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Appendix B Solving Recurrence Equations ： With Applications to Analysis of Recursive Algorithms

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B.1 Solving Recurrences Using Induction Algorithm B.1 Factorial Problem: Determine n!=n(n-1)(n-2)…(3)(2)(1) when n>=1. 0!=1 Input: a nonnegative integer n. Output: n!. int fact(int n){ if(n==0) return 1; else return n*fact(n-1); }

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B.1 Solving Recurrences Using Induction

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Example B.2

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B.1 Solving Recurrences Using Induction

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B.2 Solving Recurrences Using The Characteristic Equation B.2.1 Homogeneous Linear Recurrences Definition A recurrence of the form a 0 t n + a 1 t n-1 + ··· + a k t n-k = 0 where k and the a i terms are constants, is called a homogeneous linear recurrence equation with constant coefficients.

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B.2 Solving Recurrences Using The Characteristic Equation Example B.4 The following are homogeneous linear recurrence equations with constant coefficients: 7t n - 3t n-1 = 0 6t n - 5t n-1 + 8t n-2 = 0 8t n - 4t n-3 = 0

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B.2 Solving Recurrences Using The Characteristic Equation

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Example B.10 We solve the recurrence

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B.2 Solving Recurrences Using The Characteristic Equation

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Example B.11

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B.2 Solving Recurrences Using The Characteristic Equation

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B.2.2 Nonhomogeneous Linear Recurrences Definition: A recurrence of the form a 0 t n + a 1 t n-1 + ··· + a k t n-k = f(n) where k and the a i terms are constants and f(n) is a function other than the zero function, is called a nonhomogeneous linear recurrence equation with constant coefficients.

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B.2 Solving Recurrences Using The Characteristic Equation Example B.14

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B.2 Solving Recurrences Using The Characteristic Equation

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Example B.15

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B.2 Solving Recurrences Using The Characteristic Equation

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Example B.16

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B.2 Solving Recurrences Using The Characteristic Equation

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Example B.17

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B.2.3 Change of Variables (Domain Transformations) Example B.18

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B.2.3 Change of Variables (Domain Transformations)

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Example B.19

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B.2.3 Change of Variables (Domain Transformations)

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Example B.20

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B.2.3 Change of Variables (Domain Transformations)

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B.3 Solving Recurrences By Substitution

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