Python program to Compute a Polynomial Equation

The following article contains programs to compute a polynomial equation given that the coefficients of the polynomial are stored in a List.

Examples:

# Evaluate value of 2x3 - 6x2 + 2x - 1 for x = 3
Input: poly[] = {2, -6, 2, -1}, x = 3
Output: 5

# Evaluate value of 2x3 + 3x + 1 for x = 2
Input: poly[] = {2, 0, 3, 1}, x = 2
Output: 23

# Evaluate value of 2x + 5 for x = 5
Input: poly[] = {2, 5}, x = 5
Output: 15

The equation will be of type:

We will be provided with the value of variable and we have to compute the value of the polynomial at that point. To do so we have two approaches.

Approach

• Naive Method: Using for loop to compute the value.
• Optimised Method: Using Horner’s Method for computing the value.

Naive method:

In this approach, the following methodology will be followed. This is the most naive approach to do such questions.

• First coefficient c_n will be multiplied with xⁿ
• Then coefficient c_n-1 will be multiplied with x^n-1
• The results produced in the above two steps will be added
• This will go on till all the coefficient are covered.

Example:

Python3

# 2x3 - 6x2 + 2x - 1 for x = 3
poly = [2, -6, 2, -1]
x = 3
n = len(poly)

# Declaring the result
result = 0

# Running a for loop to traverse through the list
for i in range(n):

    # Declaring the variable Sum
    Sum = poly[i]

    # Running a for loop to multiply x (n-i-1)
    # times to the current coefficient
    for j in range(n - i - 1):
        Sum = Sum * x

    # Adding the sum to the result
    result = result + Sum

# Printing the result
print(result)

Output:

5

Time Complexity: O(n²)

Optimized method:

Horner’s method can be used to evaluate polynomial in O(n) time. To understand the method, let us consider the example of 2x³ – 6x² + 2x – 1. The polynomial can be evaluated as ((2x – 6)x + 2)x – 1. The idea is to initialize result as the coefficient of x_n which is 2 in this case, repeatedly multiply the result with x and add the next coefficient to result. Finally, return the result. 

Python3

# + poly[1]x(n-2) + .. + poly[n-1]
def horner(poly, n, x):

    # Initialize result
    result = poly[0]

    # Evaluate value of polynomial
    # using Horner's method
    for i in range(1, n):

        result = result*x + poly[i]

    return result

# Driver program to
# test above function.


# Let us evaluate value of
# 2x3 - 6x2 + 2x - 1 for x = 3
poly = [2, -6, 2, -1]
x = 3
n = len(poly)

print("Value of polynomial is:", horner(poly, n, x))

Output:

Value of polynomial is: 5

Time Complexity: O(n)