Module Descriptors

Computing

MM246: Numerical Analysis II (5 ECTS)

This course builds on Numerical Analysis I. Topics covered include: Newton cotes formulae and error analysis, categorisation of direct and iterative numerical methods, Gauss Seidel iterative scheme and its convergence, eigenvalues and eigenvectors, Gerschgorin circle theoerm, power method and its varieties, Euler's method, Improved Euler's method, Modified (mid-point) Euler's method, fourth order Runge-Kutta method, iterative solution of algebraic equations and convergence analysis.

Taught in Semester(s) 2. Examined in Semester(s) 2.

Workload: 106 hours (24 Lecture hours, 12 Lab hours, 70 Self study hours).

Module Learning Outcomes. On successful completion of this module the learner should be able to:

  1. Derive Simpson's composite rule and use it to approximate integrals. Compute an upper bound on this error. Obtain an estimate of this error by halving the strip width.
  2. Explain the concept of convergence of iterative schemes. Compute the 2 norms for the difference between two sucessive iteration results.
  3. Compute a number of iterations of the Gauss Seidal iterative scheme. Explain the conditions for convergence of the Gauss Seidal iterative scheme.
  4. Define eigenvalue, dominant eigenvalue, eigenvector and characteristic equation of a matrix. Use Gerschgorin's circle theorem and its corollary to put bounds on the eigenvalues. Prove Gerschgorin's circle theorem.
  5. Define linear dependence, linear independence, basis and know when eigenvectors form a basis. Prove and implement the power method, inverse power method, a method to find eigenvalues closest to a given number.
  6. Define orthonormal and orthogonal vectors, orthogonal matrix. Define similar matricies and know their properties. Apply Jacobi's method for eigenvalues of certain matrices.
  7. Define initial value problem (IVP), single step method, Taylor's theorem, truncation error and global error. Implement Euler's, Improved Euler's method, Modified Euler's method and the fourth order Runge-Kutta method on IVP including the following ordinary differential equations: first order, systems of first order and higher order problems.
  8. Know the requirements for convergence of a zero finding iterative scheme. Implement the bisection method, Newton's method and the secant method.

Indicative Content

This numerical analysis course builds on an earlier numerical analysis course. The material covered includes:

  1. Newton Cotes formulae: Simpson's rule, Lagrange interpolation formulae, derivation of the composite Simpson's rule, calculating approximate integrals using the Composite Simpson's rule, analysis of the error, deriving an error estimate formula for Trapezoidal and Simpson's rule by halving the strip width,
  2. Numerical methods in linear algebra 1: categorising problems into direct methods and iterative methods, norms, convergence of iterative schemes, Gauss Seidel iterative scheme and numerical examples, conditions for convergence of the Gauss Seidel scheme.
  3. Numerical analysis in linear algebra 2: Eigenvectors and eigenvalues, characteristic equation, dominant eigenvalue, Gerschgorin's circle theorem and its corollary, proof and uses of Gerschgorin's circle theorem, linear dependence and independence of a set of vectors, basis, theoerm on when eigenvectors form a basis, normalisation, power method steps and proof, inverse power method, method to find eigenvalues closest to a given number, numerical method to find other eigenvalues, Jacobi's method to find all eigenvalues of certain matrices, orthonormal and orthogonal vectors, orthogonal matrix, similar matrices and their properties.
  4. Numerical methods for approximately solving ODE's: IVP definition, single step method, Taylor's theoerm, Euler's method, truncation and global error, Improved Euler's method, Modified (Midpoint) Euler method, fourth order Runge-Kutta method, using the above methods to solve systems of first order and higher order differential equations IVP.
  5. Iterative solutions to algebraic equations: fixed point, scheme requirements for convergence, Newton's method, bisection method, secant method, relaxation.

Module Resources

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