Polynomial Evaluation

Goal: Evaluate a polynomial . We use Horner's rule, given as:

Lets apply it to .

What would happen if we tried to find a zero of using a simple zero finder? The root finder is based on the bisection algorithm:

function bisect(xl, xu, tol, p)

/* find a zero of p(x) in [xl, xu] assuming  p(xl) . p(xu) < 0 */
/* stop if zero found to within +/- tol  */

if (xu-xl) <= (2 * tol)
    return ((xl + xu)/2)
else 
    mid = (xl + xu)/2
    pm = p(mid)  /* the polynomial P(x) is evaluated a x = mid */
    if p(xl) . pm < 0 then
        return bisect(xl,mid,tol,p)
    else
        return bisect(mid xu,tol,p)
    endif
endif

Let us apply to with to . What will happen? Look at the graphs of shown in Fig. 4 and Fig. 5.

  
Figure 4: Graph of p(x)

  
Figure 5: Graph of in the neighborhood of x =2 evaluated using IEEE double precision arithmetic