The polynomial p n ( ) = 0 + 1 +···+ n n can be separated into the sum of two polynomials, one which contains even powers of  and the other involving odd powers. This problem explores the...

The polynomial
p_n
() =

₀+
₁
+···+

_n



ⁿ

can be separated into the sum of two polynomials, one which contains even powers of
 and the other involving odd powers. This problem explores the computational benefits of this. To make things simple, you can assume
 is even, so
 = 2, where
 is a positive integer.

(a) Setting
 =
2, find
() and
() so that


_n
() =
() +


_g
().

(b) What is the minimum flop count to compute the expression in part (a)? Also, explain why it is about half-way between the flop count for the direct method and the count using Horner’s method.

(c) Evaluate (1.4) using the formula in part (a), and then plot the values for 0.98 ≤
 ≤ 1.02 (use 1000 points in this interval). In comparison to the plot obtained using the direct method, does the reduced flop count reduce the error in the calculation?