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Projects in Scientific Computation

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This interdisciplinary book provides a compendium of projects, plus numerous example programs for readers to study and explore. Designed for advanced undergraduates or graduates of science, mathematics and engineering who will deal with scientific computation in their future studies and research, it also contains new and useful reference materials for researchers. The problem sets range from the tutorial to exploratory and, at times, to "the impossible". The projects were collected from research results and computational dilemmas during the author's tenure as Chief Scientist at NeXT Computer, and from his lectures at Reed College. The content assumes familiarity with such college topics as calculus, differential equations, and at least elementary programming. Each project focuses on computation, theory, graphics, or a combination of these, and is designed with an estimated level of difficulty. The support code for each takes the form of either C or Mathematica, and is included in the appendix and on the bundled diskette. The algorithms are clearly laid out within the projects, such that the book may be used with other symbolic numerical and algebraic manipulation products.

Table of contents

Chapter 1:Numbers Everywhere: Selected topics in numerical analysis

Numerical evaluation
› evaluation of famous constants
› evaluation of elementary functions
› special functions

Equation solving
› matrix algebra
› nonlinear equation systems
› differential equations

Random numbers and Monte Carlo
› generating random numbers
› numerical integration and Monte Carlo
› evaluation of elementary functions
› evaluation of elementary functions

Chapter 2:Exploratory Computation: Collected intra- and interdisciplinary projects

Mathematical problems
› planar geometry problems
› symbolic manipulation
› real and complex analysis

Nature-motivated models
› neural network experiments
› genetic algorithms and artificial life

Projects from biology
› population models
› physiology, neurobiology and medicine
› molecular biology

Projects from physics and chemistry
› classical physics
› quantum theory
› molecules and structure
› relativity

Chapter 3:The Lure of Large Numbers: Projects in number theory

Large-integer arithmetic
› testing the operations
› Prime numbers
› Mersenne primes
› primes in general

Fast algorithms
› fast multiplication
› fast mod, division, inversion
› other fast algorithms

Factoring
› factoring algorithms
› status of Fermat numbers

Chapter 4:The FFT Forest: The ubiquitous FFT and its relatives

Discrete Fourier transform
› fundamental DFT manipulations
› algebraic aspects of the DFT
› DFT test signals
› direct DFT software

FFT algorithms
› recursive FFTs
› FFT indexing and butterflies
› complex FFTs, N a power of 2
› real-signal FFTs
› FFTs for other radices
› FFTs in higher dimensions
› applications of FFT

Real-valued transforms
› Hartley transform
› discrete cosine transform
› Walsh-Hadamard transform
› square-wave transform

Number-theoretic transforms
› exploring number theoretic transforms

Chapter 5:Wavelets: Young arrivals in the transform family

Chords, notes and little waves
› windowed Fourier transform
› continuous wavelet transform

Discrete wavelet bases
› example wavelet expansions
› mother function and its wavelet
› wavelets of compact support

Discrete wavelet transform
› fast wavelet transform algorithms
› applications of fast wavelet transforms

Chapter 6:Complexity Reigns: Chaos & fractals & such

Chaos
› quadratic map algebra
› bifurcation and chaos
› chaos models
› chaos, stability and Lyapunov exponents
› applications of chaos theory

Fractals
› theory of fractals
› visualization of fractals
› fractal Brownian noise
› measurement of fractal dimension

Chapter 7:Signals from the Real World: Projects in signal processing

Data compression
› tour of lossless data compressors

Sound
› examples of sound processing
› examples of sound compression

Images
› examples of image processing
› image compression

Appendix
Support code for the book projects