On Modern Matrix Iteration Processes of Bernoulli and Graeffe Type

are valid in a certain neighborhood of x. This definition states nothing concerning the behavior in the large, which is, in practice a t least, as impor tan t as the asymptot ic behavior. For matr ix problems, however, there exist special classes of i terative procedures which are linearly or quadratically convergent in the large, tha t is, for a lmost any choice of the trial solution; the word "a lmos t" is used in its classical meaning. These procedures, as we shall show explicitly later on, are based theoret ica l ly-but not numerically---on the step-by-step forward solution of certain linear difference equations with constant coefficients, like Bernoulli 's equation; therefore we call them procedures of Bernoulli type. As is well known, these solutions are weighted power sums of certain characteristic quantit ies dp :