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version 1.4, 2010/08/06 15:32:36 version 1.21, 2023/08/07 08:39:12
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   *> \brief \b DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at
   *            http://www.netlib.org/lapack/explore-html/
   *
   *> \htmlonly
   *> Download DTGEX2 + dependencies
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgex2.f">
   *> [TGZ]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgex2.f">
   *> [ZIP]</a>
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgex2.f">
   *> [TXT]</a>
   *> \endhtmlonly
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
   *                          LDZ, J1, N1, N2, WORK, LWORK, INFO )
   *
   *       .. 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, * )
   *       ..
   *
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> 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)**T = Q(out) * A(out) * Z(out)**T
   *>        Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T
   *>
   *> \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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDQ,N)
   *>          On entry, if WANTQ = .TRUE., the orthogonal 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 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..
   *> \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). 1 <= J1 <= N.
   *> \endverbatim
   *>
   *> \param[in] N1
   *> \verbatim
   *>          N1 is INTEGER
   *>          The order of the first block (A11, B11). N1 = 0, 1 or 2.
   *> \endverbatim
   *>
   *> \param[in] N2
   *> \verbatim
   *>          N2 is INTEGER
   *>          The order of the second block (A22, B22). N2 = 0, 1 or 2.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)).
   *> \endverbatim
   *>
   *> \param[in] LWORK
   *> \verbatim
   *>          LWORK is INTEGER
   *>          The dimension of the array WORK.
   *>          LWORK >=  MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 )
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is 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).
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee
   *> \author Univ. of California Berkeley
   *> \author Univ. of Colorado Denver
   *> \author NAG Ltd.
   *
   *> \ingroup doubleGEauxiliary
   *
   *> \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:
   *  ================
   *>
   *> \verbatim
   *>
   *>  [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.
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,        SUBROUTINE DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z,
      $                   LDZ, J1, N1, N2, WORK, LWORK, INFO )       $                   LDZ, J1, N1, N2, WORK, LWORK, INFO )
 *  *
 *  -- LAPACK auxiliary routine (version 3.2) --  *  -- LAPACK auxiliary routine --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *     November 2006  
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       LOGICAL            WANTQ, WANTZ        LOGICAL            WANTQ, WANTZ
Line 15 Line 232
      $                   WORK( * ), Z( LDZ, * )       $                   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 (MAX(1,LWORK)).  
 *  
 *  LWORK   (input) INTEGER  
 *          The dimension of the array WORK.  
 *          LWORK >=  MAX( 1, 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.  
 *  
 *  =====================================================================  *  =====================================================================
 *  Replaced various illegal calls to DCOPY by calls to DLASET, or by DO  *  Replaced various illegal calls to DCOPY by calls to DLASET, or by DO
 *  loops. Sven Hammarling, 1/5/02.  *  loops. Sven Hammarling, 1/5/02.
Line 134 Line 239
 *     .. Parameters ..  *     .. Parameters ..
       DOUBLE PRECISION   ZERO, ONE        DOUBLE PRECISION   ZERO, ONE
       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )        PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
       DOUBLE PRECISION   TEN        DOUBLE PRECISION   TWENTY
       PARAMETER          ( TEN = 1.0D+01 )        PARAMETER          ( TWENTY = 2.0D+01 )
       INTEGER            LDST        INTEGER            LDST
       PARAMETER          ( LDST = 4 )        PARAMETER          ( LDST = 4 )
       LOGICAL            WANDS        LOGICAL            WANDS
       PARAMETER          ( WANDS = .TRUE. )        PARAMETER          ( WANDS = .TRUE. )
 *     ..  *     ..
 *     .. Local Scalars ..  *     .. Local Scalars ..
