RREF vs LU Decomposition: Which is Better for Big Data?

The Big Data Matrix Challenge

In modern data science, matrices are huge. Whether training neural networks or processing millions of user recommendations, algorithms must solve systems of linear equations of the form Ax = b where A contains thousands or millions of dimensions.

To solve these systems, we have two primary matrix reduction methods: Gauss-Jordan elimination (RREF) and LU Decomposition. Let's compare their computational characteristics to see why industrial code libraries avoid RREF.

Analyzing RREF (Gauss-Jordan)

To solve a system using RREF, we construct the augmented matrix [A | b] and perform row operations to reduce it to [I | x]. The computational complexity of this reduction is:

Complexity = O(n³) operations

This works fine for a single system. But what if you get a new set of targets bnew? Because the vector b is blended into the row reduction steps, you must run the entire row reduction algorithm again from scratch, costing another O(n3) operations. This is highly inefficient.

Analyzing LU Decomposition

LU Decomposition solves this issue by factoring the matrix A into a product of two triangular matrices: a Lower triangular matrix L and an Upper triangular matrix U, such that A = LU. This factorization is performed once and costs O(n3) operations.

Once factored, solving Ax = b is split into two steps:

1. Solve Ly = b for y using forward substitution. Since L is lower triangular, this is fast: O(n2).
2. Solve Ux = y for x using back substitution. Since U is upper triangular, this is also fast: O(n2).

If you get a new target vector bnew, you do not need to re-factor the matrix. You simply run the forward and back-substitutions using the existing L and U matrices. This costs only O(n²) operations, making it thousands of times faster than RREF for multiple solves.

RREF vs LU Decomposition Comparison