What is the first derivative of this function?

Question

I am trying to implement a computer simulation of the algorithm described in this paper.

I need to find a root using Newton's method. The problem is that I need to know the derivative of the complex modulus function and my math knowledge is not helping me a lot. The function is described at p. 4 of the PDF file:

I mention that the function contains complex values, excepting the absolute values (|.|) and lambda.

Can anybody help me with the first derivative of this function?

Answer 1

Don't bother with Newton's method. You need to choose an initial $\lambda$, so choose two and use the secant method.

A good discussion is here:

https://en.wikipedia.org/wiki/Secant_method

Answer 2

Write each term of the sum as $$f_j(\lambda)=\left|{a_j\over b_j+\lambda}\right|^2= {a_j\over b_j+\lambda} {\overline{a_j}\over \overline{b_j}+\lambda}$$ Then we just differentiate by the product rule: $$\begin{align} f'_j(\lambda)&=-{a_j\over b_j+\lambda}{\overline{a_j}\over (\overline{b_j}+\lambda)^2}-{a_j\over( b_j+\lambda)^2}{\overline{a_j}\over \overline{b_j}+\lambda}\\ &={-|a_j|^2\over|b_j+\lambda|^2}\left({1\over b_j+\lambda}+{1\over \overline{b_j}+\lambda}\right)\\ &={-|a_j|^2\over|b_j+\lambda|^2}\left({2\Re(b_j+\lambda)\over|b_j+\lambda|^2}\right)\\ &=-2|a_j|^2{\lambda+\Re(b_j)\over|b_j+\lambda|^4} \end{align}$$

Answer 3

To avoid nightmares and to keep the power of Newton method, compute the derivative by central finite differences.