       LOGICAL            DTRONG, WEAK        LOGICAL            STRONG, WEAK
       INTEGER            I, IDUM, LINFO, M        INTEGER            I, IDUM, LINFO, M
       DOUBLE PRECISION   BQRA21, BRQA21, DDUM, DNORM, DSCALE, DSUM, EPS,        DOUBLE PRECISION   BQRA21, BRQA21, DDUM, DNORMA, DNORMB, DSCALE,
      $                   F, G, SA, SB, SCALE, SMLNUM, SS, THRESH, WS       $                   DSUM, EPS, F, G, SA, SB, SCALE, SMLNUM,
        $                   THRESHA, THRESHB
 *     ..  *     ..
 *     .. Local Arrays ..  *     .. Local Arrays ..
       INTEGER            IWORK( LDST )        INTEGER            IWORK( LDST )
Line 185 Line 291
       END IF        END IF
 *  *
       WEAK = .FALSE.        WEAK = .FALSE.
       DTRONG = .FALSE.        STRONG = .FALSE.
 *  *
 *     Make a local copy of selected block  *     Make a local copy of selected block
 *  *
Line 202 Line 308
       DSUM = ONE        DSUM = ONE
       CALL DLACPY( 'Full', M, M, S, LDST, WORK, M )        CALL DLACPY( 'Full', M, M, S, LDST, WORK, M )
       CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )        CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )
         DNORMA = DSCALE*SQRT( DSUM )
         DSCALE = ZERO
         DSUM = ONE
       CALL DLACPY( 'Full', M, M, T, LDST, WORK, M )        CALL DLACPY( 'Full', M, M, T, LDST, WORK, M )
       CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )        CALL DLASSQ( M*M, WORK, 1, DSCALE, DSUM )
       DNORM = DSCALE*SQRT( DSUM )        DNORMB = DSCALE*SQRT( DSUM )
       THRESH = MAX( TEN*EPS*DNORM, SMLNUM )  *
   *     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*DNORMA, SMLNUM )
         THRESHB = MAX( TWENTY*EPS*DNORMB, SMLNUM )
 *  *
       IF( M.EQ.2 ) THEN        IF( M.EQ.2 ) THEN
 *  *
Line 216 Line 335
 *  *
          F = S( 2, 2 )*T( 1, 1 ) - T( 2, 2 )*S( 1, 1 )           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 )           G = S( 2, 2 )*T( 1, 2 ) - T( 2, 2 )*S( 1, 2 )
          SB = ABS( T( 2, 2 ) )           SA = ABS( S( 2, 2 ) ) * ABS( T( 1, 1 ) )
          SA = ABS( S( 2, 2 ) )           SB = ABS( S( 1, 1 ) ) * ABS( T( 2, 2 ) )
          CALL DLARTG( F, G, IR( 1, 2 ), IR( 1, 1 ), DDUM )           CALL DLARTG( F, G, IR( 1, 2 ), IR( 1, 1 ), DDUM )
          IR( 2, 1 ) = -IR( 1, 2 )           IR( 2, 1 ) = -IR( 1, 2 )
          IR( 2, 2 ) = IR( 1, 1 )           IR( 2, 2 ) = IR( 1, 1 )
Line 239 Line 358
          LI( 2, 2 ) = LI( 1, 1 )           LI( 2, 2 ) = LI( 1, 1 )
          LI( 1, 2 ) = -LI( 2, 1 )           LI( 1, 2 ) = -LI( 2, 1 )
 *  *
 *        Weak stability test:  *        Weak stability test: |S21| <= O(EPS F-norm((A)))
 *           |S21| + |T21| <= O(EPS * F-norm((S, T)))  *                           and  |T21| <= O(EPS F-norm((B)))
 *  *
          WS = ABS( S( 2, 1 ) ) + ABS( T( 2, 1 ) )           WEAK = ABS( S( 2, 1 ) ) .LE. THRESHA .AND.
