# Forelesninger (Lectures)

Forelesningene vil dekke følgende områder, ikke nødvendigvis i denne rekkefølge:
(The lectures will cover the following topics, not necessarily in this sequence).

• Introduksjon (Introduction)
• Polynominterpolasjon (Polynomial interpolation)
• Numerisk derivasjon og integrasjon (Numerical differentiation and integration)
• Ordinære differensialligninger (Ordinary differential equations)
• Lineære og ikke-lineære ligninger (Linear and nonlinear equations)
• Splines
Week Subject Reading Key concepts Files
34 Introduction 1.1, 1.2, 3.1 1.1-1.2 is background material, supposed known.
The bisection algorithm.
bisection.m
Polynomial interpolation 6.1 (to Cheb. pol.) State the problem. Existence and uniqueness, Lagrange interpolation and the error formula. cardinal.m, lagrange.m
35 Polynomial interpolation (cont). 6.1 Chebyshev polynomials, definition and properties. Use of Chebyshev nodes for interpolation. Theorem 6, 7 and 8 without proofs.
Divided differences 6.2 (minus the Hermite-Genocchi formula)The algorithm for divided differences and Newton interpolation formula.
Newtons forward and backward difference formula (Exercise 2) and the error formula for equidistant grids (Exercise 1).
36 Hermite interpolation.
Started with numerical differentiation
6.3, 7.1 Hermite interpolation with divided differences (General Newton interpolation formula). How to find it and how to use it.
From 7.1: Simple approximations to f'(x). Error analysis by truncated Taylor series.
Numerical differentiation 7.1. (2.1 for those who are interested in rounding errors). Derive approximations by Taylor expansions and Polynomial interpolation. Richardson Extrapolation. Error plot. The effect of rounding errors.
37 Numerical integration The first few pages of 7.2. (the remaining parts will be covered later).
Note on Euler-Maclaurin formula (or 7.7).
Composite Trapezoidal rule with error term. Euler-Maclaurins formula.
Romberg integration 7.4 Romberg algorithm and recursive Trapezoid rule
38 No lectures (project work)
39 Simpson's rule, composite Simpson, adaptive quadrature 7.2, to "Error analysis", p. 486, and 7.5 Be able to derive an adaptive quadrature algorithm. Error estimates.
Gauss quadrature 7.4 and Note on orthogonal polynomials Be able to find orthogonal polynomials for a given inner product, and to derive Gaussian quadrature from them.
Error formula for the Gaussian quadrature.
simpson.m
40 Numerical solution of ordinary differential equations.Lecture1, Lecture2 (corr.) Existence and uniqueness of solutions of ODEs. Lipschitz conditions. Autonomous and nonautonomous systems of ODEs. Convergence of Eulers method.
Runge-Kutta methods.Lecture3, sec. 3. and 4.1. The general definition of a Runge-Kutta method. Some examples. Explicit and implicit schemes. euler.m, vdp.m
41 Runge-Kutta methods (cont.) Lecture4, sec. 4.1.
The B-series tutorial (not part of the curriculum).
How to find order conditions from rooted trees and B-series.
Adaptive RK-methods. Lecture notes, sec. 4.2. Embedded Runge-Kutta pair, error estimation, stepsize selection
42 Stiff equations Lecture notes, sec. 5 Linear test equation, stability function, stability domain, A-stability. stab.m
Collocation methods.
Nonlinear equations
Lecture5 (corr)
K&C 3.2
How to derive collocation metods.
Newtons method
To check the order of a RK-method: ordcond.m
newton.m, newton_ex.m
43 Nonlinear equations. 3.4 + Note on nonlinear equations Fixed point iterations, including the fixed point theorem (and how it can be applied). Newtons method, and steepest descent.
Order of convergence.
steep.m, steep_ex.m
44 Linear equations. 4.2-4.3
(4.1 is supposed known).
Gauss elimination with and without pivoting. Diagonal dominant matrices. LU factorizations. gauss.m, gauss_piv.m
4.4, 4.6 Vector and matrix norms. Condition numbers.
Iterative techniqes,
Jacobi, Gauss-Seidel and SOR.
Necessary and sufficient conditions for convergence (on Monday).
heat.m
45 Eigenvalues. 5.1-5.2 The power method, including the inverse and shifted inverse method.
Deflation, Gershgorin's circle theorem.
gershgorin.m
46 Linear multistep methods for ODEsLecture6 The methods. Local discretization error, consistency and order. Difference equations. Zero-stability. The convergence theorem. lmm.m
Predictor corrector methods. Lecture7 Construction of Adams methods. Predictor-corrector pairs. Milne's device predcorr.m
47 Splines 6.4 The definition of a spline. Cubic spline interpolation. ncs.m, ncs.m