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. \[ x_n = \frac{y_n}{L_{nn}} \qquad x_{n-1} = \frac{y_{n-1} - L_{n-1,n}x_n}{L_{n-1,n-1}} \qquad x_{n-2} = \frac{y_{n-2} - L_{n-2,n-1}x_{n-1} - L_{n-2,n}x_n}{L_{n-2,n-2}} \ldots \]