Computational Methods for Numerical Analysis with R

James P Howard, II

• Verlag: Chapman and Hall/CRC
• Jahr: 2017
• ISBN:
  9781351646505
• Auflage: 1
• Thema: Mathematics
• Thema: Number Theory

Computational Methods for Numerical Analysis with R is an overview of traditional numerical analysis topics presented using R. This guide shows how common functions from linear algebra, interpolation, numerical integration, optimization, and differential equations can be implemented in pure R code. Every algorithm described is given with a complete function implementation in R, along with examples to demonstrate the function and its use.

Computational Methods for Numerical Analysis with R is intended for those who already know R, but are interested in learning more about how the underlying algorithms work. As such, it is suitable for statisticians, economists, and engineers, and others with a computational and numerical background.

Contents

List of Figures

List of Tables

List of R Functions

Preface

1 Introduction to Numerical Analysis

1.1 Numerical Analysis

1.1.1 The Goals of Numerical Analysis

1.1.2 Numerical Analysis in R

1.1.3 Efficiency

1.2 Data Types in R

1.2.1 Data Types

1.2.2 Data Structures

1.3 Elementary Problems

1.3.1 Summation Algorithms

1.3.2 Evaluating Polynomials

1.3.3 The nth Root Algorithm

Comments

Exercises

2 Error Analysis

2.1 True Values

2.1.1 Accuracy

2.1.2 Precision

2.2 Internal Data Storage

2.2.1 Binary Numbers

2.2.2 Floating Point Numbers

2.3 Numerical Error

2.3.1 Round-Off Error and Machine ϵ

2.3.2 Loss of Significance

2.3.3 Overflow and Underflow

2.3.4 Error Propagation and Stability

2.4 Applications

2.4.1 Simple Division Algorithms

2.4.2 Binary Long Division

Comments

Exercises

3 Linear Algebra

3.1 Vectors and Matrices

3.1.1 Vector and Matrix Operations

3.1.2 Elementary Row Operations

3.2 Gaussian Elimination

3.2.1 Row Echelon Form

3.2.2 Tridiagonal Matrices

3.3 Matrix Decomposition

3.3.1 LU Decomposition

3.3.2 Cholesky Decomposition

3.4 Iterative Methods

3.4.1 Jacobi Iteration

3.4.2 Gauss–Seidel Iteration

3.5 Applications

3.5.1 Least Squares

Comments

Exercises

4 Interpolation and Extrapolation

4.1 Polynomial Interpolation

4.1.1 Linear Interpolation

4.1.2 Higher-Order Polynomial Interpolation

4.2 Piecewise Interpolation

4.2.1 Piecewise Linear Interpolation

4.2.2 Cubic Spline Interpolation

4.2.3 Bézier Curves

4.3 Multidimensional Interpolation

4.3.1 Bilinear Interpolation

4.3.2 Nearest Neighbor Interpolation

4.4 Applications

4.4.1 Time Series Interpolation

4.4.2 Computer Graphics

Comments

Exercises

5 Differentiation and Integration

5.1 Numerical Differentiation

5.1.1 Finite Differences

5.1.2 The Second Derivative

5.2 Newton–Cotes Integration

5.2.1 Multipanel Interpolation Rules

5.2.2 Newton–Cotes Errors

5.2.3 Newton–Cotes Forms, Generally

5.3 Gaussian Integration

5.3.1 The Gaussian Method

5.3.2 Implementation Details

5.4 More Techniques

5.4.1 Adaptive Integrators

5.4.2 Romberg’s Method

5.4.3 Monte Carlo Methods

5.5 Applications

5.5.1 Revolved Volumes

5.5.2 Gini Coefficients

Comments

Exercises

6 Root Finding and Optimization

6.1 One-Dimensional Root Finding

6.1.1 Bisection Method

6.1.2 Newton–Raphson Method

6.1.3 Secant Method

6.2 Minimization and Maximization

6.2.1 Golden Section Search

6.2.2 Gradient Descent

6.3 Multidimensional Optimization

6.3.1 Multidimensional Gradient Descent

6.3.2 Hill Climbing

6.3.3 Simulated Annealing

6.4 Applications

6.4.1 Least Squares

6.4.2 The Traveling Salesperson

Comments

Exercises

7 Differential Equations

7.1 Initial Value Problems

7.1.1 Euler Method

7.1.2 Runge–Kutta Methods, Generally

7.1.3 Linear Multistep Methods

7.2 Systems of Ordinary Differential Equations

7.2.1 Solution Systems and Initial Value Problems

7.2.2 Boundary Value Problems

7.3 Partial Differential Equations

7.3.1 The Heat Equation

7.3.2 The Wave Equation

7.4 Applications

7.4.1 Carbon Dating

7.4.2 Lotka–Volterra Equations

Comments

Exercises

Suggested Reading

Index

List of Figures

1.1 Assembly of partial sums in piecewise addition.

2.1 Newark is near New York (Stamen Design)

2.2 At least we got to New York, this time (Stamen Design)

2.3 A journey of a thousand miles starts with a good map (Stamen Design)

2.4 Error in the measurement of length

2.5 Additive error in measurement of length

2.6 Multiplicative error in measurement of length

4.1 Linear interpolation of two points

4.2 Quadratic interpolation of three points

4.3 Cubic interpolation of y=tan((x-1/4))+1

4.4 Piecewise linear interpolation of four points

4.5 Cubic spline interpolation of four points

4.6 Cubic spline interpolation of five points

4.7 Quadratic Bézier interpolation of three points

4.8 Cubic Bézier interpolation of four points

4.9 Cubic Bézier int...