN + 1
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C
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IF( NOTRNA ) THEN
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TRANAT = 'T'
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ELSE
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TRANAT = 'N'
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END IF
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C
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C Fill in the remaining triangle of the symmetric residual matrix.
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C
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CALL MA02ED( UPLO, N, R, LDR )
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C
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KASE = 0
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C
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C REPEAT
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10 CONTINUE
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CALL DLACON( NN, DWORK( ITMP ), DWORK, IWORK, EST, KASE )
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IF( KASE.NE.0 ) THEN
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C
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C Select the triangular part of symmetric matrix to be used.
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C
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IF( DLANSY( '1-norm', 'Upper', N, DWORK, N, DWORK( ITMP ) )
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$ .GE.
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$ DLANSY( '1-norm', 'Lower', N, DWORK, N, DWORK( ITMP ) )
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$ ) THEN
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UPLOW = 'U'
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LOWER = .FALSE.
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ELSE
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UPLOW = 'L'
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LOWER = .TRUE.
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END IF
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C
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IF( KASE.EQ.2 ) THEN
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IJ = 0
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IF( LOWER ) THEN
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C
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C Scale the lower triangular part of symmetric matrix
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C by the residual matrix.
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C
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DO 30 J = 1, N
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DO 20 I = J, N
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IJ = IJ + 1
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DWORK( IJ ) = DWORK( IJ )*R( I, J )
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20 CONTINUE
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IJ = IJ + J
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30 CONTINUE
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ELSE
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C
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C Scale the upper triangular part of symmetric matrix
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C by the residual matrix.
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C
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DO 50 J = 1, N
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DO 40 I = 1, J
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IJ = IJ + 1
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DWORK( IJ ) = DWORK( IJ )*R( I, J )
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40 CONTINUE
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IJ = IJ + N - J
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50 CONTINUE
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END IF
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END IF
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C
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IF( UPDATE ) THEN
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C
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C Transform the right-hand side: RHS := U'*RHS*U.
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C
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CALL MB01RU( UPLOW, 'Transpose', N, N, ZERO, ONE, DWORK, N,
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$ U, LDU, DWORK, N, DWORK( ITMP ), NN, INFO2 )
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CALL DSCAL( N, HALF, DWORK, N+1 )
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END IF
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CALL MA02ED( UPLOW, N, DWORK, N )
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C
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IF( KASE.EQ.2 ) THEN
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C
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C Solve op(T)'*Y*op(T) - Y = scale*RHS.
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C
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CALL SB03MX( TRANA, N, T, LDT, DWORK, N, SCALE,
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$ DWORK( ITMP ), INFO2 )
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ELSE
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C
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C Solve op(T)*W*op(T)' - W = scale*RHS.
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C
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CALL SB03MX( TRANAT, N, T, LDT, DWORK, N, SCALE,
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$ DWORK( ITMP ), INFO2 )
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END IF
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C
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IF( INFO2.GT.0 )
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$ INFO = N + 1
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C
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IF( UPDATE ) THEN
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C
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C Transform back to obtain the solution: Z := U*Z*U', with
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C Z = Y or Z = W.
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C
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CALL MB01RU( UPLOW, 'No transpose', N, N, ZERO, ONE, DWORK,
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$ N, U, LDU, DWORK, N, DWORK( ITMP ), NN, INFO2 )
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CALL DSCAL( N, HALF, DWORK, N+1 )
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END IF
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C
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IF( KASE.EQ.1 ) THEN
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IJ = 0
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IF( LOWER ) THEN
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C
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C Scale the lower triangular part of symmetric matrix
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C by the residual matrix.
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C
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DO 70 J = 1, N
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DO 60 I = J, N
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IJ = IJ + 1
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DWORK( IJ ) = DWORK( IJ )*R( I, J )
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60 CONTINUE
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IJ = IJ + J
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70 CONTINUE
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ELSE
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C
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C Scale the upper triangular part of symmetric matrix
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C by the residual matrix.
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C
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DO 90 J = 1, N
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DO 80 I = 1, J
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IJ = IJ + 1
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DWORK( IJ ) = DWORK( IJ )*R( I, J )
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80 CONTINUE
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IJ = IJ + N - J
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90 CONTINUE
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END IF
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END IF
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C
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C Fill in the remaining triangle of the symmetric matrix.
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C
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CALL MA02ED( UPLOW, N, DWORK, N )
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GO TO 10
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END IF
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C
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C UNTIL KASE = 0
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C
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C Compute the estimate of the relative error.
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C
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TEMP = XANORM*SCALE
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IF( TEMP.GT.EST ) THEN
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FERR = EST / TEMP
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ELSE
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FERR = ONE
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END IF
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C
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RETURN
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C
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C *** Last line of SB03SX ***
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END
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