1st Edition
An Introduction to Scientific, Symbolic, and Graphical Computation
This down-to-earth introduction to computation makes use of the broad array of techniques available in the modern computing environment. A self-contained guide for engineers and other users of computational methods, it has been successfully adopted as a text in teaching the next generation of mathematicians and computer graphics majors.
Preface
Mathematical Computation
Scientific, Symbolic, and Graphical Computation
Themes of this Book
Symbolic Computation
An Example
A More Complex Example
The Representation of Functions
Sets and Number Systems
Vectors
Functions
Representation of Functions, Curves and Surfaces
Explicit and Implicit Representation
Parametric Representations
Polynomial Representations
Procedural Representations
Discretisation and Computation of Functions
Line Segments and Circles
Appendix A. Raster Graphics Fundamentals
Appendix B. Simple Maple Examples
Appendix C. Matrix Representations
Supplementary Exercises
Interpolation
A Motivating Problem
Properties of Polynomials
Lagrange Interpolation
Piecewise Polynomial Interpolation
Pricewise Linear Interpolation
Representations for Polynomial Curves
Putting the Pieces Together
General Space Curves
Computational Methods for Polynomial Evaluation
Matrix computation
Direct Polynomial Evaluation
Horner’s Rule
Table Look-Up
Forward Differencing Techniques
Transforming Curves
Motivation
Formulation
An Introduction to Polynomial Surfaces
Appendix A. Computing the Change-of-Basis Matrix
Supplementary Exercises
Approximation and Sampling
Problems with Interpolation
Ringing
Noise
Undersampling
Divergence
Summary
Types of Approximation
Approximation Using Uniform Cubic B-Splines
Signals and Filters
Sample Filters and Their Effect
Sampling, Filtering, and Reconstruction
The Sampling Theorem: An Intuitive View
Reconstruction
Filtering
Supplementary Exercises
Computational Integration
Introduction
Basic Numerical Quadrature
Riemann Sums
Integration Based on Pricewise Polynomial Interpolation
Formulae for Compound Integration
Adaptive Numerical Integration
Comparison of Results
Monte Carlo Methods
Summary
Appendix A. Maple Code to Model Quadrature Rules
Series Approximations
Representations for the Real Numbers
The Representation of Integers and Fixed-Point Numbers
The Representation of Floating-Point Numbers
Polynomial Series
Taylor Polynomials
Error Analysis of Quadrature Algorithms
Non-Polynomial Series: Trigonometric Fourier Series
Definition
Examples
Generalised Fourier Series and the Fourier Transform
Changing the Domain of a Fourier Series
The Fourier Transform
Convolution and Frequency Domain Representations
Frequency-Domain Filtering
The Sampling Theorem Revisited
Appendix A. Maple Code to Compute Quadrature Rules
Finding the Zeroes of a Function
Motivation: Intersection Problems
Symbolic Computation of the Roots of Polynomials
Numerical Methods for Computing Zeroes
Pricewise Approximation
Bisection
The Newton-Raphson Method
The Secant Method
Index
Biography
Fiume , Eugene
