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)
|34||Introduction||1.1, 1.2, 3.1||1.1-1.2 is background material, supposed known.|
The bisection algorithm.
|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).
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.
|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.|
| How to derive collocation metods.|
|To check the order of a RK-method: ordcond.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.
|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.|
Jacobi, Gauss-Seidel and SOR.
Necessary and sufficient conditions for convergence (on Monday).
|45||Eigenvalues.||5.1-5.2||The power method, including the inverse and shifted inverse method.|
Deflation, Gershgorin's circle theorem.
|46||Linear multistep methods for ODEs||Lecture6||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|