Here's a detailed description of the Cholesky method for solving the system $Ax = b$.

Decompose $A$

Given a symmetric positive definite matrix $A$, we can decompose it to find $L$ in $A = L L^T$, where $L$ is a lower triangular matrix.

Forward Substitution

Solve the system $Ly = b$ for $y$. This can be done using forward substitution since $L$ is a lower triangular matrix. $$y_1 = \frac{b_1}{L_{11}} \qquad y_2 = \frac{b_2 - L_{21}y_1}{L_{22}} \qquad y_3 = \frac{b_3 - L_{31}y_1 - L_{32}y_2}{L_{33}} \ldots$$

Backward Substitution

Once $y$ is found, solve the system $L^T x = y$ for $x$ using backward substitution since $L^T$ is an upper triangular matrix. $$x_n = \frac{y_n}{L_{nn}} \qquad x_{n-1} = \frac{y_{n-1} - L_{n-1,n}x_n}{L_{n-1,n-1}} \qquad x_{n-2} = \frac{y_{n-2} - L_{n-2,n-1}x_{n-1} - L_{n-2,n}x_n}{L_{n-2,n-2}} \ldots$$