MATLAB - Factorial

MATLAB - Factorial Calculation and Its Nuances

The factorial of a non-negative integer ‘n’, denoted as n!, is the product of all positive integers up to n. It is a fundamental function in combinatorics, probability, and advanced mathematics.

The Recommended Method: `factorial()` Function

The most efficient, accurate, and recommended way to calculate a factorial in MATLAB is by using the built-in factorial() function. It is optimized for performance and handles various input types gracefully.

Example 1: Factorial of a Scalar

Example 2: Factorial of an Array

The factorial function is vectorized, meaning it can operate on each element of an array simultaneously.

Important Considerations and Limitations

Input Data Types and Constraints

The input n must be a non-negative real integer. If you provide a negative integer or a non-integer, MATLAB will throw an error.

Floating-Point Precision and Overflow

Factorial values grow extremely rapidly. MATLAB uses double-precision floating-point numbers by default. This has two consequences:

Large Numbers: For n > 21, the result is too large to be represented exactly as an integer and is returned as a floating-point approximation.
Overflow: For n > 170, the result of factorial(n) exceeds the largest possible double-precision number and overflows to Inf.

Solution for Very Large Numbers: Symbolic Math

To calculate factorials for n > 170 with perfect precision, you can use the Symbolic Math Toolbox™. Convert the number to a symbolic object first.

While you should always use the built-in factorial function in practice, writing your own version is a great programming exercise. Here are two common approaches.

1. Iterative Approach (Using a `for` loop)

This method builds the result by looping and multiplying.

2. Recursive Approach

A recursive function is one that calls itself. This is an elegant but often less efficient way to define factorial.

If you’re using factorials, you might also be interested in:

gamma(x): The Gamma function, which is a generalization of the factorial. For an integer n, gamma(n+1) = factorial(n).
nchoosek(n, k): Calculates the binomial coefficient “n choose k”, which is the number of ways to choose k items from a set of n. It is defined using factorials: n! / (k! * (n-k)!).