Recursions and Their Stability

Abstract

Recursion formulae are algorithms in which an initial value is used to produce a new value, which in turn is inserted into the algorithm to produce again a new value, the process being repeated as many times as is desired. Jacobi’s iteration is just such a recursion, whose algorithm is the basis for solving a set of linear algebraic equations. We consider the simplest case of two such equations in two unknowns,

$$\begin{array}{*{20}{c}} {{{a}_{{11}}} \cdot {{x}_{1}} + {{a}_{{12}}} + {{x}_{2}} = {{b}_{1}}} \\ {{{a}_{{21}}} \cdot {{x}_{1}} + {{a}_{{22}}} \cdot {{x}_{2}} = {{b}_{2}}} \\ \end{array}$$