## MATH2601 Numerical Analysis with Computation

### 15 creditsClass Size: 80

Module manager: Dr Evy Kersale
Email: E.Kersale@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2022/23

### Pre-requisite qualifications

(MATH1920 or MATH2920) and {MATH1010 or MATH1005 or [MATH1050 and (MATH1060 or MATH1331)]}, or equivalent.

### This module is mutually exclusive with

 MATH2600 Numerical Analysis

This module is not approved as a discovery module

### Objectives

Most of the problems that students meet when they are introduced to, for example, integration or differential equations, will have nice analytic solutions. In real life though this is typically not the case and so solutions have to be evaluated numerically (ie with the aid of a computer). This module explains how to express mathematical operations in terms of operations that can be performed on a computer. It is a good preparation for the Level 3 module in Numerical Methods (MATH 3474). This module also involves a practical implementation of the algorithmic ideas. Students should improve their programming skills begun in MATH1920, and gain confidence and facility with computational mathematics.

In order to pass the module, students must pass the MATH2600 component (which is at least 40% on exam and in-course assessment combined) and score at least 40% on the Computer Exercises.

Learning outcomes
On completion of this module, students should be able to:
- describe how errors arise in computations;
- solve simple non-linear equations by root-finding techniques;
- calculate the interpolating polynomial through discrete data points;
- derive and use quadrature formulae based on integration of polynomial interpolates;
- write down suitable numerical schemes for solving first order ordinary differential equations;
- solve linear systems of algebraic equations using Gaussian elimination and LU factorisation;
- implement numerical algorithms computationally and interpret the results of programs;
- understand some of the practical issues associated with mathematical programming.

### Syllabus

- Introduction. Computer arithmetic. Errors; round-off error, truncation error.
- Solution of nonlinear equations in one variable. Bisection method; fixed point iteration; Newton-Raphson iteration; secant method. Order of convergence.
- Interpolation. Lagrange interpolation; error term. cubic splines.
- Numerical integration. Trapezoidal rule. Method of undetermined coefficients. Simpson's rule. Newton-Cotes formulae. Composite integration methods. Richardson extrapolation; Romberg integration.
- Ordinary differential equations (initial value problems). Euler's method; errors. Runge-Kutta methods. Multi-step methods. Stability.
- Linear systems of algebraic equations. Gaussian elimination. Pivoting. LU factorisation.
- Practical implementation of theoretical principles in Python.
- Investigation of computational aspects of numerical analysis.

### Teaching methods

 Delivery type Number Length hours Student hours Workshop 10 1.00 10.00 Lecture 22 1.00 22.00 Practical 5 2.00 10.00 Tutorial 5 1.00 5.00 Private study hours 103.00 Total Contact hours 47.00 Total hours (100hr per 10 credits) 150.00

### Private study

Students should spend 68 hours in the same way as for MATH2600. The remaing 35 hours are spent writing computer code, analysing results, and writing up work.

### Opportunities for Formative Feedback

Five fortnightly pieces of coursework are marked and feedback given. Of these, 2 are assessed towards the final module grade.

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment Computer Exercise In order to pass the module, students must pass the MATH2600 component (which is at least 40% on exam and in-course assessment combined) and score at least 40% on the Computer Exercises. 30.00 In-course Assessment * 10.00 Total percentage (Assessment Coursework) 40.00

Students who have failed the MATH2600 component will need to resit the exam; students who have failed the computer exercises will need to resubmit these. There is no resit available for the 10% in-course assessment component of this module. If the module is failed, the mark for this component will be carried forward and added to the resit exam and/or computer exercises mark with the same weighting as listed above.

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 00 mins 60.00 Total percentage (Assessment Exams) 60.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated