해피레포트 - 대학원 이산수학 레포트(영문)

카테고리 : 레포트 > 공학,기술계열 파일이름 :Reccurrence_equation.pdf 문서분량 : 7 page 등록인 : dreamappvard 문서뷰어 : 아크로뱃리더프로그램 등록/수정일 : 13.05.28 / 13.05.28 구매평가 : 다운로드수 : 0 판매가격 : 4,000 원

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대학원 과정 이산수학 Discrete Mathmatics 리써치를베이스로 한 레포트입니다.

본문일부/목차
In this project, solutions to second order linear recurrence equations with constant coeffi- cients have been investigated. We have used generating functions to derive the general solution to the homogeneous equation and we show that in general the particular solution is complicated to find. By limiting the right hand side (RHS) in the equation to a polynomial-exponential family of functions we can however find the particular solution in a closed form. We show that the homogeneous solution is a linear combination of exponential functions and the particular solution is of the same form as the RHS of the equation with an increase in polynomial order if any part of the RHS can be expressed in terms of the homogeneous solution, so called resonance. Using generating functions to solve such problems require a lot of computations and par- tial fractions expansions. Therefore a more hands on approach is presented and discussed where the forms of the homogeneous and particular solutions are assumed, based on the pre- viously derived solutions. The homogeneous solution is determined by solving a characteristic equation, and using the characteristic roots together with the assumed form of the solution the solution is given with two undetermined coefficients. The particular solution is found by substituting the assumed form of the particular solution into the equations and solving a linear system of equations. Finally the unknown coefficients are determined from the initial conditions. 1 Introduction Recurrence equations are equations that, given initial conditions, recursively defines a sequence. The set of linear recursive equations is a subset of the set of recurrence equations and linear equations have many properties that enables more explicit theoretical analyses of the problem. A general linear recurrence equation can be written as [1, 3],  ∑K dk(n)an−k = f(n), n ≥ K,  k=0 ai =βi,i=0,1,...,K−1 (1) for arbitrary coefficients dk(n), initial conditions βi and a function f(n). There is a wide range of applications for linear recurrence equation where the field of computer algorithms is a large application area. Even if the problem is not linear the problem can sometimes be approximated as linear and it can therefore be solved using the methods developed for linear problems. In this report we focus on linear second order recurrence equations with constant coefficients hence K = 2 and dk,k = 0,1,2 are constants. This report aims to give a semi-rigorous introduction to the field of linear second order recurrence equations with constant coefficients where the solutions are derived and easy-to-use methods are presented. In Section 2 we are first given a short overview to second order linear recurrence equations with constant coefficients where to the two main solution methods are introduced. Then in Section 2.1 we are given a brief introduction to the method of homogeneous and particular solutions. In Section 2.2 the method of generating functions is described where the solution to the homogeneous equation is derived in Section 2.2.1 and ditto for the particular solution is derived in Section 2.2.2 for a particular family of functions f(n) (polynomial-exponential functions). In Section 2.3 a hands-on solution method derived from the generating function results is introduced arriving at a final solution to the problem. References [1] Tang M., Tang V.T. Using Generating Functions to Solve Linear Inhomogeneous Recurrence Equa- tions, Proceedings of the 6th WSEAS International Conference on Simulation, Modelling and Opti- mization, Lisbon, Portugal, September 22-24, 2006. [2] Parag H. Dave; Himanshu B. Dave, Design and Analysis of Algorithms, p.709, Pearson Education India, 2007, ISBN 978-81-775-8595-7 [3] Kauers, M., Paule P., The Concrete Tetrahedron, Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates, p.66 Texts and Monographs in Symbolic Computation, 2011, ISBN: 978-3-7091-0445-3 [4] Epp, Susanna, Discrete Mathematics with Applications, 4th ed., p317-319, DePaul University, BROOKS/COLE CENGANGE Learning, 2011 [5] Cull P.; Flahive M.E, Robson, R.O., Difference equations: from rabbits to chaos, p.74, New York : Springer, c2005, ISBN:0387232338 REFERENCES ∑N i=0 The solution is found for f(n) ∈ F but For other forms of f(n), other forms of apn have to be assumed which may be very complicated if f(n) is a complicated expression. Just note that cos n, sin n ∈ F since they can be expressed in terms of e±in. 2.3.3 Solution to the full problem Having found the homogeneous and particular solutions to the problem the solution is given as a sum of the two , an = ahn + apn. This solution has two unknown parameters A, B, see (25), which are easily determined by the initial conditions a0,a1. The parameters A,B are given by (31).

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카테고리 : 레포트 > 공학,기술계열
파일이름 :Reccurrence_equation.pdf
문서분량 : 7 page	등록인 : dreamappvard
문서뷰어 : 아크로뱃리더프로그램	등록/수정일 : 13.05.28 / 13.05.28
구매평가 :	다운로드수 : 0

대학원 이산수학 레포트(영문)

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