Applied Numerical Analysis

Applied Numerical Analysis (Weeks 1-7, Dr. R.P. Dwight)

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A foundational course in numerical analysis. We assume good knowledge of real analysis and calculus, but no previous experience in numerical analysis. We introduce numerical methods for solving non-linear equations, ODEs, optimization problems and integrals. We analyze these in detail, with regard to consistency, accuracy, stability and convergence, theoretically and in practice.

1. Taylor Series and Solutions of non-linear equations.

- Computer arithmetic and round-off errors.

- Concepts of accuracy, consistency, stability and convergence.

- Uses and limitations of numerical methods.

- Solution methods for non-linear equations:

2. Polynomial interpolation

- Statement of the interpolation conditions.

- Important properties of polynomials, the Weierstrass approximation theorem.

- Choice of basis: monomial, Newton, Lagrange.

- Uniqueness and existence of polynomial interpolants.

- Gibbs phenomena.

3. Advanced interpolation methods and regression

- Interpolation on triangles and rectangles (patches).

- Splines.

- Radial basis functions and volume splines.

- Least-squares regression

4. Numerical differentiation and integration

- Finite difference formulas, truncation error, rounding error.

- Quadrature based on polynomial interpolation.

- Derivation of integration conditions.

- Newton-Cotes and Gauss rules

- 2D integration based on product rules.

5. Numerical methods for solving ordinary differential equations (ODEs)

- Properties and classification of ODEs, standard form of ODE.

- Definition of a consistent ODE solver.

- Linear stability analysis (von Neumann analysis), time-step criteria.

- Convergence analysis.

- Single and multistep methods:

6. Numerical optimization

- Define the unconstrained optimization problem

- Local and global optima, design space, cost function, derivatives.

- Quadratic forms as an example problem.

- Common optimisation methods