Mathematics
MA378: Numerical Analysis II (5 ECTS)
Polynomial interpolation and its applications in numerical integration, numerical differentiation, splines, and finite element methods for ODEs.
Taught in Semester(s) 2. Examined in Semester(s) 2.
Workload: 100 hours (24 Lecture hours, 10 Tutorial hours, 6 Lab hours, 60 Self study hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- Construct Lagrange and Hermite interpolating polynomials to a function/set of points.
- Bound the error in polynomial interpolation.
- Derive Cauchy's theorem.
- Construct piecewise linear and cubic splines.
- Derive formulas for Newton-Cotes quadrature in low degrees.
- Derive formulas for Gaussian quadrature in low degrees.
- Bound the error in Newton-Cotes and Gaussian quadrature.
- Use the FEM to approximately solve ODEs.
- Derive the system of equations of the FEM solution with piecewise linear basis functions.
Indicative Content
Polynomial interpolation and its applications:
- Lagrange and Hermite interpolation with error formulas. Runge's example.
- Linear and cubic splines (natural and variations).
- Newton-Cotes quadrature with error formulas.
- Orthogonal polynomials.
- Gaussian Quadrature.
- Finite element methods for ODEs.
Module Resources
E. Süli and D. Mayers, "An introduction to Numerical Analysis", Cambridge University Press.
J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis", Springer.
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