520 lines
18 KiB
Fortran
520 lines
18 KiB
Fortran
SUBROUTINE MB04TS( TRANA, TRANB, N, ILO, A, LDA, B, LDB, G, LDG,
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$ Q, LDQ, CSL, CSR, TAUL, TAUR, DWORK, LDWORK,
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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 compute a symplectic URV (SURV) decomposition of a real
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C 2N-by-2N matrix H:
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C
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C [ op(A) G ] T [ op(R11) R12 ] T
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C H = [ ] = U R V = U * [ ] * V ,
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C [ Q op(B) ] [ 0 op(R22) ]
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C
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C where A, B, G, Q, R12 are real N-by-N matrices, op(R11) is a real
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C N-by-N upper triangular matrix, op(R22) is a real N-by-N lower
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C Hessenberg matrix and U, V are 2N-by-2N orthogonal symplectic
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C matrices. Unblocked version.
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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 ) as follows:
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C = 'N': op( A ) = A;
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C = 'T': op( A ) = A';
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C = 'C': op( A ) = A'.
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C
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C TRANB CHARACTER*1
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C Specifies the form of op( B ) as follows:
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C = 'N': op( B ) = B;
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C = 'T': op( B ) = B';
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C = 'C': op( B ) = B'.
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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 A. N >= 0.
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C
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C ILO (input) INTEGER
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C It is assumed that op(A) is already upper triangular,
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C op(B) is lower triangular and Q is zero in rows and
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C columns 1:ILO-1. ILO is normally set by a previous call
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C to MB04DD; otherwise it should be set to 1.
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C 1 <= ILO <= N, if N > 0; ILO=1, if N=0.
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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 matrix A.
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C On exit, the leading N-by-N part of this array contains
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C the triangular matrix R11, and in the zero part
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C information about the elementary reflectors used to
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C compute the SURV decomposition.
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C
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C LDA INTEGER
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C The leading dimension of the array A. LDA >= MAX(1,N).
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C
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C B (input/output) DOUBLE PRECISION array, dimension (LDB,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 matrix B.
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C On exit, the leading N-by-N part of this array contains
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C the Hessenberg matrix R22, and in the zero part
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C information about the elementary reflectors used to
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C compute the SURV decomposition.
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C
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C LDB INTEGER
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C The leading dimension of the array B. LDB >= MAX(1,N).
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C
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C G (input/output) DOUBLE PRECISION array, dimension (LDG,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 matrix G.
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C On exit, the leading N-by-N part of this array contains
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C the matrix R12.
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C
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C LDG INTEGER
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C The leading dimension of the array G. LDG >= MAX(1,N).
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C
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C Q (input/output) DOUBLE PRECISION array, dimension (LDQ,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 matrix Q.
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C On exit, the leading N-by-N part of this array contains
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C information about the elementary reflectors used to
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C compute the SURV decomposition.
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C
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C LDQ INTEGER
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C The leading dimension of the array Q. LDG >= MAX(1,N).
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C
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C CSL (output) DOUBLE PRECISION array, dimension (2N)
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C On exit, the first 2N elements of this array contain the
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C cosines and sines of the symplectic Givens rotations
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C applied from the left-hand side used to compute the SURV
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C decomposition.
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C
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C CSR (output) DOUBLE PRECISION array, dimension (2N-2)
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C On exit, the first 2N-2 elements of this array contain the
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C cosines and sines of the symplectic Givens rotations
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C applied from the right-hand side used to compute the SURV
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C decomposition.
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C
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C TAUL (output) DOUBLE PRECISION array, dimension (N)
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C On exit, the first N elements of this array contain the
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C scalar factors of some of the elementary reflectors
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C applied from the left-hand side.
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C
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C TAUR (output) DOUBLE PRECISION array, dimension (N-1)
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C On exit, the first N-1 elements of this array contain the
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C scalar factors of some of the elementary reflectors
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C applied from the right-hand side.
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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
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C value of LDWORK.
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C On exit, if INFO = -16, DWORK(1) returns the minimum
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C value of LDWORK.
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C
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C LDWORK INTEGER
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C The length of the array DWORK. LDWORK >= MAX(1,N).
