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SUBROUTINE CTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
$ LDZ, J1, INFO )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
LOGICAL WANTQ, WANTZ
INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, N
* ..
* .. Array Arguments ..
COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
$ Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* CTGEX2 swaps adjacent diagonal 1 by 1 blocks (A11,B11) and (A22,B22)
* in an upper triangular matrix pair (A, B) by an unitary equivalence
* transformation.
*
* (A, B) must be in generalized Schur canonical form, that is, A and
* B are both upper triangular.
*
* Optionally, the matrices Q and Z of generalized Schur vectors are
* updated.
*
* Q(in) * A(in) * Z(in)' = Q(out) * A(out) * Z(out)'
* Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)'
*
*
* Arguments
* =========
*
* WANTQ (input) LOGICAL
* .TRUE. : update the left transformation matrix Q;
* .FALSE.: do not update Q.
*
* WANTZ (input) LOGICAL
* .TRUE. : update the right transformation matrix Z;
* .FALSE.: do not update Z.
*
* N (input) INTEGER
* The order of the matrices A and B. N >= 0.
*
* A (input/output) COMPLEX arrays, dimensions (LDA,N)
* On entry, the matrix A in the pair (A, B).
* On exit, the updated matrix A.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* B (input/output) COMPLEX arrays, dimensions (LDB,N)
* On entry, the matrix B in the pair (A, B).
* On exit, the updated matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* Q (input/output) COMPLEX array, dimension (LDZ,N)
* If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit,
* the updated matrix Q.
* Not referenced if WANTQ = .FALSE..
*
* LDQ (input) INTEGER
* The leading dimension of the array Q. LDQ >= 1;
* If WANTQ = .TRUE., LDQ >= N.
*
* Z (input/output) COMPLEX array, dimension (LDZ,N)
* If WANTZ = .TRUE, on entry, the unitary matrix Z. On exit,
* the updated matrix Z.
* Not referenced if WANTZ = .FALSE..
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1;
* If WANTZ = .TRUE., LDZ >= N.
*
* J1 (input) INTEGER
* The index to the first block (A11, B11).
*
* INFO (output) INTEGER
* =0: Successful exit.
* =1: The transformed matrix pair (A, B) would be too far
* from generalized Schur form; the problem is ill-
* conditioned.
*
*
* Further Details
* ===============
*
* Based on contributions by
* Bo Kagstrom and Peter Poromaa, Department of Computing Science,
* Umea University, S-901 87 Umea, Sweden.
*
* In the current code both weak and strong stability tests are
* performed. The user can omit the strong stability test by changing
* the internal logical parameter WANDS to .FALSE.. See ref. [2] for
* details.
*
* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
* M.S. Moonen et al (eds), Linear Algebra for Large Scale and
* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
*
* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
* Eigenvalues of a Regular Matrix Pair (A, B) and Condition
* Estimation: Theory, Algorithms and Software, Report UMINF-94.04,
* Department of Computing Science, Umea University, S-901 87 Umea,
* Sweden, 1994. Also as LAPACK Working Note 87. To appear in
* Numerical Algorithms, 1996.
*
* =====================================================================
*
* .. Parameters ..
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ),
$ CONE = ( 1.0E+0, 0.0E+0 ) )
REAL TEN
PARAMETER ( TEN = 10.0E+0 )
INTEGER LDST
PARAMETER ( LDST = 2 )
LOGICAL WANDS
PARAMETER ( WANDS = .TRUE. )
* ..
* .. Local Scalars ..
LOGICAL STRONG, WEAK
INTEGER I, M
REAL CQ, CZ, EPS, SA, SB, SCALE, SMLNUM, SS, SUM,
$ THRESH, WS
COMPLEX CDUM, F, G, SQ, SZ
* ..
* .. Local Arrays ..
COMPLEX S( LDST, LDST ), T( LDST, LDST ), WORK( 8 )
* ..
* .. External Functions ..
REAL SLAMCH
EXTERNAL SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL CLACPY, CLARTG, CLASSQ, CROT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, CONJG, MAX, REAL, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.LE.1 )
$ RETURN
*
M = LDST
WEAK = .FALSE.
STRONG = .FALSE.
*
* Make a local copy of selected block in (A, B)
*
CALL CLACPY( 'Full', M, M, A( J1, J1 ), LDA, S, LDST )
CALL CLACPY( 'Full', M, M, B( J1, J1 ), LDB, T, LDST )
*
* Compute the threshold for testing the acceptance of swapping.
*
EPS = SLAMCH( 'P' )
SMLNUM = SLAMCH( 'S' ) / EPS
SCALE = REAL( CZERO )
SUM = REAL( CONE )
CALL CLACPY( 'Full', M, M, S, LDST, WORK, M )
CALL CLACPY( 'Full', M, M, T, LDST, WORK( M*M+1 ), M )
CALL CLASSQ( 2*M*M, WORK, 1, SCALE, SUM )
SA = SCALE*SQRT( SUM )
THRESH = MAX( TEN*EPS*SA, SMLNUM )
*
* Compute unitary QL and RQ that swap 1-by-1 and 1-by-1 blocks
* using Givens rotations and perform the swap tentatively.
