dtrsen(3S)

NAME
     DTRSEN - reorder 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,

SYNOPSIS
     SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, M, S,
                        SEP, WORK, LWORK, IWORK, LIWORK, INFO )

         CHARACTER      COMPQ, JOB

         INTEGER        INFO, LDQ, LDT, LIWORK, LWORK, M, N

         DOUBLE         PRECISION S, SEP

         LOGICAL        SELECT( * )

         INTEGER        IWORK( * )

         DOUBLE         PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( *
                        ), WR( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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 elemnts equal and its off-diagonal
     elements of opposite sign.

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

     SELECT  (input) 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 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 either both
             included in the cluster or both excluded.

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

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

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

     WR      (output) DOUBLE PRECISION array, dimension (N)
             WI      (output) 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       (output) INTEGER
             The dimension of the specified invariant subspace.  0 < = M <= N.

     S       (output) 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) 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    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
             On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

     LWORK   (input) INTEGER
             The dimension of the array WORK.  If JOB = 'N', LWORK >=
             max(1,N); if JOB = 'E', LWORK >= M*(N-M); if JOB = 'V' or 'B',
             LWORK >= 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   (workspace) INTEGER array, dimension (LIWORK)
             IF JOB = 'N' or 'E', IWORK is not referenced.

     LIWORK  (input) INTEGER
             The dimension of the array IWORK.  If JOB = 'N' or 'E', LIWORK >=
             1; if JOB = 'V' or 'B', LIWORK >= 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    (output) 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.

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

     where N = n1+n2 and Z' 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', then the reordered real Schur factorization of A is given by A =
     (Q*Z)*(Z'*T*Z)*(Q*Z)', 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


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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