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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
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C METHOD
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C
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C The matrices U and V are represented as products of symplectic
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C reflectors and Givens rotators
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C
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C U = diag( HU(1),HU(1) ) GU(1) diag( FU(1),FU(1) )
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C diag( HU(2),HU(2) ) GU(2) diag( FU(2),FU(2) )
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C ....
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C diag( HU(n),HU(n) ) GU(n) diag( FU(n),FU(n) ),
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C
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C V = diag( HV(1),HV(1) ) GV(1) diag( FV(1),FV(1) )
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C diag( HV(2),HV(2) ) GV(2) diag( FV(2),FV(2) )
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C ....
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C diag( HV(n-1),HV(n-1) ) GV(n-1) diag( FV(n-1),FV(n-1) ).
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C
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C Each HU(i) has the form
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C
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C HU(i) = I - tau * v * v'
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C
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C where tau is a real scalar, and v is a real vector with
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C v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in
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C Q(i+1:n,i), and tau in Q(i,i).
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C
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C Each FU(i) has the form
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C
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C FU(i) = I - nu * w * w'
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C
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C where nu is a real scalar, and w is a real vector with
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C w(1:i-1) = 0 and w(i) = 1; w(i+1:n) is stored on exit in
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C A(i+1:n,i), if op(A) = 'N', and in A(i,i+1:n), otherwise. The
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C scalar nu is stored in TAUL(i).
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C
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C Each GU(i) is a Givens rotator acting on rows i and n+i,
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C where the cosine is stored in CSL(2*i-1) and the sine in
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C CSL(2*i).
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C
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C Each HV(i) has the form
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C
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C HV(i) = I - tau * v * v'
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C
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C where tau is a real scalar, and v is a real vector with
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C v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in
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C Q(i,i+2:n), and tau in Q(i,i+1).
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C
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C Each FV(i) has the form
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C
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C FV(i) = I - nu * w * w'
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C
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C where nu is a real scalar, and w is a real vector with
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C w(1:i) = 0 and w(i+1) = 1; w(i+2:n) is stored on exit in
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C B(i,i+2:n), if op(B) = 'N', and in B(i+2:n,i), otherwise.
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C The scalar nu is stored in TAUR(i).
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C
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C Each GV(i) is a Givens rotator acting on columns i+1 and n+i+1,
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C where the cosine is stored in CSR(2*i-1) and the sine in
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C CSR(2*i).
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C
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C NUMERICAL ASPECTS
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C
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C The algorithm requires 80/3 N**3 + 20 N**2 + O(N) floating point
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C operations and is numerically backward stable.
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C
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C REFERENCES
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C
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C [1] Benner, P., Mehrmann, V., and Xu, H.
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C A numerically stable, structure preserving method for
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C computing the eigenvalues of real Hamiltonian or symplectic
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C pencils. Numer. Math., Vol 78 (3), pp. 329-358, 1998.
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C
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C CONTRIBUTORS
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C
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C D. Kressner, Technical Univ. Berlin, Germany, and
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C P. Benner, Technical Univ. Chemnitz, Germany, December 2003.
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C
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C REVISIONS
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C
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C V. Sima, June 2008 (SLICOT version of the HAPACK routine DGESUV).
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C
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C KEYWORDS
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C
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C Elementary matrix operations, Matrix decompositions, Hamiltonian
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C matrix
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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 TRANA, TRANB
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INTEGER ILO, INFO, LDA, LDB, LDG, LDQ, LDWORK, N
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C .. Array Arguments ..
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DOUBLE PRECISION A(LDA,*), B(LDB,*), CSL(*), CSR(*), DWORK(*),
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$ G(LDG,*), Q(LDQ,*), TAUL(*), TAUR(*)
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C .. Local Scalars ..
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LOGICAL LTRA, LTRB
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INTEGER I
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DOUBLE PRECISION ALPHA, C, NU, S, TEMP
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C .. External Functions ..
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LOGICAL LSAME
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EXTERNAL LSAME
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C .. External Subroutines ..
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EXTERNAL DLARF, DLARFG, DLARTG, DROT, 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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C Check the scalar input parameters.
