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1. Finite Difference Calculus
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Introduction to Computational Physics
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Preamble
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I. Basic Tools of Our Trade
Most of the methods used by computational physicists are drawn from three areas of numerical mathematics, namely from
CALCULUS OF
DIFFERENCES
LINEAR ALGEBRA
STOCHASTICS
Subsections
1. Finite Difference Calculus
1.1 Definitions
1.2 Interpolation Formulae
1.2.1 NGF Interpolation
1.2.2 NGB Interpolation
1.2.3 ST Interpolation
1.3 Difference Quotients
1.3.1 First Derivatives
1.3.2 Second Derivatives
1.3.3 First derivatives in 2 dimensions
1.3.4 Second derivatives in 2 dimensions
1.4 Sample Applications
2. Linear Algebra
2.1 General remarks
2.2 Exact Methods
2.2.1 Gauss Elimination and Back Substitution
2.2.2 Householder Transformation
2.2.3 LU Decomposition
2.2.4 Recursion Method
2.3 Iterative Methods
2.3.1 Iterative Improvement
2.3.2 Jacobi Relaxation
2.3.3 Gauss-Seidel Relaxation (GSR)
2.3.4 Successive Over-Relaxation (SOR)
2.3.5 Conjugate Gradient Method
2.4 Eigenvalues and Eigenvectors
2.4.1 Largest Eigenvalue and Related Eigenvector
2.4.2 Arbitrary Eigenvalue/-vector: Inverse Iteration
2.5 Sample Applications
2.5.1 Thermal Conduction in 1D
2.5.2 Potential Equation in 2D
3. Stochastics
3.1 Equidistribution
3.1.1 Linear Congruential Generators
3.1.2 Shift Register Generators
3.2 Other Distributions
3.2.1 Transformation of probability densities
3.2.2 Transformation Method
3.2.3 Generalized Transformation Method:
3.2.4 Rejection Method
3.2.5 Multivariate Gaussian Distribution
3.2.6 Homogeneous distributions in Orientation Space
3.3 Random Sequences
3.3.1 Basics
3.3.2 Generating a stationary Gaussian Markov sequence
3.3.3 Wiener-Lévy Process (Unbiased Random Walk)
3.3.4 Markov Chains (Biased Random Walk)
3.3.5 Monte Carlo Method
3.4 Stochastic Optimization
3.4.1 Simulated Annealing
3.4.2 Genetic Algorithms
Franz J. Vesely Oct 2005
See also:
"Computational Physics - An Introduction," Kluwer-Plenum 2001
