Solution of Algebraic and Transcendental Equations5-9
Example 5.5
Using the method of iteration find a positive root of
the equation xe
x
= 1 that lies between 0 and 1.
Solution We write the equation
f ( x) =xe
x
− 1 = 0 (1)
in the form x=g(x) =e
−x
(2)
Iteratative formula is
1
=()=
+
n
x
nn
xgxe
(3)
We find g(x) =e
−x
and g¢(x) =−e
−x
so that |g¢(x)| < 1 for x< 1. Thus, the convergence of
iterative formula (3) is assured.
Let us take the starting value x
0
= 1. We find the
successive iterates in Table 5.2, which are accurate
up to four decimal places.
By the Intermediate Value Theorem, a root of
(1) lies in
1
, 1.
2
⎛⎞
⎜⎟
⎝⎠
From equation (1) we hav
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