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<HTML>
<HEAD><TITLE>MB04RZ - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB04RZ">MB04RZ</A></H2>
<H3>
Reduction of a complex matrix pencil in generalized complex Schur form 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 a complex matrix pair (A,B) in generalized complex
Schur form to a block-diagonal form using well-conditioned
non-unitary equivalence transformations. The condition numbers
of the left and right transformations used for the reduction
are roughly bounded by PMAX, where PMAX is a given value.
The transformations are optionally postmultiplied in the given
matrices X and Y. The generalized Schur form is optionally
ordered, so that clustered eigenvalues are grouped in the same
pair of blocks.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB04RZ( JOBX, JOBY, SORT, N, PMAX, A, LDA, B, LDB, X,
$ LDX, Y, LDY, NBLCKS, BLSIZE, ALPHA, BETA,
$ TOL, IWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBX, JOBY, SORT
INTEGER INFO, LDA, LDB, LDX, LDY, N, NBLCKS
DOUBLE PRECISION PMAX, TOL
C .. Array Arguments ..
INTEGER BLSIZE(*), IWORK(*)
COMPLEX*16 A(LDA,*), ALPHA(*), B(LDB,*), BETA(*), X(LDX,*),
$ Y(LDY,*)
</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 left transformations are
accumulated, as follows:
= 'N': The left transformations are not accumulated;
= 'U': The left transformations are accumulated in X
(the given matrix X is updated).
JOBY CHARACTER*1
Specifies whether or not the right transformations are
accumulated, as follows:
= 'N': The right transformations are not accumulated;
= 'U': The right transformations are accumulated in Y
(the given matrix Y is updated).
SORT CHARACTER*1
Specifies whether or not the diagonal elements of the
generalized 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 pair of blocks.
= '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, B, X and Y. 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 and FURTHER COMMENTS). 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 in the
generalized complex Schur form, as returned by the LAPACK
Library routine ZGGES. The strictly lower triangular part
is used as workspace.
On exit, the leading N-by-N upper triangular part of
this array contains the computed block-diagonal matrix,
corresponding to the given matrix A. The remaining part
is set to zero.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) COMPLEX*16 array, dimension (LDB,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix B in the
generalized complex Schur form, as returned by the LAPACK
Library routine ZGGES. The diagonal elements of B are real
non-negative, hence the imaginary parts of these elements
are zero. The strictly lower triangular part is used as
workspace. The matrix B is assumed nonzero.
On exit, the leading N-by-N upper triangular part of this
array contains the computed upper triangular block-
diagonal matrix, corresponding to the given matrix B. The
remaining part is set to zero. The diagonal elements of B
are real non-negative.
LDB INTEGER
The leading dimension of the array B. LDB >= 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, for instance the left
transformation matrix VSL returned by the LAPACK Library
routine ZGGES.
On exit, if JOBX = 'U', the leading N-by-N part of this
array contains the product of the given matrix X and the
left transformation matrix that reduced (A,B) to block-
diagonal form. The local transformation matrix is itself a
product of non-unitary equivalence 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 the array X.
LDX >= 1, if JOBX = 'N';
LDX >= MAX(1,N), if JOBX = 'U'.
Y (input/output) COMPLEX*16 array, dimension (LDY,*)
On entry, if JOBY = 'U', the leading N-by-N part of this
array must contain a given matrix Y, for instance the
right transformation matrix VSR returned by the LAPACK
Library routine ZGGES.
On exit, if JOBY = 'U', the leading N-by-N part of this
array contains the product of the given matrix Y and the
right transformation matrix that reduced (A,B) to block-
diagonal form. The local transformation matrix is itself a
product of non-unitary equivalence transformations having
elements with magnitude less than or equal to PMAX.
If JOBY = 'N', this array is not referenced.
LDY INTEGER
The leading dimension of the array Y.
LDY >= 1, if JOBY = 'N';
LDY >= MAX(1,N), if JOBY = 'U'.
NBLCKS (output) INTEGER
The number of diagonal blocks of the matrices A and B.
BLSIZE (output) INTEGER array, dimension (N)
The first NBLCKS elements of this array contain the orders
of the resulting diagonal blocks of the matrices A and B.
