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SUBROUTINE CTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
$ LDZ, J1, INFO ) * * -- LAPACK auxiliary routine (version 3.3.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * -- April 2011 -- * * .. 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)**H = Q(out) * A(out) * Z(out)**H * Q(in) * B(in) * Z(in)**H = Q(out) * B(out) * Z(out)**H * * * 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 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, 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 ) * * 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 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**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 = 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 |