Mathematical Methods for Physicists: A concise introduction - Site Map

NUMERICAL METHODS
Figure 13.2.
course, also equation 13.4). Now setting x ˆ x 1 on the right hand side of Eq.
(13.4) we obtain the equation
g…x† ˆh…x 1 †;
…13:6†
which by hypothesis we can solve. If the solution is x 2 , we set x ˆ x 2 on the right
hand side of Eq. (13.4) and obtain
g…x† ˆh…x 2 †:
By repeating this process, we obtain the nth approximation
g…x† ˆh…x n1 †:
…13:7†
…13:8†
From geometric considerations or interpretation of this procedure, we can see
that the sequence x 1 ; x 2 ; ...; x n converges to the root x ˆ 0 if, in the interval
2jx 1 x 0 j centered at x 0 , the following conditions are met:
…1† jg 0 …x†j > jh 0 )
…x†j; and
…13:9†
…2† The derivatives are bounded:
Example 13.1
Find the approximate values of the real roots of the transcendental equation
e x 4x ˆ 0:
Let g…x† ˆx and h…x† ˆe x =4; so the original equation can be rewrit-
Solution:
ten as
x ˆ e x =4:
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