**Unformatted text preview: **t violations to [email protected] y (k + 1) = αm y (k ) + αm−1 y (k − 1) · · · + α1 y (k − m + 1) + α0 (7.10.1) in which α0 , α1 , . . . , αm along with initial conditions y (0), y (1), . . . , y (m − 1)
are known constants, and y (m), y (m + 1), y (m + 2) . . . are unknown. Diﬀerence
equations are the discrete analogs of diﬀerential equations, and, among other
ways, they arise by discretizing diﬀerential equations. For example, discretizing
a second-order linear diﬀerential equation results in a system of second-order
diﬀerence equations as illustrated in Example 1.4.1, p 19. The theory of linear
diﬀerence equations parallels the theory for linear diﬀerential equations, and
a technique similar to the one used to solve linear diﬀerential equations with
constant coeﬃcients produces the solution of (7.10.1) as D
E T
H m y (k ) = α0
+
βi λk ,
i
1 − α1 − · · · − αm i=1 IG
R for k = 0, 1, . . . (7.10.2) in which the λi ’s are the roots of λm − αm λm−1 − · · · − α0 = 0, and the βi ’s
are constants determined by the initial conditions y (0), y (1), . . . , y (m − 1). The
ﬁrst term on the right-hand side of (7.10.2) is a particular solution of (7.10.1),
and the summation term in (7.10.2) is the general solution of the associated
homogeneous equation deﬁned by setting α0 = 0.
This section focuses on systems of ﬁrst-order linear diﬀerence equations with
constant coeﬃcients, and such systems can be written in matrix form as Y
P O
C x(k + 1) = Ax(k ) (a homogeneous system) x(k + 1) = Ax(k ) + b(k ) or (a nonhomogeneous system), (7.10.3) where matrix An×n , the initial vector x(0), and vectors b(k ), k = 0, 1, . . . , are
known. The problem is to determine the unknown vectors x(k ), k = 1, 2, . . . ,
along with an expression for the limiting vector limk→∞ x(k ). Such systems are
used to model linear discrete-time evolutionary processes, and the goal is usually
to predict how (or to where) the pr...

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