082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

220 Chapter 6 Sequences and series
whenthesevaluesaresubstitutedintotheequationbothsidesareequal.Equationswhere
the unknown quantity, x, occurs only to the first power are called linear equations.
Otherwise an equation is non-linear. A simple way of finding the roots of an equation
f (x) = 0 istosketch a graph ofy = f (x)as shown inFigure 6.8.
The roots are those values of x where the graph cuts or touches the x axis.
Generally, thereisnoanalytical wayofsolvingtheequation f (x) = 0andsoitisoften
necessary to resort to approximate or numerical techniques of solution. An iterative
technique is one which produces a sequence of approximate solutions which may converge
to a root. Iterative techniques can fail in that the sequence produced can diverge.
Whetherornotthishappensdependsupontheequationtobesolvedandtheavailability
ofagoodestimateoftheroot.Suchanestimatecouldbeobtainedbysketchingagraph.
The technique we shall describe here is known as simple iteration. It requires that the
equation berewrittenintheformx =g(x). Anestimateoftherootismade,sayx 0
,and
this value is substituted into the r.h.s. ofx =g(x). This yields another estimate,x 1
. The
process isthen repeated. Formally weexpress this as
x n+1
=g(x n
)
This is a recurrence relation which produces a sequence of estimates x 0
,x 1
,x 2
,....
Under certain circumstances the sequence will converge to a root of the equation. It
is particularly simple to program this technique on a computer. A check would be built
into the program totestwhether ornot successive estimates areconverging.
Example6.20 Solve the equation f (x) = e −x −x = 0 by simple iteration.
Solution Theequationmustfirstbearrangedintotheformx =g(x),andsowewritee −x −x = 0
as
x=e −x
In this example we see that g(x) = e −x . The recurrence relation which will produce
estimates of the rootis
x n+1
= e −x n
Table6.1
Iterativesolutionof
e −x −x=0.
n
x n
f(x)
Figure6.8
A rootof f (x) = 0 occurswhere the
graph touches orcrossesthexaxis.
x
0 0
1 1
2 0.368
3 0.692
4 0.501
5 0.606
6 0.546
.
.
. 0.567