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Unformatted text preview: t violations to meyer@ncsu.edu 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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