dtgex2.f

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      SUBROUTINE DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
     $                   LDZ, J1, N1, N2, WORK, LWORK, INFO )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      LOGICAL            WANTQ, WANTZ
      INTEGER            INFO, J1, LDA, LDB, LDQ, LDZ, LWORK, N, N1, N2
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
     $                   WORK( * ), Z( LDZ, * )
*     ..
*
*  Purpose
*  =======
*
*  DTGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22)
*  of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair
*  (A, B) by an orthogonal equivalence transformation.
*
*  (A, B) must be in generalized real Schur canonical form (as returned
*  by DGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
*  diagonal blocks. B is 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDZ,N)
*          On entry, if WANTQ = .TRUE., the orthogonal 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) DOUBLE PRECISION array, dimension (LDZ,N)
*          On entry, if WANTZ =.TRUE., the orthogonal 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). 1 <= J1 <= N.
*
*  N1      (input) INTEGER
*          The order of the first block (A11, B11). N1 = 0, 1 or 2.
*
*  N2      (input) INTEGER
*          The order of the second block (A22, B22). N2 = 0, 1 or 2.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (LWORK).
*
*  LWORK   (input) INTEGER
*          The dimension of the array WORK.
*          LWORK >=  MAX( N*(N2+N1), (N2+N1)*(N2+N1)*2 )
*
*  INFO    (output) INTEGER
*            =0: Successful exit
*            >0: If INFO = 1, the transformed matrix (A, B) would be
*                too far from generalized Schur form; the blocks are
*                not swapped and (A, B) and (Q, Z) are unchanged.
*                The problem of swapping is too ill-conditioned.
*            <0: If INFO = -16: LWORK is too small. Appropriate value
*                for LWORK is returned in WORK(1).
*
*  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 ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
      DOUBLE PRECISION   TEN
      PARAMETER          ( TEN = 1.0D+01 )
      INTEGER            LDST
      PARAMETER          ( LDST = 4 )
      LOGICAL            WANDS
      PARAMETER          ( WANDS = .TRUE. )
*     ..
*     .. Local Scalars ..
      LOGICAL            DTRONG, WEAK
      INTEGER            I, IDUM, LINFO, M
      DOUBLE PRECISION   BQRA21, BRQA21, DDUM, DNORM, DSCALE, DSUM, EPS,
     $                   F, G, SA, SB, SCALE, SMLNUM, SS, THRESH, WS
*     ..
*     .. Local Arrays ..
      INTEGER            IWORK( LDST )
      DOUBLE PRECISION   AI( 2 ), AR( 2 ), BE( 2 ), IR( LDST, LDST ),
     $                   IRCOP( LDST, LDST ), LI( LDST, LDST ),
     $                   LICOP( LDST, LDST ), S( LDST, LDST ),
     $                   SCPY( LDST, LDST ), T( LDST, LDST ),
     $                   TAUL( LDST ), TAUR( LDST ), TCPY( LDST, LDST )
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DGEMM, DGEQR2, DGERQ2, DLACPY, DLAGV2,
     $                   DLARTG, DLASSQ, DORG2R, DORGR2, DORM2R, DORMR2,
     $                   DROT, DSCAL, DTGSY2
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
      INFO = 0
*
*     Quick return if possible
*
      IF( N.LE.1 .OR. N1.LE.0 .OR. N2.LE.0 )
     $   RETURN
      IF( N1.GT.N .OR. ( J1+N1 ).GT.N )
     $   RETURN
      M = N1 + N2
      IF( LWORK.LT.MAX( N*M, M*M*2 ) ) THEN
         INFO = -16
         WORK( 1 ) = MAX( N*M, M*M*2 )
         RETURN
      END IF
*
      WEAK = .FALSE.
      DTRONG = .FALSE.
*
*     Make a local copy of selected block
*
      CALL DCOPY( LDST*LDST, ZERO, 0, LI, 1 )
      CALL DCOPY( LDST*LDST, ZERO, 0, IR, 1 )
      CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, S, LDST )
      CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, T, LDST )
*
*     Compute threshold for testing acceptance of swapping.
