 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/functional-programming-design-and-analysis-of-algorithms/2/g/solution

The derivative fixed-point equation is unsuitable as an executable definition: requesting its first coefficient asks for the second coefficient of the same stream, then the third, indefinitely. It is not productive. As an algebraic equation it is also missing the constant coefficient: every scalar multiple of the formal power series exponential, including zero, satisfies it.

expx :: Fractional a => [a]
expx = plus one (integral expx)

The initial coefficient is available as 1 + 0; each subsequent coefficient needs only a previously determined coefficient. The recurrence relation is

a_{0} = 1

and

a_{n + 1} = a_{n} / (n + 1)

, so $a_{n} = 1/ n!$ . This proves both uniqueness of the formal solution and productivity of every finite coefficient list prefix.

 New to topics? Read the docs here!