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; 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.
