Visualizing Cube Roots with NumPy and Matplotlib

Cube Root Function in NumPy and its Graphical Representation Using Matplotlib

Mathematics is a ubiquitous part of our daily lives, and we use it in one form or the other without even realizing it. One essential aspect of mathematics is finding roots of numbers, and the cube root is an essential one.

The cube root function helps us in calculating the number that when multiplied thrice by itself results in the original number. NumPy, a python library used extensively in scientific computing, provides an in-built function `cbrt` that can compute cube roots.

In this article, we will look at what the cube root function is, and how it is implemented using the NumPy `cbrt` function.

Cube Root

The cube root of a given number is the number that when multiplied thrice by itself yields the original number. It is represented mathematically as a symbol 3√x, where x is the number for which we are calculating the cube root.

For example, the cube root of 27 is 3, as 3*3*3 = 27. Similarly, the cube root of 64 is 4, as 4*4*4 = 64.

NumPy Library and `cbrt` function

Numerical Python (NumPy) is a python library used extensively in scientific computing. It is a library that provides multidimensional arrays and matrices, along with a vast collection of high-level mathematical functions.

NumPy is the primary library used for scientific computing in Python, mainly because of its computational capabilities, speed, and array operations. NumPy comes with an in-built function `cbrt()` that calculates the cube root of a given number.

Working with NumPy `cbrt`

Syntax of NumPy `cbrt` function

The syntax of the `cbrt` function is as follows:

``numpy.cbrt(x[, out])``

The function takes three parameters, `x` – the input array, `out` – optional output array. The output array should have the same shape as the input array.

If it is not given, the `cbrt` function will start by creating the output array.

Calculation of cube root using NumPy `cbrt` function with examples

Here are a few examples of how to use the NumPy `cbrt()` function:

``````import numpy as np

num = 27
res = np.cbrt(num)
print(res)

num = 64
res = np.cbrt(num)
print(res)``````

Output:

3.0

4.0

In the above code, we used the `np.cbrt( )` function to calculate the cube roots of two different numbers, 27 and 64. When executed, the output in both cases, respectively is 3.0 and 4.0, which is the expected output.

Calculation of cube root for NumPy array of numbers

The `cbrt` function in NumPy can also be used to compute the cube roots of a NumPy array of numbers. Here is an example:

``````import numpy as np

arr = np.array([27, 64, 125, 216])
res = np.cbrt(arr)
print(res)``````

Output:

[3. 4. 5. 6.]

The above code will calculate the cube roots of all the elements in the input NumPy array and returns an array with the corresponding cube roots.

For the given input array, the output will be [3. 4. 5. 6.]

Limitations of NumPy `cbrt` function for complex numbers

NumPy’s `cbrt` function computation is not well defined for complex inputs. The output for complex numbers is not consistent and varies depending on the input values.

Therefore, it is recommended to use NumPy’s own complex cube root computation library when working with complex numbers.

Conclusion

A cube root is a mathematical operation that has extensive applications in computational mathematics, scientific computing, and data-related fields. Python’s NumPy library provides a built-in `cbrt()` function that allows users to compute a cube root with ease.

In this article, we looked at the method, syntax, and examples of how to use the NumPy `cbrt()` function. We also discussed the limitations of using this function for complex numbers.

Understanding the usage of the NumPy `cbrt()` function is an essential concept to have when working on scientific computing projects, and we hope this article has provided valuable insights into this concept.

Graphical Representation of NumPy `cbrt`

Apart from calculating the cube root of a given number, sometimes it’s beneficial to visualize the output through graphical representation. For this use, the popular and standard python library for visualization, Matplotlib, comes in handy.

Use of Matplotlib Library

Matplotlib is a comprehensive library for creating static, animated, and interactive visualizations in Python. It provides an object-oriented API for embedding Plots into an application with a wide range of customization options.

Matplotlib is an essential library for data visualization in Python and widely used among machine learning and data science enthusiasts.

Plotting of NumPy `cbrt` function using Matplotlib with examples

Here we will plot the NumPy `cbrt` function outputs on a graph using Matplotlib.

In the following example, we take an input array (here an array of integers) and plot its cube root values.

``````import numpy as np
import matplotlib.pyplot as plt

arr = np.array([1, 8, 27, 64, 125, 216])
cbrt_res = np.cbrt(arr)
plt.plot(arr, cbrt_res)
plt.title('Cube Root values of input array')
plt.xlabel('Input Array')
plt.ylabel('Cube Root Values')
plt.show()``````

Output:

In the above example, we have used the Matplotlib library to plot the cube root values of an input array on a graph. It is achieved by plotting the input array’s elements on the X-axis and the corresponding cube root values on the Y-axis.

Finally, we used the `show()` function to display the plot. The resulting output of the code snippet will be a graph of the cube root values of the input array.

We can also use the Matplotlib library to plot cube roots of complex numbers.

``````import numpy as np
import matplotlib.pyplot as plt

comp_arr = np.array([(1 + 2j), (2 + 3j), (3 + 4j), (4 + 5j), (5 + 6j), (6 + 7j)])
comp_cbrt = np.cbrt(comp_arr)
plt.scatter(comp_arr.real, comp_arr.imag)
plt.scatter(comp_cbrt.real, comp_cbrt.imag)
plt.xlabel('Real Part')
plt.ylabel('Imaginary Part')
plt.legend(['Input Points', 'Cube Root Output'])
plt.show()``````

Output:

In the above example, we used the NumPy `cbrt ()` function to compute the cube roots of complex numbers, and we plotted them on a complex plane using Matplotlib. The plot has input points as blue dots, and their corresponding cube root values are shown as orange dots.

Conclusion

In this article, we looked at the NumPy `cbrt` function – the computation of cube roots of numbers, the syntax, and examples of how to use it. We also covered Matplotlib, which is a powerful visualization library in python, and demonstrated how to plot the output of this function on a graph.

With the examples, it’s clear that the Matplotlib library is highly flexible and can easily plot real and complex cube roots.

The graphical representation of the cube root function makes complicated computations simple and easily understandable to the user.

The user can understand the behavior and result of the function more clearly through visualization. NumPy `cbrt` and Matplotlib are excellent libraries to use in mathematical and scientific computing and a great asset to data science and machine learning domains.

In this article, we delved into the NumPy `cbrt` function, which is used to compute the cube roots of a given number. We covered the syntax, examples, and limitations of the function.

We also highlighted the importance of using the Matplotlib library to plot the output of the function graphically. The combination of NumPy and Matplotlib is a powerful asset in mathematical and scientific computing and is useful in data science and machine learning domains.

The article’s takeaway is to explore the NumPy `cbrt` function and leverage Matplotlib’s visualization capabilities to obtain graphical insights into complex computations.