Numerical Recipes Eq. 9.4.6!


by burnt
Tags: numerical, recipes
burnt
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#1
Sep22-10, 02:54 PM
P: 2
1. The problem statement, all variables and given/known data

I want to derive equation Eq. 9.4.6 in Numerical Recipes from the expressions given, as stated in the book!
The equation represents the next (i+1 th) deviation [tex]\epsilon[/tex] from the true root.
Eq. 9.4.6:

[tex]
\epsilon_{i+1} = -\epsilon_i^2 \frac{f''(x)}{2f'(x)}
[/tex]

2. Relevant equations

Eq. 9.4.5:

[tex]
\epsilon_{i+1} = \epsilon_i + \frac{f(x_i)}{f'(x_i)}
[/tex]

[tex]\epsilon_i[/tex] represents deviation from true root.


General Taylor expansion:

Eq. 9.4.3:
[tex]
f(x+\epsilon) = f(x) + \epsilon f'(x) + ...
[/tex]

[tex]
f'(x+\epsilon) = f'(x) + \epsilon f''(x) + ...
[/tex]



3. The attempt at a solution

[tex]
\epsilon_{i+1} = \epsilon_i^2 \frac{f''(x)}{f'(x) + \epsilon_i f''(x)}
[/tex]

but this is not equation 9.4.6! Please help!
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