Triangular linear system

A triangular linear system is solved by forward substitution for a lower triangular coefficient matrix or backward substitution for an upper triangular coefficient matrix, using $O(n^2)$ operations for a dense $n$ by $n$ matrix.

Forward substitution solves a lower-triangular system from its first row downward, substituting each newly determined variable into the rows below.

Backward substitution solves an upper-triangular system from its last row upward.