Mathematical Methods for Physics and Engineering : A Comprehensive Guide

NUMERICAL METHODS

Of the four methods mentioned, no single one is ideal, and, in practice, some

mixture of them is usually to be preferred. The particular combination of methods

selected will depend a great deal on how easily the progress of the calculation

may be monitored, but some combination of the first three methods mentioned,

followed by the NR scheme if great accuracy were required, would be suitable

for most situations.

27.2 Convergence of iteration schemes

For iteration schemes in whichxn+1can be expressed as a differentiable function

ofxn, for example the rearrangement or NR methods of the previous section, a

partial analysis of the conditions necessary for a successful scheme can be made

as follows.

Suppose the general iteration formula is expressed as

xn+1=F(xn) (27.13)

((27.7) and (27.12) are examples). Then the sequence of valuesx 1 ,x 2 ,...,xn,...is

required to converge to the valueξthat satisfies both

f(ξ)=0 and ξ=F(ξ). (27.14)

If the error in the solution at thenth stage isn,i.e.xn=ξ+n,then

ξ+n+1=xn+1=F(xn)=F(ξ+n). (27.15)

For the iteration process to converge, a decreasing error is required, i.e.|n+1|<

|n|. To see what this implies aboutF, we expand the right-hand term of (27.15)

by means of a Taylor series and use (27.14) to replace (27.15) by

ξ+n+1=ξ+nF′(ξ)+^12 ^2 nF′′(ξ)+···. (27.16)

This shows that, for smalln,

n+1≈F′(ξ)n

and that a necessary (but not sufficient) condition for convergence is that

|F′(ξ)|< 1. (27.17)

It should be noted that this is a condition onF′(ξ) and not onf′(ξ), which

may have any finite value. Figure 27.3 illustrates in a graphical way how the

convergence proceeds for the case 0<F′(ξ)<1.

Equation (27.16) suggests that ifF(x) can be chosen so thatF′(ξ) = 0 then the

ratio|n+1/n|could be made very small, of ordernin fact. To go even further,

if it can be arranged that the first few derivatives ofFvanish atx=ξthen the