Example 6.  Compare the Horner method and ordinary method of polynomial evaluation for
  

Solution 6.

Construct a the polynomial and two version of Horner's method.

Aside.  This polynomial has known roots  .  

For illustration we can evaluate .

We are done.

Aside.  Consider the computational time for vector arguments for evaluating  P(x)  and the nested multiplication form  Q(x)  and Horner form  H(x).  The latter two take roughly  1/2  the time to do the evaluations

We are really done.

Aside.  The product form of the polynomial,    is the "best computational form."  We can compare the "error" with it and the other three forms of this polynomial.  

An analysis of the error has been done by G. W. Stewart in the article:
Error Analysis of the Algorithm for Shifting the Zeros of a Polynomial by Synthetic Division  
G. W. Stewart  
Mathematics of Computation, Vol. 25, No. 113. (Jan., 1971), pp. 135-139, Jstor.  

(c) John H. Mathews 2004