Horner Rule)

Horner Rule is a common Rule, but there are not many related information on the Internet. I mainly found two.
1. http://blog.csdn.net/lwj1396/archive/2008/07/18/2669993.aspx

2. http://flynoi.blog.hexun.com/31272178_d.html
In the second article, the reason for the emergence of the Horna law is described: it provides an efficient method for polynomial evaluation.
"The most common algorithm for polynomial evaluation is to find the value of each item and accumulate the value. This algorithm is inefficient at both time and space, because the data size is not large, it is intuitive and simple and easy to use. However, once the data size is too large, this algorithm is powerless"

This is a more detailed concept of the Horna law:
Suppose there are n + 2 real numbers a0, a1 ,..., An, and x sequence, to polynomial Pn (x) = anxn + an-1xn-1 +... + A1x + a0 evaluate the value. The direct method is to evaluate each item separately and accumulate the value of each item. This method is very inefficient and requires n + (n-1) +... + 1 = n (n + 1)/2 multiplication and n addition operations. Is there any more efficient algorithm? The answer is yes. Through the following transformation we can get a much faster algorithm, that is, Pn (x) = anxn + an-1xn-1 +... + A1x + a0 = ((... (Anx + an-1) x + an-2) x + an-3 )...) X + a1) x + a0, which is called the Horna rule.

Here I wrote a small program of the Horna rule:

 

 

// Author: Tanky Woo
// Blog: www.WuTianQi.com
# Include <iostream>
Using namespace std;
 
// Calculate the value of an * x ^ n + an-1 * x ^ (n-1) +... a2 * x ^ 2 + a1 * x + a0
Double HornerRule (double a [], int n, double x );
 
Int main ()
{
Double *;
Int n;
Double x;
 
Cout <"Input the n (a0, a1, a2... ):";
Cin> n;
 
A = new double [n + 1];
Cout <"input the A0, A1, A2,..., an (n + 1 Numbers ):";
For (INT I = 0; I <= N; ++ I)
Cin> A [I];
 
Cout <"input the X :";
Cin> X;
 
For (INT I = N; I> = 0; -- I)
{
If (I! = N)
Cout <"+ ";
Cout <a [I] <"* x ^" <I;
}
Cout <"=" <HornerRule (a, n, x) <endl;
 
Return 0;
}
 
 
Double HornerRule (double a [], int n, double x)
{
Double res = 0.0;
 
For (int I = n; I> = 0; -- I)
Res = x * res + a [I];
 
Return res;
}