Fundamentals of Numerical Mathematics for Physicists and Engineers

Meseguer , Alvaro 지음

Wiley 2020년 05월 14일 출간

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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 pro

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About the Author ix Preface xi Acknowledgments xv Part I 1 1 Solution Methods for Scalar Nonlinear Equations 3 1.1 Nonlinear Equations in Physics 3 1.2 Approximate Roots: Tolerance 5 1.2.1 The Bisection Method 6 1.3 Newton’s Method 10 1.4 Order of a Root-Finding Method 13 1.5 Chord and Secant Methods 16 1.6 Conditioning 18 1.7 Local and Global Convergence 20 Problems and Exercises 24 2 Polynomial Interpolation 29 2.1 Function Approximation 29 2.2 Polynomial Interpolation 30 2.3 Lagrange’s Interpolation 33 2.3.1 Equispaced Grids 37 2.4 Barycentric Interpolation 39 2.5 Convergence of the Interpolation Method 43 2.5.1 Runge’s Counterexample 46 2.6 Conditioning of an Interpolation 49 2.7 Chebyshev’s Interpolation 54 Problems and Exercises 60 3 Numerical Differentiation 63 3.1 Introduction 63 3.2 Differentiation Matrices 66 3.3 Local Equispaced Differentiation 72 3.4 Accuracy of Finite Differences 75 3.5 Chebyshev Differentiation 80 Problems and Exercises 84 4 Numerical Integration 87 4.1 Introduction 87 4.2 Interpolatory Quadratures 88 4.2.1 Newton Cotes Quadratures 92 4.2.2 Composite Quadrature Rules 95 4.3 Accuracy of Quadrature Formulas 98 4.4 Clenshaw Curtis Quadrature 104 4.5 Integration of Periodic Functions 112 4.6 Improper Integrals 115 4.6.1 Improper Integrals of the First Kind 116 4.6.2 Improper Integrals of the Second Kind 119 Problems and Exercises 125 Part II 129 5 Numerical Linear Algebra 131 5.1 Introduction 131 5.2 Direct Linear Solvers 132 5.2.1 Diagonal and Triangular Systems 133 5.2.2 The Gaussian Elimination Method

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저자(글) Meseguer

ALVARO MESEGUER, PHD, is Associate Professor at the Department of Physics at Polytechnic University of Catalonia (UPC BarcelonaTech), Barcelona, Spain, where he teaches Numerical Methods, Fluid Dynamics, and Mathematical Physics to advanced undergraduates in Engineering Physics and Mathematics. He has published more than 30 articles in peer-reviewed journals within the fields of computational fluid dynamics, and nonlinear physics.

저자(글) Alvaro