Horner Method

HORNER METHOD:
It is one of the most preferred methods to evaluate a polynomial because of its simple approach and lesser arithmetic operations.
                                                      DESIGN:
We can give the general form of the polynomial equation by:
    a0x^0 + a1x^1 + a2x^2 +......anx^n
In general,given the polynomial of the form,
  a0x^0 + a1x^1 + a2x^2 + .......+ an-2x^n-2 + an-1x^n-1 + anx^n
  We can rearrange this as shown below:
    a0 + x(a1 + x(a2 + x(a3 + x(a4+...............x(an-4 + x(an-3 + x(an-2 + x(an-1 + xan))))....))))
The above representation requires only 'n'  multiplications and 'n' additions there by number of computations can be reduced and efficiency 
can be improved.This method of solving the polynomial is called HORNER'S method of evaluating polynomial.
The equation can be written in reverse as shown below:
((((anx + an-1)x + an-2)x + an-3)x....+ a3)x + a2)x + a1)x + a0
This can be generally written as :
sum + a0
In general,we can write sum =(sum + ai)x for i=n-1,n-2,.....3,2,1
The partial algorithm can be written as:

                        sum = a[n]*x
                        for i= n-1 down to 1
                          sum = (sum + a[i]) * x
                        end for
                        sum = sum + a[0];
                        
The complete c function can be written as:
                         
                         double evaluate(double a[],int n,double x)
                         {
                                double sum;
                                int i;
                                ..................................
                                sum = a[n] * x;
                                ..................................
                                for(i=n-1;i>=1;i--)
                                {
                                      sum = (sum + a[i]) * x;
                                }
                                ..................................
                                sum = sum + a[0];
                                ..................................
                                return sum;
                         }