An Introduction to Numerical Methods: A MATLAB Approach, Fifth Edition continues to offer readers an accessible and practical introduction to numerical analysis. It presents a wide range of useful and important algorithms for scientific and engineering applications, using MATLAB to illustrate each numerical method with full details of the computed results so that the main steps are easily visualized and interpreted. This edition also includes new chapters on Approximation of Continuous Functions and Dealing with Large Sets of Data.
In each chapter, we have attempted to present clear examples in every section followed by a good number of related exercises at the end of each section with answers to some exercises. It is the purpose of this book to implement various important numerical methods on a personal computer and to provide not only a source of theoretical information on the methods covered but also to allow the student to easily interact with the computer and the algorithm for each method using MATLAB.
MATLAB (MATrix LABoratory) is a powerful interactive system for matrix-based computation designed for scientific and engineering use. It is good for many forms of numeric computation and visualization. MATLAB language is a high-level matrix/array language with control flow statements, functions, data structures, input/output, and object-oriented programming features. To fully use the power and computing capabilities of this software program in classrooms and laboratories by teachers and students in science and engineering, part of this text is intended to introduce the computational power of MATLAB to modern numerical methods.
MATLAB has several advantages. There are three major elements that have contributed to its immense popularity. First, it is extremely easy to use since data can be easily entered, especially for algorithms that are adaptable to a table format. This is an important feature because it allows students to experiment with many numerical problems in a short period of time. Second, it includes high-level commands for two-dimensional and three-dimensional data visualization and presentation graphics. Plots are easily obtained from within a script or in command mode. Third, the most evident power of a MATLAB is its speed of calculation. The program gives instantaneous feedback. Because of their popularity, MATLAB and other software such as Mathematica are now available in more university microcomputer laboratories.
Features:
Covers the most common numerical methods encountered in science and engineering
Illustrates the methods using MATLAB
Ideal as an undergraduate textbook for numerical analysis
Presents numerous examples and exercises, with selected answers provided at the back of the book
Accompanied by downloadable MATLAB code hosted at site
Author(s): Abdelwahab Kharab, Ronald B. Guenther
Series: Numerical Analysis and Scientific Computing Series
Publisher: CRC Press
Year: 2023
Language: english
Pages: 631
Cover
Half Title
Series Page
Title Page
Copyright Page
Table of Contents
Preface
Author Biographies
1 Introduction
1.1 About MATLAB and MATLAB Gui (Graphical User Interface)
1.2 An Introduction to MATLAB
1.2.1 Matrices and Matrix Computation
1.2.2 Polynomials
1.2.3 Output Format
1.2.4 Planar Plots
1.2.5 3-D Mesh Plots
1.2.6 Function Files
1.2.7 Defining Functions
1.2.8 Relations and Loops
1.3 Taylor Series
2 Number System and Errors
2.1 Floating-Point Arithmetic
2.2 Round-Off Errors
2.3 Truncation Error
2.4 Interval Arithmetic
3 Roots of Equations
3.1 The Bisection Method
3.2 Fixed Point Iteration
3.3 The Secant Method
3.4 Newton’s Method
3.4.1 MATLAB’s Methods
3.5 Convergence of the Newton and Secant Methods
3.6 Multiple Roots and the Modified Newton Method
3.7 Newton’s Method for Nonlinear Systems
Applied Problems
4 System of Linear Equations
