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.

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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

April 2012

  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