**Course co-ordinator(s):** David Bourne (Edinburgh).

**Aims:**

When solving problems in science, engineering or economics, a real-life situation is first converted into a mathematical model. This is often called the formulation of the problem and it is given in terms of mathematical equations. Only a handful of model equations can be solved in a neat analytical form. Hence we need numerical analysis, comprising a set of techniques for finding approximate solutions of these equations. This course provides an introduction to very basic methods in numerical analysis both from a theoretical and a practical perspective. It also provides an introduction to programming the scientific computing package Matlab.

## Detailed Information

**Pre-requisite course(s):** F17CA Calculus A & F17CB Calculus B .

**Location: **Edinburgh.

**Semester: **2.

**Syllabus:**

**Computer arithmetics:** Number bases. Floating point numbers. Significant figures and digits. Round off errors. Absolute, relative and percentage errors. Floating point arith- metics.

(3 lectures)

** The elements of Matlab:** The different Matlab environments. M-files and M-functions. Variables, data types and internal Matlab functions. Text strings. The "If" conditional and logical pipelines. "For" loops.

(3 lectures)

**The basics of Matlab Programming:** The bisection method. Convergence of the bisection method. The regula falsi methods. Bisection and regula falsi algorithms with different stopping criteria in Matlab.

(3 lectures)

**Solving algebraic equations numerically:** The bisection method. Convergence of the bisection method. The regula falsi methods. Bisection and regula falsi algorithms with different stopping criteria in Matlab.

(3 lectures)

**Solving smooth non-linear scalar equations:** Newton's method. Fixed point iteration. Taylor's Theorem. Order of convergence. Order of convergence for fixed point iteration.

(3 lectures)

**Analysis of orders of covergence for fixed point iteration:** How to terminate fixed point iterations in practice. Multiple roots of a functional equation. The Newton method for multiple roots. The Newton method in Matlab. Fixed point iterations in Matlab and the logistic map.

(3 lectures)

**Interpolation:** Polynomial interpolation. Direct method. Lagrange polynomials. Newton polynomials. Interpolation error and Chevyshev points. Interpolation in Matlab.

(3 lectures)

**Numerical Integration:** Simple trapezoidal, midpoint and Simpson's rules. Derivation of the simple integration rules. The composite trapezoidal, midpoint and Simpson's rules. Matlab integration with applications to solving different practical problems.

(3 lectures)

**Convergence of the numerical integration methods:** Convergence of the simple rules. Convergence of the composite rules. Minimal number of quadrature points for accuracy. Order of magnitude of the error in computing integrals.

(3 lectures)

**Numerical differentiation:** Forward, backward, central and five-point differentiation. Error formulas for differentiation methods. Higher order derivatives. Differentiating in Matlab and applications.

(3 lectures)

**Assessment Methods:**

30% by class tests or other continuous assessment

70% by end of module 2-hour exam

Resit Type: resit exam in semester 3

**Contact Hours:** 3 lectures + 1 tutorial or lab session per weeks.

**SCQF Level: **8.

**Credits:** 15.