LU Factorization. What to do with L and U

Question

I have a general knowledge question.

I have a problem to solve Ax = b using LU Factorization (A and b are provided).

I solved for L and U, but now I have no clue how to use these to find x.

I've tried to look online, but I can't seem to find anything. If someone could explain this, it would be great.

Answer 1

For solving a linear system of equations by LU decomposition you need to split the process in two parts:

You now that:

$A\cdot x = b$

So your guess is that:

$LU\cdot x = b$

Well, you are almost right! Now here's come the aforementioned split:

Calculate $y$ such that:

$L\cdot y = b$ (you have $L$, you have $b$, just calculate $y$)

And now let's solve for $x$

$U\cdot x = y$ (you have $U$ and $y$, solve for $x$).

And we are done!