Most current prevalent iterative methods can be classified into the so-calledextended Krylov subspace methods, a class of iterative methods which do notfall into this category are also proposed in this paper. Comparing withtraditional Krylov subspace methods which always depend on the matrix-vectormultiplication with a fixed matrix, the newly introduced methods(the so-called(progressively) accumulated projection methods, or AP (PAP) for short) use aprojection matrix which varies in every iteration to form a subspace from whichan approximate solution is sought. More importantly an accelerativeapproach(called APAP) is introduced to improve the convergence of PAP method.Numerical experiments demonstrate some surprisingly improved convergencebehavior. Comparison between benchmark extended Krylov subspace methods(BlockJacobi and GMRES) are made and one can also see remarkable advantage of APAP insome examples. APAP is also used to solve systems with extremelyill-conditioned coefficient matrix (the Hilbert matrix) and numericalexperiments shows that it can bring very satisfactory results even when thesize of system is up to a few thousands.
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