An Adaptive Multi-step Levenberg–Marquardt Method

We propose an adaptive multi-step Levenberg–Marquardt (LM) method for nonlinear equations. The adaptive scheme can decide automatically whether an iteration should evaluate the Jacobian matrix at the current iterate to compute an LM step, or use the latest evaluated Jacobian to compute an approximate LM step, so that not only the Jacobian evaluation but also the linear algebra work can be saved. It is shown that the adaptive multi-step LM method converges superlinearly under the local error bound condition, which does not require the full column rank of the Jacobian at the solution. Numerical experiments demonstrate the efficiency of the adaptive multi-step LM method.