tgexc(3)

NAME
tgexc - tgexc: reorder generalized Schur form

SYNOPSIS

  Functions

subroutine ctgexc (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, ifst, ilst, info)
CTGEXC
subroutine dtgexc (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, ifst, ilst, work, lwork, info)
DTGEXC
subroutine stgexc (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, ifst, ilst, work, lwork, info)
STGEXC
subroutine ztgexc (wantq, wantz, n, a, lda, b, ldb, q, ldq, z, ldz, ifst, ilst, info)
ZTGEXC

Detailed Description

Function Documentation

  subroutine ctgexc (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 ifst, integer ilst, integer info)

CTGEXC

Purpose:

    !>
    !> CTGEXC reorders the generalized Schur decomposition of a complex
    !> matrix pair (A,B), using an unitary equivalence transformation
    !> (A, B) := Q * (A, B) * Z**H, so that the diagonal block of (A, B) with
    !> row index IFST is moved to row ILST.
    !>
    !> (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 upper triangular 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 upper triangular 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)
    !>          On entry, if WANTQ = .TRUE., the unitary matrix Q.
    !>          On exit, the updated matrix Q.
    !>          If WANTQ = .FALSE., Q is not referenced.
    !>

    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)
    !>          On entry, if WANTZ = .TRUE., the unitary matrix Z.
    !>          On exit, the updated matrix Z.
    !>          If WANTZ = .FALSE., Z is not referenced.
    !>

    LDZ

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

    IFST

    !>          IFST is INTEGER
    !>

    ILST

    !>          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.
    !>

    INFO

    !>          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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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.
     [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

  subroutine dtgexc (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 ifst, integer ilst, double precision, dimension( * ) work, integer lwork, integer info)

DTGEXC

Purpose:

    !>
    !> 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
    !>
    !>

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, 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.
    !>

    LDA

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

    B

    !>          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).
    !>

    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.
    !>          If WANTQ = .FALSE., Q is not referenced.
    !>

    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.
    !>          If WANTZ = .FALSE., Z is not referenced.
    !>

    LDZ

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

    IFST

    !>          IFST is INTEGER
    !>

    ILST

    !>          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.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          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.
    !>

    INFO

    !>          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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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.
    !>

  subroutine stgexc (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 ifst, integer ilst, real, dimension( * ) work, integer lwork, integer info)

STGEXC

Purpose:

    !>
    !> STGEXC 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 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 generalized real Schur canonical
    !>          form.
    !>          On exit, the updated matrix A, again in generalized
    !>          real Schur canonical form.
    !>

    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 generalized real Schur canonical
    !>          form (A,B).
    !>          On exit, the updated matrix B, again in generalized
    !>          real Schur canonical form (A,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.
    !>          If WANTQ = .FALSE., Q is not referenced.
    !>

    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.
    !>          If WANTZ = .FALSE., Z is not referenced.
    !>

    LDZ

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

    IFST

    !>          IFST is INTEGER
    !>

    ILST

    !>          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.
    !>

    WORK

    !>          WORK is REAL array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          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.
    !>

    INFO

    !>          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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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.
    !>

  subroutine ztgexc (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 ifst, integer ilst, integer info)

ZTGEXC

Purpose:

    !>
    !> ZTGEXC reorders the generalized Schur decomposition of a complex
    !> matrix pair (A,B), using an unitary equivalence transformation
    !> (A, B) := Q * (A, B) * Z**H, so that the diagonal block of (A, B) with
    !> row index IFST is moved to row ILST.
    !>
    !> (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, dimension (LDA,N)
    !>          On entry, the upper triangular 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, dimension (LDB,N)
    !>          On entry, the upper triangular 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)
    !>          On entry, if WANTQ = .TRUE., the unitary matrix Q.
    !>          On exit, the updated matrix Q.
    !>          If WANTQ = .FALSE., Q is not referenced.
    !>

    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)
    !>          On entry, if WANTZ = .TRUE., the unitary matrix Z.
    !>          On exit, the updated matrix Z.
    !>          If WANTZ = .FALSE., Z is not referenced.
    !>

    LDZ

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

    IFST

    !>          IFST is INTEGER
    !>

    ILST

    !>          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.
    !>

    INFO

    !>          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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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.
     [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.

Author

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