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dtrsen.f(3)				      LAPACK				      dtrsen.f(3)

NAME
       dtrsen.f -

SYNOPSIS
   Functions/Subroutines
       subroutine dtrsen (JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, M, S, SEP, WORK, LWORK,
	   IWORK, LIWORK, INFO)
	   DTRSEN

Function/Subroutine Documentation
   subroutine dtrsen (characterJOB, characterCOMPQ, logical, dimension( * )SELECT, integerN,
       double precision, dimension( ldt, * )T, integerLDT, double precision, dimension( ldq, *
       )Q, integerLDQ, double precision, dimension( * )WR, double precision, dimension( * )WI,
       integerM, double precisionS, double precisionSEP, double precision, dimension( * )WORK,
       integerLWORK, integer, dimension( * )IWORK, integerLIWORK, integerINFO)
       DTRSEN

       Purpose:

	    DTRSEN reorders the real Schur factorization of a real matrix
	    A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
	    the leading diagonal blocks of the upper quasi-triangular matrix T,
	    and the leading columns of Q form an orthonormal basis of the
	    corresponding right invariant subspace.

	    Optionally the routine computes the reciprocal condition numbers of
	    the cluster of eigenvalues and/or the invariant subspace.

	    T must be in Schur canonical form (as returned by DHSEQR), that is,
	    block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
	    2-by-2 diagonal block has its diagonal elements equal and its
	    off-diagonal elements of opposite sign.

       Parameters:
	   JOB

		     JOB is CHARACTER*1
		     Specifies whether condition numbers are required for the
		     cluster of eigenvalues (S) or the invariant subspace (SEP):
		     = 'N': none;
		     = 'E': for eigenvalues only (S);
		     = 'V': for invariant subspace only (SEP);
		     = 'B': for both eigenvalues and invariant subspace (S and
			    SEP).

	   COMPQ

		     COMPQ is CHARACTER*1
		     = 'V': update the matrix Q of Schur vectors;
		     = 'N': do not update Q.

	   SELECT

		     SELECT is LOGICAL array, dimension (N)
		     SELECT specifies the eigenvalues in the selected cluster. To
		     select a real eigenvalue w(j), SELECT(j) must be set to
		     .TRUE.. To select a complex conjugate pair of eigenvalues
		     w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
		     either SELECT(j) or SELECT(j+1) or both must be set to
		     .TRUE.; a complex conjugate pair of eigenvalues must be
		     either both included in the cluster or both excluded.

	   N

		     N is INTEGER
		     The order of the matrix T. N >= 0.

	   T

		     T is DOUBLE PRECISION array, dimension (LDT,N)
		     On entry, the upper quasi-triangular matrix T, in Schur
		     canonical form.
		     On exit, T is overwritten by the reordered matrix T, again in
		     Schur canonical form, with the selected eigenvalues in the
		     leading diagonal blocks.

	   LDT

		     LDT is INTEGER
		     The leading dimension of the array T. LDT >= max(1,N).

	   Q

		     Q is DOUBLE PRECISION array, dimension (LDQ,N)
		     On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
		     On exit, if COMPQ = 'V', Q has been postmultiplied by the
		     orthogonal transformation matrix which reorders T; the
		     leading M columns of Q form an orthonormal basis for the
		     specified invariant subspace.
		     If COMPQ = 'N', Q is not referenced.

	   LDQ

		     LDQ is INTEGER
		     The leading dimension of the array Q.
		     LDQ >= 1; and if COMPQ = 'V', LDQ >= N.

	   WR

		     WR is DOUBLE PRECISION array, dimension (N)

	   WI

		     WI is DOUBLE PRECISION array, dimension (N)

		     The real and imaginary parts, respectively, of the reordered
		     eigenvalues of T. The eigenvalues are stored in the same
		     order as on the diagonal of T, with WR(i) = T(i,i) and, if
		     T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
		     WI(i+1) = -WI(i). Note that if a complex eigenvalue is
		     sufficiently ill-conditioned, then its value may differ
		     significantly from its value before reordering.

	   M

		     M is INTEGER
		     The dimension of the specified invariant subspace.
		     0 < = M <= N.

	   S

		     S is DOUBLE PRECISION
		     If JOB = 'E' or 'B', S is a lower bound on the reciprocal
		     condition number for the selected cluster of eigenvalues.
		     S cannot underestimate the true reciprocal condition number
		     by more than a factor of sqrt(N). If M = 0 or N, S = 1.
		     If JOB = 'N' or 'V', S is not referenced.

	   SEP

		     SEP is DOUBLE PRECISION
		     If JOB = 'V' or 'B', SEP is the estimated reciprocal
		     condition number of the specified invariant subspace. If
		     M = 0 or N, SEP = norm(T).
		     If JOB = 'N' or 'E', SEP is not referenced.

	   WORK

		     WORK is DOUBLE PRECISION 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.
		     If JOB = 'N', LWORK >= max(1,N);
		     if JOB = 'E', LWORK >= max(1,M*(N-M));
		     if JOB = 'V' or 'B', LWORK >= max(1,2*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.
		     If JOB = 'N' or 'E', LIWORK >= 1;
		     if JOB = 'V' or 'B', LIWORK >= max(1,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 T failed because some eigenvalues are too
			  close to separate (the problem is very ill-conditioned);
			  T may have been partially reordered, and WR and WI
			  contain the eigenvalues in the same order as in T; S and
			  SEP (if requested) are set to zero.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   April 2012

       Further Details:

	     DTRSEN first collects the selected eigenvalues by computing an
	     orthogonal transformation Z to move them to the top left corner of T.
	     In other words, the selected eigenvalues are the eigenvalues of T11
	     in:

		     Z**T * T * Z = ( T11 T12 ) n1
				    (  0  T22 ) n2
				       n1  n2

	     where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns
	     of Z span the specified invariant subspace of T.

	     If T has been obtained from the real Schur factorization of a matrix
	     A = Q*T*Q**T, then the reordered real Schur factorization of A is given
	     by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span
	     the corresponding invariant subspace of A.

	     The reciprocal condition number of the average of the eigenvalues of
	     T11 may be returned in S. S lies between 0 (very badly conditioned)
	     and 1 (very well conditioned). It is computed as follows. First we
	     compute R so that

				    P = ( I  R ) n1
					( 0  0 ) n2
					  n1 n2

	     is the projector on the invariant subspace associated with T11.
	     R is the solution of the Sylvester equation:

				   T11*R - R*T22 = T12.

	     Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
	     the two-norm of M. Then S is computed as the lower bound

				 (1 + F-norm(R)**2)**(-1/2)

	     on the reciprocal of 2-norm(P), the true reciprocal condition number.
	     S cannot underestimate 1 / 2-norm(P) by more than a factor of
	     sqrt(N).

	     An approximate error bound for the computed average of the
	     eigenvalues of T11 is

				    EPS * norm(T) / S

	     where EPS is the machine precision.

	     The reciprocal condition number of the right invariant subspace
	     spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
	     SEP is defined as the separation of T11 and T22:

				sep( T11, T22 ) = sigma-min( C )

	     where sigma-min(C) is the smallest singular value of the
	     n1*n2-by-n1*n2 matrix

		C  = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )

	     I(m) is an m by m identity matrix, and kprod denotes the Kronecker
	     product. We estimate sigma-min(C) by the reciprocal of an estimate of
	     the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
	     cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).

	     When SEP is small, small changes in T can cause large changes in
	     the invariant subspace. An approximate bound on the maximum angular
	     error in the computed right invariant subspace is

				 EPS * norm(T) / SEP

       Definition at line 313 of file dtrsen.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

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