NUMERICAL METHODS
nxn+1 n
18.5 4.5
2 5.191 1.19
3 4.137 1. 4 × 10 −^1
4 4.002 257 2. 3 × 10 −^3
5 4.000 000 637 6. 4 × 10 −^7
64 —Table 27.5 Successive approximations to√
16 using the iteration scheme
(27.20).The following is an iteration scheme for finding the square root ofX:xn+1=1
2
(
xn+X
xn)
. (27.20)
Show that it has second-order convergence and illustrate its efficiency by finding, say,√
16
starting with a very poor guess,√
16 = 1.
If this scheme does converge toξthenξwill satisfy
ξ=1
2
(
ξ+X
ξ)
⇒ ξ^2 =X,as required. The iteration functionFis given by
F(x)=1
2
(
x+X
x)
,
and so, sinceξ^2 =X,
F′(ξ)=1
2
(
1 −
X
x^2)
x=ξ=0,
whilst
F′′(ξ)=(
X
x^3)
x=ξ=
1
ξ=0.
Thus the procedure has second-order, but not third-order, convergence.
We now show the procedure in action. Table 27.5 gives successive values ofxnand of
n, the difference betweenxnand the true value, 4. As we can see, the scheme is crude
initially, but oncexngets close toξ, it homes in on the true value extremely rapidly.
27.3 Simultaneous linear equationsAs we saw in chapter 8, many situations in physical science can be described
approximately or exactly by a set ofNsimultaneous linear equations inN