## Transcribed Text

Exercise 1
100%
The goal of this project is to implement the fixed point iteration
in matlab. Here x7 € R" is a column vector, DF(x) is the Jacobian matrix evaluated at PER",
i.e.
The interface to your rostine should be of the form
function x FICAMD TOL)
where
x is a columan vector of leagth n. On the imput ==10 and on the exit x 2' where I is the
first index such that |F(x¹) TOL or I = 2N1 it the tolerance has mot been met in less
than NI iterations.
Fisa function which evaluates F(x), i.e.
function y - TIVAL(x)
FP is a function which evaluates FP(x) : DF(x), i.e.
function re
A few remarks are in order:
1. The function routines F, FP must be consistant with the length of - in the call PICARD,
i.e. if length(x) = n then the argament to F and FP is a column vector of length
n
a
produce, respectively. a columan vector of length n and a n x n matrix.
2. The routine PICARD should generate the subsequent iterations in the same vector - (not
in an array of vectors). At most, you should need 2 other vectors of length n in its imple-
mentation.
3. At each step of the iteration, print (use fprintf) j x, F(x), norm(F(x)), where i is the
iteration number (and x:==7). Stop the iteration when normaF(x)) TOL and report the
values of j. r, F(x), norm(F(x)) when the tolerance is achieved
Use your implementation on the following two problema.
1. Compute the roots of is = 1 by writing this ass a 2 x2 system involving the retal and imaginary
parts (see lecture motes). This gives a 2x 2 system of the form
F
(:))((ARB)-()). = =
Write the corresponding routines for F and FP. Rum the PICARD iteration with starting
iterates
using the tolerance TOL = 10-8.
2. Repent the above computations for f(x) = 24+22²+1

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Q1

clear; clc; close all;

TOL = 1e-6;

x0 = [1 1 -1 -2 2;

1 -1 -1 0 0];

NI = 1000;

for i=1:length(x0(1,:))

fprintf("\n================================================================\n");

disp(['(',num2str(i),'.)Executing with initial guess ',num2str(x0(1,i)),'+',num2str(x0(2,i)),'i']);

x = PICARD(x0, NI, @Q1f, @Jacobian, TOL);

end

fprintf("\n================================================================\n");...