Saturday, June 20, 2015

Recursive Programming: The General Principle

Recursion, where a function calls itself, is a fundamental programming method. Recursion is a different way of iterating over a List than using a loop. Recurse is from the Latin recurrere, to run back (I picture a dog retrieving a stick over and over).

There are two principles of writing a recursive function, which is piecewise:
  1. Recursive case: Define a function f(n+1) recursively in terms of f(n), or equivalently, f(n) in terms of f(n-1)
  2. Base case: Define one or more base cases, which are stopping points for the recursion, so that it doesn't become infinite. 
An elementary example is exponentiation defined this way:

x= x•x(n-1)

First we define the base case, so that Mathematica will match it to prevent an infinite recursive loop. Let's just consider positive integers and zero. The base case is any integer raised to the 0th power equals 1.


The recursive case is:

exponentiation[x_,n_Integer/;x>0]:=x exponentiation[x,n-1]

Start testing a recursive function with the simplest examples:




Let's make sure the Condition requiring x > 0 works; it does.


Trace shows the exponentiation hitting the base case and stopping.

{exponentiation[1,1],1 exponentiation[1,1-1],{{1-1,0},exponentiation[1,0],1},1 1,1}

To compute 22, the function calls itself to computer 22-1 i.e. 21, which in turn calls itself to compute 20, which is the base case, 1. Once it has recursed down to the base case, it has a value to substitute in the preceding computations and the recursion 'unwinds.' All recursive functions work this way.

{exponentiation[2,2],2 exponentiation[2,2-1],{{2-1,1},exponentiation[2,1],2 exponentiation[2,1-1],{{1-1,0},exponentiation[2,0],1},2 1,2},2 2,4}




Use Block for Large Recursions and Debugging

You may run into the built-in default recursion limit, $RecursionLimit, of 1024 when performing a large recursion.

In[584]:= exponentiation[2,10000]
During evaluation of In[584]:= $RecursionLimit::reclim: Recursion depth of 1024 exceeded. >>
Out[584]= Hold[exponentiation[2,8978-1]]

The safe way to exceed the limit is using Block. Block temporarily changes the value of a Global or System variable. I've suppressed the output here since it is 3,011 digits long, but you can try this safely at home without "//IntegerDigits//Length".


Mathematica will warn you and stops an infinite recursion with a default recursion limit specified by $RecursionLimit ($IterationLimit is used similarly to stop infinite loops).


Just as you can use Block to temporarily exceed $RecursionLimit, you can use Block to temporarily 'tighten up' $RecursionLimit while writing a recursive function.

In[592]:= Block[{$RecursionLimit=20},factorial@x_:=x*factorial[x-1];factorial@2]
During evaluation of In[592]:= $RecursionLimit::reclim: Recursion depth of 20 exceeded. >>
Out[592]= Hold[factorial[-14-1]]

Always remember to write the base case rule!

In[593]:= Block[{$RecursionLimit=20},factorial@x_:=x*factorial[x-1];factorial@0=1;factorial@2]
Out[593]= 2

Now we may safely compute the factorial of larger numbers:



Note all the zeroes at the end. They must be from each time a multiple of 10 is incorporated into the factorial.

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