I would like to solve a set of N = 300 independent 1D nonlinear equations of the type f_i(x_i) = a_i where the a_i are given and each function f_i is a univariate strictly increasing bijection. There is a unique solution for each nonlinear equation. So far, I am using NLsolve.jl with the Newton method to solve simultaneously the equations. I am already using the fact that the Jacobian is diagonal (all the problems are independent).
However the f_i have some bumps (see Figure) that are sometimes challenging for the Newton method if the initial condition is far from the root. How can I improve the convergence?
Is there a way to exploit that the Jacobian has only strictly positive values?