Applied Scientific Computing: With Python

Preface
Contents
1 Motivation and Background
1.1 Mathematical Modeling and Applications
1.2 Applied Scientific Computing
1.3 Python Programming
1.4 Background
1.4.1 Series Expansions
1.5 Modeling Errors Versus Errors
1.6 Conclusions and Connections: Motivation and Background
2 Number Representations and Errors
2.1 Introduction
2.2 Floating-Point Numbers
2.2.1 Python Number Representation
2.3 Sources of Errors
2.3.1 Rounding Errors
2.3.2 Truncation Error
2.3.3 Ill-Conditioning
2.4 Measures of Error and Precision
2.5 Floating-Point Arithmetic
2.6 Conclusions and Connections: Number Representation and Errors
3 Numerical Calculus
3.1 Introduction
3.2 Numerical Differentiation
3.3 Numerical Integration
3.4 Composite Formulas
3.5 Practical Numerical Integration
3.6 Conclusions and Connections: Numerical Calculus
3.7 Python Functions for Numerical Calculus
4 Linear Equations
4.1 Introduction
4.2 Gauss Elimination
4.2.1 Pivoting in Gauss Elimination
4.2.2 Tridiagonal Systems
4.3 LU Factorization and Applications
4.4 Iterative Methods
4.5 Linear Least Squares Approximation
4.6 Eigenvalues
4.7 Conclusions and Connections: Linear Equations
4.8 Python's Linear Algebra Functions
4.8.1 Linear Equations
4.8.2 Linear Least Squares
4.8.3 Eigenvalues
4.8.4 Basic Linear Algebra Functions
5 Iterative Solution of Nonlinear Equations
5.1 Introduction
5.1.1 Summary of Convergence of Sequences
5.2 The Bisection Method
5.3 Fixed Point Iteration
5.4 Newton's Method
5.5 The Secant Method
5.6 Newton's Method in Higher Dimensions
5.6.1 Newton's Method: Two Equations in Two Unknowns
5.7 Conclusions and Connections: Iterative Solution of Nonlinear Equations
5.8 Python Functions for Equation Solving
6 Interpolation
6.1 Introduction
6.2 Lagrange Interpolation
6.3 Difference Representations
6.3.1 Divided Difference Interpolation
6.4 Splines
6.5 Conclusions and Connections: Interpolation
6.6 Python Interpolation Functions
7 Differential Equations
7.1 Introduction and Euler's Method
7.2 Runge–Kutta Methods
7.3 Multistep Methods
7.4 Systems of Differential Equations
7.5 Boundary Value Problems: Shooting Methods
7.6 Conclusions and Connections: Differential Equations
7.7 Python Functions for Ordinary Differential Equations
Further Reading and Bibliography
Index