Solve polynomial equation in python

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

Example:

Python3

poly = [2, -6, 2, -1]

x = 3

n = len(poly)

result = 0

for i in range(n):

    Sum = poly[i]

    for j in range(n - i - 1):

        Sum = Sum * x

    result = result + Sum

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 xn which is 2 in this case, repeatedly multiply the result with x and add the next coefficient to result. Finally, return the result. 

Python3

def horner(poly, n, x):

    result = poly[0]

    for i in range(1, n):

        result = result*x + poly[i]

    return result

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)