# Chapter 2 - Solving Recurrences

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• Chapter 2Solving Recurrences

2.1 Types of Recurrences2.2 Finding Generating Functions2.3 Partial Fractions2.4 Characteristic Roots2.5 Simultaneous Recursions2.6 Fibonacci Number Identities2.7 Non-Constant Coefficients2.8 Divide-and-Conquer Relations

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• 2 Chapter 2 Solving Recurrences

• Section 2.1 Types of Recurrences 3

2.1 TYPES OF RECURRENCES

• 4 Chapter 2 Solving Recurrences

Some Linear Recursions

Recurrences without Fixed Degree

• Section 2.1 Types of Recurrences 5

Variable Coefficients

• 6 Chapter 2 Solving Recurrences

Fig 2.1.1

Table 2.1.1

• Section 2.1 Types of Recurrences 7

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Some Non-linear Recurrences

• Section 2.2 Finding an OGF 9

2.2 FINDING AN OGF

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• Section 2.2 Finding an OGF 11

• 12 Chapter 2 Solving Recurrences

• Section 2.2 Finding an OGF 13

• 14 Chapter 2 Solving Recurrences

• Section 2.3 Partial Fractions 15

2.3 PARTIAL FRACTIONS

• 16 Chapter 2 Solving Recurrences

• Section 2.3 Partial Fractions 17

• 18 Chapter 2 Solving Recurrences

• Section 2.4 Characteristic Roots 19

2.4 CHARACTERISTIC ROOTS

Characteristic Equation

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• Section 2.4 Characteristic Roots 21

Alternative Initial Values

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Repeated Roots

• Section 2.4 Characteristic Roots 23

Non-homogeneous Equations

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• Section 2.4 Characteristic Roots 25

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• Section 2.4 Characteristic Roots 27

• 28 Chapter 2 Solving Recurrences

Complex Roots

• Section 2.4 Characteristic Roots 29

• 30 Chapter 2 Solving Recurrences

2.5 SIMULTANEOUS RECURSIONS

Fig 2.5.1

• Section 2.5 Simultaneous Recursions 31

Fibonacci Rabbits

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Ubiquitousness of the Fibonacci Seq

Fig 2.5.2

• Section 2.5 Simultaneous Recursions 33

Fig 2.5.3

Solving the Fibonacci Recurrence

• 34 Chapter 2 Solving Recurrences

• Section 2.5 Simultaneous Recursions 35

• 36 Chapter 2 Solving Recurrences

Some Tiling Problems

Fig 2.5.4

• Section 2.5 Simultaneous Recursions 37

• 38 Chapter 2 Solving Recurrences

2.6 FIBO NUMBER IDENTITIES

Forward-Shift and Subscript Multipliers

• Section 2.6 Fibo Number Identities 39

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Cassinis Identity

• Section 2.6 Fibo Number Identities 41

• 42 Chapter 2 Solving Recurrences

Fibonacci Number System

• Section 2.6 Fibo Number Identities 43

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2.7 NON-CONSTANT COEFFICIENTS

A Reduction Strategy

• Section 2.7 Non-Constant Coefficients 45

• 46 Chapter 2 Solving Recurrences

• Section 2.7 Non-Constant Coefficients 47

Sum in a Recurrence: Quicksort

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Algorithm 2.7.1:

• Section 2.7 Non-Constant Coefficients 49

Analysis of the Time to Quicksort

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• Section 2.7 Non-Constant Coefficients 51

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Confirming Small Cases

• Section 2.8 Divide-&-Conquer Relations 53

2.8 DIVIDE-&-CONQUER RELATIONS

Binary Search

• 54 Chapter 2 Solving Recurrences

• Section 2.8 Divide-&-Conquer Relations 55

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Algorithm 2.8.1:

Analysis of the Time for a Binary Search

• Section 2.8 Divide-&-Conquer Relations 57

Merging

Algorithm 2.8.2:

• 58 Chapter 2 Solving Recurrences

• Section 2.8 Divide-&-Conquer Relations 59

Iterative Mergesort

• 60 Chapter 2 Solving Recurrences

Recursive Mergesort

Algorithm 2.8.3:

Analysis of the Time or a Mergesort

• Section 2.8 Divide-&-Conquer Relations 61

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The Josephus Recurrence

Fig 2.8.1

• Section 2.8 Divide-&-Conquer Relations 63

Fig 2.8.2

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Fig 2.8.3

• Section 2.8 Divide-&-Conquer Relations 65

Fig 2.8.4

• 66 Chapter 2 Solving Recurrences

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