# Theory

Goal:
Understand Lipschitz continuity, the Bisection algoritm, the concepts sequence, Cauchy sequence, convergence and real number and the intermediate value theorem.

AMB&S chap. 12 - 16

Exercises:
13.1ac, 13.2, 13.3acd, 13.4ac, 13.5, 13.9, 13.10, 13.11, 13.14, 13.15, 13.16acfg, 13.19,
14.2a, 14.6ac, 14.7,
15.2, 15.3, (15.4,) 15.5, 15.6, 15.11, (15.13,) 15.14, 15.15, 15.17, 15.18, 15.19, 15.23, 15.24,
12.1, 12.3, 12.4, 12.5, 12.7, 12.9, 12.10, 12.11, 12.12, 12.14, 12.16, 12.17, 12.18, 12.19,
13.7, 13.8, 13.16bde, 13.17ac,

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

# Matlab

Goal:
A matlab function implementation of the Bisection algorithm (part of project work) for the solution of an arbitrary algebraic equation f(x)=0, with input arguments f, int and tol, where f is the name of a matlab function file defining y=f(x), int is a given interval containing a root, and tol is a given tolerance for the length of the final interval containing a root, and with output argument int, where int is the final interval containing the root.

Recall from Matlab intro II how to use while loops and conditional code (if-elseif-else-end), and from the first ALA Matlab session about how to write functions in Matlab. Also consult your Matlab Handbook on these issues.

Excercises:
13.21,

# Activity plan:

L=Lecture, S=Studio, G=Group work
L1: The Bisection algorithm, sequences, Cauchy sequence, convergence, real number (AMB&S ch.13 - 15),
S1: Experiment in the Bisection lab, finish your intro project if not done yet. Continue with S2 program if time permits.
G1: Work on problems from the book.
S2: Complete the assignment under S1 by making your own implementation of the Bisection algorithm starting from the bisection code shell.
L2: More on convergence, Lipschitz continuity, and the question if the equation really got solved, Bolzano's theorem and The Intermediate Value theorem.
L3: Convergence of fixed point iteration.
S3: Presentation of intro projects. Experiment in the Fixed Point lab, implement a Fixed Point solver according to fixed point iteration code specification/documentation