General description
It is often impossible to find the exact solution of a mathematical problem using standard techniques. In these situations one has to resort to numerical techniques. Numerical analysis is concerned with the development and analysis of mehtods for the numerical solution of practical problems. This course will provide an introduction to the subject.

Aims
To introduce students to the techniques for the numerical approximation of mathematical problems, and to the analysis of these techniques.

Learning outcomes
On completion of this module, students should be able to:

• construct an interpolating polynomial using either the Lagrange or Newton formula, and describe their relative advantages and disadvantages;
• prove the error formula for Lagrange interpolation;
• construct divided and forward difference tables for prescribed data;
• construct a cubic spline interpolant and discuss the advantage of this approach over the use of the Lagrange interpolant;
• derive the trapezoidal and Simpson's rules for approximating an integral;
• derive the error term for the trapezoidal rule;
• generate a set of orthogonal polynomials;
• derive the formula for Romberg integration;
• derive Gauss-type integration rules;
• determine the root(s) of a nonlinear equation using the bisection method, functional iteration and Newton's method;
• state and prove the conditions under which the sequence x_{r+1}=g(x_[r}) converges to a unique root of the equation x=g(x);
• determine the order of an iterative process for computing the root of an equation;
• give a geometrical interpretation of Newton's method;
• state the conditions under which an initial value problem possesses a unique solution;
• define the concepts of consistency, convergence and stability for one-step for solving initial value problems;
• compute numerical approximations to the solution of initial value problems using one-step methods including predictor-corrector methods;
• determine the consistency, convergence and stability of given one-step methods.

Syllabus
1. POLYNOMIAL INTERPOLATION: Lagrange's formula. Newton's formula and divided differences. The forward difference formula. Interpolation error. Cubic spline interpolation.
2. NUMERICAL INTEGRATION: Trapezoidal rule, Simpson's rule. Composite integration rules. Quadrature errors. Romberg integration. Orthogonal polynomials and Gaussian quadrature rules.
3. SOLUTION OF NONLINEAR EQUATION IN A SINGLE VARIABLE: Bisection method. Fixed point methods and contraction mappings. Newton's method. Order of convergence.
4. INITIAL VALUE PROBLEMS: Existence and uniqueness of solutions. Euler's method. Local truncation error.
Consistency. Convergence. Stability. General one-step methods. Trapezoidal method. Predictor-corrector methods.