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ctgsen.f(3)				      LAPACK				      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
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      ctgsen.f(3)
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