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 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 |