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C
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INFO = 0
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LTRA = LSAME( TRANA, 'T' ) .OR. LSAME( TRANA, 'C' )
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LTRB = LSAME( TRANB, 'T' ) .OR. LSAME( TRANB, 'C' )
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IF ( .NOT.LTRA .AND. .NOT.LSAME( TRANA, 'N' ) ) THEN
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INFO = -1
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ELSE IF ( .NOT.LTRB .AND. .NOT.LSAME( TRANB, 'N' ) ) THEN
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INFO = -2
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ELSE IF ( N.LT.0 ) THEN
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INFO = -3
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ELSE IF ( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
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INFO = -4
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ELSE IF ( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -6
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ELSE IF ( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -8
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ELSE IF ( LDG.LT.MAX( 1, N ) ) THEN
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INFO = -10
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ELSE IF ( LDQ.LT.MAX( 1, N ) ) THEN
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INFO = -12
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ELSE IF ( LDWORK.LT.MAX( 1, N ) ) THEN
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DWORK(1) = DBLE( MAX( 1, N ) )
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INFO = -18
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END IF
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C
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C Return if there were illegal values.
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C
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IF ( INFO.NE.0 ) THEN
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CALL XERBLA( 'MB04TS', -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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IF ( N.EQ.0 ) THEN
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DWORK(1) = ONE
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RETURN
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END IF
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C
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DO 10 I = ILO, N
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ALPHA = Q(I,I)
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IF ( I.LT.N ) THEN
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C
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C Generate elementary reflector HU(i) to annihilate Q(i+1:n,i)
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C
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CALL DLARFG( N-I+1, ALPHA, Q(I+1,I), 1, NU )
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C
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C Apply HU(i) from the left.
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C
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Q(I,I) = ONE
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CALL DLARF( 'Left', N-I+1, N-I, Q(I,I), 1, NU, Q(I,I+1),
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$ LDQ, DWORK )
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IF ( LTRA ) THEN
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CALL DLARF( 'Right', N-I+1, N-I+1, Q(I,I), 1, NU, A(I,I),
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$ LDA, DWORK )
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ELSE
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CALL DLARF( 'Left', N-I+1, N-I+1, Q(I,I), 1, NU, A(I,I),
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$ LDA, DWORK )
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END IF
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IF ( LTRB ) THEN
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CALL DLARF( 'Right', N, N-I+1, Q(I,I), 1, NU, B(1,I),
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$ LDB, DWORK )
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ELSE
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CALL DLARF( 'Left', N-I+1, N, Q(I,I), 1, NU, B(I,1), LDB,
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$ DWORK )
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END IF
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CALL DLARF( 'Left', N-I+1, N, Q(I,I), 1, NU, G(I,1), LDG,
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$ DWORK )
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Q(I,I) = NU
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ELSE
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Q(I,I) = ZERO
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END IF
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C
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C Generate symplectic Givens rotator GU(i) to annihilate Q(i,i).
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C
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TEMP = A(I,I)
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CALL DLARTG( TEMP, ALPHA, C, S, A(I,I) )
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C
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C Apply G(i) from the left.
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C
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IF ( LTRA ) THEN
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CALL DROT( N-I, A(I+1,I), 1, Q(I,I+1), LDQ, C, S )
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ELSE
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CALL DROT( N-I, A(I,I+1), LDA, Q(I,I+1), LDQ, C, S )
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END IF
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IF ( LTRB ) THEN
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CALL DROT( N, G(I,1), LDG, B(1,I), 1, C, S )
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ELSE
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CALL DROT( N, G(I,1), LDG, B(I,1), LDB, C, S )
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END IF
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CSL(2*I-1) = C
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CSL(2*I) = S
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C
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IF ( I.LT.N ) THEN
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IF ( LTRA ) THEN
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C
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C Generate elementary reflector FU(i) to annihilate
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C A(i,i+1:n).
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C
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CALL DLARFG( N-I+1, A(I,I), A(I,I+1), LDA, TAUL(I) )
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C
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C Apply FU(i) from the left.
