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

Appendix B

Solving Recurrence Equations：With Applications to Analysis of R

ecursive Algorithms

B.1 Solving Recurrences Using Induction

Algorithm B.1 FactorialProblem: Determine n!=n(n-1)(n-2)…(3)(2)(1) whe

n n>=1. 0!=1Input: a nonnegative integer n.Output: n!. int fact(int n){ if(n==0) return 1; else return n*fact(n-1); }




Example B.2



B.2 Solving Recurrences Using The Characteristic Equation

B.2.1 Homogeneous Linear Recurrences

Definition

A recurrence of the form

a0tn + a1tn-1 + ··· + aktn-k = 0

where k and the ai terms are constants, is called a homogeneous linear recurrence equation with constant coefficients.


Example B.4 The following are homogeneous linear recurrence equations with constant coefficients:

7tn - 3tn-1 = 0

6tn - 5tn-1 + 8tn-2 = 0

8tn - 4tn-3 = 0













Example B.10 We solve the recurrence



Example B.11



B.2.2 Nonhomogeneous Linear Recurrences

Definition: A recurrence of the form

a0tn + a1tn-1 + ··· + aktn-k = f(n)

where k and the ai terms are constants and f(n) is a function other than the zero function, is called a nonhomogeneous linear recurrence equation with constant coefficients.


Example B.14




Example B.15



Example B.16



Example B.17

B.2.3 Change of Variables (Domain Transformations)

Example B.18



Example B.19



Example B.20


B.3 Solving Recurrences By Substitution




Documents

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