Programming Languages and Paradigms
Please do all three questions using the LearnOCaml system as explained in class. There is a fourth
question which you don’t have to do or even look at. This is a so-called “spiritual growth” question.
It will not earn you any extra credit for the course.
Q1. [30 points] Here is a way to compute the square root of a positive real number say x. We make
a wild guess that the square root is some number g. Then we check if g
is close to x. If it is we
return g as our answer; if not we refine our guess using the formula
0 = (g + x/g)/2.0
is the new guess. We keep repeating until we are close enough. Write an OCaml program to
implement this idea. Since we are working with floating point numbers we cannot check for equality;
we need to check if the numbers are close enough. I have written a function called close for you
as well as a function called square. In OCaml floating point operations need special symbols: you
need to put a dot after the operation. Thus you need to write, for example, 2.0 + .3.0. All the
basic arithmetic operations have dotted versions: +., ∗., /. and must be used with floating point
numbers. You cannot write expressions like 2 + .3.5 or 2.0 + 3.0. We will not test your program on
negative inputs so we are not requiring you to deal with the possibility that you may get a negative
answer. There are of course built-in functions to compute square roots. We have written the tester
program to check if you used this; you will get a zero if you use a built-in function.
Q2. [30 points] This is similar to the question above except that we are computing cube roots. The
formula to refine the guess is
0 = (2.0 ∗ g + x/g2
where I am using mathematical notation with arithmetic operation symbols overloaded. In your
code you must use the floating-point versions of the arithmetic operations.
Q3. [40 points] In lecture 1 I quickly explained the Russian peasant algorithm for fast exponentiation. In this exercise you have to implement a tail-recursive version of the algorithm. Everything is
with integers in this question so please do not use floating point operations. Our tester will detect
whether you used built-in functions, so please don’t try to use them.
Q4. [0 points] This question is for your spiritual growth only. Do not think it will give you
extra credit or help you learn the material better. It will however stretch your brain in other
directions. Do not attempt it if you have not yet finished the required homework. Do not submit
a solution; please talk to me if you have solved it. Do not worry about it if you don’t understand
Why do the algorithms for square root and cube root above converge? Will this kind of idea work
for any function? What is the correct update formula for fifth roots?