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<HTML>
<HEAD><TITLE>MB05MY - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB05MY">MB05MY</A></H2>
<H3>
Computation of the orthogonal matrix reducing a given matrix to real Schur form T, of the eigenvalues, and of the upper triangular matrix of right eigenvectors of T
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To compute, for an N-by-N real nonsymmetric matrix A, the
orthogonal matrix Q reducing it to real Schur form T, the
eigenvalues, and the right eigenvectors of T.
The right eigenvector r(j) of T satisfies
T * r(j) = lambda(j) * r(j)
where lambda(j) is its eigenvalue.
The matrix of right eigenvectors R is upper triangular, by
construction.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB05MY( BALANC, N, A, LDA, WR, WI, R, LDR, Q, LDQ,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER BALANC
INTEGER INFO, LDA, LDQ, LDR, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), DWORK( * ), Q( LDQ, * ),
$ R( LDR, * ), WI( * ), WR( * )
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
BALANC CHARACTER*1
Indicates how the input matrix should be diagonally scaled
to improve the conditioning of its eigenvalues as follows:
= 'N': Do not diagonally scale;
= 'S': Diagonally scale the matrix, i.e. replace A by
D*A*D**(-1), where D is a diagonal matrix chosen
to make the rows and columns of A more equal in
norm. Do not permute.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the given matrix A.
On exit, the leading N-by-N upper quasi-triangular part of
this array contains the real Schur canonical form of A.
LDA INTEGER
The leading dimension of array A. LDA >= max(1,N).
WR (output) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION array, dimension (N)
WR and WI contain the real and imaginary parts,
respectively, of the computed eigenvalues. Complex
conjugate pairs of eigenvalues appear consecutively
with the eigenvalue having the positive imaginary part
first.
R (output) DOUBLE PRECISION array, dimension (LDR,N)
The leading N-by-N upper triangular part of this array
contains the matrix of right eigenvectors R, in the same
order as their eigenvalues. The real and imaginary parts
of a complex eigenvector corresponding to an eigenvalue
with positive imaginary part are stored in consecutive
columns. (The corresponding conjugate eigenvector is not
stored.) The eigenvectors are not backward transformed
for balancing (when BALANC = 'S').
LDR INTEGER
The leading dimension of array R. LDR >= max(1,N).
Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
The leading N-by-N part of this array contains the
orthogonal matrix Q which has reduced A to real Schur
form.
LDQ INTEGER
The leading dimension of array Q. LDQ >= MAX(1,N).
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
If BALANC = 'S' and LDWORK > 0, DWORK(2),...,DWORK(N+1)
return the scaling factors used for balancing.
LDWORK INTEGER
The length of the array DWORK. LDWORK >= max(1,4*N).
For good performance, LDWORK must generally be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
> 0: if INFO = i, the QR algorithm failed to compute all
the eigenvalues, and no eigenvectors have been
computed; elements i+1:N of WR and WI contain
eigenvalues which have converged.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
This routine uses the QR algorithm to obtain the real Schur form
T of matrix A. Then, the right eigenvectors of T are computed,
but they are not backtransformed into the eigenvectors of A.
MB05MY is a modification of the LAPACK driver routine DGEEV.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., and Sorensen, D.
LAPACK Users' Guide: Second Edition.
SIAM, Philadelphia, 1995.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE> 3
The algorithm requires 0(N ) operations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
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