From 3e51bd7e53194b1065e83623775fd633a55f27eb Mon Sep 17 00:00:00 2001 From: Gustav Behm Date: Sun, 16 Jun 2024 18:31:32 +0200 Subject: [PATCH] Correct implementation of Laguerre's method (#27) * Correct implementation of Laguerre's method * Update time-since-periapsis-and-keplers-equation/universal-variables.md --------- Co-authored-by: Bryan Weber --- .../universal-variables.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/time-since-periapsis-and-keplers-equation/universal-variables.md b/time-since-periapsis-and-keplers-equation/universal-variables.md index c889ec5..c8ef22a 100644 --- a/time-since-periapsis-and-keplers-equation/universal-variables.md +++ b/time-since-periapsis-and-keplers-equation/universal-variables.md @@ -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.