Iterative methods are often used for solving discretized partial differential equations. In that context a rigorous analysis of the convergence of simple methods such as the Jacobi method can be given.
As an example, consider the boundary value problem
The eigenfunctions of the and operator are the same: for the function is an eigenfunction corresponding to . The eigenvalues of the Jacobi iteration matrix are then .
From this it is easy to see that the high frequency modes (i.e., eigenfunction with large) are damped quickly, whereas the damping factor for modes with small is close to . The spectral radius of the Jacobi iteration matrix is , and it is attained for the eigenfunction .
Spectral radius: The spectral radius of a matrix is . Spectrum: The set of all eigenvalues of a matrix.
The type of analysis applied to this example can be generalized to higher dimensions and other stationary iterative methods. For both the Jacobi and Gauss-Seidel method (below) the spectral radius is found to be where is the discretization mesh width, i.e., where is the number of variables and is the number of space dimensions.