Mathematical Methods for Physics and Engineering - Matematica.NET

NUMERICAL METHODSOf the four methods mentioned, no single one is ideal, and, in practice, somemixture of them is usually to be preferred. The particular combination of methodsselected will depend a great deal on how easily the progress of the calculationmay be monitored, but some combination of the first three methods mentioned,followed by the NR scheme if great accuracy were required, would be suitablefor most situations.27.2 Convergence of iteration schemesFor iteration schemes in which x n+1 can be expressed as a differentiable functionof x n , for example the rearrangement or NR methods of the previous section, apartial analysis of the conditions necessary for a successful scheme can be madeas follows.Suppose the general iteration formula is expressed asx n+1 = F(x n ) (27.13)((27.7) and (27.12) are examples). Then the sequence of values x 1 ,x 2 ,...,x n ,... isrequired to converge to the value ξ that satisfies bothf(ξ) =0 and ξ = F(ξ). (27.14)If the error in the solution at the nth stage is ɛ n ,i.e.x n = ξ + ɛ n ,thenξ + ɛ n+1 = x n+1 = F(x n )=F(ξ + ɛ n ). (27.15)For the iteration process to converge, a decreasing error is required, i.e. |ɛ n+1 |