Two-layer iteration schemes

Two-layer iteration schemes. The problem statement. In what follows it is required to solve a first kind equation of the form 

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Before going further, we recall that any linear one-step iterative method can be written as 

The usual practice involves the numerical solution of problem (5) with a prescribed accuracy e 0 (a relative accuracy — w / j/o ID) being understood that the calculations should be terminated if 

Two-layer iteration schemes. The problem statement. In what follows 

In connection with inconvenience caused by the unknown vector u, it seems reasonable to replace this condition by the inequality for the residual 

In the general case the accepted view is the termination condition of the type 

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Factorized iteration schemes and ADM. The main idea behind this approach is connected with the equivalence between ADM from the preceding sections and the two-layer iteration scheme... 

Let us stress here that the applications of the above framework to noncommutative operators Ay and A2 could result in wrong reasoning in light of the property that operator (27) is non-self-adjoint and scheme (26) does not fall within the category of two-layer iteration schemes lying in the fundamentals of the general theory. 

Three-layer iteration schemes. So far we have considered two-layer iteration schemes available for solving operator equations of the form Au = / with a self-adjoint operator A under the assumption that the spectral bounds and for the operator A are known in advance either in a space H or in a space Hb, where B = B > 0 is some stabilizator. Other iterative methods find a wide range of applications in some or other aspects. 

Further reduction of a two-layer iteration scheme with the accompanying self-adjoint operators to an explicit one by replacing the variable w by R w leads to... 

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