# Need help implementing an algorithm to solve roots of a transcendental equation

I'm trying to implement this algorithm but I'm having problems reproducing the exemple that it gives a solution to.

The general method that I tried is:

1. Make a grid $$\theta \in [0,2\pi)$$, with say N=256 points in it
2. Create a complex vector $$G(\theta) = \frac{1}{F(e^{i\theta})}$$, where $$F(z) = z\tan(z) - 1$$ is the complex extension of the transcendental equation in question.
3. Solve $$G_{k} = \int\limits_{0}^{2\pi} G(\theta)e^{ik\theta}d\theta$$ for as many $$k$$ as I need.

If I understood correctly, there'll be a $$nroot\times nroot$$ symmetric matrix with $$2\times nroot$$ independent elements. E.g for $$nroot = 3$$ I will have to solve for $$G_{1},G_{2},G_{3},G_{4},G_{5},G_{6}$$. Eventually $$G_{6}$$ will go to the right side of the equation as when we fix the value of $$c_{3}$$.

1. Solve the system $$A\cdot X = -B$$

The problem that I'm getting is the following:

The eventual values of $$c_{k}$$ are complex, and in the system that the author gets to in (18) there are just real numbers. The author mentions the use of the routine cfft which I'm unaware of, I solved the integral explicitly and with ordinary FFT and in neither I get real numbers. Even the real part of the $$c_{k}$$'s I'm getting, even though are of the same order of the solution it is far from it.

My solution is just wrong, I get the wrong system therefore the wrong $$c_{k}$$'s and I'm unable to get to the solution.

• Your title doesn't reflects the topic rather than your actual question. – Yuval Filmus Oct 7 '18 at 19:07
• I'll change to a more appropriate one . – Matheus S. Oct 7 '18 at 19:13