.TH "SRC/strsen.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
SRC/strsen.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBstrsen\fP (job, compq, select, n, t, ldt, q, ldq, wr, wi, m, s, sep, work, lwork, iwork, liwork, info)"
.br
.RI "\fBSTRSEN\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "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)"

.PP
\fBSTRSEN\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
!>          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)\&.
!> 
.fi
.PP
.br
\fICOMPQ\fP 
.PP
.nf
!>          COMPQ is CHARACTER*1
!>          = 'V': update the matrix Q of Schur vectors;
!>          = 'N': do not update Q\&.
!> 
.fi
.PP
.br
\fISELECT\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrix T\&. N >= 0\&.
!> 
.fi
.PP
.br
\fIT\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDT\fP 
.PP
.nf
!>          LDT is INTEGER
!>          The leading dimension of the array T\&. LDT >= max(1,N)\&.
!> 
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
!>          LDQ is INTEGER
!>          The leading dimension of the array Q\&.
!>          LDQ >= 1; and if COMPQ = 'V', LDQ >= N\&.
!> 
.fi
.PP
.br
\fIWR\fP 
.PP
.nf
!>          WR is REAL array, dimension (N)
!> 
.fi
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.br
\fIWI\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIM\fP 
.PP
.nf
!>          M is INTEGER
!>          The dimension of the specified invariant subspace\&.
!>          0 < = M <= N\&.
!> 
.fi
.PP
.br
\fIS\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fISEP\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
!>          WORK is REAL array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
!> 
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
!>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
!>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK\&.
!> 
.fi
.PP
.br
\fILIWORK\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
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\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  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
!> 
.fi
.PP
 
.RE
.PP

.PP
Definition at line \fB312\fP of file \fBstrsen\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.