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C
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TEMP = A(I,I)
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A(I,I) = ONE
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CALL DLARF( 'Right', N-I, N-I+1, A(I,I), LDA, TAUL(I),
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$ A(I+1,I), LDA, DWORK )
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CALL DLARF( 'Left', N-I+1, N-I, A(I,I), LDA, TAUL(I),
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$ Q(I,I+1), LDQ, DWORK )
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IF ( LTRB ) THEN
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CALL DLARF( 'Right', N, N-I+1, A(I,I), LDA, TAUL(I),
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$ B(1,I), LDB, DWORK )
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ELSE
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CALL DLARF( 'Left', N-I+1, N, A(I,I), LDA, TAUL(I),
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$ B(I,1), LDB, DWORK )
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END IF
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CALL DLARF( 'Left', N-I+1, N, A(I,I), LDA, TAUL(I),
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$ G(I,1), LDG, DWORK )
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A(I,I) = TEMP
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ELSE
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C
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C Generate elementary reflector FU(i) to annihilate
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C A(i+1:n,i).
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C
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CALL DLARFG( N-I+1, A(I,I), A(I+1,I), 1, TAUL(I) )
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C
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C Apply FU(i) from the left.
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C
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TEMP = A(I,I)
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A(I,I) = ONE
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CALL DLARF( 'Left', N-I+1, N-I, A(I,I), 1, TAUL(I),
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$ A(I,I+1), LDA, DWORK )
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CALL DLARF( 'Left', N-I+1, N-I, A(I,I), 1, TAUL(I),
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$ Q(I,I+1), LDQ, DWORK )
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IF ( LTRB ) THEN
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CALL DLARF( 'Right', N, N-I+1, A(I,I), 1, TAUL(I),
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$ B(1,I), LDB, DWORK )
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ELSE
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CALL DLARF( 'Left', N-I+1, N, A(I,I), 1, TAUL(I),
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$ B(I,1), LDB, DWORK )
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END IF
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CALL DLARF( 'Left', N-I+1, N, A(I,I), 1, TAUL(I), G(I,1),
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$ LDG, DWORK )
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A(I,I) = TEMP
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END IF
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ELSE
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TAUL(I) = ZERO
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END IF
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IF ( I.LT.N )
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$ ALPHA = Q(I,I+1)
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IF ( I.LT.N-1 ) THEN
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C
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C Generate elementary reflector HV(i) to annihilate Q(i,i+2:n)
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C
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CALL DLARFG( N-I, ALPHA, Q(I,I+2), LDQ, NU )
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C
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C Apply HV(i) from the right.
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C
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Q(I,I+1) = ONE
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CALL DLARF( 'Right', N-I, N-I, Q(I,I+1), LDQ, NU,
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$ Q(I+1,I+1), LDQ, DWORK )
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IF ( LTRA ) THEN
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CALL DLARF( 'Left', N-I, N, Q(I,I+1), LDQ, NU,
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$ A(I+1,1), LDA, DWORK )
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ELSE
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CALL DLARF( 'Right', N, N-I, Q(I,I+1), LDQ, NU,
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$ A(1,I+1), LDA, DWORK )
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END IF
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IF ( LTRB ) THEN
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CALL DLARF( 'Left', N-I, N-I+1, Q(I,I+1), LDQ, NU,
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$ B(I+1,I), LDB, DWORK )
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ELSE
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CALL DLARF( 'Right', N-I+1, N-I, Q(I,I+1), LDQ, NU,
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$ B(I,I+1), LDB, DWORK )
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END IF
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CALL DLARF( 'Right', N, N-I, Q(I,I+1), LDQ, NU,
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$ G(1,I+1), LDG, DWORK )
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Q(I,I+1) = NU
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ELSE IF ( I.LT.N ) THEN
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Q(I,I+1) = ZERO
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END IF
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|
IF ( I.LT.N ) THEN
|
|
C
|
|
C Generate symplectic Givens rotator GV(i) to annihilate
|
|
C Q(i,i+1).
