# Theory

Goal:
Understand the definition of x^r, and the concepts fixed point, fixed point iteration, and contraction mapping. Understand the the idea of coupled systems of equations.
Understand elementary vector algebra, including scaling, addition, scalar multiplication and projection of vectors.

AMB&S ch. 20 (and also ch. 15-19)

Exercises:
18.1, 18.3abd but simplify by considering 1/(2+x^2) instead of 1/(1+x^2),
19.1, 19.2a, 19.3b, 19.4, 19.10, 19.11, 19.12, 19.13, 19.14, 19.16, (19.17,)
19.18 but simplify by considering g(x)=x^2/(10-x) with L=21/81 on [-1 1] instead of g(x)=x^4/(10-x)^2,
(19.19, 19.21,) 19.22ab

Project:
Project f(x)=0 , including modeling, solver implementation, application to model equations, analysis and conclusions, presentation.

# Matlab

Goal:
Make sure you have functioning and fully understood bisection and fixed point solvers (part of project f(x)=0). Seek to extend your fixed point solver so as to be applicable also to systems of equations. Recall how vectors (lists) and matrices are created and manipulated in matlab.

Excercises:
19.5 (part of project), some of application problems 19.6, 19.7, 19.8, 19.9 (possible application parts of project).

# Activity plan:

L=Lecture, S=Studio, G=Group work
L1: vectors, vector algerbra, scalar product, projection, rotation,
S1: Finish your bisection and fixed point solvers. Make sure they work, and that you fully undestand how!
Consider example systems of equations of the form f1(x1,x2)=0, f2(x1,x2)=0 and g1(x1,x2)=x1, g2(x1,x2)=x2, for example x1=(x2-1)/2, x2=x1^2. Plot the two curves in a x1-x2 coordinate system and seek to identify the solutions graphically. Seek to extend your fixed point solver to systems of this form and seek the solutions usin fixed point iteration!
G1: Finish the extra problems handed out last week. Continue with problem solving from the book.
L2: linear dependence/independence, basis and change of basis, systems of (two linear) equations, the vector product, matrix, inverse matrix.
S2: Experiment with the Vector Calculator and in the Euclids plane lab.
Get comfortable with matlabs syntax for creating and manipulating vectors and matrices, and methods for solving linear systems of equations.
G2: Continue working on the problems in the book.

/Nils