Fundamentals of Numerical Mathematics for Physicists and Engineers

Introduces the fundamentals of numerical mathematics and illustrates its applications to a wide variety of disciplines in physics and engineering Applying numerical mathematics to solve scientific problems, this book helps readers understand the mathematical and algorithmic elements that lie beneath numerical and computational methodologies in order to determine the suitability of certain techniques for solving a given problem. It also contains examples related to problems arising in classical mechanics, thermodynamics, electricity, and quantum physics. Fundamentals of Numerical Mathematics for Physicists and Engineers is presented in two parts. Part I addresses the root finding of univariate transcendental equations, polynomial interpolation, numerical differentiation, and numerical integration. Part II examines slightly more advanced topics such as introductory numerical linear algebra, parameter dependent systems of nonlinear equations, numerical Fourier analysis, and ordinary differential equations (initial value problems and univariate boundary value problems). Chapters cover: Newton s method, Lebesgue constants, conditioning, barycentric interpolatory formula, Clenshaw-Curtis quadrature, GMRES matrix-free Krylov linear solvers, homotopy (numerical continuation), differentiation matrices for boundary value problems, Runge-Kutta and linear multistep formulas for initial value problems. Each section concludes with Matlab hands-on computer practicals and problem and exercise sets. This book: Provides a modern perspective of numerical mathematics by introducing top-notch techniques currently used by numerical analysts Contains two parts, each of which has been designed as a one-semester course Includes computational practicals in Matlab (with solutions) at the end of each section for the instructor to monitor the student's progress through potential exams or short projects Contains problem and exercise sets (also with solutions) at the end of each section Fundamentals of Numerical Mathematics for Physicists and Engineers is an excellent book for advanced undergraduate or graduate students in physics, mathematics, or engineering. It will also benefit students in other scientific fields in which numerical methods may be required such as chemistry or biology.

Introduces the fundamentals of numerical mathematics and illustrates its applications to a wide variety of disciplines in physics and engineeringApplying numerical mathematics to solve scientific problems, this book helps readers understand the mathematical and algorithmic elements that lie beneath numerical and computational methodologies in order to determine the suitability of certain techniques for solving a given problem. It also contains examples related to problems arising in classical mechanics, thermodynamics, electricity, and quantum physics.Fundamentals of Numerical Mathematics for Physicists and Engineers is presented in two parts. Part I addresses the root finding of univariate transcendental equations, polynomial interpolation, numerical differentiation, and numerical integration. Part II examines slightly more advanced topics such as introductory numerical linear algebra, parameter dependent systems of nonlinear equations, numerical Fourier analysis, and ordinary differential equations (initial value problems and univariate boundary value problems). Chapters cover: Newton's method, Lebesgue constants, conditioning, barycentric interpolatory formula, Clenshaw-Curtis quadrature, GMRES matrix-free Krylov linear solvers, homotopy (numerical continuation), differentiation matrices for boundary value problems, Runge-Kutta and linear multistep formulas for initial value problems. Each section concludes with Matlab hands-on computer practicals and problem and exercise sets. This book:* Provides a modern perspective of numerical mathematics by introducing top-notch techniques currently used by numerical analysts* Contains two parts, each of which has been designed as a one-semester course* Includes computational practicals in Matlab (with solutions) at the end of each section for the instructor to monitor the student's progress through potential exams or short projects* Contains problem and exercise sets (also with solutions) at the end of each sectionFundamentals of Numerical Mathematics for Physicists and Engineers is an excellent book for advanced undergraduate or graduate students in physics, mathematics, or engineering. It will also benefit students in other scientific fields in which numerical methods may be required such as chemistry or biology.

About the Author ixPreface xiAcknowledgments xvPart I 11 Solution Methods for Scalar Nonlinear Equations 31.1 Nonlinear Equations in Physics 31.2 Approximate Roots: Tolerance 51.2.1 The Bisection Method 61.3 Newton's Method 101.4 Order of a Root-Finding Method 131.5 Chord and Secant Methods 161.6 Conditioning 181.7 Local and Global Convergence 20Problems and Exercises 242 Polynomial Interpolation 292.1 Function Approximation 292.2 Polynomial Interpolation 302.3 Lagrange's Interpolation 332.3.1 Equispaced Grids 372.4 Barycentric Interpolation 392.5 Convergence of the Interpolation Method 432.5.1 Runge's Counterexample 462.6 Conditioning of an Interpolation 492.7 Chebyshev's Interpolation 54Problems and Exercises 603 Numerical Differentiation 633.1 Introduction 633.2 Differentiation Matrices 663.3 Local Equispaced Differentiation 723.4 Accuracy of Finite Differences 753.5 Chebyshev Differentiation 80Problems and Exercises 844 Numerical Integration 874.1 Introduction 874.2 Interpolatory Quadratures 884.2.1 Newton-Cotes Quadratures 924.2.2 Composite Quadrature Rules 954.3 Accuracy of Quadrature Formulas 984.4 Clenshaw-Curtis Quadrature 1044.5 Integration of Periodic Functions 1124.6 Improper Integrals 1154.6.1 Improper Integrals of the First Kind 1164.6.2 Improper Integrals of the Second Kind 119Problems and Exercises 125Part II 1295 Numerical Linear Algebra 1315.1 Introduction 1315.2 Direct Linear Solvers 1325.2.1 Diagonal and Triangular Systems 1335.2.2 The Gaussian Elimination Method 1355.3 LU Factorization of a Matrix 1405.3.1 Solving Systems with LU 1455.3.2 Accuracy of LU 1475.4 LU with Partial Pivoting 1505.5 The Least Squares Problem 1605.5.1 QR Factorization 1625.5.2 Linear Data Fitting 1735.6 Matrix Norms and Conditioning 1785.7 Gram-Schmidt Orthonormalization 1835.7.1 Instability of CGS: Reorthogonalization 1875.8 Matrix-Free Krylov Solvers 193Problems and Exercises 2046 Systems of Nonlinear Equations 2096.1 Newton's Method for Nonlinear Systems 2106.2 Nonlinear Systems with Parameters 2206.3 Numerical Continuation (Homotopy) 224Problems and Exercises 2327 Numerical Fourier Analysis 2357.1 The Discrete Fourier Transform 2357.1.1 Time-Frequency Windows 2437.1.2 Aliasing 2467.2 Fourier Differentiation 251Problems and Exercises 2588 Ordinary Differential Equations 2618.1 Boundary Value Problems 2628.1.1 Bounded Domains 2628.1.2 Periodic Domains 2758.1.3 Unbounded Domains 2778.2 The Initial Value Problem 2798.2.1 Runge-Kutta One-Step Formulas 2818.2.2 Linear Multistep Formulas 2878.2.3 Convergence of Time-Steppers 2978.2.4 A-Stability 3018.2.5 A-Stability in Nonlinear Systems: Stiffness 315Problems and Exercises 330Solutions to Problems and Exercises 335Glossary of Mathematical Symbols 367Bibliography 369Index 373