Semiconvergence criteria of iterations and extrapolated iterations and constructive methods of semiconvergent iteration matrices

In this paper we simultaneously study iterations and extrapolated iterations for the consistent rectangular or sigular linear systems(1.1)Ax=b. In the case that A is rectangular, let A=M-N be a subproper splitting of A and let the generalized inverse M"T","S^(^1^,^2^) exist, and in the case that A is singular, let A=M-N is a usual splitting of A. Consider the iteration(2.3)x"j"+"1=Gx"j+c,where G=M"T","S^(^1^,^2^)Nandc=M"T","S^(^1^,^2^)b if A is rectangular, and G=M^-^1N and c=M^-^1b if A is singular, and the extrapolated iteration(2.19)x"j"+"1=G"@wx"[email protected],where @[email protected]?R,@w 0,1, and(2.20)G"@w=([email protected])[email protected] We establish the semiconcovergence criteria of the iterations (2.3) and (2.19), and present the efficient and convenient construction methods of the semiconvergent iteration matrices G and G"@w, and the choices of the extrapolated parameter @w.

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