Question-64
There is a more efficient algorithm (in terms of the number of multiplications and additions used) for evaluating polynomials than the conventional algorithm. It is called Horner’s method. This pseudocode shows how to use this method to find the value of anxn+an−1xn−1+⋅⋅⋅+a1x+a0 at x=c.
procedure Horner(c,a0,a1,a2,…,an: real numbers)
y≔an
fori≔1ton
y≔y* c+an−i
returny{y=ancn+an−1cn−1+⋅⋅⋅+a1c+a0}
a) Evaluate 9x2+2x+3 at x=2 by working through each step of the algorithm showing the values assigned at each assignment step.
y = ?
i = ?
y = ?
i = ?
y = ?
b) Exactly how many multiplications and additions are used to evaluate a polynomial of degree n at x=c? (Do not count additions used to increment the loop variable.)
There are = ?
There are = ?
Solution
a)
y = 9
i = 1
y = 20
i = 2
y =43
b)
There are = n multiplications.
There are = n additions.