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OpenBLAS/lapack-netlib/SRC/ztgex2.f

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*> \brief \b ZTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZTGEX2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ztgex2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ztgex2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ztgex2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
* LDZ, J1, INFO )
*
* .. Scalar Arguments ..
* LOGICAL WANTQ, WANTZ
* INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, N
* ..
* .. Array Arguments ..
* COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZTGEX2 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)**H = Q(out) * A(out) * Z(out)**H
*> Q(in) * B(in) * Z(in)**H = Q(out) * B(out) * Z(out)**H
*>
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] WANTQ
*> \verbatim
*> WANTQ is LOGICAL
*> .TRUE. : update the left transformation matrix Q;
*> .FALSE.: do not update Q.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*> WANTZ is LOGICAL
*> .TRUE. : update the right transformation matrix Z;
*> .FALSE.: do not update Z.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX*16 array, dimensions (LDA,N)
*> On entry, the matrix A in the pair (A, B).
*> On exit, the updated matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is COMPLEX*16 array, dimensions (LDB,N)
*> On entry, the matrix B in the pair (A, B).
*> On exit, the updated matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*> Q is COMPLEX*16 array, dimension (LDQ,N)
*> If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit,
*> the updated matrix Q.
*> Not referenced if WANTQ = .FALSE..
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*> LDQ is INTEGER
*> The leading dimension of the array Q. LDQ >= 1;
*> If WANTQ = .TRUE., LDQ >= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> Z is COMPLEX*16 array, dimension (LDZ,N)
*> If WANTZ = .TRUE, on entry, the unitary matrix Z. On exit,
*> the updated matrix Z.
*> Not referenced if WANTZ = .FALSE..
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> LDZ is INTEGER
*> The leading dimension of the array Z. LDZ >= 1;
*> If WANTZ = .TRUE., LDZ >= N.
*> \endverbatim
*>
*> \param[in] J1
*> \verbatim
*> J1 is INTEGER
*> The index to the first block (A11, B11).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> =0: Successful exit.
*> =1: The transformed matrix pair (A, B) would be too far
*> from generalized Schur form; the problem is ill-
*> conditioned.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2017
*
*> \ingroup complex16GEauxiliary
*
*> \par Further Details:
* =====================
*>
*> 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.
*
*> \par Contributors:
* ==================
*>
*> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*> Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
* ================
*>
*> [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.
*> \n
*> [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.
*>
* =====================================================================
SUBROUTINE ZTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
$ LDZ, J1, INFO )
*
* -- LAPACK auxiliary routine (version 3.7.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2017
*
* .. Scalar Arguments ..
LOGICAL WANTQ, WANTZ
INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, N
* ..
* .. Array Arguments ..
COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
$ Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX*16 CZERO, CONE
PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
$ CONE = ( 1.0D+0, 0.0D+0 ) )
DOUBLE PRECISION TWENTY
PARAMETER ( TWENTY = 2.0D+1 )
INTEGER LDST
PARAMETER ( LDST = 2 )
LOGICAL WANDS
PARAMETER ( WANDS = .TRUE. )
* ..
* .. Local Scalars ..
LOGICAL DTRONG, WEAK
INTEGER I, M
DOUBLE PRECISION CQ, CZ, EPS, SA, SB, SCALE, SMLNUM, SS, SUM,
$ THRESH, WS
COMPLEX*16 CDUM, F, G, SQ, SZ
* ..
* .. Local Arrays ..
COMPLEX*16 S( LDST, LDST ), T( LDST, LDST ), WORK( 8 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL ZLACPY, ZLARTG, ZLASSQ, ZROT
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DCONJG, MAX, SQRT
* ..
* .. Executable Statements ..
*
INFO = 0
*
* Quick return if possible
*
IF( N.LE.1 )
$ RETURN
*
M = LDST
WEAK = .FALSE.
DTRONG = .FALSE.
*
* Make a local copy of selected block in (A, B)
*
CALL ZLACPY( 'Full', M, M, A( J1, J1 ), LDA, S, LDST )
CALL ZLACPY( 'Full', M, M, B( J1, J1 ), LDB, T, LDST )
*
* Compute the threshold for testing the acceptance of swapping.
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
SCALE = DBLE( CZERO )
SUM = DBLE( CONE )
CALL ZLACPY( 'Full', M, M, S, LDST, WORK, M )
CALL ZLACPY( 'Full', M, M, T, LDST, WORK( M*M+1 ), M )
CALL ZLASSQ( 2*M*M, WORK, 1, SCALE, SUM )
SA = SCALE*SQRT( SUM )
*
* THRES has been changed from
* THRESH = MAX( TEN*EPS*SA, SMLNUM )
* to
* THRESH = MAX( TWENTY*EPS*SA, SMLNUM )
* on 04/01/10.
* "Bug" reported by Ondra Kamenik, confirmed by Julie Langou, fixed by
* Jim Demmel and Guillaume Revy. See forum post 1783.
*
THRESH = MAX( TWENTY*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 ZLARTG( G, F, CZ, SZ, CDUM )
SZ = -SZ
CALL ZROT( 2, S( 1, 1 ), 1, S( 1, 2 ), 1, CZ, DCONJG( SZ ) )
CALL ZROT( 2, T( 1, 1 ), 1, T( 1, 2 ), 1, CZ, DCONJG( SZ ) )
IF( SA.GE.SB ) THEN
CALL ZLARTG( S( 1, 1 ), S( 2, 1 ), CQ, SQ, CDUM )
ELSE
CALL ZLARTG( T( 1, 1 ), T( 2, 1 ), CQ, SQ, CDUM )
END IF
CALL ZROT( 2, S( 1, 1 ), LDST, S( 2, 1 ), LDST, CQ, SQ )
CALL ZROT( 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**H*S*QR, B-QL**H*T*QR)) <= O(EPS*F-norm((A, B)))
*
CALL ZLACPY( 'Full', M, M, S, LDST, WORK, M )
CALL ZLACPY( 'Full', M, M, T, LDST, WORK( M*M+1 ), M )
CALL ZROT( 2, WORK, 1, WORK( 3 ), 1, CZ, -DCONJG( SZ ) )
CALL ZROT( 2, WORK( 5 ), 1, WORK( 7 ), 1, CZ, -DCONJG( SZ ) )
CALL ZROT( 2, WORK, 2, WORK( 2 ), 2, CQ, -SQ )
CALL ZROT( 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 = DBLE( CZERO )
SUM = DBLE( CONE )
CALL ZLASSQ( 2*M*M, WORK, 1, SCALE, SUM )
SS = SCALE*SQRT( SUM )
DTRONG = SS.LE.THRESH
IF( .NOT.DTRONG )
$ 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 ZROT( J1+1, A( 1, J1 ), 1, A( 1, J1+1 ), 1, CZ,
$ DCONJG( SZ ) )
CALL ZROT( J1+1, B( 1, J1 ), 1, B( 1, J1+1 ), 1, CZ,
$ DCONJG( SZ ) )
CALL ZROT( N-J1+1, A( J1, J1 ), LDA, A( J1+1, J1 ), LDA, CQ, SQ )
CALL ZROT( 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 ZROT( N, Z( 1, J1 ), 1, Z( 1, J1+1 ), 1, CZ,
$ DCONJG( SZ ) )
IF( WANTQ )
$ CALL ZROT( N, Q( 1, J1 ), 1, Q( 1, J1+1 ), 1, CQ,
$ DCONJG( 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 ZTGEX2
*
END
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