8.1.1 Arrays and mathematical operations

One of the primary strengths of NumPy is its ability to handle arrays and perform a wide range of mathematical operations efficiently. NumPy arrays allow you to perform element-wise operations, matrix operations, and mathematical computations without the need for explicit loops, which improves both performance and readability.

1. Creating NumPy Arrays

Before diving into mathematical operations, let's first look at how to create NumPy arrays.

import numpy as np

# Creating 1D array
arr1 = np.array([1, 2, 3, 4])

# Creating 2D array (Matrix)
arr2 = np.array([[1, 2], [3, 4]])

# Creating 3D array
arr3 = np.array([[[1, 2], [3, 4]], [[5, 6], [7, 8]]])

print(arr1)
print(arr2)
print(arr3)

2. Element-wise Mathematical Operations

NumPy allows you to apply mathematical operations element-wise. This means the operation is performed for each element in the array individually.

a. Addition

arr1 = np.array([1, 2, 3])
arr2 = np.array([4, 5, 6])

# Element-wise addition
result = arr1 + arr2
print(result)  # Output: [5 7 9]

b. Subtraction

result = arr1 - arr2
print(result)  # Output: [-3 -3 -3]

c. Multiplication

result = arr1 * arr2
print(result)  # Output: [4 10 18]

d. Division

result = arr2 / arr1
print(result)  # Output: [4.  2.5 2.]

e. Exponentiation

result = arr1 ** 2
print(result)  # Output: [1 4 9]

3. Mathematical Functions in NumPy

NumPy provides a rich set of mathematical functions to operate on arrays. These functions are optimized for performance.

a. Sum, Mean, and Standard Deviation

arr = np.array([1, 2, 3, 4, 5])

# Sum of elements
print(np.sum(arr))  # Output: 15

# Mean of elements
print(np.mean(arr))  # Output: 3.0

# Standard deviation of elements
print(np.std(arr))  # Output: 1.4142135623730951

b. Square Root, Logarithm

arr = np.array([4, 9, 16])

# Square root of each element
print(np.sqrt(arr))  # Output: [2. 3. 4.]

# Logarithm of each element
print(np.log(arr))  # Output: [1.38629436 2.19722458 2.77258872]

c. Trigonometric Functions

NumPy also supports trigonometric operations.

arr = np.array([0, np.pi/2, np.pi])

# Sine of each element
print(np.sin(arr))  # Output: [0. 1. 0.]

# Cosine of each element
print(np.cos(arr))  # Output: [1. 0. -1.]

4. Matrix Operations

NumPy also provides support for matrix operations that are widely used in fields such as linear algebra and machine learning.

a. Dot Product

The dot product of two arrays can be computed using np.dot(). It is commonly used for matrix multiplication.

arr1 = np.array([1, 2, 3])
arr2 = np.array([4, 5, 6])

result = np.dot(arr1, arr2)
print(result)  # Output: 32 (1*4 + 2*5 + 3*6)

b. Matrix Multiplication

For matrix multiplication, NumPy offers the @ operator (also known as the matrix multiplication operator) or np.matmul().

arr1 = np.array([[1, 2], [3, 4]])
arr2 = np.array([[5, 6], [7, 8]])

result = arr1 @ arr2
print(result)
# Output:
# [[19 22]
#  [43 50]]

c. Transpose

You can transpose a matrix using the .T attribute.

arr = np.array([[1, 2], [3, 4]])

# Transposing the matrix
result = arr.T
print(result)
# Output:
# [[1 3]
#  [2 4]]

5. Broadcasting in NumPy

Broadcasting is a powerful feature in NumPy that allows you to perform operations on arrays of different shapes. The smaller array is "broadcast" over the larger array, which allows for element-wise operations between arrays of different sizes.

arr1 = np.array([1, 2, 3])
arr2 = np.array([[1], [2]])

# Broadcasting arr2 to match the shape of arr1
result = arr1 + arr2
print(result)
# Output:
# [[2 3 4]
#  [3 4 5]]

6. Vectorized Operations (No Loops)

In standard Python, you would use loops to perform operations on lists or arrays. However, NumPy allows for vectorized operations, meaning the operation can be applied to the entire array without the need for explicit looping, which is both faster and more readable.

arr = np.array([1, 2, 3, 4])

# Multiply each element by 2 (vectorized operation)
result = arr * 2
print(result)  # Output: [2 4 6 8]

7. Conclusion

NumPy’s powerful array manipulation and mathematical functions allow for fast and efficient handling of numerical data. By leveraging vectorization and broadcasting, you can perform complex mathematical and matrix operations with ease. Whether you are working with small datasets or large-scale scientific computations, NumPy provides a flexible and optimized framework for all your mathematical needs in Python.