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Can we approximate the complex roots of nonlinear equations by using Bisection or Newton Raphson mehods? Is there any numerical example where these methods were applied to approximate complex roots of nonlinear equations?

Thank you very much for your kind help.

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    If you give the command `fsolve(@(x) cos(x) - 0.5 * log(x), -1)` to either MATLAB or Octave, they numerically compute the complex root `-1.3962 + 1.0575i`. – 2017-01-07
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    @FabioSomenzi Thanks for the reply. What algorithm matlab uses to determine complex roots? – 2017-01-07
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    If memory serves, `fsolve` is essentially Newton-Raphson. – 2017-01-07
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    As long as you have good starting estimates for your roots, Newton-Raphson will work. Bisection proper is really only intended for real roots bracketed by an interval with sign change in the function values. There are $n$-dimensional analogues of bisection, but I haven't seen any practical implementations of them. – 2017-01-07

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Just for illustration purposes.

Considering the example given by Fabio Somenzi in his comment, let us consider $$f(x)=\cos(x)-\frac 12 \log(x)$$ $$f'(x)=-\sin (x)-\frac{1}{2 x}$$ and using Newton method $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ Using $x_0=-1$, the iterates are then $$\left( \begin{array}{cc} n & x_n \\ 0 & -1.\\ 1 & -1.402768537+1.170950654\, i \\ 2 & -1.395966461+1.061622499\, i \\ 3 & -1.396202450+1.057509098\, i \\ 4 & -1.396204006+1.057504104\, i \end{array} \right)$$

Using $x_0=-5$, the iterates would be $$\left( \begin{array}{cc} n & x_n \\ 0 & -5 \\ 1 & -5.606638776-1.828794893\, i \\ 2 & -6.116952531-0.724676302\, i \\ 3 & -8.118034658+0.066008890\, i \\ 4 & -6.821533247+1.504217990\, i \\ 5 & -7.228499045+1.210834478\, i \\ 6 & -7.333452336+1.311166271\, i \\ 7 & -7.324671179+1.309558263\, i \\ 8 & -7.324653929+1.309589814\, i \end{array} \right)$$

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    Thank you very much for your answer. Have you computed it by using some codes? If yes, could you please provide me. Thank you again. – 2017-01-11
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    @srijan. I just coded what I wrote ! That's all. Cheers. – 2017-01-11