As an example of the simplest of Tikhonov's methods, consider minimizing the following functional for the solution vector z:
This is the Tikhonov method with zeroth order regularization with regularization parameter . The zeroth order corresponds to the inclusion of the identity matrix in equation (9).
Other matrices that are used in practice include numerical approximations of the gradient or Laplacian, depending on which characteristic of the solution one is trying to constrain.
(See exercise 8.)