349 lines
12 KiB
Fortran
349 lines
12 KiB
Fortran
SUBROUTINE TB01LD( DICO, STDOM, JOBA, N, M, P, ALPHA, A, LDA, B,
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$ LDB, C, LDC, NDIM, U, LDU, WR, WI, DWORK,
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$ 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 reduce the system state matrix A to an ordered upper real
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C Schur form by using an orthogonal similarity transformation
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C A <-- U'*A*U and to apply the transformation to the matrices
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C B and C: B <-- U'*B and C <-- C*U.
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C The leading block of the resulting A has eigenvalues in a
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C suitably defined domain of interest.
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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 DICO CHARACTER*1
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C Specifies the type of the system as follows:
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C = 'C': continuous-time system;
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C = 'D': discrete-time system.
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C
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C STDOM CHARACTER*1
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C Specifies whether the domain of interest is of stability
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C type (left part of complex plane or inside of a circle)
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C or of instability type (right part of complex plane or
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C outside of a circle) as follows:
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C = 'S': stability type domain;
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C = 'U': instability type domain.
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C
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C JOBA CHARACTER*1
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C Specifies the shape of the state dynamics matrix on entry
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C as follows:
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C = 'S': A is in an upper real Schur form;
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C = 'G': A is a general square dense matrix.
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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 state-space representation,
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C i.e. the order of the matrix A. N >= 0.
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C
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C M (input) INTEGER
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C The number of system inputs, or of columns of B. M >= 0.
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C
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C P (input) INTEGER
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C The number of system outputs, or of rows of C. P >= 0.
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C
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C ALPHA (input) DOUBLE PRECISION.
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C Specifies the boundary of the domain of interest for the
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C eigenvalues of A. For a continuous-time system
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C (DICO = 'C'), ALPHA is the boundary value for the real
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C parts of eigenvalues, while for a discrete-time system
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C (DICO = 'D'), ALPHA >= 0 represents the boundary value
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C for the moduli of eigenvalues.
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C
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C A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
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C On entry, the leading N-by-N part of this array must
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C contain the unreduced state dynamics matrix A.
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C If JOBA = 'S' then A must be a matrix in real Schur form.
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C On exit, the leading N-by-N part of this array contains
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C the ordered real Schur matrix U' * A * U with the elements
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C below the first subdiagonal set to zero.
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C The leading NDIM-by-NDIM part of A has eigenvalues in the
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C domain of interest and the trailing (N-NDIM)-by-(N-NDIM)
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C part has eigenvalues outside the domain of interest.
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C The domain of interest for lambda(A), the eigenvalues
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C of A, is defined by the parameters ALPHA, DICO and STDOM
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C as follows:
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C For a continuous-time system (DICO = 'C'):
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C Real(lambda(A)) < ALPHA if STDOM = 'S';
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C Real(lambda(A)) > ALPHA if STDOM = 'U';
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C For a discrete-time system (DICO = 'D'):
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C Abs(lambda(A)) < ALPHA if STDOM = 'S';
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C Abs(lambda(A)) > ALPHA if STDOM = 'U'.
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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,N).
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C
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C B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
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C On entry, the leading N-by-M part of this array must
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C contain the input matrix B.
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C On exit, the leading N-by-M part of this array contains
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C the transformed input matrix U' * B.
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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 P-by-N part of this array must
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C contain the output matrix C.
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C On exit, the leading P-by-N part of this array contains
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C the transformed output matrix C * U.
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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,P).
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C
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C NDIM (output) INTEGER
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C The number of eigenvalues of A lying inside the domain of
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C interest for eigenvalues.
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C
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C U (output) DOUBLE PRECISION array, dimension (LDU,N)
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C The leading N-by-N part of this array contains the
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C orthogonal transformation matrix used to reduce A to the
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C real Schur form and/or to reorder the diagonal blocks of
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C real Schur form of A. The first NDIM columns of U form
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C an orthogonal basis for the invariant subspace of A
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C corresponding to the first NDIM eigenvalues.
