CDS: Numerical Methods Assignments¶

See lecture notes and documentation on Brightspace for Python and Jupyter basics. If you are stuck, try to google or get in touch via Discord.
Solutions must be submitted via the Jupyter Hub.
Make sure you fill in any place that says YOUR CODE HERE or "YOUR ANSWER HERE".

Submission¶

Name all team members in the the cell below
make sure everything runs as expected
restart the kernel (in the menubar, select Kernel$\rightarrow$Restart)
run all cells (in the menubar, select Cell$\rightarrow$Run All)
Check all outputs (Out[*]) for errors and resolve them if necessary
submit your solutions in time (before the deadline)

team_members = ""

Polynomial Interpolation¶

In the following you will construct the interpolating polynomial for the pairs $x_k = [2,3,4,5,6]$ and $y_k = [2,5,5,5,6]$ using the "inversion method" via the Vandermonde matrix, as discussed in the lecture.

In [12]:

import numpy as np
import math as m
import scipy.special
from matplotlib import pyplot as plt

Task 1¶

Set up the Vandermonde matrix and calculate its determinant using $\text{numpy.linalg.det()}$.

In [20]:

xk = np.array([2,3,4,5,6])
yk = np.array([2,5,5,5,6])
def GetVDMMatrix(xk):
    """Function generates a VDM matrix with the same format as the lecture"""
    VDM = np.vander(xk,len(xk))
    return VDM

In [21]:

# print the Vandermonde matrix here in an appropriate format and calculate the determinant
"""VDM matrix is printed and determinant is calculated"""
print(GetVDMMatrix(xk))

detVDM = np.linalg.det(GetVDMMatrix(xk))
print(detVDM)

[[  16    8    4    2    1]
 [  81   27    9    3    1]
 [ 256   64   16    4    1]
 [ 625  125   25    5    1]
 [1296  216   36    6    1]]
288.0000000000136

In [22]:

"""Check that GetVDMMatrix returns the correct output"""
assert np.allclose( GetVDMMatrix([2.0, 4.0]), [[1.0, 2.0], [1.0, 4.0]] )

---------------------------------------------------------------------------
AssertionError                            Traceback (most recent call last)
/tmp/ipykernel_93712/3791331615.py in <module>
      1 """Check that GetVDMMatrix returns the correct output"""
----> 2 assert np.allclose( GetVDMMatrix([2.0, 4.0]), [[1.0, 2.0], [1.0, 4.0]] )

AssertionError:

Task 2¶

Write a function that constructs the interpolating polynomial for $\text{x}$ (a user-defined array of $x$ values) using the Vandermonde matrix from the previous task and $\text{numpy.linalg.inv()}$.

In [59]:

def interpVDM(xk, yk, x):
    """Interpolating polynomial is generated with x values and corresponding y_values (xk and yk respectively).
    Returns an array with y values generated using this polynomial corresponding with given input array x"""
    VDM = GetVDMMatrix(xk)
    Invert = np.linalg.inv(VDM)
    a_values = np.matmul(Invert,yk)
    y = np.zeros(len(x))
    for i in range(len(x)):
        for j in range(len(a_values)):
            y[i] += a_values[len(a_values)-1-j]*(x[i]**j)
    return y

In [60]:

"""Check that interpVDM returns the correct output"""
assert np.allclose( interpVDM([1.0, 3.0], [4.0, 6.0], [2.5]), [5.5] )
assert np.allclose( interpVDM([5.0, 14.0], [12.0, -3.0], [6.5]), [9.5] )

Task 3¶

Plot the interpolating polynomial from $x=2$ to $x=6$ using $x$-step-sizes of $0.01$.

In [64]:

x_values = np.linspace(2,6,400)
def generate_y(x_values):
    """Generates y values to plot by using function interpVDM from previous exercise with corresponding x values"""
    y_values = interpVDM(xk,yk,x_values)
    return y_values
y_values = generate_y(x_values)
    
plt.figure()
plt.xlabel("x")
plt.ylabel("y")
plt.title("Interpolating polynomial")
plt.plot(x_values, y_values)
plt.show()

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