Abstract
In different fields in space researches, Scientists are in need to deal with the product of matrices. In this paper, we develop conditions under which a product Пi=0∞ of matrices chosen from a possibly infinite set of matrices M={Pj, j∈J} converges. There exists a vector norm such that all matrices in M are no expansive with respect to this norm and also a subsequence {ik}k=0∞ of the sequence of nonnegative integers such that the corresponding sequence of operators {Pik}k=0∞ converges to an operator which is paracontracting with respect to this norm. The continuity of the limit of the product of matrices as a function of the sequences {ik}k=0∞ is deduced. The results are applied to the convergence of inner-outer iteration schemes for solving singular consistent linear systems of equations, where the outer splitting is regular and the inner splitting is weak regular.
Highlights
Let the standard iterative method for solving the system of linear equations Ax b (1)Txk 1 Qxk b (2)is used to compute a sequence of iterations whose limit should be the solution to Equation (1).If A is a nonsingular matrix, to obtain a good approximation to the solution of Equation (1), one need not to even solve the system (2) exactly for each xk 1
There exists a vector norm such that all matrices in M are no expansive with respect to this norm and a subsequence
Txk 1 Qxk b is used to compute a sequence of iterations whose limit should be the solution to Equation (1)
Summary
Let the standard iterative method for solving the system of linear equations Ax b (1)Txk 1 Qxk b (2)is used to compute a sequence of iterations whose limit should be the solution to Equation (1).If A is a nonsingular matrix, to obtain a good approximation to the solution of Equation (1), one need not to even solve the system (2) exactly for each xk 1. Let the standard iterative method for solving the system of linear equations where A Rn,n and x, b are n-vectors [1], be induced by the splitting of A into A T Q , where T is a nonsingular matrix. Pj , j J , and there exists such that each matrix in M is a vector norm no expansive with on respect to .
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