What values should I use to obtain the non-real roots using Muller's method? The equation I am working on is: $$f(x) = x^4+2x^3+5x^2+5x-3$$
The equation has 2 real 2 non-real roots. I would like to solve for one of the non-real roots but I don't know what values I should give as a guess. I thought I must give 3 non-real numbers as a guess to make it converge to a non-real root. However I cannot really construct a parabola with them.
What values should I give? If they need to be real, how can I make it converge to a non-real root? If they need to be non-real, how can I construct a parabola or just its equation with non-real numbers?
Due to the square root from the quadratic solution formula, even if you start with real numbers the iteration can spontaneously leave the real axis. Of course you can also force this by setting one or multiple of the initial points to have an imaginary part. Note that this is no guarantee to find a complex root, the iteration may still converge to the real axis.
The roots of the given polynomial have magnitudes via root radius bounds between $\frac12$ and $5$, so you could for instance start with $x_0=2$, $x_1=2i$ and $x_2=-2$. Or you could construct points with a random radius in $[2,3]$ and a random angle.
Remember that Muller's method finds the roots of the quadratic Newton interpolation polynomial \begin{align} p(x)&=f(x_2)+f[x_1,x_2](x-x_2)+f[x_0,x_1,x_2](x-x_1)(x-x_2) \\ &=f(x_2)+\Bigl(f[x_1,x_2]+f[x_0,x_2]-f[x_0,x_1]\Bigr)(x-x_2)+f[x_0,x_1,x_2](x-x_2)^2 \end{align} and sets the next point $x_3$ as the root that is closest to $x_2$.
The divided differences can be computed for complex points the same way as for real points.
Introducing $w=f[x_1,x_2]+f[x_0,x_2]-f[x_0,x_1]$ gives the root computation \begin{align} 0&=4f_2^2+2\,(2f_2)\,w(x_2-x_1)+4f_2f_{012}(x-x_2)^2\\ &=(2f_2+w(x_2-x_1))^2-(w^2-4f_2f_{012})(x-x_2)^2\\ x&=x_2-\frac{2f_2}{w\pm\sqrt{w^2-4f_2f_{012}}} \end{align} In a complex version you will have to take the complex square root. It is easiest to follow if you compute both possible denominators and select the one with the larger absolute value, see for instance here.