A note on convergence of wilkinson and hilbert matrix with pre conditioned conjugate gradient method

Abstract

The present work is an attempt to make a study of Pre Conditioned Conjugate Gradient Method for almost Ill-conditioned matrices with computational sensitivity. Such matrices play very important role in solving physical problems arising in engineering and science. It is well known that the ill conditioning of a matrix system plays quite a dominant role adversely affect the performance of technique and the accuracy of the solution. In order to test validation of programmes 4 by 4 Wilkinson Matrix and 10 by 10 Hilbert Matrix are tested. Various computational aspects like as CPU time, number of iterations and convergence criteria are analyzed for this numerical solver. It is observed that Pre Conditioned Conjugate Gradient method is fast converging and gives very good results in both cases.