| 1 Approximate Numbers 1.1 Sources of errors in numerical calculations 1.2 Absolute and relative errors 1.3 Representation of numbers 1.4 Significant digits 1.5 Errors of elementary operations 1.5.1 Error of a sum 1.5.2 Error of a difference 1.5.3 Error of a product 1.5.4 Error of a quotient References 2 Basic Programming Techniques 2.1 Programming concepts 2.2 Functions and parameters 2.3 Passing arguments to Python functions 2.4 Passing arguments to C/C++ functions 2.5 Arrays in Python 2.6 Dynamic array allocation in C/C++ 2.7 Basic matrix operations References 3 Elements of Scientific Graphics 3.1 The Tkinter package 3.2 The Canvas widget 3.2.1 Commonly used Canvas items 3.2.2 Methods for updating Canvas widgets 3.3 Simple Tkinter applications 3.4 Plotting functions of one variable 3.5 The graphics library graphlib.py 3.5.1 Auxiliary functions 3.5.2 Routines for plotting functions 3.5.3 Simple 3D rendering routines 3.5.4 Histograms 3.6 Creating plots in C++ using the library graphlib.py References 4 Sorting and Indexing 4.1 Introduction 4.2 Bubble sort 4.3 Insertion sort 4.4 Quicksort 4.5 Indexing and ranking 4.6 Implementations in C/C++ 4.7 Problems References 5 Evaluation of Functions 5.1 Evaluation of polynomials by Horner's scheme 5.2 Evaluation of analytic functions 5.3 Continued fractions 5.4 Orthogonal polynomials 5.5 Spherical harmonics. Associated Legendre functions 5.6 Spherical Bessel functions 5.7 Implementations in C/C++ 5.8 Problems References 6 Algebraic and Transcendental Equations 6.1 Root separation 6.2 Bisection method 6.3 Method of false position 6.4 Method of successive approximations 6.5 Newton's method 6.6 Secant method 6.7 Birge-Vieta method 6.8 Newton's method for systems of nonlinear equations 6.9 Implementations in C/C++ 6.10 Problems References 7 Systems of Linear Equations 7.1 Introduction 7.2 Gaussian elimination with backward substitution 7.3 Gauss-Jordan elimination 7.4 LU factorization 7.5 Inversion of triangular matrices 7.6 Cholesky factorization 7.7 Tridiagonal systems of linear equations 7.8 Block tridiagonal systems of linear equations 7.9 Complex matrix equations 7.10 Jacobi and Gauss-Seidel iterative methods 7.11 Implementations in C/C++ 7.12 Problems References 8 Eigenvalue problems 8.1 Introduction 8.2 Diagonalization of matrices by similarity transformations 8.3 Jacobi method 8.4 Generalized eigenvalue problems for symmetric matrices 8.5 Implementations in C/C++ 8.6 Problems References |
9 Modeling of Tabulated Functions 9.1 Interpolation and regression 9.2 Lagrange interpolation polynomial 9.3 Neville's interpolation method 9.4 Cubic spline interpolation 9.5 Linear regression 9.6 Multilinear regression models 9.7 Nonlinear regression. The Levenberg-Marquardt method 9.8 Implementations in C/C++ 9.9 Problems References 10 Integration of Functions 10.1 Introduction 10.2 Trapezoidal Rule; A Heuristic Approach 10.3 The Newton-Cotes Quadrature Formulas 10.4 Trapezoidal Rule 10.5 Simpson's Rule 10.6 Adaptive Quadrature Methods 10.7 Romberg's Method 10.8 Improper Integrals: Open Formulas 10.9 Midpoint Rule 10.10 Gaussian Quadratures 10.11 Multidimensional integration 10.12 Adaptive multidimensional integration 10.13 Implementations in C/C++ 10.14 Problems References 11 Monte Carlo Method 11.1 Introduction 11.2 Integration of functions 11.3 Importance sampling 11.4 Multidimensional integrals 11.5 Generation of random numbers 11.6 Implementations in C/C++ 11.7 Problems References 12 Ordinary Differential Equations 12.1 Introduction 12.2 Taylor series method 12.3 Euler's method 12.4 Runge-Kutta methods 12.5 Adaptive step size control 12.6 Methods for second-order ODEs 12.7 Numerov's method 12.8 Shooting methods for two-point problems 12.9 Finite-difference methods for linear two-point problems 12.10 Implementations in C/C++ 12.11 Problems References 13 Partial Differential Equations 13.1 Introduction 13.2 Boundary-value problems for elliptic differential equations 13.3 Initial-value problems for parabolic differential equations 13.3.1 Explicit finite-difference method 13.3.2 von Neumann stability analysis 13.3.3 Implicit finite-difference method 13.3.4 Crank-Nicolson method 13.3.5 Spatially variable diffusion coefficient 13.4 Time-dependent Schrödinger equation 13.5 Initial-value problems for hyperbolic differential equations 13.6 Implementations in C/C++ 13.7 Problems References A Dynamic array allocation in C/C++ B Basic operations with vectors and matrices C Embedding Python in C/C++ References D The numerical libraries numxlib.py and numxlib.h E The graphics library graphlib.py based on Tkinter F The C++ interface to the graphics library graphlib.py G List of programs by chapter Index |