1 Approximations, Errors and the Taylor Series 1.1 Approximations and Errors 1.1.1 Approximations 1.1.2 Round-off Errors 1.1.3 Principles to Hold During the Numerical Calculations 1.2 Truncation Errors and the Taylor Series 1.2.1 Truncation Errors 1.2.2 The Taylor Series 1.2.3 Some Techniques in Numerical Computation 1.3 Control of Total Numerical Error 1.3.1 Truncation Error 1.3.2 Total Numerical Error 1.3.3 Control of Numerical Errors 1.4 Problems for Chapter 1 1.5 Computer Work for Chapter 1 2 Interpolation and Finite Differences 2.1 Finite Difference 2.2 Newton Interpolation 2.2.1 Basis Functions 2.2.2 Newton Interpolation 2.2.3 Newton's Divided-difference Interpolating Polynomial 2.2.4 Errors of Newton Interpolation 2.3 Interpolation Formulae 2.3.1 NGF Interpolation 2.3.2 NGB Interpolation 2.3.3 ST Interpolation 2.4 Difference Quotients 2.4.1 DNGF Formulae 2.4.2 DNGB Formulae 2.4.3 DST Formulae 2.5 Problems for Chapter 2 2.6 Computer Work for Chapter 2 3 Numerical Integration 3.1 Numerical Integration Methods 3.2 Newton-Cotes Quadrature Rules 3.2.1 The Trapezoid Rule 3.2.2 Simpson's Rule 3.2.3 Error Estimation 3.3 Composite and Adaptive Quadrature 3.3.1 Composite Quadrature Rules 3.3.2 Automatic and Adaptive Quadrature 3.4 Numerical Integration of Multi-dimensional Integrals 3.5 Problems for Chapter 3 3.6 Computer Work for Chapter 3 4 Matrix Algebra 4.1 Types of Matrices 4.2 Gauss Elimination and Back Substitution 4.2.1 The Elimination of Unknowns 4.2.2 The Algorithm of Gauss Elimination and Back Substitution 4.2.3 Techniques for Improving Solutions 4.3 LU Decomposition and Matrix Inversion 4.3.1 Overview of LU Decomposition 4.3.2 LU Decomposition Algorithm 4.3.3 Procedure from Gauss Elimination 4.3.4 The Matrix Inverse and Error Analysis 4.4 Tridiagonal Matrices and Recursion Method 4.4.1 Tri-diagonal Systems 4.4.2 Recursion Method 4.5 Iterative Methods 4.6 Jacobi Method 4.6.1 The Algorithm 4.6.2 The Convergency 4.7 Gauss-Seidel Method(GS) 4.7.1 The Algorithm 4.7.2 The Convergency 4.8 Successive Over-Relaxation Method(SOR) 4.9 Conjugate Gradient Method(CG) 4.9.1 The Gradient 4.9.2 Steepest Descent Method 4.9.3 CG Method 4.10 Problems for Chapter 4 4.11 Computer Work for Chapter 4 5 Ordinary Differential Equations 5.1 Types of Differential Equations 5.2 Euler Method 5.2.1 Error Analysis 5.2.2 Stability 5.2.3 Application to Vector Equations 5.3 The Leapfrog Methods 5.3.1 Stability Analysis 5.3.2 GeneraliZation to Multi-step Scheme 5.4 Implicit Methods 5.4.1 The Most Fundamental Scheme 5.4.2 Implicit Scheme of Second Order -- Improved Euler Method 5.5 The Runge-Kutta Method 5.5.1 The Basic Idea of Runge-Kutta Method 5.5.2 Stability Analysis 5.5.3 Adaptive RK Method 5.6 Predictor-Corrector(PC) Method 5.7 Boundary Value Problems and Initial Value Problems of Second Order 5.7.1 Shooting Method 5.7.2 Numerov's Method 5.8 Problems for Chapter 5 5.9 Computer Work for Chapter 5 6 Partial Differential Equations 6.1 Types of Equations 6.2 Elliptic Equations 6.2.1 Two or More Dimensions 6.2.2 ADI (alternating direction implicit) Method 6.3 Hyperbolic Equations 6.3.1 The FTCS Scheme 6.3.2 The Lax Scheme 6.3.3 Leapfrog Scheme 6.4 Parabolic Equations 6.4.1 A Simple Method -- FTCS Scheme 6.4.2 Implicit Scheme of First Order 6.4.3 Crank-Nicholson (CN) Scheme 6.5 Five-point Stencil for 2D Poisson Equation in Electromagnetic Field 6.6 Problems for Chapter 6 6.7 Computer Work for Chapter 6 7 Monte Carlo Methods and Simulation 7.1 Probability 7.1.1 Chance and Probability 7.1.2 A One-dimensional Random Walk 7.1.3 Probability Distribution 7.1.4 Random Variables 7.2 Random Number Generators 7.2.1 Linear Gongruential Generators 7.2.2 Shift Register Generators 7.3 Non-uniform Probability Distribution 7.3.1 Inverse Transform Method 7.3.2 Generalized Transformation Method -- Box-Muller Teehnique 7.4 Monte Carlo Integration 7.4.1 Splash Method (Hit or Miss Method) 7.4.2 Sample Mean Method 7.4.3 Two Theorems in Probability Theory 7.4.4 MC Error Analysis 7.4.5 Importanee Sampling Technique 7.5 Stoehastie Dynamics 7.5.1 Random Sequences 7.5.2 Stoehastie Dynamics 7.6 Monte Carlo Simulation and Ising Model 7.6.1 Simulation Methods 7.6.2 Random Walk Methods 7.6.3 The Ising Model 7.6.4 The Metropolis Algorithm 7.7 Problems for Chapter 7 7.8 Computer Work for Chapter 7 Bibliography
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