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A Structure-Preserving Method for Generalized Algebraic RiccatiEquations Based on Pencil Arithmetic
(2004)
This paper describes a numerical method for extracting the stable
right deflating subspace of a matrix pencil Z Y using
a spectral projection method. It has several advantages compared
to other spectral projection methods like the sign function
method. In particular it avoids the rounding error induced
loss of accuracy associated with matrix inversions. The new algorithm
is particularly well adapted to solving continuous-time
algebraic Riccati equations. In numerical examples, it solves
Riccati equations to high accuracy.
We investigate the condition number for a complex eigenvalue of a real matrix under
real perturbations. Based on an explicit formula, it is shown that this number is never
smaller than 1/
p
2 times the corresponding condition number with respect to complex
perturbations. This result can be generalized to the condition number of an arbitrary
complex-valued function under real perturbations. This extends to related condition
numbers.
This paper introduces arithmetic-like operations on matrix pencils. The pencil-arithmetic
operations extend elementary formulas for sums and products of rational numbers and
include the algebra of linear transformations as a special case. These operation give an
unusual perspective on a variety of pencil related computations. We derive generalizations of
monodromy matrices and the matrix exponential. A new algorithm for computing a pencilarithmetic
generalization of the matrix sign function does not use matrix inverses and gives
an empirically forward numerically stable algorithm for extracting deflating subspaces.
We present structure preserving algorithms for the numerical com-
putation of structured staircase forms of skew-symmetric/symmetric
matrix pencils along with the Kronecker indices of the associated skew-
symmetric/symmetric Kronecker-like canonical form. These methods
allow deflation of the singular structure and deflation of infinite eigenvalues with index greater than one. Two algorithms are proposed: one
for general skew-symmetric/symmetric pencils and one for pencils in
0
which the skew-symmetric matrix is a direct sum of 0 and J = −I I .
0
We show how to use the structured staircase form to solve boundary
value problems arising in control applications and present numerical
examples.
We discuss the eigenvalue problem for
general and structured matrix polynomials which may
be singular and may have eigenvalues at infinity.
We derive staircase
condensed forms that allow deflation of the infinite eigenvalue and
singular structure of the matrix polynomial.
The remaining reduced order staircase form leads to
new types of linearizations which determine the finite eigenvalues and
and corresponding eigenvectors. The new linearizations
also simplify the construction of structure preserving linearizations.