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Mathematical Methods by Pearson
book

Mathematical Methods by Pearson

by E. Rukmangadachari
May 2024
Intermediate to advanced content levelIntermediate to advanced
437 pages
17h 41m
English
Pearson India
Content preview from Mathematical Methods by Pearson
Solution of Algebraic and Transcendental Equations 5-9
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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Publisher Resources

ISBN: 9781299446557Publisher Website