Write a C program, polynomial.h, containing the following polynomial operation functions:

1- float horner(float p[], int n, float x), which computes and returns the value of the following (n-1)-th degree polynomial p(x) of coefficients p[0], , p[n-1]. p(x) = p[0]x^n-1 + p[1]x^n-2 ++ p[n-2]x¹ + p[n-1]x⁰

It is required to use Horners algorithm (supplementary link).

2- void derivative(float p[], int n, float d[]), which computes the derivative of input (n-1)-th degree polynomial by p[], output the derivative of (n-2)-th degree polynomial to array d[]. The derivative of the above polynomial p(x) is as follows. p(x) = (n-1)* p[0]x^n-2+ (n-2)p[1]x^n-3 ++ p[n-2]x⁰

3- float newton(float p[], int n, float x0), which finds an approximate real root x of polynomial p(x) using the Newtons method with start position x0. Use the fault tolerant 1e-6 (or 0.000001) as a stop condition, i.e., if r is the actual root, stop the iteration if |x-r|<1e-6 or |p(x)| < 1e-6.

Use the provided the main function program polynomial_main.c to test the above functions. The output should be like the following.

Public test

command: gcc polynomial_main.c -o polynomial command: polynomial p(0.00)=1.00*0.00^3+2.00*0.00^2+3.00*0.00^1+4.00*0.00^0=4.00 p(1.00)=1.00*1.00^3+2.00*1.00^2+3.00*1.00^1+4.00*1.00^0=10.00 p(10.00)=1.00*10.00^3+2.00*10.00^2+3.00*10.00^1+4.00*10.00^0=1234.00 d(0.00)=3.00*0.00^2+4.00*0.00^1+3.00*0.00^0=3.00 d(1.00)=3.00*1.00^2+4.00*1.00^1+3.00*1.00^0=10.00 d(10.00)=3.00*10.00^2+4.00*10.00^1+3.00*10.00^0=343.00 root=-1.65 p(-1.65)=0.00

polynomial_main.c:

*/ #include #include #include #include "polynomial.h"

void display_polynomial(float p[], int n, float x) { int i; for (i=0; i EPSILON && i !=0) printf("+"); printf("%.2f*%.2f^%d", p[i], x, n-i-1); } } int main(int argc, char *argv[]) { int n = 4; float p[] = {1, 2, 3, 4}; int m = 3; float x[] = {0,1,10}; // test display and horner functions int i; for (i=0; i

return 0; }

polynomial.h:

/* * your program signature */ #ifndef POLYNOMIAL_H #define POLYNOMIAL_H #include #include #define EPSILON 1e-6 float horner(float p[], int n, float x) { // your implementation } // compute the derivative of polynomial p[], and output to d[] void derivative(float p[], int n, float d[]) { // your implementation } // Use Newton's method to find and return a root of polynomial of p[] float newton(float p[], int n, float x0) { // your implementation } #endif