ztgex2.f

          SUBROUTINE ZTGEX2( 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*16         A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
         $                   Z( LDZ, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  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)' = 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*16 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*16 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*16 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*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..
    *
    *  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*16         CZERO, CONE
          PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ),
         $                   CONE = ( 1.0D+0, 0.0D+0 ) )
          DOUBLE PRECISION   TEN
          PARAMETER          ( TEN = 10.0D+0 )
          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 )
          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 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'*S*QR, B-QL'*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