Numerical Analysis and Graphic Visualization with MATLAB

Now, there is a complete introduction to numerical methods and visualization with the latest, most powerful version of MATLAB, Version 6.0. Dr. Shoichiro Nakamura introduces the skills and knowledge needed to solve numerical equations with MATLAB, understand the computational results, and present them graphically.

This book brings together all four cornerstones of numerical analysis with MATLAB: the fundamental techniques of MATLAB programming; the mathematical basis of numerical methods; the application of numerical analysis to engineering, scientific, and mathematical problems; and the creation of scientific graphics. Coverage includes:

• Complete introductory tutorials for both MATLAB 6.0 programming and professional-quality 3D graphics
• Linear algebra applications: matrices, vectors, Gauss elimination, Gauss-Jordan elimination, LU decomposition, and more
• Polynomials and interpolation, including interpolation with Chebyshev points; cubic hermite, 2D and transfinite interpolation; and M-files
• Numerical integration, differentiation, and roots of nonlinear equations
• Advanced techniques, including curve fitting, spline functions, and boundary value problems

Whether you are a student, engineer, scientist, researcher, or economic analyst, MATLAB 6 offers you unprecedented power for defining and solving problems. Put that power to work — with Numerical Analysis and Graphical Visualization with MATLAB, second edition.

Contents

1: MATLAB Primer

Before Starting Calculations

How to Do Calculations

Branch Statements

Loops with for or while

Reading and Writing

Array Variables

Unique Aspect of Numbers in MATLAB

Mathematical Functions of MATLAB

Functions that Do Chores

Developing a Program as M-file

How to Write User's Own Functions

Saving and Loading of Data

How to Make Hard Copies

2: Graphics with MATLAB

Simple Plotting

Interactive Editing of Figures

How to Print or Record Graphs

Plotting of Two-dimensional Functions

Triangular Grid and Contours

Curvilinear Grid and Contours

Plotting Curved Surfaces

MATLAB as Drawing Board

Interactive Graphics

M-files

3: Linear Algebra

Matrices and Vectors

Matrix and Vector Operations in MATLAB

Inverse Matrix

Linear Equations

Unsolvable Problems

Determinant

Ill-conditioned Problems

Gauss Elimination

Gauss-Jordan Elimination and Matrix Inversion

LU Decomposition

Iterative Solution

Matrix Eigenvalues

4: Polynomials and Interpolation

MATLAB Commands for Polynomials

Linear Interpolation

Polynomial Interpolation with Power Series

Lagrange Interpolation Polynomial

Error of Interpolation Polynomials

Differentiation and Integration of Lagrange Interpolation Formula

Interpolation with Chebyshev Points

Cubic Hermite Interpolation

Two Dimensional Interpolation

Transfinite Interpolation

M-files

5: Numerical Integration

Trapezoidal Rule

Simpson's Rules

Other Quadratures

Numerical Integration with Infinite Limits or Singularities

MATLAB Commands for Integrations

Numerical Integration on a Two-dimensional Domain

M-files

6: Numerical Differentiation

Derivatives of Interpolation Polynomials

Difference Approximations

Taylor Expansion Method

Algorithms to Automate Derivation

Difference Approximation for Partial Derivatives

Numerical Evaluation of High Order Derivatives

M-Files

7: Roots of Nonlinear Equations

Graphical Method

Bisection Method

Newton Iteration

Secant Method

Successive Substitution Method

Simultaneous Nonlinear Equations

M-files

8: Curve Fitting to Measured Data

Line Fitting

Nonlinear Curve Fitting with a Power Function

Curve Fitting with a Higher-Order Polynomial

Curve Fitting by a Linear Combination of Known Functions

9: Spline Functions and Nonlinear Interpolation

C-spline Interpolation

Cubic B-Spline

Interpolation with a Nonlinear Function

M-files

10: Initial-Value Problems of Ordinary Differential Equations

First-Order ODEs

Euler Methods

Runge-Kutta Methods

Shooting Method

Method of Lines

11: Boundary-Value Problems of Ordinary Differential Equations

Introduction

Boundary-Value Problems for Rods and Slabs

Solution of Tridiagonal Equation

Variable Coefficients and Nonuniform Grid Spacing

Cylinders and Spheres

Nonlinear Ordinary Differential Equations

Appendices

A: Colors

B: Drawing of Three-Dimensional Objects

C: Movie

D: Image Processing

E: Graphical User Interface