Horner’s Method for Polynomial Evaluation

Given a polynomial of the form c_nxⁿ + c_n-1x^n-1 + c_n-2x^n-2 + … + c₁x + c₀ and a value of x, find the value of polynomial for a given value of x. Here c_n, c_n-1, .. are integers (may be negative) and n is a positive integer.
Input is in the form of an array say poly[] where poly[0] represents coefficient for xⁿ and poly[1] represents coefficient for x^n-1 and so on.
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

A naive way to evaluate a polynomial is to one by one evaluate all terms. First calculate xⁿ, multiply the value with c_n, repeat the same steps for other terms and return the sum. Time complexity of this approach is O(n²) if we use a simple loop for evaluation of xⁿ. Time complexity can be improved to O(nLogn) if we use O(Logn) approach for evaluation of xⁿ.
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 coefficient of xⁿ which is 2 in this case, repeatedly multiply result with x and add next coefficient to result. Finally return result.
Following is implementation of Horner’s Method. 

C++

#include <iostream>
using namespace std;

// returns value of poly[0]x(n-1) + poly[1]x(n-2) + .. + poly[n-1]
int horner(int poly[], int n, int x)
{
    int result = poly[0]; // Initialize result

    // Evaluate value of polynomial using Horner's method
    for (int i=1; i<n; i++)
        result = result*x + poly[i];

    return result;
}

// Driver program to test above function.
int main()
{
    // Let us evaluate value of 2x3 - 6x2 + 2x - 1 for x = 3
    int poly[] = {2, -6, 2, -1};
    int x = 3;
    int n = sizeof(poly)/sizeof(poly[0]);
    cout << "Value of polynomial is " << horner(poly, n, x);
    return 0;
}

Java

// Java program for implementation of Horner Method
// for Polynomial Evaluation
import java.io.*;

class HornerPolynomial
{
    // Function that returns value of poly[0]x(n-1) +
    // poly[1]x(n-2) + .. + poly[n-1]
    static int horner(int poly[], int n, int x)
    {
        // Initialize result
        int result = poly[0];

        // Evaluate value of polynomial using Horner's method
        for (int i=1; i<n; i++)
            result = result*x + poly[i];

        return result;
    }

    // Driver program
    public static void main (String[] args)
    {
        // Let us evaluate value of 2x3 - 6x2 + 2x - 1 for x = 3
        int[] poly = {2, -6, 2, -1};
        int x = 3;
        int n = poly.length;
        System.out.println("Value of polynomial is "
                                        + horner(poly,n,x));
    }
}

// Contributed by Pramod Kumar

Python3

# Python program for
# implementation of Horner Method
# for Polynomial Evaluation

# returns value of poly[0]x(n-1)
# + 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))

# This code is contributed
# by Anant Agarwal.

C#

// C# program for implementation of
// Horner Method  for Polynomial Evaluation.
using System;

class GFG
{
    // Function that returns value of poly[0]x(n-1) +
    // poly[1]x(n-2) + .. + poly[n-1]
    static int horner(int []poly, int n, int x)
    {
        // Initialize result
        int result = poly[0];

        // Evaluate value of polynomial
        // using Horner's method
        for (int i = 1; i < n; i++)
            result = result * x + poly[i];

        return result;
    }

    // Driver Code
    public static void Main()
    {
        // Let us evaluate value of
        // 2x3 - 6x2 + 2x - 1 for x = 3
        int []poly = {2, -6, 2, -1};
        int x = 3;
        int n = poly.Length;
        Console.Write("Value of polynomial is "
                            + horner(poly,n,x));
    }
}

// This code Contributed by nitin mittal.

PHP

<?php
// PHP program for implementation
// of Horner Method for Polynomial
// Evaluation.

// returns value of poly[0]x(n-1) +
// poly[1]x(n-2) + .. + poly[n-1]
function horner($poly, $n, $x)
{
    // Initialize result
    $result = $poly[0];

    // Evaluate value of polynomial
    // using Horner's method
    for ($i = 1; $i < $n; $i++)
        $result = $result *
                  $x + $poly[$i];

    return $result;
}

// Driver Code

// Let us evaluate value of
// 2x3 - 6x2 + 2x - 1 for x = 3
$poly = array(2, -6, 2, -1);
$x = 3;
$n = sizeof($poly) / sizeof($poly[0]);

echo "Value of polynomial is ".
         horner($poly, $n, $x);

// This code is contributed by mits.
?>

Javascript

<script>
// Javascript program for implementation
// of Horner Method for Polynomial
// Evaluation.

// returns value of poly[0]x(n-1) +
// poly[1]x(n-2) + .. + poly[n-1]
function horner(poly, n, x)
{

    // Initialize result
    let result = poly[0];

    // Evaluate value of polynomial
    // using Horner's method
    for (let i = 1; i < n; i++)
        result = result *
                  x + poly[i];

    return result;
}

// Driver Code

// Let us evaluate value of
// 2x3 - 6x2 + 2x - 1 for x = 3
let poly = new Array(2, -6, 2, -1);
let x = 3;
let n = poly.length

document.write("Value of polynomial is " +
         horner(poly, n, x));

// This code is contributed by _saurabh_jaiswal.
</script>

Output:

Value of polynomial is 5

Time Complexity: O(n)

Auxiliary Space: O(1)
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