SRC/ctgsen.f(3)

NAME
SRC/ctgsen.f

SYNOPSIS

  Functions/Subroutines
subroutine ctgsen (ijob, wantq, wantz, select, n, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork, info)
CTGSEN

Function/Subroutine Documentation

  subroutine ctgsen (integer ijob, logical wantq, logical wantz, logical, dimension( * ) select, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( ldz, * ) z, integer ldz, integer m, real pl, real pr, real, dimension( * ) dif, complex, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

CTGSEN

Purpose:

    !>
    !> CTGSEN reorders the generalized Schur decomposition of a complex
    !> matrix pair (A, B) (in terms of an unitary equivalence trans-
    !> formation Q**H * (A, B) * Z), so that a selected cluster of eigenvalues
    !> appears in the leading diagonal blocks of the pair (A,B). The leading
    !> columns of Q and Z form unitary bases of the corresponding left and
    !> right eigenspaces (deflating subspaces). (A, B) must be in
    !> generalized Schur canonical form, that is, A and B are both upper
    !> triangular.
    !>
    !> CTGSEN also computes the generalized eigenvalues
    !>
    !>          w(j)= ALPHA(j) / BETA(j)
    !>
    !> of the reordered matrix pair (A, B).
    !>
    !> Optionally, the routine computes estimates of reciprocal condition
    !> numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
    !> (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
    !> between the matrix pairs (A11, B11) and (A22,B22) that correspond to
    !> the selected cluster and the eigenvalues outside the cluster, resp.,
    !> and norms of  onto left and right eigenspaces w.r.t.
    !> the selected cluster in the (1,1)-block.
    !>
    !>

Parameters
    IJOB

    !>          IJOB is INTEGER
    !>          Specifies whether condition numbers are required for the
    !>          cluster of eigenvalues (PL and PR) or the deflating subspaces
    !>          (Difu and Difl):
    !>           =0: Only reorder w.r.t. SELECT. No extras.
    !>           =1: Reciprocal of norms of  onto left and right
    !>               eigenspaces w.r.t. the selected cluster (PL and PR).
    !>           =2: Upper bounds on Difu and Difl. F-norm-based estimate
    !>               (DIF(1:2)).
    !>           =3: Estimate of Difu and Difl. 1-norm-based estimate
    !>               (DIF(1:2)).
    !>               About 5 times as expensive as IJOB = 2.
    !>           =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
    !>               version to get it all.
    !>           =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
    !>

    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.
    !>

    SELECT

    !>          SELECT is LOGICAL array, dimension (N)
    !>          SELECT specifies the eigenvalues in the selected cluster. To
    !>          select an eigenvalue w(j), SELECT(j) must be set to
    !>          .TRUE..
    !>

    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 generalized
    !>          Schur canonical form.
    !>          On exit, A is overwritten by the reordered 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 generalized
    !>          Schur canonical form.
    !>          On exit, B is overwritten by the reordered matrix B.
    !>

    LDB

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

    ALPHA

    !>          ALPHA is COMPLEX array, dimension (N)
    !>

    BETA

    !>          BETA is COMPLEX array, dimension (N)
    !>
    !>          The diagonal elements of A and B, respectively,
    !>          when the pair (A,B) has been reduced to generalized Schur
    !>          form.  ALPHA(i)/BETA(i) i=1,...,N are the generalized
    !>          eigenvalues.
    !>

    Q

    !>          Q is COMPLEX array, dimension (LDQ,N)
    !>          On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
    !>          On exit, Q has been postmultiplied by the left unitary
    !>          transformation matrix which reorder (A, B); The leading M
    !>          columns of Q form orthonormal bases for the specified pair of
    !>          left eigenspaces (deflating subspaces).
    !>          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., Z is an N-by-N matrix.
    !>          On exit, Z has been postmultiplied by the left unitary
    !>          transformation matrix which reorder (A, B); The leading M
    !>          columns of Z form orthonormal bases for the specified pair of
    !>          left eigenspaces (deflating subspaces).
    !>          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.
    !>

    M

    !>          M is INTEGER
    !>          The dimension of the specified pair of left and right
    !>          eigenspaces, (deflating subspaces) 0 <= M <= N.
    !>

    PL

    !>          PL is REAL
    !>

    PR

    !>          PR is REAL
    !>
    !>          If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
    !>          reciprocal  of the norm of  onto left and right
    !>          eigenspace with respect to the selected cluster.
    !>          0 < PL, PR <= 1.
    !>          If M = 0 or M = N, PL = PR  = 1.
    !>          If IJOB = 0, 2 or 3 PL, PR are not referenced.
    !>

    DIF

    !>          DIF is REAL array, dimension (2).
    !>          If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
    !>          If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
    !>          Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
    !>          estimates of Difu and Difl, computed using reversed
    !>          communication with CLACN2.
    !>          If M = 0 or N, DIF(1:2) = F-norm([A, B]).
    !>          If IJOB = 0 or 1, DIF is not referenced.
    !>

    WORK

    !>          WORK is COMPLEX 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
    !>          If IJOB = 1, 2 or 4, LWORK >=  2*M*(N-M)
    !>          If IJOB = 3 or 5, LWORK >=  4*M*(N-M)
    !>
    !>          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.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
    !>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
    !>

