الفهرس 
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Abstract
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PREFACE
The matrix exponential plays a central role in linear systems and
control theory. Mathematical models of many physical, biological, and
economic processes involve systems of linear, constant coefficient ordinary
differential equations X (t) =A x(t) where A is a given fixed, real or
complex nx n matrix. The solution to this equation is given by
At At OCJ (At)k .
X (t) = e x(O) , where e = L denotes the exponential of
k::::O k!
the matrix At and x(O) is the initial solution. In this thesis:
*We give explicit formulas for computing the exponential of some special
matrices and of some special block matrices.
*Also , we present the use and modification of a recent technique, so
called restrictive Pade approximation, used to approximate the exponential
matrix and compute the transient response vectors for the above system.
*Also, we consider the nonlinear matrix equation
x -A* F (X) A =1 , F(X)=~X-l .
This type of matrix equation often arises in the analysis of ladder,
networks, optimal control theory, dynamic programming, stochastic
filtering, and statistics. The iteration process Xk+1=I+A* F (Xi) A ,
k=O,1,2,... is investigated (under some conditions) to obtain the positive
definite solution for this equation. Some numerical examples which
describe the performance of the algorithm are presented.
.._-- .- --_. -- - - .__ .
._.--- ----- -- ._-- ,-_. -----_.
_’ ._ •• •• • ,0’ --’-
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This thesis consists of five chapters where the first two chapters begin
with an introduction, in which are given some information and ideas on its
contents.
Chapter one:
We summarize some basic concepts, definitions and theorems, for
computing the exponential matrix eA
.
Chapter two :
In this chapter we will introduce the linear and nonlinear matrix
equations, we present the Lyapunov matrix equation as an example to the
linear matrix equation. Also, some properties of the positive definite
solution to the nonlinear matrix equation X+AT X·I A = I are discussed,
where the smallest and largest positive definite solutions are mentioned.
Chapter three:
In this chapter, we introduce some explicit formulas for some special
types of block matrices that appear very often in control theory. Such
block matrices are related to the second-order mechanical vibration
equation M x + ex + Kx = 0 where M, C and K are real or complex
n x n matrices.
Chapter four:
In this chapter, a hybrid technique is presented to compute the
transient response vectors in such an iterative explicit form. A recent
method ,so called restrictive Pade approximation, is used and extended to
approximate the exponential matrix [14]. Numerical test example is
given to illustrate the accuracy of the suggested method compared with
previous works as well as the exact solution.