*> \brief \b DTGEXC * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DTGEXC + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, * LDZ, IFST, ILST, WORK, LWORK, INFO ) * * .. Scalar Arguments .. * LOGICAL WANTQ, WANTZ * INTEGER IFST, ILST, INFO, LDA, LDB, LDQ, LDZ, LWORK, N * .. * .. Array Arguments .. * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), * \$ WORK( * ), Z( LDZ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DTGEXC reorders the generalized real Schur decomposition of a real *> matrix pair (A,B) using an orthogonal equivalence transformation *> *> (A, B) = Q * (A, B) * Z**T, *> *> so that the diagonal block of (A, B) with row index IFST is moved *> to row ILST. *> *> (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, dimension (LDA,N) *> On entry, the matrix A in generalized real Schur canonical *> form. *> On exit, the updated matrix A, again in generalized *> real Schur canonical form. *> \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, dimension (LDB,N) *> On entry, the matrix B in generalized real Schur canonical *> form (A,B). *> On exit, the updated matrix B, again in generalized *> real Schur canonical form (A,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. *> If WANTQ = .FALSE., Q is not referenced. *> \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. *> If WANTZ = .FALSE., Z is not referenced. *> \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,out] IFST *> \verbatim *> IFST is INTEGER *> \endverbatim *> *> \param[in,out] ILST *> \verbatim *> ILST is INTEGER *> Specify the reordering of the diagonal blocks of (A, B). *> The block with row index IFST is moved to row ILST, by a *> sequence of swapping between adjacent blocks. *> On exit, if IFST pointed on entry to the second row of *> a 2-by-2 block, it is changed to point to the first row; *> ILST always points to the first row of the block in its *> final position (which may differ from its input value by *> +1 or -1). 1 <= IFST, ILST <= N. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK. *> LWORK >= 1 when N <= 1, otherwise LWORK >= 4*N + 16. *> *> If LWORK = -1, then a workspace query is assumed; the routine *> only calculates the optimal size of the WORK array, returns *> this value as the first entry of the WORK array, and no error *> message related to LWORK is issued by XERBLA. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> =0: successful exit. *> <0: if INFO = -i, the i-th argument had an illegal value. *> =1: The transformed matrix pair (A, B) would be too far *> from generalized Schur form; the problem is ill- *> conditioned. (A, B) may have been partially reordered, *> and ILST points to the first row of the current *> position of the block being moved. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup doubleGEcomputational * *> \par Contributors: * ================== *> *> Bo Kagstrom and Peter Poromaa, Department of Computing Science, *> Umea University, S-901 87 Umea, Sweden. * *> \par References: * ================ *> *> \verbatim *> *>  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. *> \endverbatim *> * ===================================================================== SUBROUTINE DTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, \$ LDZ, IFST, ILST, WORK, LWORK, INFO ) * * -- LAPACK computational routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. Scalar Arguments .. LOGICAL WANTQ, WANTZ INTEGER IFST, ILST, INFO, LDA, LDB, LDQ, LDZ, LWORK, N * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), \$ WORK( * ), Z( LDZ, * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO PARAMETER ( ZERO = 0.0D+0 ) * .. * .. Local Scalars .. LOGICAL LQUERY INTEGER HERE, LWMIN, NBF, NBL, NBNEXT * .. * .. External Subroutines .. EXTERNAL DTGEX2, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX * .. * .. Executable Statements .. * * Decode and test input arguments. * INFO = 0 LQUERY = ( LWORK.EQ.