"Provides a clear and comprehensive overview of the fundamental theories, numerical methods, and iterative processes encountered in difference calculus. Explores classical problems such as orthological polynomials, the Euclidean algorithm, roots of polynomials, and well-conditioning. "

Preface

iii

Discrete Calculus

1

(34)

Introduction

1

(1)

Discrete Calculus

1

(3)

Summation and Negative Powers of Δ

4

(5)

Equations reducible to simple form

6

(3)

Factorial Powers and Stirling Numbers

9

(2)

Bernoulli Numbers and Polynomials

11

(2)

Matrix Form

13

(11)

Pascal matrix and combinatorics

14

(5)

Pascal matrix and Bernoulli polynomials

19

(1)

Pascal matrix and Bernstein polynomials

20

(2)

Pascal matrix and Stirling numbers

22

(2)

Comparison Principle

24

(5)

Problems and Remarks

29

(5)

Notes

34

(1)

Linear Difference Equations

35

(40)

Introduction

35

(1)

Preliminaries

35

(3)

Fundamental Theory

38

(5)

Adjoint and transposed equations

42

(1)

The Method of Variation of Constants

43

(1)

Linear Equations with Constant Coefficients

44

(6)

Use of Operators Δ and E

50

(3)

Method of Generating Functions

53

(6)

Stability of Solutions

59

(2)

Absolute Stability

61

(6)

Boundary Value Problems

67

(2)

Problems and Remarks

69

(3)

Notes

72

(3)

Linear Systems of Difference Equations

75

(28)

Introduction

75

(1)

Basic Theory

75

(4)

Method of Variation of Constants

79

(3)

Autonomous Systems

82

(1)

Systems Representing High-Order Equations

82

(6)

One-sided Green's functions

85

(3)

Poincare Theorem

88

(2)

Periodic Solutions

90

(5)

Boundary Value Problems

95

(4)

Problems

99

(2)

Notes

101

(2)

Stability Theory

103

(40)

Introduction

103

(1)

Stability Notions

103

(4)

The Linear Case

107

(1)

Autonomous Linear Systems

108

(2)

Linear Equations with Periodic Coefficients

110

(2)

Use of the Comparison Principle

112

(4)

Variation of Constants

116

(3)

Stability by First Approximation

119

(2)

Liapunov Functions

121

(8)

Domain of Asymptotic Stability

129

(3)

Converse Theorems

132

(5)

Total and Practical Stability

137

(3)

Problems

140

(2)

Notes

142

(1)

Difference Equations as Banded Matrices

143

(22)

Introduction

143

(1)

Initial Value Problems

143

(3)

Boundary Values Problems

146

(11)

Invertibility of tridiagonal matrices

149

(5)

Sufficient conditions for well-conditioning

154

(3)

Cyclic Reduction

157

(6)

The case of Toeplitz tridiagonal matrices

159

(4)

Problems and Remarks

163

(1)

Notes

163

(2)

Applications to Numerical Analysis

165

(30)

Introduction

165

(1)

Iterative Methods

165

(2)

Local Results

167

(3)

Semilocal Results

170

(10)

Newton-Kantorovich-like theorems

172

(7)

Effect of perturbations

179

(1)

Miller's, Olver's, and Clenshaw's Algorithms

180

(4)

Boundary Value Problems

184

(1)

Monotone Iterative Methods

185

(3)

Monotone Approximations

188

(4)

Problems

192

(2)

Notes

194

(1)

Numerical Methods for Differential Equations

195

(20)

Introduction

195

(1)

Linear Multistep Methods

196

(2)

Finite Interval

198

(3)

Infinite Interval

201

(2)

Nonlinear Case

203

(2)

Other Techniques

205

(1)

The Method of Lines

206

(2)

Spectrum of a Family of Matrices

208

(3)

Problems

211

(1)

Notes

212

(3)

Models of Real World Phenomena

215

(18)

Introduction

215

(1)

Linear Models for Population Dynamics

215

(4)

The Logistic Equation

219

(2)

Distillation of a Binary Liquid

221

(3)

Models from Economics

224

(3)

Models of Traffic in Channels

227

(4)

Problems

231

(1)

Notes

232

(1)

Historically Important Equations

233

(9)

Introduction

233

(1)

Combinations of Means

233

(2)

Arithmetic-harmonic mean

235

(1)

Arithmetic-Geometric (Borchard)

235

(3)

Arithmetic-geometric mean II

237

(1)

The Weierstrass Method

238

(1)

Difference Equations and Prime Numbers

239

(2)

Problems

241

(1)

Notes

241

(1)

Appendices

242

(41)

A Function of Matrices

243

(16)

A.1 Introduction

243

(3)

A.2 Properties of Component Matrices

246

(2)

A.3 Particular Matrices

248

(3)

A.4 Sequence of Matrices

251

(1)

A.5 Jordan Canonical Form

252

(2)

A.6 Norms of Matrices and Related Topics

254

(2)

A.7 Nonnegative Matrices

256

(3)

B The Schur Criteria

259

(4)

b.1 The Schur Criteria

259

(4)

C The Chebyshev Polynomials

263

(4)

C.1 Definitions

263

(1)

C.2 Properties of Tn(z) and Un(z)

264

(3)

D Solutions to the Problems

267

(16)

D.1 Chapter 1

267

(3)

D.2 Chapter 2

270

(2)

D.3 Chapter 3

272

(3)

D.4 Chapter 4

275

(2)

D.5 Chapter 5

277

(1)

D.6 Chapter 6

277

(2)

D.7 Chapter 7

279

(1)

D.8 Chapter 8

280

(1)

D.9 Chapter 9

281

(2)

Bibliography

283

(14)

Index

297

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