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## dtrsen()

subroutine dtrsen | ( | character | job,
|  | character | compq,
|  | logical, dimension( * ) | select,
|  | integer | n,
|  | double precision, dimension( ldt, * ) | t,
|  | integer | ldt,
|  | double precision, dimension( ldq, * ) | q,
|  | integer | ldq,
|  | double precision, dimension( * ) | wr,
|  | double precision, dimension( * ) | wi,
|  | integer | m,
|  | double precision | s,
|  | double precision | sep,
|  | double precision, dimension( * ) | work,
|  | integer | lwork,
|  | integer, dimension( * ) | iwork,
|  | integer | liwork,
|  | integer | info )

DTRSEN

Download DTRSEN + dependencies [TGZ] [ZIP] [TXT]

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

  [in]

  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).
      !>

  [in]

  COMPQ

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

  [in]

  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.
      !>

  [in]

  N

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

  [in,out]

  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.
      !>

  [in]

  LDT

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

  [in,out]

  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.
      !>

  [in]

  LDQ

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

  [out]

  WR

      !>          WR is DOUBLE PRECISION array, dimension (N)
      !>

  [out]

  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.
      !>

  [out]

  M

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

  [out]

  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.
      !>

  [out]

  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.
      !>

  [out]

  WORK

      !>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
      !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
      !>

  [in]

  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.
      !>

  [out]

  IWORK

      !>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
      !>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
      !>

  [in]

  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.
      !>

  [out]

  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.

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
      !>