Hello,
I would like to ask a few questions regarding the choice of equation-solving algorithms in MATLAB's fsolve function. I know the default is the 'trust-region-dogleg' (henceforth TRD) algorithm: does this mean this is preferable / 'superior' to others in some way? I am asking because I am attempting to solve a system of 554 simultaneous non-linear equations (a 'square' system, which TRD can handle); fsolve however takes a very long time to solve under TRD (assuming it ever does). I stopped it after the 10,000th iteration and the residual seems still quite high: please find the last few lines of the output below for your reference.
Norm of First-order Trust-region
Iteration Func-count f(x) step optimality radius
9996 5548335 181727 1.32714e-05 490 1
9997 5548890 181727 1.35706e-05 512 1
9998 5549445 181727 1.32713e-05 490 1
9999 5550000 181727 1.35705e-05 512 1
10000 5550555 181727 1.32711e-05 490 1
I thus decided to try an alternative algorithm, namely the Levenberg-Marquardt, by specifying 'Algorithm','levenberg-marquardt' in 'optimoptions'. The system was then solved much more efficiently: after 'only' 143 iterations (see last few lines below), the output stated:
fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.
Iteration Func-count Residual optimality Lambda step
140 78322 3.59762e-06 0.0252 1e-08 3.90698
141 78877 1.28284e-06 0.0877 1e-09 7.29042
142 79432 3.06443e-09 0.00444 1e-10 1.63511
143 79987 1.16041e-15 2.33e-06 1e-11 0.0374313
So I am wondering:
A) Am I safe picking Levenberg-Marquardt over TRD, or should the fact that TRD failed to solve effectively worry me?
B) What might be causing such a huge difference in outcomes?
I had a look at MATLAB documentation, which according to my understanding states that the choice of algorithms can ultimately be a matter of preference. However, it also suggests that one use TRD first, and that TRD "is the only algorithm that is specially designed to solve nonlinear equations". So I was wondering if I should try to make it solve more efficiently under TRD, or else if I should make do with Levenberg-Marquardt. Any ideas? Thank you very much!