|
|
C
|
|
IF ( LTRB ) THEN
|
|
TEMP = B(I+1,I)
|
|
CALL DLARTG( TEMP, ALPHA, C, S, B(I+1,I) )
|
|
S = -S
|
|
CALL DROT( N-I, Q(I+1,I+1), 1, B(I+1,I+1), LDB, C, S )
|
|
ELSE
|
|
TEMP = B(I,I+1)
|
|
CALL DLARTG( TEMP, ALPHA, C, S, B(I,I+1) )
|
|
S = -S
|
|
CALL DROT( N-I, Q(I+1,I+1), 1, B(I+1,I+1), 1, C, S )
|
|
END IF
|
|
IF ( LTRA ) THEN
|
|
CALL DROT( N, A(I+1,1), LDA, G(1,I+1), 1, C, S )
|
|
ELSE
|
|
CALL DROT( N, A(1,I+1), 1, G(1,I+1), 1, C, S )
|
|
END IF
|
|
CSR(2*I-1) = C
|
|
CSR(2*I) = S
|
|
END IF
|
|
IF ( I.LT.N-1 ) THEN
|
|
IF ( LTRB ) THEN
|
|
C
|
|
C Generate elementary reflector FV(i) to annihilate
|
|
C B(i+2:n,i).
|
|
C
|
|
CALL DLARFG( N-I, B(I+1,I), B(I+2,I), 1, TAUR(I) )
|
|
C
|
|
C Apply FV(i) from the right.
|
|
C
|
|
TEMP = B(I+1,I)
|
|
B(I+1,I) = ONE
|
|
CALL DLARF( 'Left', N-I, N-I, B(I+1,I), 1, TAUR(I),
|
|
$ B(I+1,I+1), LDB, DWORK )
|
|
CALL DLARF( 'Right', N-I, N-I, B(I+1,I), 1, TAUR(I),
|
|
$ Q(I+1,I+1), LDQ, DWORK )
|
|
IF ( LTRA ) THEN
|
|
CALL DLARF( 'Left', N-I, N, B(I+1,I), 1,
|
|
$ TAUR(I), A(I+1,1), LDA, DWORK )
|
|
ELSE
|
|
CALL DLARF( 'Right', N, N-I, B(I+1,I), 1,
|
|
$ TAUR(I), A(1,I+1), LDA, DWORK )
|
|
END IF
|
|
CALL DLARF( 'Right', N, N-I, B(I+1,I), 1, TAUR(I),
|
|
$ G(1,I+1), LDG, DWORK )
|
|
B(I+1,I) = TEMP
|
|
ELSE
|
|
C
|
|
C Generate elementary reflector FV(i) to annihilate
|
|
C B(i,i+2:n).
|
|
C
|
|
CALL DLARFG( N-I, B(I,I+1), B(I,I+2), LDB, TAUR(I) )
|
|
C
|
|
C Apply FV(i) from the right.
|
|
C
|
|
TEMP = B(I,I+1)
|
|
B(I,I+1) = ONE
|
|
CALL DLARF( 'Right', N-I, N-I, B(I,I+1), LDB, TAUR(I),
|
|
$ B(I+1,I+1), LDB, DWORK )
|
|
CALL DLARF( 'Right', N-I, N-I, B(I,I+1), LDB, TAUR(I),
|
|
$ Q(I+1,I+1), LDQ, DWORK )
|
|
IF ( LTRA ) THEN
|
|
CALL DLARF( 'Left', N-I, N, B(I,I+1), LDB, TAUR(I),
|
|
$ A(I+1,1), LDA, DWORK )
|
|
ELSE
|
|
CALL DLARF( 'Right', N, N-I, B(I,I+1), LDB,
|
|
$ TAUR(I), A(1,I+1), LDA, DWORK )
|
|
END IF
|
|
CALL DLARF( 'Right', N, N-I, B(I,I+1), LDB, TAUR(I),
|
|
$ G(1,I+1), LDG, DWORK )
|
|
B(I,I+1) = TEMP
|
|
END IF
|
|
ELSE IF ( I.LT.N ) THEN
|
|
TAUR(I) = ZERO
|
|
END IF
|
|
10 CONTINUE
|
|
DWORK(1) = DBLE( MAX( 1, N ) )
|
|
RETURN
|
|
C *** Last line of MB04TS ***
|
|
END
|