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Numerical Methods and Analysis with Python

Frontmatter

Introduction

Main

1. Root-finding
2. Linear Algebra and Simultaneous Equations
3. Polynomial Collocation and Approximation
4. Derivatives and Definite Integrals
5. Minimization
- 5.1. Finding the Minimum of a Function of One Variable Without Using Derivatives — A Brief Introduction
- 5.2. Finding the Minimum of a Function of Several Variables — Coming Soon
6. Initial Value Problems for Ordinary Differential Equations

Exercises

1. Exercises on the Bisection Method
2. Root Finding by Interval Halving, Exercise 1 template
3. Root Finding by Interval Halving, Exercise 2 template
4. Exercises on Fixed Point Iteration
5. Exercises on Error Measures and Convergence
6. Exercises on Newton’s Method
7. Exercises on Root-finding Without Derivatives
8. Exercises on Machine Numbers, Rounding Error and Error Propagation
9. Exercises on Solving Simultaneous Linear Equations
10. Exercises on Approximating Derivatives, the Method of Undetermined Coefficients and Richardson Extrapolation

Python Tutorial

1. Introduction
2. Getting Python Software for Scientific Computing
3. Suggestions and Notes on Python and Jupyter Notebook Usage
4. Python Basics
5. Notes on Python Coding Style (under construction)
6. Python Variables, Including Lists and Tuples, and Arrays from Package Numpy
7. Decision Making With if, else, and elif
8. Defining and Using Python Functions
9. Iteration with for
10. Iteration with while
11. Code Files, Modules, and an Integrated Development Environment
12. Recursion (vs iteration)
13. Plotting Graphs with Matplotlib
14. Numpy Array Operations and Linear Algebra
15. Package Scipy and More Tools for Linear Algebra
16. Summation and Integration
17. Random Numbers, Histograms, and a Simulation
18. Formatted Output and Some Text String Manipulation
19. Classes, Objects, Attributes, Methods: Very Basic Object-Oriented Programming in Python
20. Exceptions and Exception Handling

Appendices

1. Notebook for generating the module numericalMethods
2. Linear algebra algorithms using 0-based indexing and semi-open intervals
3. Revision notes and plans
4. Bibliography

.ipynb

Derivatives and Definite Integrals

4. Derivatives and Definite Integrals#

References:

Chapter … of [Sauer, 2022].
Chapter … of [Burden et al., 2016].
Chapter … of [Kincaid and Chenney, 1990].

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4.1. Approximating Derivatives by the Method of Undetermined Coefficients

By Brenton LeMesurier (College of Charleston, South Carolina), with contributions from Stephen Roberts (Australian National University).

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