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Logic and Recursion

Étude 4-1: Using case

Change the area/3 function that you wrote in Étude 3-2 so that it uses a case instead of pattern matching. Use a guard on the function definition to ensure that the numeric arguments are both greater than zero.

Étude 4-2: Recursion

This is a typical exercise for recursion: finding the greatest common divisor (GCD) of two integers. Instead of giving Euclid’s method, we’ll do this with a method devised by Edsger W. Dijkstra. The neat part about Dijkstra’s method is that you don’t need to do any division to find the result. Here is the method.

To find the GCD of integers M and N:

  • If M and N are equal, the result is M.

  • If M is greater than N, the result is the GCD of M - N and N

  • Otherwise M must be less than N, and the result is the GCD of M and N - M.

Write a function gcd/2 in a module named Dijkstra that implements the algorithm. This program is an ideal place to use Elixir’s cond construct. Here is some sample output.

iex(1)> c("dijkstra.ex")
[Dijkstra]
iex(2)> Dijkstra.gcd(2, 8)
2
iex(3)> Dijkstra.gcd(14, 21)
7
iex(4)> Dijkstra.gcd(125, 46)
1
iex(5)> Dijkstra.gcd(120, 36)
12

See a suggested solution in Appendix A. You can also use guards with multiple clauses to solve this étude; the solution for that approach is in Appendix A. In general, use of multiple clauses is considered more in the spirit of Elixir.

The next two exercises involve writing code to raise a number to an integer power (like 2.53 or 4-2) and finding the nth root of a number, such as the cube root of 1728 or the fifth root of 3.2.

These capabilities already exist with the :math.pow/2 function, so you may wonder why I’m asking you to re-invent the wheel. The reason is not to replace :math.pow/2, but to experiment with recursion by writing functions that can be expressed quite nicely that way.

Étude 4-3: Non-Tail Recursive Functions

Create a module named Powers (no relation to Francis Gary Powers), and write a function named raise/2 which takes parameters x and n and returns the value of xn.

Here’s the information you need to know to write the function:

  • Any number to the power 0 equals 1.

  • Any number to the power 1 is that number itself — that stops the recursion.

  • When n is positive, x^n^ is equal to x times x^(n - 1)^ — there’s your recursion.

  • When n is negative, x^n^ is equal to 1.0 / x^-n^

Note that this algorithm is not tail recursive.

Warning

The Elixir kernel already has a raise/2 function, so your function will cause a conflict unless you add this after the defmodule:

import Kernel, except: [raise: 2]

Here is some sample output.

iex(1)> c("powers.ex")
\[Powers\]
iex(2)> Powers.raise(5,1)
5
iex(3)> Powers.raise(2,3)
8
iex(4)> Powers.raise(1.2, 3)
1.728
iex(5)> Powers.raise(2, 0)
1
iex(6)> Powers.raise(2, -3)
0.125

If you try raising 0 to a negative power, you will get an error message. For now, just let the error occur. You will learn more about error handling in [ERRORS].

Étude 4-4: Tail Recursion with an Accumulator

Practice the "accumulator trick." Rewrite the raise/2 function for n greater than zero so that it calls a helper function raise/3 This new function has x, n, and an accumulator as its parameters.

Your raise/2 function will return 1 when n is equal to 0, and will return 1.0 / raise(x, -n) when n is less than zero.

When n is greater than zero, raise/2 will call raise/3 with arguments x, n, and 1 as the accumulator.

The raise/3 function will return the accumulator when n equals 0 (this will stop the recursion).

Otherwise, recursively call raise/3 with x, n - 1, and x times the accumulator as its arguments.

The raise/3 function is tail recursive.

Warning

Because the kernel also contains a raise/3 function, you have to change your import as follows:

  import Kernel, except: [raise: 2, raise: 3]

Étude 4-5: Recursion with a Helper Function ~~~~~~~~~~~~~~~ In this exercise, you will add a function nth_root/2 to the Powers module. This new function finds the nth root of a number, where n is an integer. For example, nth_root(36, 2) will calculate the square root of 36, and nth_root(1.728, 3) will return the cube root of 1.728.

The algorithm used here is the Newton-Raphson method for calculating roots. (See http://en.wikipedia.org/wiki/Newton%27s_method for details).

You will need a helper function nth_root/3, whose parameters are x, n, and an approximation to the result, which we will call a. nth_root/3 works as follows:

  • Calculate f as (a^n^ - x)

  • Calculate f_prime as n * a^(n - 1)^

  • Calculate your next approximation (call it next) as a - f / f_prime

  • Calculate the change in value (call it change) as the absolute value of next - a

  • If the change is less than some limit (say, 1.0e-8), stop the recursion and return next; that’s as close to the root as you are going to get.

  • Otherwise, call the nth_root/3 function again with x, n, and next as its arguments.

For your first approximation, use x / 2.0. Thus, your nth_root/2 function will simply be this:

nth_root(x, n) -> nth_root(x, n, x / 2.0)

Use IO.puts to show each new approximation as you calculate it. If your argument name is estimate, you would do something like this:

IO.puts("Current guess is #{estimate}")

Here is some sample output.

iex(1)> c("powers.ex")
\[Powers\]
iex(2)> Powers.nth_root(27, 3)
Current guess is 13.5
Current guess is 9.049382716049383
Current guess is 6.142823558176272
Current guess is 4.333725614685509
Current guess is 3.3683535855517652
Current guess is 3.038813723595138
Current guess is 3.0004936436555805
Current guess is 3.000000081210202
Current guess is 3.000000000000002
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