The Basic Abs Algorithm, Solving The Infinite Dimensional Linear Equations

The ABS class of algorithms initiated by J. Abaffy, C. G. Broyden and E. Spedi-cato(1982/1984)was originally given for solving the system of linear equations. Thebasic ABS algorithms only can be used to solve some finite dimensional linear systems,but can't be used to solve infinite dimensional linear systems. This paper extends the ABSmethod for solving finite dimensional linear systems directly to infinite linear systems in l₂space with bounded coefficient matrix in the sense of bounded linear operators. Our gen-eralized ABS methods are thus available for l₂ space. Fist, the basic ABS algorithms areextended to solving finite systems of linear equations in infinite dimensional space, called"semi-infinite"case. And then Algorithm I and some properties are given. Based on it,the case in which a complete infinite system of linear equations in an infinite dimensionalspace is to be solved is considered and a conceptual algorithm–Algorithm II is constructed.Therefore we investigate the convergence of Algorithm II for the complete infinite sys-tem. What is presented is that for computation purpose an implementable scheme, theε-truncated technique, for solving complete infinite systems of linear equations in infi-nite dimensional spaces. Finally, special algorithms, the general implicit LU algorithmand the Huang algorithm are considered. Their properties are given, too. Through the-ory prove,we can see this paper successfully extends the ABS method for solving finitedimensional linear systems directly to infinite linear systems in l₂ space.