-1 ) IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. ( LDQ.LT.MAX( 1, N ) ) ) THEN INFO = -9 ELSE IF( LDZ.LT.1 .OR. WANTZ .AND. ( LDZ.LT.MAX( 1, N ) ) ) THEN INFO = -11 ELSE IF( IFST.LT.1 .OR. IFST.GT.N ) THEN INFO = -12 ELSE IF( ILST.LT.1 .OR. ILST.GT.N ) THEN INFO = -13 END IF * IF( INFO.EQ.0 ) THEN IF( N.LE.1 ) THEN LWMIN = 1 ELSE LWMIN = 4*N + 16 END IF WORK(1) = LWMIN * IF (LWORK.LT.LWMIN .AND. .NOT.LQUERY) THEN INFO = -15 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTGEXC', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.LE.1 ) \$ RETURN * * Determine the first row of the specified block and find out * if it is 1-by-1 or 2-by-2. * IF( IFST.GT.1 ) THEN IF( A( IFST, IFST-1 ).NE.ZERO ) \$ IFST = IFST - 1 END IF NBF = 1 IF( IFST.LT.N ) THEN IF( A( IFST+1, IFST ).NE.ZERO ) \$ NBF = 2 END IF * * Determine the first row of the final block * and find out if it is 1-by-1 or 2-by-2. * IF( ILST.GT.1 ) THEN IF( A( ILST, ILST-1 ).NE.ZERO ) \$ ILST = ILST - 1 END IF NBL = 1 IF( ILST.LT.N ) THEN IF( A( ILST+1, ILST ).NE.ZERO ) \$ NBL = 2 END IF IF( IFST.EQ.ILST ) \$ RETURN * IF( IFST.LT.ILST ) THEN * * Update ILST. * IF( NBF.EQ.2 .AND. NBL.EQ.1 ) \$ ILST = ILST - 1 IF( NBF.EQ.1 .AND. NBL.EQ.2 ) \$ ILST = ILST + 1 * HERE = IFST * 10 CONTINUE * * Swap with next one below. * IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN * * Current block either 1-by-1 or 2-by-2. * NBNEXT = 1 IF( HERE+NBF+1.LE.N ) THEN IF( A( HERE+NBF+1, HERE+NBF ).NE.ZERO ) \$ NBNEXT = 2 END IF CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, \$ LDZ, HERE, NBF, NBNEXT, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE + NBNEXT * * Test if 2-by-2 block breaks into two 1-by-1 blocks. * IF( NBF.EQ.2 ) THEN IF( A( HERE+1, HERE ).EQ.ZERO ) \$ NBF = 3 END IF * ELSE * * Current block consists of two 1-by-1 blocks, each of which * must be swapped individually. * NBNEXT = 1 IF( HERE+3.LE.N ) THEN IF( A( HERE+3, HERE+2 ).NE.ZERO ) \$ NBNEXT = 2 END IF CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, \$ LDZ, HERE+1, 1, NBNEXT, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF IF( NBNEXT.EQ.1 ) THEN * * Swap two 1-by-1 blocks. * CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, \$ LDZ, HERE, 1, 1, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE + 1 * ELSE * * Recompute NBNEXT in case of 2-by-2 split. * IF( A( HERE+2, HERE+1 ).EQ.ZERO ) \$ NBNEXT = 1 IF( NBNEXT.EQ.2 ) THEN * * 2-by-2 block did not split. * CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, \$ Z, LDZ, HERE, 1, NBNEXT, WORK, LWORK, \$ INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE + 2 ELSE * * 2-by-2 block did split. * CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, \$ Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE + 1 CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, \$ Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE + 1 END IF * END IF END IF IF( HERE.LT.ILST ) \$ GO TO 10 ELSE HERE = IFST * 20 CONTINUE * * Swap with next one below. * IF( NBF.EQ.1 .OR. NBF.EQ.2 ) THEN * * Current block either 1-by-1 or 2-by-2. * NBNEXT = 1 IF( HERE.GE.3 ) THEN IF( A( HERE-1, HERE-2 ).NE.ZERO ) \$ NBNEXT = 2 END IF CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, \$ LDZ, HERE-NBNEXT, NBNEXT, NBF, WORK, LWORK, \$ INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE - NBNEXT * * Test if 2-by-2 block breaks into two 1-by-1 blocks. * IF( NBF.EQ.2 ) THEN IF( A( HERE+1, HERE ).EQ.ZERO ) \$ NBF = 3 END IF * ELSE * * Current block consists of two 1-by-1 blocks, each of which * must be swapped individually. * NBNEXT = 1 IF( HERE.GE.3 ) THEN IF( A( HERE-1, HERE-2 ).NE.ZERO ) \$ NBNEXT = 2 END IF CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, \$ LDZ, HERE-NBNEXT, NBNEXT, 1, WORK, LWORK, \$ INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF IF( NBNEXT.EQ.1 ) THEN * * Swap two 1-by-1 blocks. * CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, \$ LDZ, HERE, NBNEXT, 1, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE - 1 ELSE * * Recompute NBNEXT in case of 2-by-2 split. * IF( A( HERE, HERE-1 ).EQ.ZERO ) \$ NBNEXT = 1 IF( NBNEXT.EQ.2 ) THEN * * 2-by-2 block did not split. * CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, \$ Z, LDZ, HERE-1, 2, 1, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE - 2 ELSE * * 2-by-2 block did split. * CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, \$ Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE - 1 CALL DTGEX2( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, \$ Z, LDZ, HERE, 1, 1, WORK, LWORK, INFO ) IF( INFO.NE.0 ) THEN ILST = HERE RETURN END IF HERE = HERE - 1 END IF END IF END IF IF( HERE.GT.ILST ) \$ GO TO 20 END IF ILST = HERE WORK( 1 ) = LWMIN RETURN * * End of DTGEXC * END