*
F = S( 2, 2 )*T( 1, 1 ) - T( 2, 2 )*S( 1, 1 )
G = S( 2, 2 )*T( 1, 2 ) - T( 2, 2 )*S( 1, 2 )
SA = ABS( S( 2, 2 ) )
SB = ABS( T( 2, 2 ) )
CALL CLARTG( G, F, CZ, SZ, CDUM )
SZ = -SZ
CALL CROT( 2, S( 1, 1 ), 1, S( 1, 2 ), 1, CZ, CONJG( SZ ) )
CALL CROT( 2, T( 1, 1 ), 1, T( 1, 2 ), 1, CZ, CONJG( SZ ) )
IF( SA.GE.SB ) THEN
CALL CLARTG( S( 1, 1 ), S( 2, 1 ), CQ, SQ, CDUM )
ELSE
CALL CLARTG( T( 1, 1 ), T( 2, 1 ), CQ, SQ, CDUM )
END IF
CALL CROT( 2, S( 1, 1 ), LDST, S( 2, 1 ), LDST, CQ, SQ )
CALL CROT( 2, T( 1, 1 ), LDST, T( 2, 1 ), LDST, CQ, SQ )
*
* Weak stability test: |S21| + |T21| <= O(EPS F-norm((S, T)))
*
WS = ABS( S( 2, 1 ) ) + ABS( T( 2, 1 ) )
WEAK = WS.LE.THRESH
IF( .NOT.WEAK )
$ GO TO 20
*
IF( WANDS ) THEN
*
* Strong stability test:
* F-norm((A-QL'*S*QR, B-QL'*T*QR)) <= O(EPS*F-norm((A, B)))
*
CALL CLACPY( 'Full', M, M, S, LDST, WORK, M )
CALL CLACPY( 'Full', M, M, T, LDST, WORK( M*M+1 ), M )
CALL CROT( 2, WORK, 1, WORK( 3 ), 1, CZ, -CONJG( SZ ) )
CALL CROT( 2, WORK( 5 ), 1, WORK( 7 ), 1, CZ, -CONJG( SZ ) )
CALL CROT( 2, WORK, 2, WORK( 2 ), 2, CQ, -SQ )
CALL CROT( 2, WORK( 5 ), 2, WORK( 6 ), 2, CQ, -SQ )
DO 10 I = 1, 2
WORK( I ) = WORK( I ) - A( J1+I-1, J1 )
WORK( I+2 ) = WORK( I+2 ) - A( J1+I-1, J1+1 )
WORK( I+4 ) = WORK( I+4 ) - B( J1+I-1, J1 )
WORK( I+6 ) = WORK( I+6 ) - B( J1+I-1, J1+1 )
10 CONTINUE
SCALE = REAL( CZERO )
SUM = REAL( CONE )
CALL CLASSQ( 2*M*M, WORK, 1, SCALE, SUM )
SS = SCALE*SQRT( SUM )
STRONG = SS.LE.THRESH
IF( .NOT.STRONG )
$ GO TO 20
END IF
*
* If the swap is accepted ("weakly" and "strongly"), apply the
* equivalence transformations to the original matrix pair (A,B)
*
CALL CROT( J1+1, A( 1, J1 ), 1, A( 1, J1+1 ), 1, CZ, CONJG( SZ ) )
CALL CROT( J1+1, B( 1, J1 ), 1, B( 1, J1+1 ), 1, CZ, CONJG( SZ ) )
CALL CROT( N-J1+1, A( J1, J1 ), LDA, A( J1+1, J1 ), LDA, CQ, SQ )
CALL CROT( N-J1+1, B( J1, J1 ), LDB, B( J1+1, J1 ), LDB, CQ, SQ )
*
* Set N1 by N2 (2,1) blocks to 0
*
A( J1+1, J1 ) = CZERO
B( J1+1, J1 ) = CZERO
*
* Accumulate transformations into Q and Z if requested.
*
IF( WANTZ )
$ CALL CROT( N, Z( 1, J1 ), 1, Z( 1, J1+1 ), 1, CZ, CONJG( SZ ) )
IF( WANTQ )
$ CALL CROT( N, Q( 1, J1 ), 1, Q( 1, J1+1 ), 1, CQ, CONJG( SQ ) )
*
* Exit with INFO = 0 if swap was successfully performed.
*
RETURN
*
* Exit with INFO = 1 if swap was rejected.
*
20 CONTINUE
INFO = 1
RETURN
*
* End of CTGEX2
*
END
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