tgex2(3)

NAME
tgex2 - tgex2: reorder generalized Schur form

SYNOPSIS

  Functions

subroutine ctgex2 (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, info)
CTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.
subroutine dtgex2 (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, n1, n2, work, lwork, info)
DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.
subroutine stgex2 (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, n1, n2, work, lwork, info)
STGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.
subroutine ztgex2 (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, j1, info)
ZTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.

Detailed Description

Function Documentation

  subroutine ctgex2 (logical wantq, logical wantz, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( ldz, * ) z, integer ldz, integer j1, integer info)

CTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.

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
    !>
    !>

Parameters
    WANTQ

    !>          WANTQ is LOGICAL
    !>          .TRUE. : update the left transformation matrix Q;
    !>          .FALSE.: do not update Q.
    !>

    WANTZ

    !>          WANTZ is LOGICAL
    !>          .TRUE. : update the right transformation matrix Z;
    !>          .FALSE.: do not update Z.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices A and B. N >= 0.
    !>

    A

    !>          A is COMPLEX array, dimension (LDA,N)
    !>          On entry, the matrix A in the pair (A, B).
    !>          On exit, the updated matrix A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A. LDA >= max(1,N).
    !>

    B

    !>          B is COMPLEX array, dimension (LDB,N)
    !>          On entry, the matrix B in the pair (A, B).
    !>          On exit, the updated matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B. LDB >= max(1,N).
    !>

    Q

    !>          Q is COMPLEX array, dimension (LDQ,N)
    !>          If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit,
    !>          the updated matrix Q.
    !>          Not referenced if WANTQ = .FALSE..
    !>

    LDQ

    !>          LDQ is INTEGER
    !>          The leading dimension of the array Q. LDQ >= 1;
    !>          If WANTQ = .TRUE., LDQ >= N.
    !>

    Z

    !>          Z is 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

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z. LDZ >= 1;
    !>          If WANTZ = .TRUE., LDZ >= N.
    !>

    J1

    !>          J1 is INTEGER
    !>          The index to the first block (A11, B11).
    !>

    INFO

    !>          INFO is INTEGER
    !>           =0:  Successful exit.
    !>           =1:  The transformed matrix pair (A, B) would be too far
    !>                from generalized Schur form; the problem is ill-
    !>                conditioned.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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.

Contributors:
    Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:
    [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.

  subroutine dtgex2 (logical wantq, logical wantz, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( ldz, * ) z, integer ldz, integer j1, integer n1, integer n2, double precision, dimension( * ) work, integer lwork, integer info)

DTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.

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)**T = Q(out) * A(out) * Z(out)**T
    !>        Q(in) * B(in) * Z(in)**T = Q(out) * B(out) * Z(out)**T
    !>
    !>

Parameters
    WANTQ

    !>          WANTQ is LOGICAL
    !>          .TRUE. : update the left transformation matrix Q;
    !>          .FALSE.: do not update Q.
    !>

    WANTZ

    !>          WANTZ is LOGICAL
    !>          .TRUE. : update the right transformation matrix Z;
    !>          .FALSE.: do not update Z.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices A and B. N >= 0.
    !>

    A

    !>          A is DOUBLE PRECISION array, dimensions (LDA,N)
    !>          On entry, the matrix A in the pair (A, B).
    !>          On exit, the updated matrix A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A. LDA >= max(1,N).
    !>

    B

    !>          B is DOUBLE PRECISION array, dimensions (LDB,N)
    !>          On entry, the matrix B in the pair (A, B).
    !>          On exit, the updated matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B. LDB >= max(1,N).
    !>

    Q

    !>          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..
    !>

    LDQ

    !>          LDQ is INTEGER
    !>          The leading dimension of the array Q. LDQ >= 1.
    !>          If WANTQ = .TRUE., LDQ >= N.
    !>

    Z

    !>          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..
    !>

    LDZ

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z. LDZ >= 1.
    !>          If WANTZ = .TRUE., LDZ >= N.
    !>

    J1

    !>          J1 is INTEGER
    !>          The index to the first block (A11, B11). 1 <= J1 <= N.
    !>

    N1

    !>          N1 is INTEGER
    !>          The order of the first block (A11, B11). N1 = 0, 1 or 2.
    !>

    N2

    !>          N2 is INTEGER
    !>          The order of the second block (A22, B22). N2 = 0, 1 or 2.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)).
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.
    !>          LWORK >=  MAX( 1, N*(N2+N1), (N2+N1)*(N2+N1)*2 )
    !>

    INFO

    !>          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).
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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.

Contributors:
    Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

    !>
    !>  [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.
    !>

  subroutine stgex2 (logical wantq, logical wantz, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldq, * ) q, integer ldq, real, dimension( ldz, * ) z, integer ldz, integer j1, integer n1, integer n2, real, dimension( * ) work, integer lwork, integer info)

STGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an orthogonal equivalence transformation.

