# 5.1 Forward Substitution

The equation Ly = b is solved by forward substitution as follows.

 $y_{i}=\frac{b_{i}-\displaystyle\sum_{j=1}^{i-1}l_{ij}y_{i}}{l_{ii}},% \operatorname{where}1\leq i\leq n$ (62)

Algorithm 4 implements Equation 62.

Algorithm 4: Forward Substitution
 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 $\alpha$.