trsen(3)

NAME
trsen - trsen: reorder Schur form

SYNOPSIS

  Functions
subroutine ctrsen (job, compq, select, n, t, ldt, q, ldq, w, m, s, sep, work, lwork, info)
CTRSEN
subroutine dtrsen (job, compq, select, n, t, ldt, q, ldq, wr, wi, m, s, sep, work, lwork, iwork, liwork, info)
DTRSEN
subroutine strsen (job, compq, select, n, t, ldt, q, ldq, wr, wi, m, s, sep, work, lwork, iwork, liwork, info)
STRSEN
subroutine ztrsen (job, compq, select, n, t, ldt, q, ldq, w, m, s, sep, work, lwork, info)
ZTRSEN

Detailed Description

Function Documentation

  subroutine ctrsen (character job, character compq, logical, dimension( * ) select, integer n, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( * ) w, integer m, real s, real sep, complex, dimension( * ) work, integer lwork, integer info)

CTRSEN

Purpose:

    !>
    !> 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.
    !>

Parameters
    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
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  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
    !>

Definition at line 262 of file ctrsen.f.

  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

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
    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.
    !>

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
    !>

Definition at line 311 of file dtrsen.f.

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

STRSEN

Purpose:

    !>
    !> STRSEN 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 SHSEQR), 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
    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 REAL 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 REAL 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 REAL array, dimension (N)
    !>

    WI

    !>          WI is REAL 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 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 REAL 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  STRSEN 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
    !>

Definition at line 312 of file strsen.f.

  subroutine ztrsen (character job, character compq, logical, dimension( * ) select, integer n, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( * ) w, integer m, double precision s, double precision sep, complex*16, dimension( * ) work, integer lwork, integer info)

ZTRSEN

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.
    !>

Parameters
    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*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

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

    Q

    !>          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

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

    W

    !>          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

    !>          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 COMPLEX*16 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
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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
    !>

Definition at line 262 of file ztrsen.f.

Author

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