increasing fsolve accuracy when fval values are different in terms of order of magnitude

I have a set of six equations-six unknowns. When I solve it by fsolve, it "stops prematurely" and gives me 6 values for unknowns that are not exactly the answers that I'm expecting (I've tried different initial guesses, but it's the same). Answer set that fval gives is:

[x1=6.242, x2=2.148, x3=2.688, x4=1.563, x5=-3.359, x6=0.996]

and values of "fval" after solving are:

[-9e-9, 1e-10, -2e-6, 0.0054, -0.1208, -0.1615 ]

(The last 3 fvals are comparatively larger). Please let me know what I can do to enhance the results accuracy? Thanks

Answer 1

[John D'Errico]

The simple answer is the one you need to use.

SCALE YOUR VARIABLES. Scale them so they are all roughly the same order of magnitude. A factor of 10 is irrelevant. A dynamic range of 1e10 is a problem.

Then after the solve is done, scale them back.

[Sayed]

Dear John, Thanks for your answer. I'm not sure if you mean the same or not; however, I already multiplied appropriate coefficients by each of my unknowns in all 6 equations (e.g. all values of x1 are multiplied by 1e-3 and all x5 by 1e8). As a result, fsolve gives me all 6 unknowns while their absolute values are less than 10. Here is the answers I receive from fsolve:

x1=6.242, x2=2.148, x3=2.688, x4=1.563, x5=-3.359, x6=0.996

But, as I mentioned in my question, values of fval are not in the same order of magnitude. What causes the last three fval to be significantly larger than the first 3 ones? How we can prevent this to happen? It will be appreciated if let me know your point of view.

[Walter Roberson]

Why shouldn't the results be different orders of magnitude? Consider the following system of equations:

[x1 = .1; x2 = .2; x3 = .4; x5 = 10; x6 = 100]

now try to solve that system of equations with x5 and x6 not being very different from the other values.

[Sayed]

It's obvious that the answers might be from different order of magnitudes. I think what John has noted is that in case the answers are from various O.M., we can use an appropriate multiplier for each unknown in all equations in order to help the solver get more accurate results. I already tested this method in Mathematica; however, there is no change in answers before and after applying this method. But, still I don't know why values of "fval" are very different?

[Walter Roberson]

Why shouldn't they be very different? I showed a case where they must be very different. It is what is demanded by the system of equations.

Questions I would consider would include:

• was a mistake made in entering the equations?
• if the "known" answers are plugged into the equations then do the expected results get produced? This is a crosscheck onto whether the MATLAB expression has been implemented properly
• are there multiple solutions to the equations?
• when the solutions found are plugged into the MATLAB expression of the equations, do the equations hold to within tolerance? This tests whether MATLAB is doing a reasonable job with the function that it has been given
• when the solutions found are plugged into the equations on a different system, do the equations hold to within tolerance? This is a crosscheck on whether the equations have been expressed properly in MATLAB

What cause does fsolve() give when it stops "prematurely" ?

And I do not understand what you mean by drawing a difference between "Answer set that fval gives" and "values of "fval" after solving" ? What is "giving" the "Answer set" ? What is "solving" to give those other numbers?

[Sayed]

Dear Walter,

Thanks for commenting.

When we use fsolve, it returns two different things:

1- an answer set: that depending to your equations have different components and of course could be very different in their order of magnitude as well.

2- fval values that is the values of your equations function calculated with the final answer that Matlab has found (F==0). So, fval can't basically have very different values and all its components should be very close to zero; while in my case only the first 3 component are close to zero, but the rest are not [-9e-9, 1e-10, -2e-6, 0.0054, -0.1208, -0.1615 ]

This is fsolve message when stops:

Solver stopped prematurely.

fsolve stopped because it exceeded the function evaluation limit, options.MaxFunEvals = 9000 (the selected value).

[Walter Roberson]

Pass in a larger options.MaxFunEvals