The problem:

a) Consider the function f(x,y: 3) = (x - ₁)² + (y - 3₂)² - Bwith arguments

(x, y) = R² and parameters BER³. Given 3, the level set of all points (x, y) such

that f = 0 describes the circle centered at (81, 82) and with radius [33].

We are given the following list of 20 noisy measurements of points on a circle

(and the measurements be downloaded as a mat file from this link)

y

-2.3073 -3.5569

-1.6627 -4.5479

4.4413 -0.1823

-2.1021 -0.7938

-2.0460 -0.3176

5.1864 -3.9465

-2.6359 -4.3716

-1.0931

1.9392

4.7061 -3.2146

4.9005 -2.3397

0.5035

0.9895

0.5666 -5.9482

3.3504 -4.7778

1.4214

0.7869

1.7892

1.2574

-1.1229 -4.4437

0.3167 -6.1661

0.6024

4.7246

4.9007

0.7662

-1.2589

-3.6789

and we seek to find the circle 3 such that

f(MB) = 0

(5)

holds in least squares sense. Describe the function R(3) for the nonlinear least

squares problem for determining this circle and compute its Jacobian JR(3) €

R20x3

i = 1,..., 20

Extracting data points from file: For part b) and c) of this exercise. If you

prefer to download the "circle-measurements.mat" from the aforementioned link

rather than copying the above measurements, then, assuming you store the down-

loaded file in the same folder as you run Matlab or Pyton from, you can open it

and extract its information by the command load('circle-measurements.mat')

in Matlab, and by the following commands in Python:

import scipy.10

measurement Data = scipy.io. loadnat ('circle-measurements. mat ³)

x = measurement Data ['x']

y measurement Data [¹y¹]