1st Edition

An Introduction to Scientific, Symbolic, and Graphical Computation

By Eugene Fiume Copyright 1995
    322 Pages
    by A K Peters/CRC Press

    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