structural:summarization
Element-based formula

However, by taking advantage of the triangular form of L {\displaystyle \mathbf {L} }, the elements of x (k + 1) {\displaystyle \mathbf {x} ^{(k+1)}} can be computed sequentially for each row i {\displaystyle i} using forward substitution: x i (k + 1) = 1 a i i (b i − ∑ j = 1 i − 1 a i j x j (k + 1) − ∑ j = i + 1 n a i j x j (k)), i = 1, 2, …, n. {\displaystyle x_{i}^{(k+1)}={\frac {1}{a_{ii}}}\left(b_{i}-\sum _{j=1}^{i-1}a_{ij}x_{j}^{(k+1)}-\sum _{j=i+1}^{n}a_{ij}x_{j}^{(k)}\right),\quad i=1,2,\dots,n.}

Notice that the formula uses two summations per iteration which can be expressed as one summation ∑ j ≠ i a i j x j {\displaystyle \sum _{j\neq i}a_{ij}x_{j}} that uses the most recently calculated iteration of x j {\displaystyle x_{j}}. The procedure is generally continued until the changes made by an iteration are below some tolerance, such as a sufficiently small residual.