Assessment test 2022
Contents
Assessment test 2022¶
You may gain maximum of 2 points. To obtain the assessment, it is neccessary to gain at least 1 point.
Fill your solution to this notebook and send it to petr.valenta[at]email.com no later than 13/05/2022 9:00 CEST.
import numpy as np
import scipy.linalg as la
import matplotlib.pyplot as plt
Task 1¶
Fit the following data
\( i \) |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\( x_i \) |
0.00 |
0.45 |
0.90 |
1.35 |
1.79 |
2.24 |
2.69 |
3.14 |
3.59 |
4.04 |
4.49 |
4.93 |
5.38 |
5.83 |
6.28 |
\( y_i \) |
2.74 |
1.70 |
1.57 |
0.04 |
-1.30 |
-0.70 |
-1.36 |
-0.78 |
-0.50 |
-0.31 |
1.63 |
2.74 |
2.78 |
3.77 |
1.39 |
using the least squares method with the model function defined as
where \( \beta = \left[ \beta_0, \, \beta_1, \, \beta_2 \right] \) is the vector of adjustable parameters and \( x_{\mathrm{min}} \) and \( x_{\mathrm{max}} \) are the minimum and maximum values of \( x \), respectively.
[0.75 points] Print the values of \( \beta_0 \), \( \beta_1 \), and \( \beta_2 \).
[0.25 points] Plot the datapoints \( \left( x_i, y_i \right) \) as well as the calculated fitting function \( f \left( x; \beta \right) \) in the interval \( \left[ x_{\mathrm{min}}, \, x_{\mathrm{max}} \right] \).
x_p = np.array([0.00, 0.45, 0.90, 1.35, 1.79, 2.24, 2.69, 3.14, 3.59, 4.04, 4.49, 4.93, 5.38, 5.83, 6.28])
y_p = np.array([2.74, 1.70, 1.57, 0.04, -1.30, -0.70, -1.36, -0.78, -0.50, -0.31, 1.63, 2.74, 2.78, 3.77, 1.39])
def least_squares(x, y):
# add your code here
return lambda x: beta[0] + beta[1] * np.sin((2.0 * np.pi * (x - np.min(x)) / (np.max(x) - np.min(x)))) \
+ beta[2] * np.cos(2.0 * np.pi * (x - np.min(x)) / (np.max(x) - np.min(x)))
Task 2¶
Solve the following ordinary differential equation
with the initial conditions \( y \left( 0 \right) = 2 \) and \( y^{\ \prime} \! \left( 0 \right) = 0 \) in the interval \( x \in \left[ 0, \ 25 \right] \) using the \( 4^{\mathrm{th}} \) order Runge-Kutta method with the finite step of size \( h = 0.1 \).
[0.75 points] Print the values of \( y \left( 25 \right) \) and \( y^{\ \prime} \! \left( 25 \right) \).
[0.25 points] Plot \( y \left( x \right) \), \( y^{\ \prime} \! \left( x \right) \), and \( y^{\ \prime \prime} \! \left( x \right) \) in the interval \( x \in \left[ 0, \ 25 \right] \).
def runge_kutta_4(f, g, x_0, x_n, y_0, z_0, h):
# add your code here
return x, y, z