Unit information: Numerical Analysis 2 in 2015/16

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Unit name Numerical Analysis 2 MATH20700 20 I/5 Teaching Block 2 (weeks 13 - 24) Dr. Sieber Not open MATH11002 and MATH11003 None School of Mathematics Faculty of Science

Description including Unit Aims

This unit constitutes an introduction to numerical analysis. The topics covered are: approximation of functions; orthogonal polynomials; quadrature; discretisation of the initial value problem for ordinary differential equations; iterative solution of linear and nonlinear systems of equations.

Aims

To introduce students to the basics of numerical analysis; this is broadly the study of how to use computers to solve 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.

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 or fortnightly problems classes; theoretical and computational exercises to be done by students.

Assessment Information

The final assessment mark will be entirely based upon a 2½-hour examination in May/June (details below).

Summer Examination

The examination in May/June consists of a 2 ½-hour written examination consisting of FIVE questions; you should attempt FOUR. If you attempt more than four, your best four answers will be used for assessment. Calculators of the approved type (non-programmable, no text facility) are permitted in the 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 ***).