NumPy Polynomial Operations

NumPy provides a powerful set of tools for performing various mathematical operations on polynomials. These operations include addition, subtraction, multiplication, division, evaluation, differentiation, and integration. In NumPy, polynomials are represented as arrays of their coefficients in decreasing order of powers.

Polynomial Representation

A polynomial like $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ is represented in NumPy as an array: [a_n, a_{n-1}, ..., a_1, a_0].

For instance, the polynomial $3x^2 + 2x + 1$ would be represented as np.array([3, 2, 1]).

Core Polynomial Operations

Adding Polynomials

The numpy.polyadd() function is used to add two polynomials. It returns a new polynomial whose coefficients are the sum of the corresponding coefficients of the input polynomials.

Example: Adding $3x^2 + 2x + 1$ and $4x^2 + x$.

import numpy as np

# Polynomial 1: 3x^2 + 2x + 1
p1 = np.array([3, 2, 1])
# Polynomial 2: 4x^2 + x + 0
p2 = np.array([4, 1, 0])

# Add the polynomials
result_add = np.polyadd(p1, p2)
print("Result of polynomial addition:", result_add)

Output:

Result of polynomial addition: [7 3 1]

This corresponds to the polynomial $7x^2 + 3x + 1$.

Subtracting Polynomials

The numpy.polysub() function subtracts the second polynomial from the first.

Example: Subtracting $4x^2 + x$ from $3x^2 + 2x + 1$.

import numpy as np

p1 = np.array([3, 2, 1])
p2 = np.array([4, 1, 0])

result_sub = np.polysub(p1, p2)
print("Result of polynomial subtraction:", result_sub)

Output:

Result of polynomial subtraction: [-1  1  1]

This corresponds to the polynomial $-x^2 + x + 1$.

Multiplying Polynomials

The numpy.polymul() function multiplies two polynomials.

Example: Multiplying $3x^2 + 2x + 1$ by $4x^2 + x$.

import numpy as np

p1 = np.array([3, 2, 1])
p2 = np.array([4, 1, 0])

result_mul = np.polymul(p1, p2)
print("Result of polynomial multiplication:", result_mul)

Output:

Result of polynomial multiplication: [12  11  6   1   0]

This corresponds to the polynomial $12x^4 + 11x^3 + 6x^2 + x$. Note that the trailing zero coefficient is included if the result has a higher degree.

Evaluating Polynomials

The numpy.polyval() function evaluates a polynomial at a given value of $x$.

Example: Evaluating $3x^2 + 2x + 1$ at $x = 2$.

import numpy as np

p1 = np.array([3, 2, 1])
x_value = 2

result_eval = np.polyval(p1, x_value)
print(f"Polynomial evaluated at x = {x_value}:", result_eval)

Output:

Polynomial evaluated at x = 2: 17

Calculation: $3(2^2) + 2(2) + 1 = 3(4) + 4 + 1 = 12 + 4 + 1 = 17$.

Polynomial Differentiation

The numpy.polyder() function computes the derivative of a polynomial. You can also specify the order of the derivative.

Example: Finding the first derivative of $3x^2 + 2x + 1$.

import numpy as np

p1 = np.array([3, 2, 1])

derivative = np.polyder(p1)
print("Derivative of the polynomial:", derivative)

Output:

Derivative of the polynomial: [6 2]

The derivative of $3x^2 + 2x + 1$ is $6x + 2$.

Example: Finding the second derivative of $x^3 + 2x^2 + 3x$.

import numpy as np

p2 = np.array([1, 2, 3, 0]) # x^3 + 2x^2 + 3x + 0

second_derivative = np.polyder(p2, 2)
print("Second derivative of the polynomial:", second_derivative)

Output:

Second derivative of the polynomial: [6 4]

The first derivative of $x^3 + 2x^2 + 3x$ is $3x^2 + 4x + 3$. The second derivative is $6x + 4$.

Polynomial Integration

The numpy.polyint() function computes the indefinite integral of a polynomial. It can also accept an optional constant of integration.

Example: Finding the integral of $3x^2 + 2x + 1$.

import numpy as np

p1 = np.array([3, 2, 1])

integral = np.polyint(p1)
print("Integral of the polynomial:", integral)

Output:

Integral of the polynomial: [ 1.  2.  1.  0.]

The integral of $3x^2 + 2x + 1$ is $x^3 + x^2 + x + C$. By default, $C=0$.

Example: Finding the integral of $6x + 2$ with a constant of integration $C=5$.

import numpy as np

p_deriv = np.array([6, 2]) # Represents 6x + 2

integral_with_const = np.polyint(p_deriv, k=5)
print("Integral with constant of integration:", integral_with_const)

Output:

Integral with constant of integration: [3. 2. 5.]

The integral of $6x + 2$ is $3x^2 + 2x + C$. With $k=5$, the polynomial is $3x^2 + 2x + 5$.

Summary of Key Functions