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Correct implementation of Laguerre's method #27

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Jun 16, 2024
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Correct implementation of Laguerre's method #27

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Correct implementation of Laguerre's method
rootmos Jun 14, 2024
62b3976
Update time-since-periapsis-and-keplers-equation/universal-variables.md
bryanwweber Jun 16, 2024
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2 changes: 1 addition & 1 deletion time-since-periapsis-and-keplers-equation/universal-variables.md
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Expand Up @@ -312,7 +312,7 @@ The Laguerre algorithm can be implemented as:

:::{math}
:label:
\chi_{i + 1} = \chi_{i} - \frac{n f(\chi_i)}{f'(\chi_i) \pm \left[\left(n - 1\right)^2 \left(f'(\chi_i\right)^2 - n\left(n - 1\right) f(\chi_i)f''(\chi_i)\right]}
\chi_{i + 1} = \chi_{i} - \frac{n f(\chi_i)}{f'(\chi_i) \pm \left[\left(n - 1\right)^2 \left(f'(\chi_i)\right)^2 - n\left(n - 1\right) f(\chi_i)f''(\chi_i)\right]^{1/2}}
:::

The sign ambiguity in the denominator is determined by taking the sign of the numerical value of $f'(\chi_i)$. In addition, the solution is relatively insensitive to the choice of the value of $n$, which is an integer constant. It seems as though $n = 5$ is a reasonable value. Choosing $n = 1$ gives the standard Newton's algorithm.
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