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

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

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

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

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

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