This problem considers ways to compute n, where is a positive integer. This problem arose from trying to explain MATLAB’s rather large flop time in Table 1.3 for integer powers. (a) Compare the total...

This problem considers ways to compute


_n
, where

is a positive integer. This problem arose from trying to explain MATLAB’s rather large flop time in Table 1.3 for integer powers.

(a) Compare the total number of flops between computing


_n

=
∗∗ ∗, and computing

where

=

². As examples of the last formula,

⁶
=

∗

∗
, while

⁵
=

∗

∗.

(b) Suppose n = 28. Show that 28 = 2⁴
+ 2³
+ 2², and

What is the minimum number of flops required using this formula? Also explain why 2⁴
+ 2³
+ 2²
is the floating-point representation of 28. Note that this procedure is a version of the square-and-multiply algorithm.

(c) Suppose

= 100, so its floating-point representation is (1 +

+
)×2⁶. Explain how to use the idea in part (b) to calculate

¹⁰⁰. How does the flop count compare with the two methods in part (a)?

(d) Another approach, assuming

is positive, is to write

_n

=


^{n ln x}
. Based on the values in Table 1.3, what is the approximate flop time for this? How does it compare with the flop times found in parts (b) and (c)?