## Recursion in the form of a Recursive Func<T, T>

I gotta admit, I am kind of surprised that I didn’t realize I could do this sooner. I recently had a problem which required a recursive function call to come up with the answer. After some time messing around with a recursive method, and creating an API that I was not happy with, I was able to create an API that I enjoy, and seems intuitive.

# Introduction

To bring it to a simple example, consider the summation to n:

A mathematically identical formula is:

In a .NET function, this can be represented by a function:

Func<int, int> summation = x => x*(x+1)/2

Calling summation with an input integer will yield the summation to that number:

var sum10 = summation(4); //sum10 would be equal to 10

But what if I wanted to get a second level summation… First sum to n, and then use that argument as the input to the same function, to find the second level summation:

So as an easy example, calculate the summation to 3, which yields 6. Then calculate the summation to 6 which yields 21.

Represented as a mathematical formula -

So what if I wanted to represent this as .NET functions? I can always do:

//using the summation formula from above var sum3 = summation(3); //sets sum3 to 6 var sum3_2 = summation(sum3); //sets sum3 to 21

Func<int, int> summation = x => x*(x+1)/2; //for the interests of a smaller example, using shorthand int sumResultTo = 3; int level = 2; while(level-- > 0) sumResultTo = summation(sumResultTo); //sumResultTo is equal to 21 now.

Or express it as a for-loop, method calls, etc… I really didn’t like any of the options that I tried. Then it dawned on me – since I was using a Func<T, T> anyways, why not use the Func’s output from one call as the input to another directly.

# Some Code

So, I decided that I wanted a recursion class. Something that would be generic and reusable in case I ever wanted to do something like this again. It is limited to only the Func<T1, T2> overload of Func, and T1 must be the same as T2.

The first thing in this class is a private field for the function:

private readonly Func<T, T> _functionToRecurse;

So, since I want the function to be unchangeable, I have defined it as readonly. Therefore my constructor looks like:

public Recursion(Func<T, T> functionToRecurse) { if (functionToRecurse == null) { throw new ArgumentNullException("functionToRecurse", "The function to recurse can not be null"); } _functionToRecurse = functionToRecurse; }

Simple enough. Next, I want to be able to get the result of a function dependent on how many levels of recursion:

private Func<T, T> GetXLevel(int level) { if (level < 1) throw new ArgumentOutOfRangeException("level", level, "The level of recursion must be greater than 0");

if (level == 1) return _functionToRecurse; return _GetXLevel(level - 1, _functionToRecurse); }

So, if you pass in 1 for the level, you get just the Func<T,T> back. If you say that you want to go deeper down the rabbit hole, it calls a method which accepts the level it is at, and the function which it needs to use to recurse further:

private Func<T, T> _GetXLevel(int level, Func<T, T> prevFunc) { if (level == 1) return y => prevFunc(_functionToRecurse(y)); return _GetXLevel(level - 1, y => prevFunc(_functionToRecurse(y))); }

That is really all that is needed for this class. If I exposed the GetXLevel function publicly, I could use that to get the function for a level, and pass in the argument.. But I wanted something cleaner. So, I used the ‘this’ array operator for the class:

public Func<T,T> this[int level] { get { if (level < 1) { throw new ArgumentOutOfRangeException("level", level, "The level of recursion must be greater than 0"); } return this.GetXLevel(level); } }

So, using the same example above of finding the second recursion of the summation of 3:

var summator = new Recursion<int>(x => (x * (x + 1)) / 2); var sum_3_level2 = summator[2](3); //yields 21

You can even just store the delegate to the second level summation, and use it multiple times:

var summator = new Recursion<int>(x => (x * (x + 1)) / 2); var sum_level2 = summator[2]; var sum_3_level2 = sum_level2(3); //yields 21 var sum_4_level2 = sum_level2(4); //yields 55 var sum_5_level2 = sum_level2(5); //yields 120

# Full Code

Don’t think I was just going to hold off on the full file together and make you do the hard work… Copy this into a new class file:

public class Recursion<T> { private readonly Func<T, T> _functionToRecurse; public Recursion(Func<T, T> functionToRecurse) { if (functionToRecurse == null) { throw new ArgumentNullException("functionToRecurse", "The function to recurse can not be null"); } _functionToRecurse = functionToRecurse; } public Func<T,T> this[int level] { get { if (level < 1) { throw new ArgumentOutOfRangeException("level", level, "The level of recursion must be greater than 0"); } return this.GetXLevel(level); } } private Func<T, T> GetXLevel(int level) { if (level < 1) { throw new ArgumentOutOfRangeException("level", level, "The level of recursion must be greater than 0"); } if (level == 1) return _functionToRecurse; return _GetXLevel(level - 1, _functionToRecurse); } private Func<T, T> _GetXLevel(int level, Func<T, T> prevFunc) { if (level == 1) return y => prevFunc(_functionToRecurse(y)); return _GetXLevel(level - 1, y => prevFunc(_functionToRecurse(y))); } }

# Conclusion

The great thing about this class, is that it can be used with any function with same input/output parameters. I strived to find an implementation that I found clean and useful, and I finally settled on this. If you have feedback – good or bad, I would love to hear it!

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## # re: Recursion in the form of a Recursive Func<T, T>

left by uvw at 6/25/2012 12:05 PMGeneral idea:

http://en.wikipedia.org/wiki/Fixed-point_combinator#Y_combinator

C# implementation:

http://blogs.msdn.com/b/wesdyer/archive/2007/02/02/anonymous-recursion-in-c.aspx