Department of Mathematical Sciences

Mathematics

MATHEMATICS 354

Computational Mathematics and Approximation Theory

The approximation of functions by polynomials, trigonometric polynomials and splines play an essential role in natural sciences and enigineering. In this course we investigate the underlying mathematical theory, as well as the resulting numerical algorithms for practical applications. The course material includes:

  • Polynomial interpolation: The Vandermonde matrix, existence and uniqueness, the Lagrange formula, divided differences, Hermite interpolation, error estimates, Chebyshev polynomials and the Chebyshev interpolation points;
  • Bernstein polynomials and the Weierstrass theorem;
  • Existence and uniqueness of best approximations in normed linear spaces and inner product spaces;
  • Characterisation and computation of best uniform polynomial approximations;
  • Caracterisation and computation of best approximation with respect to inner product norms, orthogonal polynomials,  Legendre and Chebyshev series;
  • Interpolating quadrature: Newton-Cotes and Gauss quadrature formulas;
  • Fourier analysis: Convergence of Fourier series, the Bessel inequality, the Parseval identity;
  • Splines: The truncated power basis, B-splines, the Schoenberg-Whitney theorem for splines interpolation, local spline approximation operators.

Module information

  • 21539 354 (16) Mathematics 354
  • Year 3, semester 2 of the Programme in the Mathematical Sciences as well of the PMA (Physics Mathematics Analysis) Programme.
  • Teaching load: Three lectures and one tutorial per week.
  • Prerequisite modules: Mathematics 214 and Mathematics 244, or second year Engineering Mathematics.
  • Language specification : A

Lecturer

  • Prof JM de Villiers:  Office 1020, Bedryfsielkunde Building (e-mail: jmdv@sun.ac.za)

Study materiaal

  • Class notes

Study opportunities

The course material is covered completely during the lectures. During the tutorials there will be opportunity to solve problems. Both during the lectures and the tutorials there will be opportunity for open discussions on the work.

Assessment

The class mark is based on the test mark. The final performance mark is composed from the class mark (weight  0.4) and the examination mark (weight 0.6). To get entrance into the examination, students need a class mark of at least 40%. For students whose test mark is such that their projected class mark is less than 40%, a parachute test will be given, by virtue of which they can attain a class mark of at most 40%. To pass the module, students need a final performance mark of at least 50%.

Rationale

The module is presented within the Programme of Mathematical Sciences, as well as the PMA (Physics Mathematics Analysis) Programme. It provides a mathematical basis for related practice orientated courses at Applied Mathematics, Physics, Computer Science and Engineering.

Outcomes

Students who pass the course will be equiped with the basic mathematical techniques which are needed in practical application areas of Computational Mathematics and Approximation Theory, like geometric modelling, image processing, signal analysis and statistics. A strong focus is placed on the mathematical analysis which is needed for the development of efficient numerical algorithms. The development of abilities with respect to problem solving is strongly emphasized. For students who are interested in the follow-up Mathematical Honours Focus in Computational Mathematics and Approximation Theory, this course provides important background knowledge.