4.1 Matrices and Matrix Operations
4.2 Naive Gaussian Elimination
4.3 Gaussian Elimination with Scaled Partial Pivoting
4.4 LU Decomposition
4.4.1 Crout’s and Cholesky’s Methods
4.4.2 Gaussian Elimination Method
4.5 Iterative Methods
4.5.1 Jacobi Iterative Method
4.5.2 Gauss-Seidel Iterative Method
4.5.3 Convergence
Applied Problems
5 Interpolation
5.1 Polynomial Interpolation Theory
5.2 Newton’s Divided-Difference Interpolating Polynomial
5.3 The Error of the Interpolating Polynomial
5.4 Lagrange Interpolating Polynomial
Applied Problems
6 Interpolation with Spline Functions
6.1 Piecewise Linear Interpolation
6.2 Natural Cubic Splines
6.2.1 Smoothness Property
6.2.2 MATLAB’s Methods
Applied Problems
7 Approximation of Continuous Functions
7.1 Review of Binomial Theorem
7.2 Some Identities and the Bernstein Polynomials
8 Numerical Optimization
8.1 Analysis of Single-Variable Functions
8.2 Line Search Methods
8.2.1 Bracketing the Minimum
8.2.2 Golden Section Search
8.2.3 Fibonacci Search
8.2.4 Parabolic Interpolation
8.3 Minimization Using Derivatives
8.3.1 Newton’s Method
8.3.2 Secant Method
Applied Problems
9 Numerical Differentiation
9.1 Numerical Differentiation
9.2 Richardson’s Formula
Applied Problems
10 Numerical Integration
10.1 Trapezoidal Rule
10.2 Simpson’s Rule
10.3 Romberg Algorithm
10.4 Gaussian Quadrature
Applied Problems
11 Numerical Methods for Linear Integral Equations
11.1 Introduction
11.2 Quadrature Rules
11.2.1 Trapezoidal Rule
11.2.2 The Gauss-Nyström Method
11.3 The Successive Approximation Method
11.4 Schmidt’s Method
11.5 Volterra-Type Integral Equations
11.5.1 Euler’s Method
11.5.2 Heun’s Method
Applied Problems
12 Numerical Methods for Ordinary Differential Equations
12.1 Euler’s Method
12.2 Error Analysis
12.3 Higher-Order Taylor Series Methods
12.4 Runge-Kutta Methods
12.5 Multistep Methods
12.6 Adams-Bashforth Methods
12.7 Predictor-Corrector Methods
12.8 Adams-Moulton Methods
12.9 Numerical Stability
12.10 Higher-Order Equations and Systems of Differential Equations
12.11 Implicit Methods and Stiff Systems
12.12 Phase Plane Analysis: Chaotic Differential Equations
Applied Problems
13 Boundary-Value Problems
13.1 Finite-Difference Methods
13.2 Shooting Methods
13.2.1 The Nonlinear Case
13.2.2 The Linear Case
Applied Problems
14 Eigenvalues and Eigenvectors
14.1 Basic Theory
14.2 The Power Method
14.3 The Quadratic Method
14.3.1 MATLAB’s Methods
14.4 Eigenvalues for Boundary-Value Problems
14.5 Bifurcations in Differential Equations
Applied Problems
15 Dynamical Systems and Chaos
15.1 A Review of Linear Ordinary Differential Equations
15.2 Definitions and Theoretical Considerations
15.3 Two-Dimensional Systems
15.4 Chaos
15.5 Lagrangian Dynamics
Applied Problems
16 Partial Differential Equations
16.1 Parabolic Equations
16.1.1 Explicit Methods
16.1.2 Implicit Methods
16.2 Hyperbolic Equations
16.3 Elliptic Equations
16.4 Nonlinear Partial Differential Equations
16.4.1 Burger’s Equation
16.4.2 Reaction-Diffusion Equation
16.4.3 Porous Media Equation
16.4.4 Hamilton-Jacobi-Bellman Equation
16.5 Introduction to Finite-Element Method
16.5.1 Theory
16.5.2 The Finite-Element Method
Applied Problems
17 Dealing with Large Sets of Data
17.1 Introduction
17.2 Regression
17.3 Project
17.4 General Linear Regression
17.4.1 MATLAB’s Methods
17.5 Nonlinear Regression
17.6 Probability and Statistics
17.7 Problems Arising from Overdetermined Data
Applied Problems
Bibliography and References
A Calculus Review
A.1 Limits and Continuity
A.2 Differentiation
A.3 Integration
B MATLAB Built-in Functions
C Text MATLAB Functions
D MATLAB GUI
D.1 Roots of Equations
D.2 System of Linear Equations
D.3 Interpolation
D.4 Integration
D.5 Differentiation
D.6 Numerical Methods for Differential Equations
D.7 Boundary-Value Problems
D.8 Numerical Methods for PDEs
Answers to Selected Exercises
Index
