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Can the complex step derivative (https://mdolab.engin.umich.edu/wiki/guide-complex-step-derivative-approximation) be used for functions of complex variables ?

$f: \mathbb{C^n} \mapsto \mathbb{C^n} $ or $f: \mathbb{C^n} \mapsto \mathbb{R^n}$

I read the papers, but couldn't figure it out.

My problem is this: I have 4 equations.

  1. $e = m - y$,
  2. $y = W_3 z$
  3. $z = e^{t\, ( A + W_2 )} b $
  4. $A = \text{diag}(i \, W_1 \, x) $, where

$y, m \in \mathbb{R}$,

$z,b,x \in \mathbb{C^n}$

$W_1, W_2, A \in \mathbb{C^{nxn}} $, $W_3 \in \mathbb{C^{1xn}}$

I need to update matrices $W_1$, $W_2$, and $W_3$ by using gradient descent in order to minimize the cost function: $J = e^T e$

Equation 3) is very hard. So I was thinking about using the complex step derivative method.

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    $\begingroup$ The whole point of complex step is that $f(x)$ is real but $f(x+h)$ is not so taking the imaginary part can extract the perturbation accurately. So I doubt it $\endgroup$
    – whpowell96
    Commented Apr 19 at 13:49

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