ctgsen.f(3)

NAME
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 (integerIJOB, logicalWANTQ, logicalWANTZ, logical, dimension( * )SELECT, integerN, complex, dimension( lda, * )A, integerLDA, complex, dimension( ldb, * )B, integerLDB, complex, dimension( * )ALPHA, complex, dimension( * )BETA, complex, dimension( ldq, * )Q, integerLDQ, complex, dimension( ldz, * )Z, integerLDZ, integerM, realPL, realPR, real, dimension( * )DIF, complex, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerLIWORK, integerINFO)

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 "projections" 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 "projections" 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 "projections" 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.

Date:
    November 2011

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
      conjuguate 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 432 of file ctgsen.f.

Author

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