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

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.ti -1c
.RI "subroutine \fBztrsen\fP (job, compq, select, n, t, ldt, q, ldq, w, m, s, sep, work, lwork, info)"
.br
.RI "\fBZTRSEN\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "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)"

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\fBZTRSEN\fP  
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\fBPurpose:\fP
.RS 4

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.nf
!>
!> 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\&.
!> 
.fi
.PP
 
.RE
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\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 
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.nf
!>          COMPQ is CHARACTER*1
!>          = 'V': update the matrix Q of Schur vectors;
!>          = 'N': do not update Q\&.
!> 
.fi
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.br
\fISELECT\fP 
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.nf
!>          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\&.\&.
!> 
.fi
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.br
\fIN\fP 
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!>          N is INTEGER
!>          The order of the matrix T\&. N >= 0\&.
!> 
.fi
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\fIT\fP 
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!>          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\&.
!> 
.fi
.PP
.br
\fILDT\fP 
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!>          LDT is INTEGER
!>          The leading dimension of the array T\&. LDT >= max(1,N)\&.
!> 
.fi
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\fIQ\fP 
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!>          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\&.
!> 
.fi
.PP
.br
\fILDQ\fP 
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!>          LDQ is INTEGER
!>          The leading dimension of the array Q\&.
!>          LDQ >= 1; and if COMPQ = 'V', LDQ >= N\&.
!> 
.fi
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.br
\fIW\fP 
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.nf
!>          W is COMPLEX*16 array, dimension (N)
!>          The reordered eigenvalues of T, in the same order as they
!>          appear on the diagonal of T\&.
!> 
.fi
.PP
.br
\fIM\fP 
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!>          M is INTEGER
!>          The dimension of the specified invariant subspace\&.
!>          0 <= M <= N\&.
!> 
.fi
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\fIS\fP 
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!>          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\&.
!> 
.fi
.PP
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\fISEP\fP 
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!>          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\&.
!> 
.fi
.PP
.br
\fIWORK\fP 
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!>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&.
!> 
.fi
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\fILWORK\fP 
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!>          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\&.
!> 
.fi
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\fIINFO\fP 
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!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value
!> 
.fi
.PP
 
.RE
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\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

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Univ\&. of California Berkeley 

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Univ\&. of Colorado Denver 

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

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.nf
!>
!>  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
!> 
.fi
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.RE
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Definition at line \fB262\fP of file \fBztrsen\&.f\fP\&.
.SH "Author"
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Generated automatically by Doxygen for LAPACK from the source code\&.