I.3.6.1: LU-form

In block relaxation methods, including generalized block methods, we update to by the rule Differentiation gives It should be emphasized that in many cases of interest in does not depend on so that for all . It is also important to realize that the derivatives, which we write without arguments in this section, are generally evaluated at points of the form . At fixed points, however, for all , and we can just write without ambiguity. And for our purposes the derivatives at fixed points are the interesting ones.

Now define and Also and so that . From or This is the Lower-Upper or LU form of the derivative of the algorithmic map.

For two blocks is equal to and if and this is Thus the non-zero eigenvalues are the eigenvalues of .