On a successive transformation of probability distribution and its application to the analysis of the optimum gradient method

In this paper we define a type of transformation of probability distribution and analyze the limiting behavior of the result of successive applications of the transformation to some initial probability distribution. By using the results of this analysis we can get a fairly general insight into the so-called optimum-gradient method in numerical analysis. We can prove the conjecture which was stated by Forsythe and Motzkin [7] and was used as the logical basis of an acceleration procedure for the optimum gradient method [4][5][6]. It was stated by Forsythe [4] that this conjecture seems to be hard to prove as the related transformation is rather complicated. But our present proof is rather simple. Further, we can see the relation between the condition-number of the related matrix and the convergence rate of the optimum gradient method. By using the relation which according to [5] is first proved by Kantrovich [8], we can say that when the matrix is ill-conditioned the convergence rate tends near to its worst possible value. Using the same data as those treated by Forsythes in paper [5], we give some numerical examples.