On The Convergence Of General Parallel Multi-splitting Itera-tive Methods For Semi-definite Linear Systems

First, this paper proposes the general non-stationary multi-splittingalgorithm. whenR_j~T+R_j R_j~T AR_j,(j= 1,2,···, J) are symmetric positivedefinite matrix onR(A), the quotient convergence and convergence areequivalent for this algorithm. we can also prove that general stationarymulti-splitting algorithm must be semi-norm convergence, furthermore, itis convergence on this conditions.The above results are the generalizationof the result in [18] to Cao. Then we extend the algorithm to general non-stationary multi-splitting algorithm. Because for each iterative numberk, inner iterative number is diferent, we have not discuss the equivalentof quotient convergence and convergence, we can only prove a relativelyweak results:If R_j~T+Rj R_j~T AR_j,(j= 1,2,···, J) are symmetric positivedefinite matrix onR(A), and the sequence of number of iterations satisfiesμk_j> K, then general non-stationary multi-splitting algorithm is semi-norm convergent, furthermore, it is quotient convergent. Finally, this paperproposes the general two-stage non-stationary multi-splitting algorithm. IfjTR_j~T+R_j~T AR_j,(j= 1,2,···, J) are symmetric positive definite matrixonR(A),R(A) R(M), and the sequence of number of iterations satisfiesμk,j> K, then general non-stationary two-stage multi-splitting algorithmis quotient convergent.