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I have a matrix equation $$ \textrm{det}(\mathbb{M}) - \textrm{tr}(\mathbb{M}) + 1 = 0 $$ where $\mathbb{M}(z)$ is a matrix function of a complex number $z$ that I want to solve for. Because I have really two equations with two unknowns (real and imaginary parts), I tried Broyden's First Method as implemented in SciPy and cannot get it to converge.

To do the math with Broyden's method requiring real-valued things, I took a guess at $z_0$ as a 2-element array, broke the equation up into real and imaginary parts, with the $0 + 0i$ being the root. Trying to figure out the value of $z$ that satisfies this.

Any pointers for using multivariate root finding schemes?

webb
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  • How many initial values did you try? How did the iterations behave? – marty cohen Oct 09 '13 at 03:19
  • I've tried a number of initial values -- physically I know what the answer should be close to. As for behavior, it seemed to just hover around the initial value and never move anywhere. I'm still in the process of debugging my parameters though, so I may have simply settled somewhere weird. What kind of conditions keep this algorithm from converging? This is my first pass at using it. – webb Oct 09 '13 at 15:27
  • Yes, I can safely say that $f(x, y)$ just stays 1 regardless of the iterations. I'm digging in the Python a bit more but I seem to be using the API correctly. – webb Oct 09 '13 at 15:38

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