ZTRSEN(3)      LAPACK routine of NEC Numeric Library Collection      ZTRSEN(3)



NAME
       ZTRSEN

SYNOPSIS
       SUBROUTINE ZTRSEN (JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S, SEP,
           WORK, LWORK, INFO)



PURPOSE
            ZTRSEN 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.




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

           SELECT    (input)
                     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         (input)
                     N is INTEGER
                     The order of the matrix T. N >= 0.

           T         (input/output)
                     T is COMPLEX*16 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       (input)
                     LDT is INTEGER
                     The leading dimension of the array T. LDT >= max(1,N).

           Q         (input/output)
                     Q is COMPLEX*16 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       (input)
                     LDQ is INTEGER
                     The leading dimension of the array Q.
                     LDQ >= 1; and if COMPQ = 'V', LDQ >= N.

           W         (output)
                     W is COMPLEX*16 array, dimension (N)
                     The reordered eigenvalues of T, in the same order as they
                     appear on the diagonal of T.

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

           S         (output)
                     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       (output)
                     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      (output)
                     WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK     (input)
                     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      (output)
                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  if INFO = -i, the i-th argument had an illegal value






FURTHER DETAILS
             ZTRSEN 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



LAPACK routine                  31 October 2017                      ZTRSEN(3)