phynexis-deps/lapack
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity
Files
e2428ec4cbbbfb86fb48a67a0696347f73aebc21
lapack/SRC/ctgex2.f
T
Julie e18d437924 Making LAPACK's code eternal... no more version and date in source files. 
GitHub is now enabling us to track accurately version and date.
No need for this anymore.
2021-03-25 10:16:58 -07:00

371 lines
12 KiB
FortranFixed
Raw Blame History

*> \brief \b CTGEX2 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 CTGEX2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ctgex2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ctgex2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ctgex2.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CTGEX2( 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 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> 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)**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 array, dimension (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 array, dimension (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 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 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.
*
*> \ingroup complexGEauxiliary
*
*> \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 CTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
$ LDZ, J1, INFO )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
LOGICAL WANTQ, WANTZ
INTEGER INFO, J1, LDA, LDB, LDQ, LDZ, N
* ..
* .. Array Arguments ..
COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
$ Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ),
$ CONE = ( 1.0E+0, 0.0E+0 ) )
REAL TWENTY
PARAMETER ( TWENTY = 2.0E+1 )
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, SUM,
$ THRESHA, THRESHB
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( M*M, WORK, 1, SCALE, SUM )
SA = SCALE*SQRT( SUM )
SCALE = DBLE( CZERO )
SUM = DBLE( CONE )
CALL CLASSQ( M*M, WORK(M*M+1), 1, SCALE, SUM )
SB = 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.
*
THRESHA = MAX( TWENTY*EPS*SA, SMLNUM )
THRESHB = MAX( TWENTY*EPS*SB, 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 ) ) * ABS( T( 1, 1 ) )
SB = ABS( S( 1, 1 ) ) * 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| <= O(EPS F-norm((A)))
* and |T21| <= O(EPS F-norm((B)))
*
WEAK = ABS( S( 2, 1 ) ).LE.THRESHA .AND.
$ ABS( T( 2, 1 ) ).LE.THRESHB
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 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 = DBLE( CZERO )
SUM = DBLE( CONE )
CALL CLASSQ( M*M, WORK, 1, SCALE, SUM )
SA = SCALE*SQRT( SUM )
SCALE = DBLE( CZERO )
SUM = DBLE( CONE )
CALL CLASSQ( M*M, WORK(M*M+1), 1, SCALE, SUM )
SB = SCALE*SQRT( SUM )
STRONG = SA.LE.THRESHA .AND. SB.LE.THRESHB
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
Reference in New Issue View Git Blame Copy Permalink