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C
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C LDU INTEGER
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C The leading dimension of array U. LDU >= max(1,N).
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C
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C WR, WI (output) DOUBLE PRECISION arrays, dimension (N)
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C WR and WI contain the real and imaginary parts,
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C respectively, of the computed eigenvalues of A. The
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C eigenvalues will be in the same order that they appear on
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C the diagonal of the output real Schur form of A. Complex
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C conjugate pairs of eigenvalues will appear consecutively
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C with the eigenvalue having the positive imaginary part
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C first.
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C
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C Workspace
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C
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C DWORK DOUBLE PRECISION array, dimension (LDWORK)
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C On exit, if INFO = 0, DWORK(1) returns the optimal value
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C of LDWORK.
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C
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C LDWORK INTEGER
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C The dimension of working array DWORK.
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C LDWORK >= MAX(1,N) if JOBA = 'S';
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C LDWORK >= MAX(1,3*N) if JOBA = 'G'.
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C For optimum performance LDWORK should be larger.
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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 = 1: the QR algorithm failed to compute all the
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C eigenvalues of A;
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C = 2: a failure occured during the ordering of the real
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C Schur form of A.
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C
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C METHOD
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C
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C Matrix A is reduced to an ordered upper real Schur form using an
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C orthogonal similarity transformation A <-- U'*A*U. This
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C transformation is determined so that the leading block of the
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C resulting A has eigenvalues in a suitably defined domain of
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C interest. Then, the transformation is applied to the matrices B
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C and C: B <-- U'*B and C <-- C*U.
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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 about 14N floating point operations.
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C
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C CONTRIBUTOR
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C
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C A. Varga, German Aerospace Center,
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C DLR Oberpfaffenhofen, March 1998.
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C Based on the RASP routine SRSFOD.
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C
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C REVISIONS
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C
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C V. Sima, Research Institute for Informatics, Bucharest, Oct. 2001.
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C
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C KEYWORDS
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C
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C Invariant subspace, orthogonal transformation, real Schur form,
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C similarity transformation.
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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
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PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
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C .. Scalar Arguments ..
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CHARACTER DICO, JOBA, STDOM
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INTEGER INFO, LDA, LDB, LDC, LDU, LDWORK, M, N, NDIM, P
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DOUBLE PRECISION ALPHA
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C .. Array Arguments ..
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DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), U(LDU,*),
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$ WI(*), WR(*)
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C .. Local Scalars ..
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LOGICAL DISCR, LJOBG
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INTEGER I, IERR, LDWP, SDIM
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DOUBLE PRECISION WRKOPT
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C .. Local Arrays ..
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LOGICAL BWORK( 1 )
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C .. External Functions ..
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LOGICAL LSAME, SELECT
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EXTERNAL LSAME, SELECT
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C .. External Subroutines ..
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EXTERNAL DCOPY, DGEES, DGEMM, DGEMV, DLACPY, DLASET,
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$ MB03QD, MB03QX, XERBLA
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C .. Intrinsic Functions ..
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INTRINSIC DBLE, MAX
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C
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C .. Executable Statements ..
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C
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INFO = 0
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DISCR = LSAME( DICO, 'D' )
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LJOBG = LSAME( JOBA, 'G' )
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C
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C Check input scalar arguments.
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C
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IF( .NOT. ( LSAME( DICO, 'C' ) .OR. DISCR ) ) THEN
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INFO = -1
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ELSE IF( .NOT. ( LSAME( STDOM, 'S' ) .OR.
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$ LSAME( STDOM, 'U' ) ) ) THEN
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INFO = -2
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ELSE IF( .NOT. ( LSAME( JOBA, 'S' ) .OR. LJOBG ) ) 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( M.LT.0 ) THEN
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INFO = -5
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ELSE IF( P.LT.0 ) THEN
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INFO = -6
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ELSE IF( DISCR .AND. ALPHA.LT.ZERO ) THEN
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INFO = -7
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -9
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -11
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ELSE IF( LDC.LT.MAX( 1, P ) ) THEN
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INFO = -13
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ELSE IF( LDU.LT.MAX( 1, N ) ) THEN
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INFO = -16
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ELSE IF( LDWORK.LT.MAX( 1, N ) .OR.
