# Lecture plan

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
slides
Introduction to MATLAB slides Exercise 1 Euler
35 Numerical solution of nonlinear equations
Existence of solutions
Simple iterations
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.
slides
Exercise 2
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
LU factorization
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.
Exercise 3
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.
6.1-6.3
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
lagrange.m
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
Orthogonal polynomials
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).
Exercise 7

Composite Gauss
The Euler-Maclauring expansion
Romberg integration
Note on orthogonal plynomials and Gauss quadrature
10.5
Theorem 7.4 (no proof required)
7.7.
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.
lmm.m
Project 2
45 No lectures
46 No lectures
47 Leftovers
Stiff ordinary differential equations