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The aims of the course and expected learning outcomes are that at the end of the course the students should be able to:

• understand the basic ideas of mathematical modelling and fundamentals of numerical computation.
• understand the mathematical ideas behind these methods in order to be able to distinguish their advantages and disadvantages for different classes of problems.
• be able to formulate a solution strategy for manual or computer implementation for common types of problems in Engineering and Physics and be able to design and write basic computer programs.
• be able to choose adequate available software for the solution of these problems.

The course starts with basic details of numerical calculations: machine representation of numbers, error sources and estimation of error bounds in algorithms. It continues with the basic techniques to solve nonlinear equations (root finding methods). Different techniques for Interpolation of discrete data and approximation by functions are covered next, including, polynomial interpolations; least squares methods for discrete data and functions; Taylor expansions and Padé approximants. Discrete approximation of derivatives are then described, setting the starting point to introduce finite differences methods for differential equations, which are covered later. Basic numerical integration methods, quadrature methods and integration methods for initial value problems, Euler, Taylor and Runge-Kutta methods are covered next. The chapter on matrix computations methods include the direct and iterative solutions of linear systems, Gauss elimination, Jacobi and Gauss-Seidel, steepest descent, conjugate gradients and other Krylov subspace methods. Preconditioning techniques are also introduced. Eigenvalue problems are solved with iterative methods, power method and the inverse, shifted inverse and Rayleigh iteration methods. The weak solution of boundary value problems is then described using the Rayleigh-Ritz approach and the weighted residual and variational methods are developed presenting finally the finite element method as an implementation of these.

Emphasis is made on the mathematical ideas and strategies underpinning the methods covered in the course rather than on the routine application of these techniques.

This course is suitable to mathematically-oriented students that would like to pursue a career including a substantial component of mathematical and computer modelling. The students are expected to have a solid understanding of algebra, linear algebra (vectors and matrices), calculus and differential equations.