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$ LDWORK.LT.MAX( 1, 3*N ) .AND. LJOBG ) THEN
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INFO = -20
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END IF
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C
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IF( INFO.NE.0 ) THEN
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C
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C Error return.
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C
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CALL XERBLA( 'TB01LD', -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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NDIM = 0
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IF( N.EQ.0 )
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$ RETURN
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C
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IF( LSAME( JOBA, 'G' ) ) THEN
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C
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C Reduce A to real Schur form using an orthogonal similarity
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C transformation A <- U'*A*U, accumulate the transformation in U
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C and compute the eigenvalues of A in (WR,WI).
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C
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C Workspace: need 3*N;
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C prefer larger.
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C
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CALL DGEES( 'Vectors', 'Not ordered', SELECT, N, A, LDA, SDIM,
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$ WR, WI, U, LDU, DWORK, LDWORK, BWORK, INFO )
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WRKOPT = DWORK( 1 )
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IF( INFO.NE.0 ) THEN
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INFO = 1
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RETURN
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END IF
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ELSE
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C
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C Initialize U with an identity matrix.
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C
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CALL DLASET( 'Full', N, N, ZERO, ONE, U, LDU )
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WRKOPT = 0
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END IF
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C
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C Separate the spectrum of A. The leading NDIM-by-NDIM submatrix of
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C A corresponds to the eigenvalues of interest.
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C Workspace: need N.
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C
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CALL MB03QD( DICO, STDOM, 'Update', N, 1, N, ALPHA, A, LDA,
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$ U, LDU, NDIM, DWORK, INFO )
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IF( INFO.NE.0 )
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$ RETURN
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C
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C Compute the eigenvalues.
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C
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CALL MB03QX( N, A, LDA, WR, WI, IERR )
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C
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C Apply the transformation: B <-- U'*B.
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C
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IF( LDWORK.LT.N*M ) THEN
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C
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C Not enough working space for using DGEMM.
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C
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DO 10 I = 1, M
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CALL DCOPY( N, B(1,I), 1, DWORK, 1 )
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CALL DGEMV( 'Transpose', N, N, ONE, U, LDU, DWORK, 1, ZERO,
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$ B(1,I), 1 )
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10 CONTINUE
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C
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ELSE
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CALL DLACPY( 'Full', N, M, B, LDB, DWORK, N )
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CALL DGEMM( 'Transpose', 'No transpose', N, M, N, ONE, U, LDU,
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$ DWORK, N, ZERO, B, LDB )
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WRKOPT = MAX( WRKOPT, DBLE( N*M ) )
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END IF
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C
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C Apply the transformation: C <-- C*U.
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C
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IF( LDWORK.LT.N*P ) THEN
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C
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C Not enough working space for using DGEMM.
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C
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DO 20 I = 1, P
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CALL DCOPY( N, C(I,1), LDC, DWORK, 1 )
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CALL DGEMV( 'Transpose', N, N, ONE, U, LDU, DWORK, 1, ZERO,
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$ C(I,1), LDC )
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20 CONTINUE
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C
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ELSE
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LDWP = MAX( 1, P )
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CALL DLACPY( 'Full', P, N, C, LDC, DWORK, LDWP )
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CALL DGEMM( 'No transpose', 'No transpose', P, N, N, ONE,
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$ DWORK, LDWP, U, LDU, ZERO, C, LDC )
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WRKOPT = MAX( WRKOPT, DBLE( N*P ) )
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END IF
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C
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DWORK( 1 ) = WRKOPT
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C
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RETURN
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C *** Last line of TB01LD ***
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END
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