Quantcast XP Math - Forums - View Single Post - Horner’s method
View Single Post
Old 03-24-2007   #15
Posts: n/a

Originally Posted by hyderman View Post
here you go guys look how easy


The Horner’s method is an algorithm that evaluates polynomials. The following pseudocode shows how to use this method to find the
value of a_nx^n + a_{n-1}x^{n-1} + . . . + a_1x + a_0 at x = c.
procedure Horner(c, a_0, a_1, a_2, . . . , a_n : real numbers)
y := a_n
for i := 1 to n
y := y * c + a_{n-i}
end {y = a_nc^n + a_{n-1}c^{n-1} + . . . + a_1c + a_0}

(a) Evaluate x^2 + 5x + 3 at x = 2 by working through each step of the algorithm.

y:=1 [since n=2 and a_2=1]
i:=1 [first time the loop runs]
y:=y*2+5 =1*2+5=2+5=7 [since c=2 and a_1=5]
i:=2 [second time the loop runs]
y:=y*2+3=7*2+3=14+3=17 [since c=2 and a_0=3]
end [loop runs n=2 times only]

Indeed, 2^2+5*2+3=4+10+3=17 !

(b) Exactly how many multiplications and additions are used by this algorithm to evaluate a polynomial of degree n at x = c? (Do not count additions used to increment the loop variable.)

Each time the loop repeats it performs 1 multiplication [y*c] and 1 addition [adds a_{n-i} to it]. Loop repeats n times [i goes from 1 to n] so this algorithm performs n multiplications and n additions !

The crazy thing is, I actually almost understood part of that.