Finding all real roots of complex function with fsolve

Hi,

Is it possible to force fsolve to return only real roots of a complex function? I have this small code;

myfun = @det_sine_strip_Epol; % function

x = NaN*ones(22,1);             % Initializes x.

for i=1:22
        % Look for the zeros in the function's current window.
        x(i)=fsolve(myfun,i);
end
x

which returns me this;

 x =

   0.8834 + 2.8654i
   2.2244 + 3.0639i
   2.9243 + 2.7614i
   3.8319 + 0.0004i
   5.1355 + 0.0003i
   6.2468 + 2.7330i
   7.0153 + 0.0000i
   8.2545 + 2.8118i
   8.7717 + 0.0003i
   9.9364 + 0.0005i
  11.0631 + 0.0017i
  12.3366 + 2.6819i
  13.0150 + 0.0000i
  14.3013 + 2.5617i
  14.9308 - 0.0000i
  16.0375 + 0.0001i
  17.0036 + 0.0000i
  17.9597 + 0.0003i
  18.9801 + 0.0001i
  19.9942 + 0.0000i
  21.0889 + 0.0038i
  22.0467 + 0.0011i

where it actually should return ;

I am not specifically looking to calculate roots of the bessel, it is just one case.

Lastly, I see that fsolve missed couple of roots like (2.4048), and can find it only when i give initial guess as 2.4. Do you think it is supposed to be?

Thanks in advance.

Answer 1

[Matt J]

Is it possible to force fsolve to return only real roots of a complex function?

Although the documentation doesn't say so explicitly (that I can see), I don't think it is legal to apply fsolve to functions that are not R^n-->R^m. In other words, using complex-valued x and f(x) as you are doing is not allowed. The optimization toolbox solvers use algorithms based on real-valued optimization theory only, I am pretty sure, and any success with complex-valued input is going to be just luck.

You could, of course, re-write your objective function f(x) in real-valued form by breaking it into its real and imaginary parts,

   x(i)=fsolve(@(z) [real(myfun(z)), imag(myfun(z))] ,i);

This would always return a real-valued root, assuming that one exists and that your initial guess is sufficiently close to it. As Walter says, however, no numerical solver can find all roots, in general.

[Star Strider]

If you give fsolve a complex initial parameter estimate, it will find complex roots.

I’ve done it.

[Alan Weiss]

There is some relatively recent documentation about the use of complex numbers in Optimization Toolboxâ„¢ Solvers. In summary, if you have analytic objective functions, then indeed you can use fsolve to produce a root starting from a complex initial point.

Alan Weiss

MATLAB mathematical toolbox documentation

[Matt J]

In summary, if you have analytic objective functions, then indeed you can use fsolve to produce a root starting from a complex initial point.

Interesting. And this is an algorithm-independent, theoretical result? It doesn't matter, say, whether you use trust-region or Levenberg-Marquardt?

Answer 2

[Star Strider]

One way would be to define the initial estimates as:

r = linspace(0, 22, 100)
for i=1:length(r)
        % Look for the zeros in the function's current window.
        x(i)=fsolve(myfun,r(i));
end

xu = uniquetol(x, 1E-6);
xr = xu(imag(xu < 1E-6));

NOTE— This is UNTESTED CODE but should work. You may need to adjust the tolerances to give the result you want.

[ttopal]

Thank you for the answer, It seems uniquetol asks for a real valued parameters;

Error using uniquetol
Input A must be a real full matrix of type single or double.

Error in zerotest (line 17)
xu = uniquetol(x, 1E-6);

[Walter Roberson]

xr = real( x(abs(imag(x))<1E-6) );
xu = uniquetol(xr, 1E-6);

Answer 3

[Walter Roberson]

"Is it possible to force fsolve to return only real roots of a complex function?"

No. You need to use some other routine. For example if you have the symbolic toolbox then you could use vpasolve() with variables constrained to be real-valued (but even then, sometimes vpasolve will ignore constraints.)

"Finding all real roots of complex function"

It has been proven that there is no algorithm that can do that in the general case: there are functions for which it can be proven that knowledge of the function value at any finite number of other locations does not give you any information about the function value at any particular location. There are other functions for which that might not be the case but none-the-less the locations of the real zeros are beyond all current mathematics, with many people having put a lot of effort into some of the situations. For example, the Reimann Hypothesis about the Zeta function is a statement about complex zeros, but for some of the interesting properties the questions can be restated in terms of positive integers.