TITLE: Solving a linear system with Cholesky factorization
QUESTION [10 upvotes]: So, I've reached the following problem:
Given $A \in R^{nxn} $  positive-definite and symmetrical, $B\in R^{nxn}$ and the vector $c\in R^n$ write an algorithm to solve to following situation:
$Ax=Bc$
After I break down A with Cholesky in $ L * L^t $ I don't know how should I proceed, should I get the inverse and find x or is there a easier method?

REPLY [16 votes]: The idea is the same of LU decomposition, i.e. use the triangular for of the matrix $L$.
For simplicity put, $Bc = b \, \in \mathbb{R}^n$, so the system is:
$$
\begin{aligned}
Ax &= b \\
L L^{T} x &= b
\end{aligned}
$$
now you call $L^{T} x = y $ and you solve the system:
$$
\left \{
    \begin{aligned}
     Ly &= b \\
     L^{T} x &= y
    \end{aligned}
\right.
$$
The matrix $L$ is triangular so you solve it directly with forward and back substitution, first $Ly =b$ and after $L^{T} x = y$.
See also wiki page.