*
      EPS = DLAMCH( 'P' )
      SMLNUM = DLAMCH( 'S' ) / EPS
      DSCALE = ZERO
      DSUM = ONE
      CALL DLACPY( 'Full', M, M, S, LDST, WORK, M )
      CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )
      CALL DLACPY( 'Full', M, M, T, LDST, WORK, M )
      CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )
      DNORM = DSCALE*SQRT( DSUM )
      THRESH = MAX( TEN*EPS*DNORM, SMLNUM )
*
      IF( M.EQ.2 ) THEN
*
*        CASE 1: Swap 1-by-1 and 1-by-1 blocks.
*
*        Compute orthogonal 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 )
         SB = ABS( T( 2, 2 ) )
         SA = ABS( S( 2, 2 ) )
         CALL DLARTG( F, G, IR( 1, 2 ), IR( 1, 1 ), DDUM )
         IR( 2, 1 ) = -IR( 1, 2 )
         IR( 2, 2 ) = IR( 1, 1 )
         CALL DROT( 2, S( 1, 1 ), 1, S( 1, 2 ), 1, IR( 1, 1 ),
     $              IR( 2, 1 ) )
         CALL DROT( 2, T( 1, 1 ), 1, T( 1, 2 ), 1, IR( 1, 1 ),
     $              IR( 2, 1 ) )
         IF( SA.GE.SB ) THEN
            CALL DLARTG( S( 1, 1 ), S( 2, 1 ), LI( 1, 1 ), LI( 2, 1 ),
     $                   DDUM )
         ELSE
            CALL DLARTG( T( 1, 1 ), T( 2, 1 ), LI( 1, 1 ), LI( 2, 1 ),
     $                   DDUM )
         END IF
         CALL DROT( 2, S( 1, 1 ), LDST, S( 2, 1 ), LDST, LI( 1, 1 ),
     $              LI( 2, 1 ) )
         CALL DROT( 2, T( 1, 1 ), LDST, T( 2, 1 ), LDST, LI( 1, 1 ),
     $              LI( 2, 1 ) )
         LI( 2, 2 ) = LI( 1, 1 )
         LI( 1, 2 ) = -LI( 2, 1 )
*
*        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 70
*
         IF( WANDS ) THEN
*
*           Strong stability test:
*             F-norm((A-QL'*S*QR, B-QL'*T*QR)) <= O(EPS*F-norm((A,B)))
*
            CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ),
     $                   M )
            CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, S, LDST, ZERO,
     $                  WORK, M )
            CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
     $                  WORK( M*M+1 ), M )
            DSCALE = ZERO
            DSUM = ONE
            CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
*
            CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ),
     $                   M )
            CALL DGEMM( 'N', 'N', M, M, M, ONE, LI, LDST, T, LDST, ZERO,
     $                  WORK, M )
            CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
     $                  WORK( M*M+1 ), M )
            CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
            SS = DSCALE*SQRT( DSUM )
            DTRONG = SS.LE.THRESH
            IF( .NOT.DTRONG )
     $         GO TO 70
         END IF
*
*        Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
*               (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)).
*
         CALL DROT( J1+1, A( 1, J1 ), 1, A( 1, J1+1 ), 1, IR( 1, 1 ),
     $              IR( 2, 1 ) )
         CALL DROT( J1+1, B( 1, J1 ), 1, B( 1, J1+1 ), 1, IR( 1, 1 ),
     $              IR( 2, 1 ) )
         CALL DROT( N-J1+1, A( J1, J1 ), LDA, A( J1+1, J1 ), LDA,
     $              LI( 1, 1 ), LI( 2, 1 ) )
         CALL DROT( N-J1+1, B( J1, J1 ), LDB, B( J1+1, J1 ), LDB,
     $              LI( 1, 1 ), LI( 2, 1 ) )
*
*        Set  N1-by-N2 (2,1) - blocks to ZERO.
*
         A( J1+1, J1 ) = ZERO
         B( J1+1, J1 ) = ZERO
*