Manual Newton's method faster than fsolve with jacobian for a nonlinear system

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CJ 2017-7-30

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https://ww2.mathworks.cn/matlabcentral/answers/350864-manual-newton-s-method-faster-than-fsolve-with-jacobian-for-a-nonlinear-system

评论： Milind Jain 2017-8-4

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I have a system of 100 nonlinear equations with 100 variables.

I use fsolve (giving it the analytical Jacobian) and it takes on average 0.033 seconds
I manually do Newton's method with the same tolerance level (1e-10) and it takes 0.009 seconds.

The quality of the results and their sensitivity to starting values are statistically the same. This difference in speed remains similar when I scale up to 1000 equations/variables [0.68 and 0.38 respectively].

Question: Am I misusing fsolve somehow? There's no way a fine-tuned optimizer like fsolve should be worse.

Note: Providing a working example here is too difficult as the function, its inputs, and the Jacobian take over a hundred lines.

I can provide the general syntax I'm using:

options = optimoptions('fsolve','SpecifyObjectiveGradient',true,...
    'MaxIter',10000,'Display','iter','TolFun',1e-6,'TolX',1e-6,'MaxFunEvals',30000);
[x]=fsolve(@(t)function(t,inputs),x0,options);

The manual method is as follows:

x0_new=x0;
dx=1;
while sum(dx.^2)>1e-6
    x0_0=x0_new;
    dx=function_taylor(x0_0,inputs);
    x0_new=x0_0+dx;
end

UPDATE: So I ran the profiler and I think it's the overhead in fsolve that causes the issue. Once I increase the number of equations, the speed differential declines a bit, and this is likely due to the shrinking share of time costs that the fsolve overhead represents.

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Milind Jain 2017-8-4

Can you please send us the code so that we can reproduce the same results at our end. This would help us in better understanding the issue and suggest improvements.

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