| Preface to the FirstEdition Preface to the Second Edition 1 Computational Methods 1-1 Numerical calculations and beyond 1-2 Integers and floating numbers 1-3 Programming language and program library 1-4 Examples of algebraic, integer and floating numbercalculations 1-5 Examples of unconventional techniques Problems 2 Integration and Differentiation 2-1 Numerical integration 2-2 Rectangular and trapezoidal rules 2-3 Simpsons rule 2-4 Gaussian quadrature 2-5 Monte Carlo integration 2-6 Multidimensional integrals and improper integrals 2-7 Numerical differentiation Problems 3 Interpolation and Extrapolation 3-1 Polynomial interpolation 3-2 Interpolation using rational functions 3-3 Continued fraction 3-4 Fourier transform 3-5 Extrapolation 3-6 Inverse interpolation 3-7 Cubic spline Problems 4 Special Functions 4-1 Hermite polynomials and harmonic oscillator 4-2 Legendre polynomials and spherical harmonics 4-3 Spherical Bessel functions 4-4 Laguerre polynomials 4-5 Error integrals and gamma functions Problems 5 Matrices 5-1 System of linear equations 5-2 Matrix inversion and LU-decomposition 5-3 Matrix approach to the eigenvalue problem 5-4 Tridiagonalization method 5-5 Eigenvalues and eigenvectors of a tridiagonal matrix 5-6 Lanczos method of constructing matrices 5-7 Nonsymmetric matrices and complex matrices Problems 6 Methods of Least Squares 6-1 Statistical description of data 6-2 Uncertainties and their propagation 6-3 The method of maximum likelihood 6-4 The method of least squares 6-5 Statistical tests of the results 6-6 Linear least-squares fit 6-7 Nonlinear least-squares fit to data Problems 7 Monte Carlo Calculations 7-1 Generation of random numbers 7-2 Molecular diffusion and Brownian motion 7-3 Data simulation and hypothesis testing 7-4 Percolation and critical phenomena 7-5 The Ising model 7-6 Path integrals in quantum mechanics 7-7 Fractals Problems 8 Finite Difference Solution of Differential Equations 8-1 Types of differential equations 8-2 Runge-Kutta methods 8-3 Solution of initial value problems by extrapolation 8-4 Boundary value problems by shooting methods 8-5 Relaxation methods 8-6 Boundary value problems in partial differential equations 8-7 Parabolic partial differential equations 8-8 Hyperbolic partial differential equations 8-9 Nonlinear differential equations 8-10 Stiffness problems Problems 9 Finite Element Solution to PDE 9-1 Background 9-2 Shape functions and finite element approximation 9-3 Assembling contributions from elements 9-4 Variational approach 9-5 Application to a two-dimensional Poisson equation Problems Appendix A A-1 Decomposition into prime numbers A-2 Bit-reversed order A-3 Gaussian elimination of a tridiagonal matrix A-4 Random bit generator A-5 Reduction of higher-order ODE to first-order Appendix B List of Fortran Program Examples Bibliography Index |
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