Elements Of Numerical Analysis With Mathematica

by John Loustau

Overview

Here we present numerical analysis to advanced undergraduate and master degree level grad students. This is to be done in one semester. The programming language is Mathematica. The mathematical foundation and technique is included. The emphasis is geared toward the two major developing areas of applied mathematics, mathematical finance and mathematical biology.

Product Details

ISBN-13:	9789813222717
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	10/13/2017
Pages:	164
Product dimensions:	5.90(w) x 9.00(h) x 0.60(d)

Table of Contents

Preface vii

1 Beginnings 1

1.1 The Programming Basics for Mathematica 2

1.2 Errors in Computation 6

1.3 Newton's Method 8

1.4 Secant Method 14

2 Linear Systems and Optimization 19

2.1 Linear Systems of Equations 20

2.2 The Norm and Spectral Radius of a Linear Transformation 27

2.3 Large Matrix Techniques 32

2.4 Functions of Several Variables: Finding Roots and Extrema 37

3 Interpolating and Fitting 43

3.1 Polynomial Interpolation 44

3.2 Bezier Interpolation 53

3.3 Least Squares Fitting 56

3.4 Cubic Splines and B-Splines 62

3.5 Hermite Interpolation 71

4 Numerical Differentiation 75

4.1 Finite Differences and Vector Fields 76

4.2 Finite Difference Method, Explicit or Forward Euler 80

4.3 Neumann Stability Analysis 85

4.4 Finite Difference Method, Implicit and Crank Nicolson 91

5 Numerical Integration 97

5.1 Trapezoid Method and Simpson's Rule 98

5.2 Midpoint Method 101

5.3 Gaussian Quadrature 105

5.4 Comments on Numerical Integration 110

6 Numerical Ordinary Differential Equations 117

6.1 First order ODE Techniques, Forward Euler and Corrector Method 119

6.2 Midpoint Method with an Application to Mathematical Oncology 122

6.3 Shooting Method with an Application to Cooling Fins 126

6.4 The Method of Lines, Revisiting the Heat Equation 129

7 Monte Carlo Method 133

7.1 Basics of Probability Theory, Terminology and Notation 135

7.2 Generating Distributed Number Sequences 137

7.3 Monte Carlo Method for Estimating the Integrals and n-Dimensional Volumes 141

7.4 Monte Carlo Method for Problems with Stochastic Elements 143

Bibliography 147

Index 149