NumPy with SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of SciPy's integration with NumPy arrays

What is the output of the following code that uses SciPy to integrate a NumPy function over a range?

NumPy

import numpy as np
from scipy.integrate import quad

f = lambda x: np.sin(x)
result, error = quad(f, 0, np.pi)
print(round(result, 5))

A 2.0

B 3.14159

C 0.0

D 1.99999

💡 Hint

Recall the integral of sin(x) from 0 to pi is 2.

The integral of sin(x) from 0 to pi is exactly 2. The quad function returns 2.0, which rounds to 2.0.

❓ data_output

intermediate

Shape of output from SciPy's hierarchical clustering

Given the following code using SciPy's hierarchical clustering on a NumPy array, what is the shape of the linkage matrix returned?

NumPy

import numpy as np
from scipy.cluster.hierarchy import linkage

X = np.array([[1, 2], [3, 4], [5, 6], [7, 8]])
Z = linkage(X, method='single')
print(Z.shape)

A (4, 3)

B (3, 3)

C (4, 4)

D (3, 4)

💡 Hint

The linkage matrix has n-1 rows and 4 columns for n samples.

For n samples, the linkage matrix has n-1 rows and 4 columns. Here n=4, so shape is (3,4).

❓ visualization

advanced

Correct dendrogram plot from hierarchical clustering

Which option shows the correct code to plot a dendrogram from a linkage matrix Z using SciPy and Matplotlib?

NumPy

import numpy as np
from scipy.cluster.hierarchy import linkage, dendrogram
import matplotlib.pyplot as plt

X = np.random.rand(5, 2)
Z = linkage(X, 'ward')

A

plt.dendrogram(Z)
plt.show()

B

plt.plot(dendrogram(Z))
plt.show()

C

dendrogram(Z)
plt.show()

D

dendrogram(X)
plt.show()

💡 Hint

The dendrogram function is from scipy.cluster.hierarchy, not matplotlib.

The dendrogram function is imported from scipy.cluster.hierarchy and called with the linkage matrix Z. Then plt.show() displays the plot.

🧠 Conceptual

advanced

Understanding output of SciPy's fftpack with NumPy arrays

What does the output array represent when applying SciPy's fftpack.fft function to a NumPy array of real numbers?

A The sorted input array in ascending order

B The frequency components of the input signal as complex numbers

C The cumulative sum of the input array elements

D The inverse Fourier transform of the input array

💡 Hint

FFT transforms a signal from time domain to frequency domain.

The fft function computes the discrete Fourier transform, returning complex numbers representing amplitude and phase of frequency components.

🔧 Debug

expert

Identify the error in SciPy optimization with NumPy function

What error will the following code raise when trying to minimize a function using SciPy's optimize.minimize with a NumPy array input?

NumPy

import numpy as np
from scipy.optimize import minimize

def f(x):
    return np.sum(x**2)

x0 = np.array([1, 2, 3])
res = minimize(f, x0, method='BFGS', jac=True)
print(res.fun)

A TypeError: The 'jac' parameter is True but the function does not return a gradient

B ValueError: jacobian function must return array_like

C TypeError: 'bool' object is not callable

D TypeError: f() missing 1 required positional argument

💡 Hint

The jac parameter expects a function or False, not True, unless the function returns gradient.

Setting jac=True tells minimize that the function returns both value and gradient, but f returns only value, causing a TypeError.