F18NA - Numerical Analysis A

Course leader(s):

Julian Braun (Edinburgh, January)

Aims

To give an introduction to some of the basic methods of numerical analysis and the scientific computing package Python.

Syllabus

1. Introduction to Numerical Analysis (1.1 1. What is numerical analysis?, 1.2 2. Approximation errors, 1.3 3. Floating-point arithmetic)

2. Algebraic Equations (2.1 1. Existence and uniqueness of solutions, 2.2 2. The bisection method, 2.3 3. The fixed-point method, 2.4 4. Newton's method)

3. Polynomial Interpolation (3.1 1. Existence and uniqueness, 3.2 2. Interpolation error, 3.3 3. Convergence and non-convergence)

4. Numerical Integration (4.1 1. The Newton-Cotes method, 4.2 2. Error estimates, 4.3 3. Composite quadrature rules, 4.4 4. Stability of Numerical Integration)

5. Numerical Differentiation (5.1 1. Finite difference approximation, 5.2 2. Error estimates, 5.3 3. Instability of Numerical Differentiation)

6. Numerical Analysis with Python (6.1 1. Basic introduction to Python, 6.2 2. Standard Programming techniques, 6.3 3. Solving algebraic equations with Python, 6.4 4. Polynomials in Python, 6.5 5. Numerical Integration and Differentiation with Python)

Learning outcomes

By the end of the course, students should be able to do the following:

apply basic concepts of numerical analysis which include approximation errors, convergence rates, and computer arithmetic.
solve algebraic equations using the Bisection Method, Fixed-Point Method, and Newton’s Method and analyse the properties of these methods.
employ Lagrange Polynomials to interpolate data points and approximate functions.
analyse basic quadrature methods and use them to approximate integrals.
analyse finite difference expressions and use them to approximate derivatives.
implement numerical methods on a computer with a programming language such as Python.

Further details

Curriculum explorer: Click here

SCQF Level: 8

Credits: 15