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RedHat 9 (Linux i386) - man page for ctgsen (redhat section l)

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 lead-
       ing 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 "projec-
       tions" 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

       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 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  postmul-
	       tiplied	by the left unitary transformation matrix which reorder (A, B); The lead-
	       ing 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 postmul-
	       tiplied by the left unitary transformation matrix which reorder (A, B); The  lead-
	       ing  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 sub-
	       spaces) 0 <= M <= N.

	       PL, PR  (output) REAL If IJOB = 1, 4 or 5, PL, PR are lower bounds on the recipro-
	       cal  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 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 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  eigen-
       values 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 (as-
       ymptotic) 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 restric-
       tion:  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.

LAPACK version 3.0			   15 June 2000 				CTGSEN(l)


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