ctgsen.f

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      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 )
*
*  -- LAPACK routine (version 3.1.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     January 2007
*
*     Modified to call CLACN2 in place of CLACON, 10 Feb 03, SJH.
*
*     .. Scalar Arguments ..
      LOGICAL            WANTQ, WANTZ
      INTEGER            IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK,
     $                   M, N
      REAL               PL, PR
*     ..
*     .. Array Arguments ..
      LOGICAL            SELECT( * )
      INTEGER            IWORK( * )
      REAL               DIF( * )
      COMPLEX            A( LDA, * ), ALPHA( * ), B( LDB, * ),
     $                   BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  Purpose
*  =======
*
*  CTGSEN reorders the generalized Schur decomposition of a complex
*  matrix pair (A, B) (in terms of an unitary equivalence trans-
*  formation Q' * (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.
*
*
*  Arguments
*  =========
*
*  IJOB    (input) 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   (input) LOGICAL
*          .TRUE. : update the left transformation matrix Q;
*          .FALSE.: do not update Q.
*
*  WANTZ   (input) LOGICAL
*          .TRUE. : update the right transformation matrix Z;
*          .FALSE.: do not update Z.
*
*  SELECT  (input) 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       (input) INTEGER
*          The order of the matrices A and B. N >= 0.
*
*  A       (input/output) 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     (input) INTEGER
*          The leading dimension of the array A. LDA >= max(1,N).
*
*  B       (input/output) 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     (input) INTEGER
*          The leading dimension of the array B. LDB >= max(1,N).
*
*  ALPHA   (output) COMPLEX array, dimension (N)
*  BETA    (output) 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       (input/output) 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     (input) INTEGER
*          The leading dimension of the array Q. LDQ >= 1.
*          If WANTQ = .TRUE., LDQ >= N.
*
*  Z       (input/output) 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     (input) INTEGER
*          The leading dimension of the array Z. LDZ >= 1.
*          If WANTZ = .TRUE., LDZ >= N.
*
*  M       (output) INTEGER
*          The dimension of the specified pair of left and right
*          eigenspaces, (deflating subspaces) 0 <= M <= N.
*
*  PL	   (output) REAL
*  PR	   (output) 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     (output) 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    (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
*          IF IJOB = 0, WORK is not referenced.  Otherwise,
*          on exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) 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   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
*          IF IJOB = 0, IWORK is not referenced.  Otherwise,
*          on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*
*  LIWORK  (input) 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    (output) 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.
*
*
*  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'*(A, B)*W = (A11 A12) (B11 B12) n1
*                              ( 0  A22),( 0  B22) n2
*                                n1  n2    n1  n2
*
*  where N = n1+n2 and U' 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'*(A, B)*W)*(Z*W)',
*
*  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', In1) ]
*            [ kron(In2, B11)  -kron(B22', In1) ].
*
*  Here, Inx is the identity matrix of size nx and A22' is the
*  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.
*
*  Based on contributions by
*     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.
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            IDIFJB