          WEAK = WS.LE.THRESH       $      ABS( T( 2, 1 ) ) .LE. THRESHB
          IF( .NOT.WEAK )           IF( .NOT.WEAK )
      $      GO TO 70       $      GO TO 70
 *  *
          IF( WANDS ) THEN           IF( WANDS ) THEN
 *  *
 *           Strong stability test:  *           Strong stability test:
 *             F-norm((A-QL'*S*QR, B-QL'*T*QR)) <= O(EPS*F-norm((A,B)))  *               F-norm((A-QL**H*S*QR)) <= O(EPS*F-norm((A)))
   *               and
   *               F-norm((B-QL**H*T*QR)) <= O(EPS*F-norm((B)))
 *  *
             CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ),              CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ),
      $                   M )       $                   M )
Line 261 Line 382
             DSCALE = ZERO              DSCALE = ZERO
             DSUM = ONE              DSUM = ONE
             CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )              CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
               SA = DSCALE*SQRT( DSUM )
 *  *
             CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ),              CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ),
      $                   M )       $                   M )
Line 268 Line 390
      $                  WORK, M )       $                  WORK, M )
             CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST, ONE,              CALL DGEMM( 'N', 'T', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
      $                  WORK( M*M+1 ), M )       $                  WORK( M*M+1 ), M )
               DSCALE = ZERO
               DSUM = ONE
             CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )              CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
             SS = DSCALE*SQRT( DSUM )              SB = DSCALE*SQRT( DSUM )
             DTRONG = SS.LE.THRESH              STRONG = SA.LE.THRESHA .AND. SB.LE.THRESHB
             IF( .NOT.DTRONG )              IF( .NOT.STRONG )
      $         GO TO 70       $         GO TO 70
          END IF           END IF
 *  *
Line 322 Line 446
      $                IR( N2+1, N1+1 ), LDST, T, LDST, T( N1+1, N1+1 ),       $                IR( N2+1, N1+1 ), LDST, T, LDST, T( N1+1, N1+1 ),
      $                LDST, LI, LDST, SCALE, DSUM, DSCALE, IWORK, IDUM,       $                LDST, LI, LDST, SCALE, DSUM, DSCALE, IWORK, IDUM,
      $                LINFO )       $                LINFO )
            IF( LINFO.NE.0 )
        $      GO TO 70
 *  *
 *        Compute orthogonal matrix QL:  *        Compute orthogonal matrix QL:
 *  *
 *                    QL' * LI = [ TL ]  *                    QL**T * LI = [ TL ]
 *                               [ 0  ]  *                                 [ 0  ]
 *        where  *        where
 *                    LI =  [      -L              ]  *                    LI =  [      -L              ]
 *                          [ SCALE * identity(N2) ]  *                          [ SCALE * identity(N2) ]
Line 344 Line 470
 *  *
 *        Compute orthogonal matrix RQ:  *        Compute orthogonal matrix RQ:
 *  *
 *                    IR * RQ' =   [ 0  TR],  *                    IR * RQ**T =   [ 0  TR],
 *  *
 *         where IR = [ SCALE * identity(N1), R ]  *         where IR = [ SCALE * identity(N1), R ]
 *  *
Line 421 Line 547
 *  *
 *        Decide which method to use.  *        Decide which method to use.
