Calculate factorials in C#

Question

How can you calculate large factorials using C#? Windows calculator in Win 7 overflows at Factorial (3500). As a programming and mathematical question I am interested in knowing how you can calculate factorial of a larger number (20000, may be) in C#. Any pointers?

[Edit] I just checked with a calc on Win 2k3, since I could recall doing a bigger factorial on Win 2k3. I was surprised by the way things worked out.

1. Calc on Win2k3 worked with even big numbers. I tried !50000 and I got an answer, 3.3473205095971448369154760940715e+213236

2. It was very fast while I did all this.

The main question here is not only to find out the appropriate data type, but also a bit mathematical. If I try to write a simple factorial code in C# [recursive or loop], the performance is really bad. It takes multiple seconds to get an answer. How is the calc in Windows 2k3 (or XP) able to perform such a huge factorial in less than 10 seconds? Is there any other way of calculating factorial programmatically in C#?

Answer 1

Have a look at the BigInteger structure:

http://msdn.microsoft.com/en-us/library/system.numerics.biginteger.aspx

Maybe this can help you implement this functionality.

CodeProject has an implementation for older versions of the framework at http://www.codeproject.com/KB/cs/biginteger.aspx.

Answer 2

If I try to write a simple factorial code in C# [recursive or loop], the performance is really bad. It takes multiple seconds to get an answer.

Let's do a quick order-of-magnitude calculation here for a naive implementation of factorial that performs n multiplications. Suppose we are on the last step. 19999! is about 2¹⁸ bits. 20000 is about 2⁵ bits; we'll assume that it is a 32 bit integer. The final multiplication therefore involves the addition of up to 2⁵ partial results each roughly 2¹⁸ bits long. The number of bit operations will therefore be on the order of 2²³.

That's for the last stage; there will be 20000 = 2¹⁶ such operations at each stage, so that is a total of about 2³⁹ operations. Some of them will of course be cheaper, but we're going for an order of magnitude here.

A modern processor does about 2³² operations per second. Therefore it will take about 2⁷ seconds to get the result.

Of course, the big integer library writers were not naive; they take advantage of the ability of the chip to do many bit operations in parallel. They're probably doing the math in 32 bit chunks, giving speedups of a factor of 2⁵. So our total order-of-magnitude calculation is that it should take about 2² seconds to get a result.

2² is 4. So your observation that it takes a few seconds to get a result is expected.

How is the calc in Windows 2k3 (or XP) able to perform such a huge factorial in less than 10 seconds?

I don't know. Extreme cleverness in exploiting the math operations on the chip probably. Or, using a non-naive algorithm for calculating factorial. Or, possibly they are using Stirling's Approximation and getting an inexact result.

Is there any other way of calculating factorial programmatically in C#?

Sure. If all you care about is the order of magnitude then you can use Stirling's Approximation. If you care about the exact value then you're going to have to compute it.

Answer 3

There exist sophisticated computational algorithms for efficiently computing the factorials of large, arbitrary precision numbers. The Schönhage–Strassen algorithm, for instance, allows you to perform asymptotically fast multiplication for arbitrarily large integers.

Case in point, Mathematica computes 22000! on my machine in less than 1 second. The Implementation Notes page at reference.wolfram.com states:

(Mathematica's) n! uses an O(log(n) M(n)) algorithm of Schönhage based on dynamic decomposition to prime powers.

Unfortunately, the implementation of such algorithms is both complicated and error prone. Rather than trying to roll your own implementation, it may be wiser for you to license a copy of Mathematica (or a similar product that meets your functional and performance needs) and either use it, or a .NET programming interface to it, to perform your computation.

Answer 4

Have you looked at System.Numerics.BigInteger?

Answer 5

Using System.Numerics BigInteger

var bi = new BigInteger(1);
var factorial = 171;
for (var i = 1; i <= factorial; i++)
{
    bi *= i;
}

will be calculated to

1241018070217667823424840524103103992616605577501693185388951803611996075221691752992751978120487585576464959501670387052809889858690710767331242032218484364310473577889968548278290754541561964852153468318044293239598173696899657235903947616152278558180061176365108428800000000000000000000000000000000000000000

For 50000! it takes a couple seconds to calculate but it seems to work and the result is a 213237 digit number and that's also what Wolfram says.

Answer 6

You will probably have to implement your own arbitrary precision numeric type.

There are various approaches. probably not the most efficient, but perhaps the simplest is to have variable length arrays of byte (unsigned char). Each element represents a digit. ideally this would be included in a class, and you can then add a method which let's you multiply the number with another arbitrary precision number. A multiply with a standard C# integer would probably also be a good idea, but a little trickier to implement.

Answer 7

You need a special big-number library for this. This link introduces the System.Numeric.BigInteger class, and incidentally has an example program that calculates factorials. But don't use the example! If you recurse like that, your stack will grow horribly. Just write a for-loop to do the multiplication.

Answer 8

Since they don't give you the result down to the last digit, they may be "cheating" using some approximation. Check out http://mathworld.wolfram.com/StirlingsApproximation.html Using Stirling's formula you can calculate (an approximation of) the factorial of n in logn time. Of course, they might as well have a dictionary with pre-calculated values of factorial(n) for every n up to one million, making the calculator show the result extremely fast.

Answer 9

This answer covers limits for basic .Net types to compute and represent n!

Basic code to calculate factorial for "SomeType" that supports multiplication:

SomeType factorial = 1;
int n = 35;
for (int i = 1; i <= n; i++)
{
   factorial *= i; 
}

Limits for built in number types:

• short - correct results up to 7!, incorrect results afterwards, code returns 0 starting 18 (similar to int)
• int - correct results up to 12!, incorrect results afterwards, code returns 0 starting at 34 (Why computing factorial of realtively small numbers (34+) returns 0)
• float - precise results up to 14!, correct but not precise afterwards, returns infinity starting at 35
• long - correct results up to 20!, incorrect results afterwards, code returns 0 starting at 66 (similar to int)
• double - precise results up to 22!, correct but not precise afterwards, returns infinity starting at 171
• BigInteger - precise and upper limit is set by memory usage only.

Note: integer types overflow pretty quickly and start producing incorrect results. Realistically if you need factorials for any practical usage long is the type to go (up to 20!), if you can't expect limited numbers - BigInteger is the only type provided in .Net Framework to provide precise results (albeit slow for large numbers as there is no built-in optimized n! method)

Answer 10

I don't know how you could do this in a language without arbitrary precision arithmetic. I guess a start could be to count factors of 5 and 2, removing them from the product, and add on these zeroes at the end.

As you can see there are many.

>>> factorial(20000)
<<non-zeroes removed>>00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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