By the end of this course, the student can:
- develop numerical algorithms for the solution of nonlinear equations, regression and interpolation tasks, initial boundary value problems,
- select an appropriate numerical method for a given mathematical problem,
- analyse the computational complexity of a numerical algorithm,
- analyse the stability of a mathematical problem and an algorithm,
- analyse the convergence behaviour of a numerical method,
- implement numerical algorithms.
Numerical methods are needed if no closed-form solution to a mathematical problem exists or if solving the problem by hand is infeasible or even impossible. Consider, for instance, the simulation of airflow around a formula one car or the fitting of large data sets. Numerically solving such real-life applications introduces several errors: modelling errors, data errors, truncation errors, and rounding errors. If the latter two errors, i.e., the computational error, can be made arbitrarily small by affording more computations, the method is convergent. In addition to convergence, the accuracy, reliability, and efficiency of a numerical method are particularly important.|
The course starts by discussing these concepts in the context of numerically solving nonlinear equations in one variable, i.e., computing zeros of certain functions. You will learn about the conditioning of mathematical problems, the stability of algorithms for their solution, and the influence of rounding errors, which are unavoidable when storing only finitely many decimals of a number. Next, the numerical solution of square linear systems by Gaussian elimination is discussed with a particular focus on the numerical stability, the computational complexity, and its implementation. When the system of linear equations is overdetermined, which frequently occurs in data fitting tasks, Gaussian elimination cannot be applied, and we investigate the approximate solution of such systems in a least-squares sense. Polynomials are fundamental objects in mathematics that can be conveniently manipulated – one can think of approximating a function by its Taylor polynomial, computing the derivative or the integral of a polynomial. These observations motivate the study of computational aspects of polynomial interpolation of a given function, from which numerical methods for the approximation of integrals and derivatives of a given function are derived and their accuracy and stability will be analysed.
In the final part of this course, the developed methods will be used for the numerical approximation of ordinary differential equations (ODEs). Depending on the analytical properties of the ODE, different numerical methods will be developed and analysed. In summary, this course treats the following topics in numerical mathematics: solution of nonlinear equations, floating-point arithmetic, conditioning of mathematical problems, stability of algorithms, solution of linear systems and least-squares and polynomial interpolation. Furthermore, numerical differentiation, as well as integration are developed and an introduction to the numerical treatment of ODEs is given.
Homework assignments (40%)
Written exam (60%)
Assumed previous knowledge
|Basic Calculus, basic linear algebra, basic programming skills (matlab/python)|
|Bachelor Applied Mathematics||Required materials|
Recommended materials-Instructional modesTests
|David E. Stewart: Numerical Analysis: A Graduate Course. Springer. Edition 1. Softcover ISBN 978-3-031-08123-1
also available freely at springer link: https://link.springer.com/book/10.1007/978-3-031-08121-7|