Recurrences

Recurrences - CSE 1400 Applied Discrete Mathematics...

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Unformatted text preview: CSE 1400 Applied Discrete Mathematics Recurrences Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Recurrence Equations 1 The Alice Recurrence Equation 1 The Gauss Recurrence Equation 2 The Triangular Recurrence Equation 3 The Mersenne Recurrence Equation 5 The Factorial Recurrence Equation 5 The Fermat Recurrence Equation 6 The Fibonacci Recurrence Equation 7 Still More Recurrences 7 Bisection 7 Newton’s Rootfinding Method 9 Dissecting Space with Hyperplanes 11 Problems on Recurrences 16 Abstract The definition of “recursive” that I like is Recursive: adj. See: Recursive “Devil’s DP Dictionary,” Stan Kelly-Bootle, English computer scientist and author ( 1929 –) Recurrence Equations R ecurrence equations are the discrete analog of differ- ential equations . Recurrence equations have two parts. Recurrence equations are similar to differential equations. Recurrence equations are studied in discrete mathe- matics, Differential equations studied in continuous mathematics. cse 1400 applied discrete mathematics recurrences 2 1 . An equation that describes how new values are computed from previous values. The form a recurrence equation is something like t n = some formula involving t n- 1 and perhaps lower terms such as t n- 2 , t n- 3 , . . . , t 2 . One or more initial values to seed the computational process. These initial values are given for the first term t and perhaps higher terms such as t 1 and t 2 . There are several basis recurrence equations that occur in discrete mathematics. The Alice Recurrence Equation “Can you do Addition?” the White Queen asked. “What’s one and one and one and one and one and one and one and one and one and one?” “I don’t know,” said Alice. “I lost count.” “She can’t do Addition,” the Red Queen interrupted. “Can you do Subtraction? Take nine from eight.” Alice terms are defined by the recurrence equation equation A simple recurrence, but Alice is fundamental to counting. a n = a n- 1 ∀ n ∈ N , n ≥ 1 with initial condition a = 1. The Alice recurrence generates terms in the unary sequence ~ A = h 1, 1, 1, 1, 1, 1, 1, 1, . . . i Terms in the Alice sequence can also be computed by the constant function a ( n ) = 1 ∀ n ∈ N = { 0, 1, 2, 3, 4, . . . } U nary notation for the natural numbers is based on the Alice sequence. Mathematical induction can be used to prove the function Using the unary alphabet { 1 } with only the character 1 can name the positive natural numbers 1 = 1 2 = 11, 3 = 111, . . . . Of course, U nary notation wastes space. To write 256 in unary notation requires a string of 256 unary 1’s, but only 3 digits or 9 bits . a ( n ) = 1 solves the recurrence equation a n = a n- 1 with initial condition a = 1....
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This note was uploaded on 02/11/2012 for the course MTH 2051 taught by Professor Shoaff during the Fall '11 term at FIT.

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Recurrences - CSE 1400 Applied Discrete Mathematics...

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