Fundamentals of Numerical Mathematics for Physicists and Engineers. Alvaro Meseguer

      

      Table of Contents

      1  Cover

      2  About the Author

      3  Preface

      4  Acknowledgments

      5  Part I: 1 Solution Methods for Scalar Nonlinear Equations 1.1 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 Interpolation 2.1 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 Integration 4.1 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 Algebra 5.1 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 Exercises Chapter 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

      List of Tables

      1 Chapter 1Table 1.1 Iterates resulting from using bisection

and Newton–Raphson

when ...

      2 Chapter 2Table 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