Forward and Back Substitution
A matrix equation in the form or is very easy to solve by an iterative process called forward substitution for lower triangular matrices and analogously back substitution for upper triangular matrices. The process is so called because for lower triangular matrices, one first computes, then substitutes that forward into the next equation to solve for, and repeats through to . In an upper triangular matrix, one works backwards, first computing, then substituting that back into the previous equation to solve for, and repeating through .
Notice that this does not require inverting the matrix.
Read more about this topic: Triangular Matrix
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