Step by Step solution about How to resolve the algorithm Horner's rule for polynomial evaluation step by step in the C programming language

The provided C code is an implementation of Horner's method to evaluate a polynomial. It takes an array of coefficients, the number of coefficients, and a value for the variable and returns the value of the polynomial.

Here's a detailed breakdown of the code:

1. horner Function:

   • This function takes three parameters:
     • coeffs: An array of doubles containing the coefficients of the polynomial, in decreasing order of their powers.
     • s: The number of coefficients in the coeffs array.
     • x: The value of the variable to evaluate the polynomial at.
   • It initializes a variable res to 0.0, which will store the result of the polynomial evaluation.
   • The function enters a loop that iterates through the coefficients in reverse order (from the highest power to the constant term).
   • Inside the loop, it updates the res variable using the Horner's method formula: res = res * x + coeffs[i].
   • Finally, the function returns the value of res, which is the result of evaluating the polynomial at x.

2. main Function:

   • This is the entry point of the program.
   • It defines an array coeffs of doubles, which contains the coefficients of the polynomial -19x³ + 7x² - 4x + 6.
   • It calls the horner function with the coeffs array, its size (sizeof(coeffs)/sizeof(double)), and the value 3.0 for x.
   • The result of the horner function is printed to the standard output with a precision of one decimal place using printf.

When you execute this program, it will output -132.0, which is the value of the polynomial -19x³ + 7x² - 4x + 6 when x is 3.0.

#include <stdio.h>

double horner(double *coeffs, int s, double x)
{
  int i;
  double res = 0.0;
  
  for(i=s-1; i >= 0; i--)
  {
    res = res * x + coeffs[i];
  }
  return res;
}


int main()
{
  double coeffs[] = { -19.0, 7.0, -4.0, 6.0 };
  
  printf("%5.1f\n", horner(coeffs, sizeof(coeffs)/sizeof(double), 3.0));
  return 0;
}