Generalized Inverse Of A, <sub> T, S </ Sub> ~ (2) Approximation, Application And Testing Of The Iteration Method And The Drazin Inverse

The representation and calculation of linear operator is an important topic in the theoy of generalized inverse. It is highly valued in the field- of both theotical research and practical use.(Refs: [1, 2, 3, 9, 10, 11, 12, 13, 17, 21, 23, 30] etc.) Many scholars ,such as Adi Ben Israel, K.Kato, M.Z.Nashed, Y.Y.Tseng, J.Ding, Jiaoxun Kuang, Guoliang Chen, Yifeng Xu, Sanzheng Qiao, Yimin Wei, etc. have done much research with good results in the existence, properties, representation, approximation, perturbation, application and computation of Moore-Penrose generalized inverse,Drazin inverse and group inverse on Hilbert spaces and Banach spaces, while the results of which are still disperse, in sharp contrast to the case of matrix. The first part of this paper is based on A(2)T,S , estabishing the unitive representation and Iteration of generalized linear operator inverse. We discussed the limit representation of the A(2)T,S on Banach spaces, Neumann series and integration as well as the evaluation of the error of approximation. On the basis of all these representations, we formed several iterations, such as Neumann iteration, Newton iteration, hyperpower iteration and the iteration based on the function interpolation and discussed the conditions of convergence, we conducted the Newmeric experiments as well.In the second part, we discussed the techques of the acceleration of convergence, and applied it to the computation of A(2)T,S of Matrix, so that the speed of convergence in Neumann iteration was raised from 1-order to 2, and the speed of hyperpower iteration from p-order to 2P, while the complexity of computation is only twice what we did before. This method was originally raised by B.Condenotti [36], Later,L.Chen, E.V.Krishnamurthy, I.Macleod(1994)[37] and Yimin Wei [38] applied it to the computation of M-P inverse and Drazin inverse. The method of block Matrix discussed in this paper generalized the case mentioned in [37, 38] and got quicker convergence series. At last, we found an unconditioned way of analysis of Markov chains and used it in the computaiton.It is always an issue concerned by the computing workers about how to evaluate the advantages and disadvantages of algorithms. G.Zielke [49] discussed the test of computation of Moore-Penrose generalized inverse first, based on which we discussed the test of computation of Drazin inverse in the third part, introduced the way to form test Matrices, tested some common methods for Drazin inverse and listed the reports of it. We discovered the calculation of group inverse, The QAQ method formed in this paper and Neumann iteration both work well,which can calcalate large scale Matrix; the Baddeev method is especially suitable for calculating small scale Matrix, but it can't deal with Matrix whose order is larger than 50, so it is with Newton iteration and hyper-power iteration. As for the calculation of Drazin inverse(group inverse excluded) , the test results is very .disappointing. The increase of index adds to the diffkuty in calculation.