# 1000 Factorial

, 27 Jul 2009 CPOL
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Optimum algorithm for calculating factorial of large number

## Introduction

Factorial or the product of positive integer numbers is one of the most usable mathematical functions (or processes). A lot of mathematical calculations need to have exact result of great number's factorial, such as 1000! to find the final response with high accuracy. But the problem is that no programming language has a variable or mechanism to store such a big figure. All of the programming languages and calculators estimate the result and then store it as scientific number. For example, calculator of Windows XP displays 1000! like this:

`4.02387260077093773543702433923 e+2567 `

So I decided to invent an algorithm for solving this problem.

## Algorithm

You know to calculate factorial of a number, you should multiply all numbers from 1 to itself, for example:

`1000! = 1*2*3*……*998*999*1000 `

There is no variable in any programming language that can store the result of this multiplication exactly; except storing in scientific notation. Because of sequential multiplication's huge result, finding a new strategy or mechanism is necessary.

In my recommended algorithm, the number is not looked at as a number, instead it is treated as sequential digits where each one has its own numerical value. In fact, an array of digits is available. Each time, the second number is multiplied by each digit of the first number (index of array) and then added with carry number up from the previous multiplication.
This process is shown, for example 12!:

`12! = 11! * 12     11! = 39916800     12! = 479001600 `

• If the result is less than 10, the result will be placed in the cell of array.
• If it is equal to or greater than 10, it will be divided by 10 and then the remainder will be placed in the array's cell and quotient placed in variable that will be added with next multiplication's response.
Note: Notice that all the numbers are stored from end of array, the same as normal calculation (real calculation).

## Programming Code

### Declarations

```int numArr[3000];
int total,rem=0,count;
register int i;```

In the first line, an array is defined such that its size depends on factorial size. "`rem`" is defined to store remainder of division.

At the end , an integer variable is defined and named "`i`" that plays the role of loop counter and array's index and due to much access, it's defined as register.

The register type modifier tells the compiler to store the variable being declared in a CPU register (if possible), to optimize access.

Note: Modern compilers have a good optimizer that decides which variables should be stored in registers. They make their own register choices when global register-allocation optimization is on.

### The Main Part of the Code

```i=2999;				//start from end of array.
numArr[2999]=1;

for(count=2;count<=1000;count++)
{
while(i>0)
{
total=numArr[i]*count+rem;
rem=0;
if(total>9)
{
numArr[i]=total%10;
rem=total/10;
}
else
numArr[i]=total;
i--;
}
rem=0;
total=0;
i=2999;
}```

According to the algorithm that was explained before, there is a loop counting 2 to 1000 and each time value of '`count`' is multiplied by array's cell and added with '`rem`' which contains carry from previous multiplication.

Finally, the result is stored in '`total`' and then placed in the array's cell.

## Another Algorithm

At first, I found another algorithm for this problem. It's based on simulating multiplication by successive additions. For example 20= 4*5 also 20= (5+5+5+5).
So putting numbers in an array and then adding it with itself 'X' times.
X for 1000! is:

X= ∑ n
n=1,2,3, … ,999,1000

Note: This algorithm is not as optimum as the first one that I explained. Also it's too dull and needs several big arrays. The second ZIP file belong to this algorithm.

## Points of Interest

I would like to point out that the algorithm has high computing speed, so it turns the program to quite an optimum state.

One of the most important features of this algorithm is using just one array, while other algorithms need more memory (using several arrays to simulate multiplication by sequential sums).

Furthermore, less coding helps to understand the program easily and you can rewrite it in any programming language.

One of the reasons for high execution speed is using register variable as loop's counter. Loop's counter is a variable with most access to it.

## Usage

As I mentioned in the introduction, factorial operation is used in a lot of mathematical calculations, especially, the computations that relate to statistics that need exact result of factorial.

So this program can be part of any program that needs factorial method, such as statistical software.

## Appreciation

I'd like to thank Mr.Fermisk Naserzade for introducing me to this problem and Mr. Iraj Safa for helping me to edit this article.

## Share

Iran (Islamic Republic Of)
Mohammad Shafienia is student of MS of IT(Information Security) in Tehran University who has been coding since 1999. He was born in Tehran, the capital of Iran.
He has started programming by BASIC, but now he is .Net developer. AI and Low-Level programming are his favorite issues. He is good at team-working and participates in research projects.
Mohammad loves sport, specially volleyball and camping. He is such outgoing person and crazy about nature.
One of his wishes is to have laboratory same as Bell Lab, How ambitious.

To contact Mohammad, email him at shafienia[AT]gmail[DOT]com
www.iranexam.net

 First PrevNext
 congratulations Andy Allinger 15-Dec-13 10:53
 Thank you Member 9451663 26-Jun-13 9:34
 good work！ kspliusa 28-Mar-12 21:40
 Also excellent jrobb229 14-Feb-12 12:02
 Re: Also excellent Mohammad Shafieenia 27-Oct-12 2:02
 excellent sir!!! kushal seth 18-May-10 6:17
 Re: excellent sir!!! Mohammad Shafieenia 27-Oct-12 1:59
 Need a break after if statement Pourya Shirazian 4-Aug-09 7:48
 Re: Need a break after if statement Mohammad Shafieenia 5-Aug-09 8:45
 very good! alejandro29A 4-Aug-09 5:48
 Re: very good! Mohammad Shafieenia 4-Aug-09 10:22
 Excellent Anil Srivastava 30-Jul-09 0:54
 Re: Excellent Mohammad Shafieenia 30-Jul-09 1:35
 Why don't you try to implement a more generalized solution zimstep 27-Jul-09 22:40
 Re: congratulation Mohammad Shafieenia 27-Jul-09 21:37
 Efficiency improvement supercat9 27-Jul-09 7:18
 Re: Efficiency improvement Mohammad Shafieenia 27-Jul-09 8:01
 Re: Efficiency improvement supercat9 27-Jul-09 19:57
 From your description of the algorithm, it should be possible to multiply your number by any number up to maxint/10-1 (i.e. 214,748,363) in the same amount of time as would be required to multiply by a smaller number. If the goal is to compute 1000!, instead of multiplying by 2, then 3, then 4, etc. it would make more sense to multiply by 1000*999*2*3*4*5, then by 998*997*6*7, then 996*995*8*9, then 994*993*10*11, etc. This would allow the desired result to be computed using about 333 full-size multiplications rather than 999 of them.   Another approach, btw, would be to use each place to hold two or three base-ten digits instead of one. This would somewhat offset the previous optimization (since the factors would have to be kept below maxint/100-1 or maxint/1000-1, but computing 1000! three digits at a time would still require less than 500 full-size multiplies, with each full-size mutltiply taking 1/3 as long as before.
 Re: Efficiency improvement [modified] brianma 5-Aug-09 0:30
 Re: Efficiency improvement supercat9 5-Aug-09 5:28
 Re: Efficiency improvement Amin Ghadirian 26-Jun-13 11:23
 "Only" up to 1142 mluri 26-Jul-09 22:14
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