# Unit information: Numerical Analysis 23 in 2015/16

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Unit name Numerical Analysis 23 MATH30010 20 H/6 Teaching Block 2 (weeks 13 - 24) Dr. Sieber Not open First year core units (MATH11006 Analysis 1 (or MATH10003 Analysis 1A and MATH10006 Analysis 1B), MATH11007 Calculus 1, MATH11005 Linear Algebra and Geometry) None School of Mathematics Faculty of Science

## Description including Unit Aims

This unit is intended to serve as a first course in numerical analysis. As such the fundamental areas of root finding, numerical differentiation, numerical integration and solving ordinary differential equations will be covered. The emphasis will be to explore numerical techniques for solving these problems theoretically. Computer programming is not required for this unit.

Unit aims

To introduce students to the basics of numerical analysis; this is broadly the study of numerical methods for solving mathematical problems.

Syllabus

• Root finding. Linear systems: Gaussian elimination and LU decomposition. Nonlinear equations: bisection, fixed point iteration, Newton-Raphson, accelerating convergence. Systems of nonlinear equations, Newton's method, steepest descent.
• Numerical differentiation and integration. Interpolation polynomial, trapezoidal rule, Simpson's rule, Richardson's extrapolation, Romberg integration, Gaussian quadrature.
• Ordinary differential equations. Initial value problems: Euler's methods, Runge-Kutta methods, multistep methods, stability, time stability, stiffness. Boundary value problems: Shooting, finite difference methods, spectral methods.

Relation to other units

This unit is a more advanced version of the Level I/5 unit, Numerical Analysis 2. The lectures for Numerical Analysis 2 and Numerical Analysis 23 are the same, but the problem sheets and examination questions for Numerical Analysis 23 are more challenging. Students may NOT take both Numerical Analysis 2 and Numerical Analysis 23.

## Intended Learning Outcomes

At the end of this unit, students should be able to

• solve nonlinear equations
• numerically differentiate
• evaluate complicated integrals and
• estimate the solutions to ordinary differential equations to any required accuracy.

Transferable Skills: Computational techniques; interpretation of computational results; relation of numerical results to mathematical theory.

## Teaching Information

Lectures; weekly problem classes; theoretical and computational exercises to be done by students.

## Assessment Information

The final assessment mark will be entirely based upon a 2 ½-hour examination

A good text which covers most of the course is:

• R.L. Burden and J.D. Faires, Numerical Analysis (PWS-Kent) (QA297 BUR)

Other texts that may be helpful to students looking for an alternative point of view on the material

• J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (QA297 STO)
• G. Dahlquist, A. Bjorck, and N. Anderson, Numerical Methods (Prentice) (QA297 DAH)
• C.F. Gerald and P.O.Wheatley, Applied Numerical Analysis (Addison-Wesley) (QA297 GER)

Many other books can be found in the numerical analysis section (books QA297 ***)