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CTGSEN(l)			       )			     CTGSEN(l)

NAME
       CTGSEN  -  reorder  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)

SYNOPSIS
       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 )

	   LOGICAL	  WANTQ, WANTZ

	   INTEGER	  IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK, M, N

	   REAL		  PL, PR

	   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 num‐
       bers  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 clus‐
	       ter 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: Com‐
	       pute 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

       WANTZ   (input) LOGICAL

       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

       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 ele‐
	       ments 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	refer‐
	       enced.

       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	refer‐
	       enced.

       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,  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  communica‐
	       tion  with  CLACON.   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 (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, dimension (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 par‐
	       tially 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 decomposi‐
       tion of a matrix pair (C, D) = Q*(A, B)*Z', then the reordered general‐
       ized 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 deflat‐
       ing 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 reorder‐
       ing.

       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 com‐
       puted  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 ei‐
       genvalue	 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 associ‐
       ated 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.

LAPACK version 3.0		 15 June 2000			     CTGSEN(l)
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