NumPy - Polynomial Representation

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Polynomial Representation in NumPy

Polynomial representation in NumPy refers to how polynomials are expressed and manipulated using arrays. A polynomial is a mathematical expression involving a sum of powers of a variable, such as ax² + bx + c, where a, b, and c are constants and x is the variable.

In NumPy, polynomials are represented as arrays, where each element corresponds to a coefficient of the polynomial.

For example, the polynomial 3x² + 2x + 1 would be represented as the array [3, 2, 1], where the first element represents the coefficient of x², the second represents the coefficient of x, and the third represents the constant term.

NumPy provides the numpy.polynomial module to create and work with polynomial objects, which allow you to perform operations like evaluation, differentiation, and integration on polynomials.

Creating Polynomials

Polynomials in NumPy can be created using the numpy.polynomial.Polynomial class, which represents polynomials in terms of their coefficients.

Example: Creating a Polynomial

In the following example, we create a polynomial 2x³ + 3x² + x + 5 using NumPy −

import numpy as np
from numpy.polynomial import Polynomial

# Define the coefficients of the polynomial
coefficients = [5, 1, 3, 2]

# Create the polynomial
p = Polynomial(coefficients)

print("Polynomial:", p)

The result shows the created polynomial −

Polynomial: 5.0 + 1.0x + 3.0x + 2.0x

Evaluating Polynomials

Polynomials can be evaluated at specific points using the __call__ method of the Polynomial class or the numpy.polyval() function, which is another method for evaluating polynomials.

Example: Evaluating a Polynomial

In the following example, we evaluate the polynomial 2x³ + 3x² + x + 5 at x = 2 using both the __call__ method and the polyval() function −

import numpy as np
from numpy.polynomial import Polynomial

# Define the polynomial
# 2x^3 + 3x^2 + x + 5
p = Polynomial([5, 1, 3, 2])

# Evaluate the polynomial at x = 2 using the __call__ method
value_call = p(2)

# Evaluate the polynomial at x = 2 using numpy.polyval
value_evaluate = np.polyval(p.coef, 2)

print("Value of the polynomial at x = 2 using __call__:", value_call)
print("Value of the polynomial at x = 2 using numpy.polyval:", value_evaluate)

The result shows the value of the polynomial at x = 2 using both methods −

Value of the polynomial at x = 2 using __call__: 35.0
Value of the polynomial at x = 2 using numpy.polyval: 52.0

Performing Operations on Polynomials

NumPy allows various operations on polynomials, such as addition, subtraction, multiplication, and division.

You can create polynomial objects by instantiating the Polynomial class and passing the coefficients of the polynomial. NumPy automatically handles the necessary calculations on the polynomial coefficients.

Example: Polynomial Addition

In the following example, we add two polynomials 2x² + 3x + 1 and x² + 4x + 2 using the + operator −

import numpy as np
from numpy.polynomial import Polynomial

# Define the polynomials
# 2x^2 + 3x + 1
p1 = Polynomial([1, 3, 2]) 
# x^2 + 4x + 2 
p2 = Polynomial([2, 4, 1])  

# Add the polynomials
result_add = p1 + p2

print("Result of polynomial addition:", result_add)

The result shows the sum of the two polynomials −

Result of polynomial addition: 3.0 + 7.0x + 3.0x

Example: Polynomial Multiplication

Here, we are multiplying two polynomials 2x + 1 and x + 2 using the * operator −

import numpy as np
from numpy.polynomial import Polynomial

# Define the polynomials
# 2x + 1
p1 = Polynomial([1, 2])  
# x + 2
p2 = Polynomial([2, 1])  

# Multiply the polynomials
result_multiply = p1 * p2

print("Result of polynomial multiplication:", result_multiply)

The result shows the product of the two polynomials −

Result of polynomial multiplication: 2.0 + 5.0x + 2.0x

Example: Polynomial Differentiation

In this example, we find the derivative of the polynomial 2x³ + 3x² + x + 5 −

import numpy as np
from numpy.polynomial import Polynomial

# Define the polynomial
# 2x^3 + 3x^2 + x + 5
p = Polynomial([5, 1, 3, 2])  

# Differentiate the polynomial
result_diff = p.deriv()

print("Result of polynomial differentiation:", result_diff)

The result shows the first derivative of the original polynomial −

Result of polynomial differentiation: 1.0 + 6.0x + 6.0x

Fitting Polynomials to Data

Fitting polynomials to data involves finding the polynomial that best matches a given set of data points. In NumPy, this is done using the numpy.fit() function, which fits a polynomial of a specified degree to the data by minimizing the error.

The result is a polynomial that approximates the data, useful for tasks like curve fitting or trend analysis.

Example: Fitting a Polynomial to Data

In the following example, we fit a polynomial of degree 2 to a set of data points −

import numpy as np
from numpy.polynomial import Polynomial

# Define data points
x = np.array([0, 1, 2, 3, 4])
y = np.array([1, 2, 0, 2, 1])

# Fit a polynomial of degree 2 to the data
p_fit = Polynomial.fit(x, y, deg=2)

print("Fitted polynomial:", p_fit)

The result shows the fitted polynomial −

Fitted polynomial: 1.2 - (7.02166694e-17)x + (1.22883961e-16)x