CTRSEN reorders the Schur factorization of a complex matrix


 A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in


 the leading positions on the diagonal of the upper 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.

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 the j-th eigenvalue, SELECT(j) must be set to .TRUE..

N

          N is INTEGER


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

T

          T is COMPLEX array, dimension (LDT,N)


          On entry, the upper triangular matrix T.


          On exit, T is overwritten by the reordered matrix T, with the


          selected eigenvalues as the leading diagonal elements.

LDT

          LDT is INTEGER


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

Q

          Q is COMPLEX 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


          unitary 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.

W

          W is COMPLEX array, dimension (N)


          The reordered eigenvalues of T, in the same order as they


          appear on the diagonal of T.

M

          M is INTEGER


          The dimension of the specified invariant subspace.


          0 <= M <= N.

S

          S is REAL


          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 REAL


          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 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.


          If JOB = 'N', LWORK >= 1;


          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.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

  CTRSEN first collects the selected eigenvalues by computing a unitary


  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**H * T * Z = ( T11 T12 ) n1


                         (  0  T22 ) n2


                            n1  n2


  where N = n1+n2. The first


  n1 columns of Z span the specified invariant subspace of T.


  If T has been obtained from the Schur factorization of a matrix


  A = Q*T*Q**H, then the reordered Schur factorization of A is given by


  A = (Q*Z)*(Z**H*T*Z)*(Q*Z)**H, 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