In Section 5.1 and Section 5.2
triangular systems are solved by techniques which use inner product accumulation on each row
of A. It is possible to formulate these algorithms in a manner that arrives
at the solution through partial sum accumulations (also known as an outer product).
Algorithm 6 solves lower triangular systems using an outer product formulation
of forward substitution. You should observe that if b_{i} is zero,
the i^{th} stage can be skipped. In this situation,
y_{i} will also be zero and the term
y_{i}l_{ki}
will not change any of the partial sums b_{k}.

for $1=1,\cdots,n$ |

$y_{i}=\displaystyle\frac{b_{i}}{l_{ii}}$ |

for $k=i+1,\cdots,n$ |

$b_{k}=b_{k}-y_{i}l_{ki}$ |

Algorithm 7 solves upper triangular systems are using an outer product
formulation of the backward substitution algorithm. Also observe that when y_{i} is zero,
the i^{th} stage can be skipped. In this situation,
x_{i} will also be zero and the term
x_{i}u_{ki}
will not change any of the partial sums y_{k}.

for $i=1,\cdots,n$ |

$x_{i}=\displaystyle\frac{y_{i}}{u_{ii}}$ |

for $k=i-1,\cdots,1$ |

$y_{k}=y_{k}-x_{i}u_{ki}$ |