Purpose:

    !>
    !> STGEX2 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 SGGES), 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
    !>
    !>

Parameters
    WANTQ

    !>          WANTQ is LOGICAL
    !>          .TRUE. : update the left transformation matrix Q;
    !>          .FALSE.: do not update Q.
    !>

    WANTZ

    !>          WANTZ is LOGICAL
    !>          .TRUE. : update the right transformation matrix Z;
    !>          .FALSE.: do not update Z.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices A and B. N >= 0.
    !>

    A

    !>          A is REAL array, dimension (LDA,N)
    !>          On entry, the matrix A in the pair (A, B).
    !>          On exit, the updated matrix A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A. LDA >= max(1,N).
    !>

    B

    !>          B is REAL array, dimension (LDB,N)
    !>          On entry, the matrix B in the pair (A, B).
    !>          On exit, the updated matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B. LDB >= max(1,N).
    !>

    Q

    !>          Q is REAL array, dimension (LDQ,N)
    !>          On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
    !>          On exit, the updated matrix Q.
    !>          Not referenced if WANTQ = .FALSE..
    !>

    LDQ

    !>          LDQ is INTEGER
    !>          The leading dimension of the array Q. LDQ >= 1.
    !>          If WANTQ = .TRUE., LDQ >= N.
    !>

    Z

    !>          Z is REAL 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

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z. LDZ >= 1.
    !>          If WANTZ = .TRUE., LDZ >= N.
    !>

    J1

    !>          J1 is INTEGER
    !>          The index to the first block (A11, B11). 1 <= J1 <= N.
    !>

    N1

    !>          N1 is INTEGER
    !>          The order of the first block (A11, B11). N1 = 0, 1 or 2.
    !>

    N2

    !>          N2 is INTEGER
    !>          The order of the second block (A22, B22). N2 = 0, 1 or 2.
    !>

    WORK

    !>          WORK is REAL array, dimension (MAX(1,LWORK)).
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.
    !>          LWORK >=  MAX( N*(N2+N1), (N2+N1)*(N2+N1)*2 )
    !>

    INFO

    !>          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).
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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.

Contributors:
    Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

    !>
    !>  [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.
    !>

  subroutine ztgex2 (logical wantq, logical wantz, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldz, * ) z, integer ldz, integer j1, integer info)

ZTGEX2 swaps adjacent diagonal blocks in an upper (quasi) triangular matrix pair by an unitary equivalence transformation.

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)**H = Q(out) * A(out) * Z(out)**H
    !>        Q(in) * B(in) * Z(in)**H = Q(out) * B(out) * Z(out)**H
    !>
    !>

Parameters
    WANTQ

    !>          WANTQ is LOGICAL
    !>          .TRUE. : update the left transformation matrix Q;
    !>          .FALSE.: do not update Q.
    !>

    WANTZ

    !>          WANTZ is LOGICAL
    !>          .TRUE. : update the right transformation matrix Z;
    !>          .FALSE.: do not update Z.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices A and B. N >= 0.
    !>

    A

    !>          A is COMPLEX*16 array, dimensions (LDA,N)
    !>          On entry, the matrix A in the pair (A, B).
    !>          On exit, the updated matrix A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A. LDA >= max(1,N).
    !>

    B

    !>          B is COMPLEX*16 array, dimensions (LDB,N)
    !>          On entry, the matrix B in the pair (A, B).
    !>          On exit, the updated matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B. LDB >= max(1,N).
    !>

    Q

    !>          Q is COMPLEX*16 array, dimension (LDQ,N)
    !>          If WANTQ = .TRUE, on entry, the unitary matrix Q. On exit,
    !>          the updated matrix Q.
    !>          Not referenced if WANTQ = .FALSE..
    !>

    LDQ

    !>          LDQ is INTEGER
    !>          The leading dimension of the array Q. LDQ >= 1;
    !>          If WANTQ = .TRUE., LDQ >= N.
    !>

    Z

    !>          Z is 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

    !>          LDZ is INTEGER
    !>          The leading dimension of the array Z. LDZ >= 1;
    !>          If WANTZ = .TRUE., LDZ >= N.
    !>

    J1

    !>          J1 is INTEGER
    !>          The index to the first block (A11, B11).
    !>

    INFO

    !>          INFO is INTEGER
    !>           =0:  Successful exit.
    !>           =1:  The transformed matrix pair (A, B) would be too far
    !>                from generalized Schur form; the problem is ill-
    !>                conditioned.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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.

Contributors:
    Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:
    [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.

Author

Generated automatically by Doxygen for LAPACK from the source code.