Numerical methods are a mainstay of researchers and professionals across the many mathematics, scientific, and engineering disciplines. The importance of these methods combined with the power and widespread availability of desktop computers virtually demands that students in these fields be well-versed not only in the numerical techniques, but also in the use of a modern computational software package.
An Introduction to Numerical Methods: A MATLAB Approach fulfills both these needs. Requiring only calculus and some basic programming concepts as background, it introduces readers to the theory and applications of the most commonly used techniques for solving numerical problems on a personal computer. It covers a wide range of useful algorithms, each presented with full details so that readers can visualize and interpret each step. The enclosed CD-ROM contains simple MATLAB functions that provide step-by-step explanations of the mechanisms behind the algorithm for each technique and guide readers through the necessary calculations.
Emphasis on understanding how the methods work, a simple, direct style, and thorough coverage make this book an outstanding initiation that allows students to see almost immediate results. It will boost their confidence in their ability to master the subject and give them valuable experience in the use of MATLAB.
Introduction About the software MATLAB
An Introduction to MATLAB
Taylor Series
Number System and Errors
Floating-Point Arithmetic
Round-Off Errors
Truncation Error
Interval Arithmetic
Roots of Equations
The Bisection Method
The Method of False Position
The Secant method
Newton’s Method
Convergence of the Newton and Secant Methods
Fixed Point Iteration
Multiple Roots and the Modified Newton Method
Newton’s Method for Nonlinear Systems
System of Linear Equations
Matrices and Matrix Operations
Naïve Gaussian Elimination
Gaussian Elimination with Scaled Partial Pivoting
Lu Decomposition
Iterative Methods
Interpolation
Polynomial Interpolation Theory
Newton’s Divided Difference Interpolating Polynomial
The Error of the Interpolating Polynomial
Lagrange Interpolating Polynomial
Interpolation with Spline Functions
Piecewise Linear Interpolation
Quadratic Spline
Natural Cubic Splines
The Method of Least Squares
Linear Least Squares
Least Squares Polynomial
Nonlinear Least Squares
Trigonometric Least Squares Polynomial
Numerical Differentiation and Integration
Numerical Differentiation
Richardson’s Formula
Trapezoidal Rule
Simpson’s Rule
Romberg Algorithm
Gaussian Quadrature
Numerical Methods for Ordinary Differential Equations
Euler’s Method
Error Analysis
Higher Order Taylor Series Methods
Runge-Kutta Methods
Multistep Methods
Adams-Bashforth Method
Predictor-Corrector Methods
Adams-Moulton Method
Numerical Stability
Higher Order Equations and Systems of Equations
Implicit Methods and Stiff Systems
Phase Plane Analysis: Chaotic Equations
Boundary Value Problems
Finite-Difference Methods
Shooting Methods
Eigenvalues and Eigenvectors
Basic Theory
The Power Method
The Quadratic Method
Eigenvalues for Boundary-Value Problems
Bifurcations in Differential Equations
Partial Differential Equations
Parabolic Equations
Hyperbolic Equations
Elliptic Equations
Bibliography and References
Appendices Calculus Review
MATLAB built-in functions
Text MATLAB functions
Answers to Selected Exercises
Index
