The Mathematical Theory of Iterative Methods

The current chapter contains results and basic concepts from the theory of iterative methods; these methods will be studied in the succeding chapters. In Section 5.1 we state the simplest concepts of functional analysis, give the basic properties of linear and non-linear operators in a Hilbert space, and also give several theorems on the solubility of operator equations. In Section 5.2, we give a systematic treatment of difference schemes as operator equations in an abstract space and indicate the properties of the corresponding operators. In Section 5.3, we look at the basic definitions and concepts from the theory of iterative processes, examine a canonical form for iterative schemes, and also the concepts of convergence and number of iterations.

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