Horner's Rule for a Polynomial and Its Derivative   
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Horner's Rule for a Polynomial and Its Derivative 
So far we have found an efficient procedure for evaluating a polynomial at any . 
Next we develop a procedure for getting its derivative  at the same . Notice 
that for any  we can divide  by  to get quotient and remainder: 

      (2)


where the quotient polynomial  has degree : 

 


It is easy to show that 

 


So if we can find the coefficients  of the quotient polynomial , we can use 
Horner's algorithm to get . 
If we substitute the polynomials into Eq. (), and equate coefficients of  on the 
lhs and rhs and solve for the coefficients  and remainder  we get a series of 
equations that can be organized as a recursion relation: 
 





These equations should look familiar, because they are nothing but Horner's 
algorithm, derived in a different way. But now we see that the intermediate 
variable  in the pseudocode actually holds the coefficients  of the quotient 
polynomial ! If we are careful, we can use them to build a dual procedure for 
evaluating both  and  without the need to store them all. 
Here is the pseudocode for the dual-purpose algorithm: 
   Step 1: Set p = a[n] and q = 0
   Step 2: Do steps 3 and 4 for i from n-1 to 0, decreasing by 1
        Step 3: set q = p + x0 * q
        Step 4: set p = a[i] + x0 * p
   Step 5: The value of P(x_0) is p and the value of P'(x_0) is q

It is left as an exercise to check this pseudocode by playing computer with a 
simple example and with the special case . 
Finally, we leave as an exercise to complete the Newton-Raphson code for finding 
the real roots of a general polynomial. 


   
Next: About this document ... Up: Horner's Rule for Polynomials Previous: 
Evaluating a polynomial: poly.cc 
Carleton DeTar 2005-10-03 
