多项式评估的霍纳法

给定形式为c _n x ⁿ + c _n-1 x ^n-1 + c _n-2 x ^n-2 +…+ c ₁ x + c ₀的多项式，并取x的值，求出a的多项式值给定x的值。在此，c _n ，c _n-1 ，…是整数(可以是负数)，而n是正整数。
输入采用数组形式，例如poly [] ，其中poly [0]代表x ^n的系数，poly [1]代表x ^n-1的系数，依此类推。
例子：

// 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

一种简单的评估多项式的方法是一个一个地评估所有项。首先计算x ⁿ ，将值乘以c _n ，对其他项重复相同的步骤并返回总和。如果我们使用一个简单的循环来评估x ⁿ ，则该方法的时间复杂度为O(n ² )。如果我们使用O(Logn)方法评估x ^n，则时间复杂度可以提高到O(nLogn)。
Horner方法可用于评估O(n)时间中的多项式。为了理解该方法，让我们考虑2x ³ – 6x ² + 2x – 1的示例。多项式可以评估为((2x – 6)x + 2)x – 1。 x ⁿ在这种情况下为2，将结果重复乘以x，然后将下一个系数相加。最终返回结果。
以下是霍纳法的实现。

C++

#include 
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

---

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

---

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.

Javascript

// 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.

输出：

Value of polynomial is 5