MAI - Matematiska institutionen

Introduction. Basic concepts. Error propagation. Basic arithmetic operations. Catastrophic cancellation.

Floating point systems. Computer arithmetic. Analysis of computational errors.
Non-linear equations. Existance. Fixed point iteration.

Equation solving. Newton-Raphsons method and the Secant method. Error estimate. Order of convergence. Applications: Ray tracing and implementation of the square root.

Basic matrix operations. Linear systems. Gaussian elimination and pivoting. The LU and Cholesky decompositions. Vector norms.

Matrix norms. Sensitivity for linear systems. The condition number. Least squares problems. The QR decomposition. Circle fitting.

The singular value decomposition. Fundamental subspaces. Ill-conditioned and rank deficient linear systems. Low rank approximation. Applications: Projections and computer graphics, Data compression, Automatic character recognition.

Polynomials. Newtons formula. Runge's phenomena. Lagrange interpolation. Spline interpolation.

B-splines. Least squares fits. Parametric curves. Bezier curves. Font design.

Extrapolation. Integration and differentiation. Adaptive methods.