Riassunto
This interdisciplinary book provides a compendium of projects, together with a large number of example programs for readers to study and explore. The book is designed for advanced undergraduate or graduate students 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." These projects were collected from research results and computational dilemmas during the author's tenure as Chief Scientist at NeXT Computer, Inc. and from his scientific computation lectures in the Department of Physics 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 some combination of these, and is designed with an estimated level of difficulty. The support code for each project takes the form of either C or Mathematica, and is printing in the Appendix. The algorithms are clearly laid out within the text projects, so the book can be used with other symbolic numerical and algebraic manipulation products.
Contenuti
1 Numbers everywhere Selected topics in numerical analysis.- 1.1 Numerical evaluation.- 1.1.1 Evaluation of famous constants.- 1.1.2 Evaluation of elementary functions.- 1.1.3 Special functions.- 1.2 Equation solving.- 1.2.1 Matrix algebra.- 1.2.2 Non-linear equation systems.- 1.2.3 Differential equations.- 1.3 Random numbers and Monte Carlo.- 1.3.1 Generating Random Numbers.- 1.3.2 Numerical integration and Monte Carlo.- 2 Exploratory computation Collected intra- and interdisciplinary projects.- 2.1 Mathematical problems.- 2.1.1 Planar geometry problems.- 2.1.2 Symbolic manipulation.- 2.1.3 Real and complex analysis.- 2.2 Nature-motivated models.- 2.2.1 Neural network experiments.- 2.2.2 Genetic algorithms and artificial life.- 2.3 Projects from biology.- 2.3.1 Population models.- 2.3.2 Physiology, neurobiology, and medicine.- 2.3.3 Molecular biology.- 2.4 Projects from physics and chemistry.- 2.4.1 Classical physics.- 2.4.2 Quantum theory.- 2.4.3 Molecules and structure.- 2.4.4 Relativity.- 3 The lure of large numbers Projects in number theory.- 3.1 Large-integer arithmetic.- 3.1.1 Testing the operations.- 3.2 Prime numbers.- 3.2.1 Mersenne primes.- 3.2.2 Primes in general.- 3.3 Fast algorithms.- 3.3.1 Fast multiplication.- 3.3.2 Fast mod, division, inversion.- 3.3.3 Other fast algorithms.- 3.4 Factoring.- 3.4.1 Factoring algorithms.- 3.4.2 Status of Fermat numbers.- 4 The FFT forest-The ubiquitous FFT and its relatives.- 4.1 Discrete Fourier transform.- 4.1.1 Fundamental DFT manipulations.- 4.1.2 Algebraic aspects of the DFT.- 4.1.3 DFT test signals.- 4.1.4 Direct DFT software.- 4.2 FFT algorithms.- 4.2.1 Recursive FFTs.- 4.2.2 FFT indexing and butterflies.- 4.2.3 Complex FFTs, N a power of 2.- 4.2.4 Real-signal FFTs.- 4.2.5 FFTs for other radices.- 4.2.6 FFTs in higher dimensions.- 4.2.7 Applications of the FFT.- 4.3 Real-valued transforms.- 4.3.1 Hartley transform.- 4.3.2 Discrete cosine transform.- 4.3.3 Walsh-Hadamard transform.- 4.3.4 Square-wave transform.- 4.4 Number-theoretic transforms.- 4.4.1 Exploring number-theoretic transforms.- 5 Wavelets Young arrivals in the transform family.- 5.1 Chords, notes, and little waves.- 5.1.1 Windowed Fourier transform.- 5.1.2 Continuous wavelet transform.- 5.2 Discrete wavelet bases.- 5.2.1 Example wavelet expansions.- 5.2.2 Mother function and its wavelet.- 5.2.3 Wavelets of compact support.- 5.3 Discrete wavelet transform.- 5.3.1 Fast wavelet transform algorithms.- 5.3.2 Applications of fast wavelet transforms.- 6 Complexity reigns Chaos & fractals & such.- 6.1 Chaos.- 6.1.1 Quadratic map algebra.- 6.1.2 Bifurcation and chaos.- 6.1.3 Chaos models.- 6.1.4 Chaos, stability, and Lyapunov exponents.- 6.1.5 Applications of chaos theory.- 6.2 Fractals.- 6.2.1 Theory of fractals.- 6.2.2 Visualization of fractals.- 6.2.3 Fractal Brownian noise.- 6.2.4 Measurement of fractal dimension.- 7 Signals from the real world Projects in signal processing.- 7.1 Data compression.- 7.1.1 Tour of lossless data compressors.- 7.2 Sound.- 7.2.1 Examples of sound processing.- 7.2.2 Examples of sound compression.- 7.3 Images.- 7.3.1 Examples of image processing.- 7.3.2 Image compression.- Appendix Support code for the book Projects.- References.
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