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<HTML>
<HEAD><TITLE>MB03RZ - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB03RZ">MB03RZ</A></H2>
<H3>
Reduction of a complex Schur form matrix to a block-diagonal form
</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 reduce an upper triangular complex matrix A (Schur form) to a
block-diagonal form using well-conditioned non-unitary similarity
transformations. The condition numbers of the transformations used
for reduction are roughly bounded by PMAX, where PMAX is a given
value. The transformations are optionally postmultiplied in a
given matrix X. The Schur form is optionally ordered, so that
clustered eigenvalues are grouped in the same block.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB03RZ( JOBX, SORT, N, PMAX, A, LDA, X, LDX, NBLCKS,
$ BLSIZE, W, TOL, INFO )C .. Scalar Arguments ..
CHARACTER JOBX, SORT
INTEGER INFO, LDA, LDX, N, NBLCKS
DOUBLE PRECISION PMAX, TOL
C .. Array Arguments ..
INTEGER BLSIZE(*)
COMPLEX*16 A(LDA,*), W(*), X(LDX,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
JOBX CHARACTER*1
Specifies whether or not the transformations are
accumulated, as follows:
= 'N': The transformations are not accumulated;
= 'U': The transformations are accumulated in X (the
given matrix X is updated).
SORT CHARACTER*1
Specifies whether or not the diagonal elements of the
Schur form are reordered, as follows:
= 'N': The diagonal elements are not reordered;
= 'S': The diagonal elements are reordered before each
step of reduction, so that clustered eigenvalues
appear in the same block;
= 'C': The diagonal elements are not reordered, but the
"closest-neighbour" strategy is used instead of
the standard "closest to the mean" strategy (see
METHOD);
= 'B': The diagonal elements are reordered before each
step of reduction, and the "closest-neighbour"
strategy is used (see METHOD).
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the matrices A and X. N >= 0.
PMAX (input) DOUBLE PRECISION
An upper bound for the absolute value of the elements of
the individual transformations used for reduction
(see METHOD). PMAX >= 1.0D0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix A to be
block-diagonalized.
On exit, the leading N-by-N upper triangular part of this
array contains the computed block-diagonal matrix, in
Schur form.
The strictly lower triangular part is used as workspace,
but it is set to zero before exit.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
X (input/output) COMPLEX*16 array, dimension (LDX,*)
On entry, if JOBX = 'U', the leading N-by-N part of this
array must contain a given matrix X.
On exit, if JOBX = 'U', the leading N-by-N part of this
array contains the product of the given matrix X and the
transformation matrix that reduced A to block-diagonal
form. The transformation matrix is itself a product of
non-unitary similarity transformations having elements
with magnitude less than or equal to PMAX.
If JOBX = 'N', this array is not referenced.
LDX INTEGER
The leading dimension of array X.
LDX >= 1, if JOBX = 'N';
LDX >= MAX(1,N), if JOBX = 'U'.
NBLCKS (output) INTEGER
The number of diagonal blocks of the matrix A.
BLSIZE (output) INTEGER array, dimension (N)
The first NBLCKS elements of this array contain the orders
of the resulting diagonal blocks of the matrix A.
W (output) COMPLEX*16 array, dimension (N)
This array contains the eigenvalues of the matrix A.
</PRE>
<B>Tolerances</B>
<PRE>
TOL DOUBLE PRECISION
The tolerance to be used in the ordering of the diagonal
elements of the upper triangular matrix.
If the user sets TOL > 0, then the given value of TOL is
used as an absolute tolerance: an eigenvalue i and a
temporarily fixed eigenvalue 1 (the first element of the
current trailing submatrix to be reduced) are considered
to belong to the same cluster if they satisfy
| lambda_1 - lambda_i | <= TOL.
If the user sets TOL < 0, then the given value of TOL is
used as a relative tolerance: an eigenvalue i and a
temporarily fixed eigenvalue 1 are considered to belong to
the same cluster if they satisfy, for j = 1, ..., N,
| lambda_1 - lambda_i | <= | TOL | * max | lambda_j |.
If the user sets TOL = 0, then an implicitly computed,
default tolerance, defined by TOL = SQRT( SQRT( EPS ) )
is used instead, as a relative tolerance, where EPS is
the machine precision (see LAPACK Library routine DLAMCH).
If SORT = 'N' or 'C', this parameter is not referenced.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
Consider first that SORT = 'N'. Let
( A A )
( 11 12 )
A = ( ),
( 0 A )
( 22 )
be the given matrix in Schur form, where initially A is the
11
first diagonal element. An attempt is made to compute a
transformation matrix X of the form
( I P )
X = ( ) (1)
( 0 I )
(partitioned as A), so that
( A 0 )
-1 ( 11 )
X A X = ( ),
( 0 A )
( 22 )
and the elements of P do not exceed the value PMAX in magnitude.
An adaptation of the standard method for solving Sylvester
equations [1], which controls the magnitude of the individual
elements of the computed solution [2], is used to obtain matrix P.
When this attempt failed, a diagonal element of A , closest to
22
the mean of those of A is selected, and moved by unitary
11
similarity transformations in the leading position of A ; the
22
moved diagonal element is then added to the block A , increasing
11
its order by 1. Another attempt is made to compute a suitable
transformation matrix X with the new definitions of the blocks A
11
and A . After a successful transformation matrix X has been
22
obtained, it postmultiplies the current transformation matrix
(if JOBX = 'U'), and the whole procedure is repeated for the
block A .
22
When SORT = 'S', the diagonal elements of the Schur form are
reordered before each step of the reduction, so that each cluster
of eigenvalues, defined as specified in the definition of TOL,
appears in adjacent elements. The elements for each cluster are
merged together, and the procedure described above is applied to
the larger blocks. Using the option SORT = 'S' will usually
provide better efficiency than the standard option (SORT = 'N'),
proposed in [2], because there could be no or few unsuccessful
attempts to compute individual transformation matrices X of the
form (1). However, the resulting dimensions of the blocks are
usually larger; this could make subsequent calculations less
efficient.
When SORT = 'C' or 'B', the procedure is similar to that for
SORT = 'N' or 'S', respectively, but the block of A whose
22
eigenvalue(s) is (are) the closest to those of A (not to their
11
mean) is selected and moved to the leading position of A . This
22
is called the "closest-neighbour" strategy.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Bartels, R.H. and Stewart, G.W. T
Solution of the matrix equation A X + XB = C.
Comm. A.C.M., 15, pp. 820-826, 1972.
[2] Bavely, C. and Stewart, G.W.
An Algorithm for Computing Reducing Subspaces by Block
Diagonalization.
SIAM J. Numer. Anal., 16, pp. 359-367, 1979.
[3] Demmel, J.
The Condition Number of Equivalence Transformations that
Block Diagonalize Matrix Pencils.
SIAM J. Numer. Anal., 20, pp. 599-610, 1983.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE> 3 4
The algorithm usually requires 0(N ) operations, but 0(N ) are
possible in the worst case, when the matrix cannot be diagonalized
by well-conditioned transformations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
The individual non-unitary transformation matrices used in the
reduction of A to a block-diagonal form have condition numbers of
the order PMAX. This does not guarantee that their product is
well-conditioned enough. The routine can be easily modified to
provide estimates for the condition numbers of the clusters of
eigenvalues.
</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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