The equation Ly = b is solved by forward substitution as follows.
| $$y_{i}=\frac{b_{i}-\displaystyle\sum_{j=1}^{i-1}l_{ij}y_{i}}{l_{ii}},\mbox{ where }1 \leq i \leq n$$ | (62) |
Algorithm 4 implements Equation 62.
| for $i=1,\cdots,n$ |
| $\alpha=b_{i}$ |
| for $j=1,\cdots,i-1$ |
| $\alpha=\alpha-l_{ij}y_{j}$ |
| $y_{i}=\displaystyle\frac{\alpha}{l_{ii}}$ |
If L is unit lower triangular, the division by lii is unnecessary (since lii is 1). Notice that the update to yi is accumulated as an inner product in $\alpha$.