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Solve nonlinear complex equations

Asked by Ed
on 10 Feb 2013
Latest activity Commented on by Meng Li
on 21 Jul 2015
Accepted Answer by Matt J


I am trying to numerically solve a nonlinear complex equation and I would like to find all the complex roots. The equation is of the type:

cot(z)*z = 1-z^2*(1+i*z)

Does a specific function exist to find all the complex roots or do I need to separate z in the real and imaginary parts?

Thanks in advance for your help!



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2 Answers

Answer by Matt J
on 10 Feb 2013
Edited by Matt J
on 10 Feb 2013
 Accepted answer

If you have the Symbolic Math Toolbox, I think SOLVE can be used to get complex-valued solutions. For the numerical solvers, I'm pretty sure you do have to reformulate the problem in terms of real and complex parts. Also, I've never heard of a numerical solver that will robustly find multiple roots for anything except polynomials.


Meng Li
on 20 Jul 2015

It seems there is no uploaded file. I will write down the equations.

Delta(eff)=4ac*Sum(1/n^2*tanh(alpha_n)/alpha_n+tanh(beta_n)/beta_n)   (n is odd integer, it starts from 1 to infinity);



You should start a new Question on this.

eff does not appear on the right hand side of your question so I do not know what the (eff) on the left relates to.

You define alpha_n and beta_n in terms of Delta(a) and Delta(c) but there is no obvious way of calculating either of those.

In your Delta(eff) formula, is 4ac = 4*a*c ?

Is alpha_n indicating alpha indexed at n?

Is the sum over odd n from 1 to infinity?

If Delta is being defined recursively (because it is defined in terms of alpha_n and beta_n that are defined in terms of Delta) then you need an initial condition.

Meng Li
on 21 Jul 2015

Delta(eff) is a experimentally measured value and this value can be separated into Delta(a) and Delta(c) through the first equation.

Yes. 4ac=4*a*c; Yes. alpha_n indicating alpha at n; Yes. the sum is over integer from 1 to infinity;

If I want to solve the equation, I should give some initial guess value for Delta(a) and Delta(c). Because the function 'fminsearch' can only give local solutions, I think the initial guess will be very important.

Answer by Azzi Abdelmalek
on 10 Feb 2013

Use fzero function

doc fzero
f=@(z)cot(z)*z -(1-z^2*(1+i*z))
sol=fzero(f,z0);  % the solution is near z0


Matt J
on 10 Feb 2013

It's interesting that this worked for z0=i, but it appears to be just a fluke. FZERO can't really handle complex-valued functions. Note,

>> sol=fzero(f,1+i)
Error using fzero (line 309)
Function value at starting guess must be finite and real.
on 10 Feb 2013

I tried the fsolve but unfortunately it seems to be related just to real functions. Thanks!

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