    LIWORK

    !>          LIWORK is INTEGER
    !>          The dimension of the array IWORK. LIWORK >= 1.
    !>          If IJOB = 1, 2 or 4, LIWORK >=  N+2;
    !>          If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M));
    !>
    !>          If LIWORK = -1, then a workspace query is assumed; the
    !>          routine only calculates the optimal size of the IWORK array,
    !>          returns this value as the first entry of the IWORK array, and
    !>          no error message related to LIWORK is issued by XERBLA.
    !>

    INFO

    !>          INFO is INTEGER
    !>            =0: Successful exit.
    !>            <0: If INFO = -i, the i-th argument had an illegal value.
    !>            =1: Reordering of (A, B) failed because the transformed
    !>                matrix pair (A, B) would be too far from generalized
    !>                Schur form; the problem is very ill-conditioned.
    !>                (A, B) may have been partially reordered.
    !>                If requested, 0 is returned in DIF(*), PL and PR.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  CTGSEN first collects the selected eigenvalues by computing unitary
    !>  U and W that move them to the top left corner of (A, B). In other
    !>  words, the selected eigenvalues are the eigenvalues of (A11, B11) in
    !>
    !>              U**H*(A, B)*W = (A11 A12) (B11 B12) n1
    !>                              ( 0  A22),( 0  B22) n2
    !>                                n1  n2    n1  n2
    !>
    !>  where N = n1+n2 and U**H means the conjugate transpose of U. The first
    !>  n1 columns of U and W span the specified pair of left and right
    !>  eigenspaces (deflating subspaces) of (A, B).
    !>
    !>  If (A, B) has been obtained from the generalized real Schur
    !>  decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the
    !>  reordered generalized Schur form of (C, D) is given by
    !>
    !>           (C, D) = (Q*U)*(U**H *(A, B)*W)*(Z*W)**H,
    !>
    !>  and the first n1 columns of Q*U and Z*W span the corresponding
    !>  deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).
    !>
    !>  Note that if the selected eigenvalue is sufficiently ill-conditioned,
    !>  then its value may differ significantly from its value before
    !>  reordering.
    !>
    !>  The reciprocal condition numbers of the left and right eigenspaces
    !>  spanned by the first n1 columns of U and W (or Q*U and Z*W) may
    !>  be returned in DIF(1:2), corresponding to Difu and Difl, resp.
    !>
    !>  The Difu and Difl are defined as:
    !>
    !>       Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
    !>  and
    !>       Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],
    !>
    !>  where sigma-min(Zu) is the smallest singular value of the
    !>  (2*n1*n2)-by-(2*n1*n2) matrix
    !>
    !>       Zu = [ kron(In2, A11)  -kron(A22**H, In1) ]
    !>            [ kron(In2, B11)  -kron(B22**H, In1) ].
    !>
    !>  Here, Inx is the identity matrix of size nx and A22**H is the
    !>  conjugate transpose of A22. kron(X, Y) is the Kronecker product between
    !>  the matrices X and Y.
    !>
    !>  When DIF(2) is small, small changes in (A, B) can cause large changes
    !>  in the deflating subspace. An approximate (asymptotic) bound on the
    !>  maximum angular error in the computed deflating subspaces is
    !>
    !>       EPS * norm((A, B)) / DIF(2),
    !>
    !>  where EPS is the machine precision.
    !>
    !>  The reciprocal norm of the projectors on the left and right
    !>  eigenspaces associated with (A11, B11) may be returned in PL and PR.
    !>  They are computed as follows. First we compute L and R so that
    !>  P*(A, B)*Q is block diagonal, where
    !>
    !>       P = ( I -L ) n1           Q = ( I R ) n1
    !>           ( 0  I ) n2    and        ( 0 I ) n2
    !>             n1 n2                    n1 n2
    !>
    !>  and (L, R) is the solution to the generalized Sylvester equation
    !>
    !>       A11*R - L*A22 = -A12
    !>       B11*R - L*B22 = -B12
    !>
    !>  Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
    !>  An approximate (asymptotic) bound on the average absolute error of
    !>  the selected eigenvalues is
    !>
    !>       EPS * norm((A, B)) / PL.
    !>
    !>  There are also global error bounds which valid for perturbations up
    !>  to a certain restriction:  A lower bound (x) on the smallest
    !>  F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
    !>  coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
    !>  (i.e. (A + E, B + F), is
    !>
    !>   x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
    !>
    !>  An approximate bound on x can be computed from DIF(1:2), PL and PR.
    !>
    !>  If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
    !>  (L', R') and unperturbed (L, R) left and right deflating subspaces
    !>  associated with the selected cluster in the (1,1)-blocks can be
    !>  bounded as
    !>
    !>   max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
    !>   max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))
    !>
    !>  See LAPACK User's Guide section 4.11 or the following references
    !>  for more information.
    !>
    !>  Note that if the default method for computing the Frobenius-norm-
    !>  based estimate DIF is not wanted (see CLATDF), then the parameter
    !>  IDIFJB (see below) should be changed from 3 to 4 (routine CLATDF
    !>  (IJOB = 2 will be used)). See CTGSYL for more 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.
     [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.

Definition at line 430 of file ctgsen.f.

Author

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