  When minimizing a nonlinear least-squares function, the Levenberg-Marquardt
algorithm can suffer from a slow convergence, particularly when it must
navigate a narrow canyon en route to a best fit. On the other hand, when the
least-squares function is very flat, the algorithm may easily become lost in
parameter space. We introduce several improvements to the Levenberg-Marquardt
algorithm in order to improve both its convergence speed and robustness to
initial parameter guesses. We update the usual step to include a geodesic
acceleration correction term, explore a systematic way of accepting uphill
steps that may increase the residual sum of squares due to Umrigar and
Nightingale, and employ the Broyden method to update the Jacobian matrix. We
test these changes by comparing their performance on a number of test problems
with standard implementations of the algorithm. We suggest that these two
particular challenges, slow convergence and robustness to initial guesses, are
complimentary problems. Schemes that improve convergence speed often make the
algorithm less robust to the initial guess, and vice versa. We provide an open
source implementation of our improvements that allow the user to adjust the
algorithm parameters to suit particular needs.
