Introduction to Numerical Methods and MATLAB: Implementations and Applications Book

(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.

Basic Ideas and Nomenclature. Newton-Cotes Rules. Gaussian Quadrature. Adaptive Quadrature. Improper Integrals and Other Complications.

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.