 *          Weak stability test:  *          Weak stability test:
 *             F-norm(S21) <= O(EPS * F-norm((S, T)))  *             F-norm(S21) <= O(EPS * F-norm((S)))
 *  *
          IF( BQRA21.LE.BRQA21 .AND. BQRA21.LE.THRESH ) THEN           IF( BQRA21.LE.BRQA21 .AND. BQRA21.LE.THRESHA ) THEN
             CALL DLACPY( 'F', M, M, SCPY, LDST, S, LDST )              CALL DLACPY( 'F', M, M, SCPY, LDST, S, LDST )
             CALL DLACPY( 'F', M, M, TCPY, LDST, T, LDST )              CALL DLACPY( 'F', M, M, TCPY, LDST, T, LDST )
             CALL DLACPY( 'F', M, M, IRCOP, LDST, IR, LDST )              CALL DLACPY( 'F', M, M, IRCOP, LDST, IR, LDST )
             CALL DLACPY( 'F', M, M, LICOP, LDST, LI, LDST )              CALL DLACPY( 'F', M, M, LICOP, LDST, LI, LDST )
          ELSE IF( BRQA21.GE.THRESH ) THEN           ELSE IF( BRQA21.GE.THRESHA ) THEN
             GO TO 70              GO TO 70
          END IF           END IF
 *  *
Line 439 Line 565
          IF( WANDS ) THEN           IF( WANDS ) THEN
 *  *
 *           Strong stability test:  *           Strong stability test:
 *              F-norm((A-QL*S*QR', B-QL*T*QR')) <= O(EPS*F-norm((A,B)))  *               F-norm((A-QL**H*S*QR)) <= O(EPS*F-norm((A)))
   *               and
   *               F-norm((B-QL**H*T*QR)) <= O(EPS*F-norm((B)))
 *  *
             CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ),              CALL DLACPY( 'Full', M, M, A( J1, J1 ), LDA, WORK( M*M+1 ),
      $                   M )       $                   M )
Line 450 Line 578
             DSCALE = ZERO              DSCALE = ZERO
             DSUM = ONE              DSUM = ONE
             CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )              CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
               SA = DSCALE*SQRT( DSUM )
 *  *
             CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ),              CALL DLACPY( 'Full', M, M, B( J1, J1 ), LDB, WORK( M*M+1 ),
      $                   M )       $                   M )
Line 457 Line 586
      $                  WORK, M )       $                  WORK, M )
             CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST, ONE,              CALL DGEMM( 'N', 'N', M, M, M, -ONE, WORK, M, IR, LDST, ONE,
      $                  WORK( M*M+1 ), M )       $                  WORK( M*M+1 ), M )
               DSCALE = ZERO
               DSUM = ONE
             CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )              CALL DLASSQ( M*M, WORK( M*M+1 ), 1, DSCALE, DSUM )
             SS = DSCALE*SQRT( DSUM )              SB = DSCALE*SQRT( DSUM )
             DTRONG = ( SS.LE.THRESH )              STRONG = SA.LE.THRESHA .AND. SB.LE.THRESHB
             IF( .NOT.DTRONG )              IF( .NOT.STRONG )
      $         GO TO 70       $         GO TO 70
 *  *
          END IF           END IF
Line 478 Line 609
 *  *
 *        Standardize existing 2-by-2 blocks.  *        Standardize existing 2-by-2 blocks.
 *  *
          DO 50 I = 1, M*M           CALL DLASET( 'Full', M, M, ZERO, ZERO, WORK, M )
             WORK(I) = ZERO  
    50    CONTINUE  
          WORK( 1 ) = ONE           WORK( 1 ) = ONE
          T( 1, 1 ) = ONE           T( 1, 1 ) = ONE
          IDUM = LWORK - M*M - 2           IDUM = LWORK - M*M - 2
Line 551 Line 680
      $                  A( J1, I ), LDA, ZERO, WORK, M )       $                  A( J1, I ), LDA, ZERO, WORK, M )
             CALL DLACPY( 'Full', M, N-I+1, WORK, M, A( J1, I ), LDA )              CALL DLACPY( 'Full', M, N-I+1, WORK, M, A( J1, I ), LDA )
             CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST,              CALL DGEMM( 'T', 'N', M, N-I+1, M, ONE, LI, LDST,
      $                  B( J1, I ), LDA, ZERO, WORK, M )       $                  B( J1, I ), LDB, ZERO, WORK, M )
             CALL DLACPY( 'Full', M, N-I+1, WORK, M, B( J1, I ), LDB )              CALL DLACPY( 'Full', M, N-I+1, WORK, M, B( J1, I ), LDB )
          END IF           END IF
          I = J1 - 1           I = J1 - 1
Removed from v.1.4  
changed lines
  Added in v.1.21

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