This schedule is tentative, changes will appear.
Textbook: Süli and Mayers, An introduction to Numerical Analysis
|34|| Introduction to the course. |
Taylor's theorem, big O -notation, rounding errors
|Introduction to MATLAB||slides||Exercise 1 Euler|
|35|| Numerical solution of nonlinear equations |
Existence of solutions
Rate of convergence
Contraction mapping theorem
|1.1-1.2 (up to Theorem 1.4)|
| Rate of convergence, asymptotic error constant, order of convergence |
Convergence of fixed point iterations.
Secant and bisection method (self study)
| 1.2 (from Theorem 1.4) - 1.8. |
|36|| Scalar equations: Convergence of Newton's method (Theorem 1.8) and max number of fixed point iterations (Theorem 1.4) with proofs. |
Newtons method for system of equations.
| A comment on an example from the lecture. |
Note on systems of equations.
| Fixed point iterations for systems of equations. |
The contraction mapping theorem in max-norm (Theorem 4.1 and 4.2).
Convergence of Newtons method (Theorem 4.4).
|4.1-4.5.||Project 1: (05.09-16.09)|
|37|| Numerical linear algebra |
Naiv Gauss elimination
Gauss-elminiation with partial pivoting.
| 2.1-2.6. |
Note on linear algebra sec. 1 and 3.
| Vector and matrix norms, sub-ordinate matrix norms |
Stability of linear system, Condition number
| 2.6 in S&M |
Section 2 the note on linear algebra.
|38|| Special matrices: Symmetric, positive-definite, diagonally dominant, sparse, tridiagonal, band. |
Cholesky factorization, LU-factorization for tridiagonal matrix.
|3.1-3.3 in S&M |
Section 5 in the note on linear algebra.
|Gershgorin's theorem. |
Iterative methods for linear systems, Jacobi method, Gauss-Seidel method, spectral radius.
|Section 6 and 7 in the note on linear algebra.|| Exercise 4 (corrected 03.10) gs.m rhoSOR.m
Solution task 2: ex4t2.m and sor.m.
|39|| Numerical interpolation |
Lagrange interpolation. Existence and uniqueness of interpolation polynomials. Error formula. All with proofs.
| The max-norm of function spaces.|
Weierstrass approximation theorem.
Minimax polynomials (existence, uniqueness, properties).
Chebyshev poynomials, their properties, why they are useful in the interpolation context.
| 8.1-8.3 (no proofs required.) |
8.4 and 8.5 (with proofs)
| Exercise 5: Set 3
|40|| Divided differences |
(including Newton forward- and backward difference formula).
Hermite interpolation, both with Lagrange polynomials and divided differences.
|Chapter 3.2 and 3.3 in Burden and Faires, "Numerical Analysis", on it's learning.|
| Numerical integration |
How Lagrange interpolation polynomials to construct numerical quadrature.
Degree of accuracy.
Composite formulas. Error formulas for the Trapezoidal rule.
|7.1-7.5 (the proof of error formula for Simpson will be given on Tuesday)||Exercise 6 solution|
|41|| Proof of the error estimate of Simpsons formula. |
|Note on Adaptive Simpsons method with Matlab code|
| Inner product space |
| 9.1-2 and 9.4 (to Theorem 9.2) |
The curriculum is covered by the note on Gauss quadrature and orthoganal polynomials (see below).
|42|| Gauss quadrature. |
The Euler-Maclauring expansion
| Note on orthogonal plynomials and Gauss quadrature |
Theorem 7.4 (no proof required)
| Splines |
Definition of a spline of degree k
Linear and cubic spline
|11.1-4. The note on splines cover the curriculum.||Exercise 8|
|43|| Ordinary differential equations |
Eulers method: \\implementation, convergence proof, how to measure the order of a method.
| Note on numerical solutions of ODEs. |
Most of the lectured material can be found in this note.
Lecture note 1-2.
| Existence and uniqueness of solutions of ODEs, |
Examples of Runge-Kutta methods
General Runge-Kutta methods
|Lecture notes 2-4 (up to 4.1) rk2.m, LotkaVolterra.m, ordertest.m|| Exercise 9:
Problem set 7: Task 1 and 2.
|44|| Order conditions for Runge-Kutta methods |
Error estimates and stepsize selection
Embedded Runge-Kutta methods
|Note, section 4.|
|Linear multistep methods|| Note, section 7.1-7.3. |
|47|| Leftovers |
Stiff ordinary differential equations
Predictor -corrector methods
|Note, section 5 and 7.4-7.5.|| Exercise 10
Solution of problem 2.