Aim:

Design, develop and execute a program in C to evaluate the given polynomial f(x) = a4x4 + a3x3 + a2x2+ a1x + a0 for given value of x and the coefficients using Horner's method.

Summary:

Horner's method of polynomial evaluation is described below

The expression is broken down in the following way

f(x) = a4x4 + a3x3 + a2x2 + a1x + a0

f(x) = x ( a4x3 + a3x2 + a2x + a1 ) + a0

f(x) = x ( x( a4x2 + a3x2 + a2 ) + a1 ) + a0

f(x) = x ( x( x ( a4x + a3 ) + a2 ) + a1 ) + a0

f(x) = x ( x( x ( x ( a4 ) + a3 ) + a2 ) + a1 ) + a0

Example:

Input values:

x=6

a0=5, a1=4, a2=3, a3=2, a4=1

sum = a4 * x -> 1 * 6 -> 6

sum = ( sum + a3 ) * x -> ( 6 + 2 ) * 6 -> 48

sum = ( sum + a2 ) * x -> ( 48 + 3 ) * 6 -> 306

sum = ( sum + a1 ) * x -> ( 306 + 4 ) *6 -> 1860

Algorithm:

1. Start
2. Input the degree and value of x
3. Input Coefficients , upper limit is provided by the value of degree
4. i= degree, sum =0
5. sum = ( sum + ai ) * x
6. decrement i by 1
7. if ( i> 0 ) goto step 5
8. add a0 to the final sum
9. display sum
10. Stop

Program: polynomial.c

#include<stdio.h>
#include<stdlib.h>
#include<math.h>

int main(void)
{
    int deg,i,Arr[10];
    float x,Sum=0;
    printf("\nEnter the degree of the polynomial and value of x\n");
    scanf("%d%f",&deg,&x); //taking input value of degree and x //
    printf("\nEnter the coefficients in descending order of degree\n");
    for(i=0;i<=deg;i++)
    {
        scanf("%d",&Arr[i]);
    }    // taking co-efficient value polynomial //

    for(i=deg;i>0;i--)
    {
        Sum=(Sum + Arr[i])*x;
    }   //evaluating polynomial using Horner's method//
    Sum = Sum + Arr[0]; //adding sum to higher degree co-efficient //
    printf("\nValue of polynomial after evaluation=%g\n",Sum);
    //printing the result//
    return 0;
}

Output:

Run the following commands in your terminal:

gcc -lm polynomial.c

./a.out

1:  Enter the degree of the polynomial and value of x
    5
    2
    Enter the coefficients in descending order of degree
    6 5 4 3 2 1
    Value of polynomial after evaluation=120.000000

gcc -lm polynomial.c

./a.out

2:  Enter the degree of the polynomial and value of x
    4 1
    Enter the coefficients in descending order of degree
    1 2 3 4 5
    Value of polynomial after evaluation=15.000000