ALPHA (output) COMPLEX*16 array, dimension (N)
BETA (output) COMPLEX*16 array, dimension (N)
If INFO = 0, then ALPHA(j)/BETA(j), j = 1, ..., N, are
the generalized eigenvalues of the matrix pair (A, B)
(the diagonals of the complex Schur form).
All BETA(j) are non-negative real numbers.
The quotients ALPHA(j)/BETA(j) may easily over- or
underflow, and BETA(j) may even be zero. Thus, the user
should avoid naively computing the ratio.
If A and B are obtained from general matrices using ZGGES,
ALPHA will be always less than and usually comparable with
norm(A) in magnitude, and BETA always less than and
usually comparable with norm(B).
</PRE>
<B>Tolerances</B>
<PRE>
TOL DOUBLE PRECISION
If SORT = 'S' or 'B', the tolerance to be used in the
ordering of the diagonal elements of the upper triangular
matrix pair.
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 (represented by the first
element of the current trailing pair of submatrices to be
reduced) are considered to belong to the same cluster if
they satisfy the following "distance" condition
| 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 finite lambda_j,
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).
The approximate symmetric chordal metric is used as
"distance" of two complex, possibly infinite numbers, x
and y. This metric is given by the formula
d(x,y) = min( |x-y|, |1/x-1/y| ),
taking into account the special cases of infinite or NaN
values.
If SORT = 'N' or 'C', this parameter is not referenced.
</PRE>
<B>Workspace</B>
<PRE>
IWORK INTEGER array, dimension (N+2)
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the matrix pencil defined by A and B is singular.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
Consider first that SORT = 'N'. Let
( A A ) ( B B )
( 11 12 ) ( 11 12 )
A = ( ), B = ( ),
( 0 A ) ( 0 B )
( 22 ) ( 22 )
be the given matrix pair in generalized Schur form, where
initially A and B are the first diagonal elements.
11 11
An attempt is made to compute the transformation matrices X and Y
of the form
( I V ) ( I W )
X = ( ), Y = ( ) (1)
( 0 I ) ( 0 I )
(partitioned as A and B ), so that (' denotes conjugate-transpose)
( A 0 ) ( B 0 )
( 11 ) ( 11 )
X' A Y = ( ), X' B Y = ( ),
( 0 A ) ( 0 B )
( 22 ) ( 22 )
and the elements of V and W do not exceed the value PMAX in
magnitude. An adaptation of the standard method for solving
generalized Sylvester equations [1], which controls the magnitude
of the individual elements of the computed solution [2], is used
to obtain V and W. When this attempt fails, an eigenvalue of
(A , B ) closest to the mean of those of (A , B ) is selected,
22 22 11 11
and moved by unitary equivalence transformations in the leading
position of (A , B ); the moved diagonal elements in A and B are
22 22
then added to the blocks A and B , respectively, increasing
11 11
their order by 1. Another attempt is made to compute suitable
transformation matrices X and Y with the new definitions of the
blocks A , A , B , and B . After successful transformation
11 22 11 22
matrices X and Y have been obtained, they postmultiply the current
transformation matrices (if JOBX = 'U' and/or JOBY = 'U') and the
whole procedure is repeated for the new blocks A and B .
22 22
When SORT = 'S', the diagonal elements of the generalized Schur
form are reordered before each step of the reduction, so that each
cluster of generalized 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 and Y 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 blocks of A and B
22 22
whose eigenvalue(s) is (are) the closest to those of (A , B )
11 11
(not to their mean) are selected and moved to the leading position
of A and B . This is called the "closest-neighbour" strategy.
22 22
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Kagstrom, B. and Westin, L.
Generalized Schur Methods with Condition Estimators for
Solving the Generalized Sylvester Equation.
IEEE Trans. Auto. Contr., 34, pp. 745-751, 1989.
[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 pencil 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 and B 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 generalized eigenvalues.
For efficiency reasons, the "absolute value" (or "magnitude") of
the complex elements x of the transformation matrices is computed
as |real(x)| + |imag(x)|.
</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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