L U factorization, how, why

by symbolipoint
Tags: factorization
 Engineering Sci Advisor HW Helper Thanks P: 7,157 L U factorization, how, why It might help to understand it a different way, when there is no pivoting. Write the matrix equation out in full for a small example: $$\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{bmatrix} = \begin{bmatrix}1 \\ l_{21} & 1 \\ l_{31} & l_{32} & 1\end{bmatrix} \begin{bmatrix} u_{11} & u_{12} & u_{13} \\ & u_{22} & u_{23} \\ & & u_{33}\end{bmatrix}$$ Now work through A term by term, and see what the LU product for that term is: ##a_{11} = u_{11}## ##a_{12} = u_{12}## ##a_{13} = u_{13}## ##a_{21} = l_{21}u_{11}##, and we already know ##u_{11}##, so we can find ##l_{21}## ##a_{22} = l_{21}u_{12} + u_{22}##, and the only unknown is ##u_{22}## and so on. This is Crout's algorithm for the LU decomposition without pivoting. Textbooks do seem to enjoy finding complcated ways to explain something that is really quite simple. Even with pivoting, it is just a systematic way of doing row operations on A, and remembering what you did, so you can do the same thing to the right hand side vector later.