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5. Introduction to Numerical Methods and MATLAB: Implementations and Applications  # Introduction to Numerical Methods and MATLAB: Implementations and Applications, 1st edition

• Gerald W. Recktenwald ###### Introduction to Numerical Methods and MATLAB: Implementations and Applications

ISBN-13:  9780201308600

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Free delivery
\$95.99 \$119.99

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## Overview

This book is an introduction to MATLAB and an introduction to numerical methods. It is written for students of engineering, applied mathematics, and science. The primary objective of numerical methods is to obtain approximate solutions to problems that are not obtainable by other means. This book teaches how the core techniques of numerical methods are used to solve otherwise unsolvable problems of modern technological significance.

The outstanding pedagogical features of this book are:

• use of numerical experiments as a means of learning why numerical methods work and how they fail
• a separate chapter reviewing the basics of applied linear algebra, and how computations involving matrices and vectors are naturally expressed in MATLAB
• use of a range of examples from those that provide a succinct illustration of a basic algorithm, to those that develop solutions to substantial problems in engineering
• consistent use of well-documented and structured code written in the MATLAB idiom
• a library of general purpose routines—the NMM Toolbox—that are readily applied to new problems
• a progressive approach to algorithm development leading the reader to an understanding of the more sophisticated routines in the built-in MATLAB toolbox.

The primary goals of the book are to provide a solid foundation in applied computing, and to demonstrate the implementation and application of standard numerical methods to practical problems. This is achieved by a systematic development of techniques beginning with the simple and ending with the sophisticated. Good programming practice is used throughout to show the reader how to clearly express and document computational ideas. By providing an extensive library of working codes, as well as an exposition of the methods used by the built-in MATLAB toolbox, the reader is challenged by the application of numerical methods to practical problems. This bypasses the ritual of forcing the reader to reinvent simple programs that fail on more technologically significant, practical problems.

(NOTE: Chapters 2-12 conclude with Summary.)

1. Introduction.

Terminology. MATLAB Overview. Organization of the Book. Rating Systems for Exercises.

I. MATLAB BASICS.

2. Interactive Computing with MATLAB.

Running MATLAB. Matrices and Vectors. Additional Types of Variables. Managing the Interactive Environment. Plotting in MATLAB.

3. MATLAB Programming.

Script m-Files. Function m-Files. Input and Output. Flow Control. Vectorization. Deus ex Machina.

4. Organizing and Debugging MATLAB Programs.

Organizing and Documenting m-Files. Organizing a Numerical Solution. Debugging.

II. NUMERICAL TECHNIQUES.

5. Unavoidable Errors in Computing.

Digital Representation of Numbers. Finite Precision Arithmetic. Truncation Error of Algorithms.

6. Finding the Roots of f(x)=0.

Preliminaries. Fixed-Point Iteration. Bisection. Newton's Method. The Secant Method. Hybrid Methods. Roots of Polynomials.

7. A Review of Linear Algebra.

Vectors. Matrices. Mathematical Properties of Vectors and Matrices. Special Matrices.

8. Solving Systems of Equations.

Basic Concepts. Gaussian Elimination. Limitations on Numerical Solutions to Ax = b. Factorization Methods. Nonlinear Systems of Equations.

9. Least-Squares Fitting of a Curve to Data.

Fitting a Line to Data. Least-Squares Fit to a Linear Combination of Functions. Multivariate Linear Least-Squares Fitting.

10. Interpolation.

Basic Ideas. Interpolating Polynomials of Arbitrary Degree. Piecewise Polynomial Interpolation. MATLAB's Built in Interpolation Functions.

11. Numerical Integration.

12. Numerical Integration of Ordinary Differential Equations.

Basic Ideas and Nomenclature. Euler's Method. Higher Order One-Step Methods. Adaptive Stepsize Algorithms. Coupled ODEs. Additional Topics.

Bibliography.

Appendix A: Eigenvalues and Eigensystems.

Eigenvectors Map onto Themselves. Mathematical Preliminaries. The Power Method. Built-in Functions for Eigenvalue Computation. Singular Value Decomposition.

Appendix B: Sparse Matrices.

Storage and Flop Savings. MATLAB Sparse Matrix Format.

MATLAB Toolbox Functions.

Listings for NMM Toolbox m-Files.

Subject Index.

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