## Table of Contents

**0. Preliminaries.**

Analysis versus Numerical Analysis.

Computers and Numerical Analysis.

An Illustrative Example.

Kinds of Errors in Numerical Procedures.

Interval Arithmetic.

Parallel and Distributed Computing.

Measuring the Efficiency of Procedures.

**1. Solving Nonlinear Equations.**

Interval Halving (Bisection).

Linear Interpolation Methods.

Newton's Method.

Muller's Method.

Fixed-Point Iteration.

Other Methods.

Nonlinear Systems.

**2. Solving Sets of Equations.**

Matrices and Vectors.

Elimination Methods.

The Inverse of a Matrix and Matrix Pathology.

Almost Singular Matrices - Condition Numbers.

Interactive Methods.

Parallel Processing.

**3. Interpolation and Curve Fitting.**

Interpolating Polynomials.

Divided Differences.

Spline Curves.

Bezier Curves and B-Splines.

Interpolating on a Surface.

Least Squares Approximations.

**4. Approximation of Functions.**

Chebyshev Polynomials and Chebyshev Series.

Rational Function Approximations.

Fourier Series.

**5. Numerical Differentiation and Integration.**

Differentiation with a Computer.

Numerical Integration - The Trapezoidal Rule.

Simpson's Rules.

An Application of Numerical Integration - Fourier Series and Fourier Transforms Adaptive Integration.

Gaussian Quadrature.

Multiple Integrals.

Applications of Cubic Splines.

**6. Numerical Solution of Ordinary Differential Equations.**

The Taylor Series Method.

The Euler Method and Its Modification.

Runge-Kutta Methods.

Multistep Methods.

Higher-Order Equations and Systems.

Stiff Equations.

Boundary-Value Problems.

Characteristic-Value Problems.

**7. Optimization.**

Finding the Minimum of

*y*=*f*(*x*).

Minimizing a Function of Several Variables.

Linear Programming.

Nonlinear Programming.

Other Optimizations.

**8. Partial Differential Equations.**

Elliptic Equations.

Parabolic Equations.

Hyperbolic Equations.

**9. Finite Element Analysis.**

Mathematical Background.

Finite Elements for Ordinary Differential Equation.

Finite Elements for Partial Differential Equation.

**Appendices.**

A. Some Basic Information from Calculus.

B. Software Resources.

**Answers to Selected Exercises.**

**References.**

**Index.**

Applied Numerical Analysis, 7th Edition

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