Alvaro Meseguer - Fundamentals of Numerical Mathematics for Physicists and Engineers

1 Cover

2 About the Author About the Author 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.

3 Preface

4 Acknowledgments

5 Part I: 1 Solution Methods for Scalar Nonlinear Equations1.1 Nonlinear Equations in Physics 1.2 Approximate Roots: Tolerance 1.3 Newton's Method 1.4 Order of a Root‐Finding Method 1.5 Chord and Secant Methods 1.6 Conditioning 1.7 Local and Global Convergence 2 Polynomial Interpolation2.1 Function Approximation 2.2 Polynomial Interpolation 2.3 Lagrange's Interpolation 2.4 Barycentric Interpolation 2.5 Convergence of the Interpolation Method 2.6 Conditioning of an Interpolation 2.7 Chebyshev's Interpolation 3 Numerical Differentiation 3.1 Introduction 3.2 Differentiation Matrices 3.3 Local Equispaced Differentiation 3.4 Accuracy of Finite Differences 3.5 Chebyshev Differentiation 4 Numerical Integration4.1 Introduction 4.2 Interpolatory Quadratures 4.3 Accuracy of Quadrature Formulas 4.4 Clenshaw–Curtis Quadrature 4.5 Integration of Periodic Functions 4.6 Improper Integrals

6 Part II: 5 Numerical Linear Algebra5.1 Introduction 5.2 Direct Linear Solvers 5.3 LU Factorization of a Matrix 5.4 LU with Partial Pivoting 5.5 The Least Squares Problem 5.6 Matrix Norms and Conditioning 5.7 Gram-Schmidt Orthonormalization 5.8 Matrix‐Free Krylov Solvers 6 Systems of Nonlinear Equations 6.1 Newton's Method for Nonlinear Systems 6.2 Nonlinear Systems with Parameters 6.3 Numerical Continuation (Homotopy) 7 Numerical Fourier Analysis 7.1 The Discrete Fourier Transform 7.2 Fourier Differentiation 8 Ordinary Differential Equations 8.1 Boundary Value Problems 8.2 The Initial Value Problem

7 1 Solutions to Problems and ExercisesChapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8

8 Glossary of Mathematical Symbols

9 Bibliography

10 Index

11 End User License Agreement

1 Chapter 1 Table 1.1 Iterates resulting from using bisection and Newton–Raphson when ...

2 Chapter 2 Table 2.1 Runge's counterexample.

3 Chapter 4Table 4.1 Coefficients of open and closed Newton–Cotes quadrature formulas (...Table 4.2 Trapezoidal and Simpson composite quadrature approximations of Table 4.3 Clenshaw–Curtis quadrature approximation of .Table 4.4 Composite trapezoidal , Clenshaw–Curtis , and composite trapezoida...Table 4.5 Cotangent quadrature rule approximation (4.88) of integral (4.91) ...Table 4.6 Approximation values of integral (4.101) using Fejér's formula (4.9...

4 Chapter 6Table 6.1 Iterates of Newton's methods applied to system (6.21).Table 6.2 Solutions and conditioning of system (6.22).Table 6.3 Iterates of Newton's methods applied to system (6.22).

5 Chapter 8Table 8.1 Leading approximated eigenvalues of boundary value problem (8.42).Table 8.2 Numerical solution of using different Runge–Kutta methods with ....Table 8.3 Adams–Bashforth normalized coefficients Table 8.4 Backward difference coefficients .

1 Chapter 1 Figure 1.1 (a) A simple exploration of reveals a change of sign within the... Figure 1.2 (a) Analysis of the order of convergence of the Newton's method: ... Figure 1.3 (a) Graph intercepting ordinates and at abscissas and , ... Figure 1.4 (a) Convergence history of Newton's and secant methods when the s...

2 Chapter 2 Figure 2.1 Equispaced grid with . Figure 2.2 (a) Trigonometric function