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The + wasn't as well formed as it is elsewhere in the example code, looking again with the knowledge that is a + I can see that it is. That makes much more sense.

When I've got time I'll have a go at finding the roots of a polynomial using the example code as I want to allow the use of complex coefficients.

Thanks for help so far.

I got this with my FreeBasic program:


I have checked the roots with Maple.

I use the eigenvalue method for degree > 4, otherwise I use the algebraic solution.

The eigenvalue codes used in both "Numerical Recipes" and my own libraries are translations of the Fortran codes in the EISPACK library (http://www.netlib.org/eispack/). This library also has routines for complex matrices which could be translated too, but this may take some time...

On the other hand, if the degree of the polynomial does not exceed 5, so that the degree of the derivative does not exceed 4, the standard formulas should be used with complex numbers I think.

Laguerre's method is an iterative method. There is no guarantee that it will find all the roots.

I can't find any sign of routines for finding the eigenvalues of a matrix of complex numbers in TNT or lapack++ and the documentation isn't much help. I resorted to EISPACK and found a routine for finding the eigenvalues of a matrix of complex numbers. Eek! It's Fortran spaghetti in Fortran IV, even Fortran 77 would've made it easier. I've disentangled the code to produce a C++ which probably has numerous bugs in int.

I've been using polynomial formulae with parameters (coefficients) for each z term. The only thing that was lacking was a means of determining the critical points. My program, Saturn, only went up as far as quintics which reduces to a quartic derivative for determining the critical values. To find critical points easily I've been zeroing out some of the terms so that the derivative factors out to zero and an nth root of a number or zero and a quadratic. I'm now in the process of developing a program called Neptune which will automatically determines all the critical points and uses all the available coefficients.


I've also find it difficult finding code for C++ for matrices of complex numbers. I've also had trouble with the EISPACK code I downloaded, I built it using gfortran but for some reason the matrix gets corrupted when passed into a subroutine the first row was duplicated for all the remaining rows, using constant array dimensions fixed that problem so that I can step through the Fortran and C++ versions to find out where the C++ version goes astray. I suspect the problem may be because the code I downloaded looks like Fortran IV and the compiler doesn 't refer to any standards prior to Fortran 95. I'll have a look at the Fortran 90 code, that will be easier to deal with.

There haven't been any pictures for while:


https://copy.com/9RUM6ytyXcG6

This is for the polynomial for the above picture,

z = z^6 + z^5 + z^4 + z^3 + z^2 + z + c

So the critical points are the roots of

f'(z) = 6z^5 + 4z^4 + 3z^3 + 2z + 1 = 0

To make this general I'll need to add a parameter for each coefficient in the fractal formula and to multiply each of them by the associated power so that they can then be used as the coefficients of the polynomial used to find the critical points.

I've also included in Neptune

z = sin(z) + c

and will include formulae such as

z = alpha*z^2 + beta*z^(-2) + c

There will probably be a number of such formulae as it's easier to optimise specific formulae than it is for a general formula that allows the powers to changed with the proviso that they are always real and integers and of opposite signs. The critical points for these formulae are always the nth roots of a number and are consequently easy to find.

I have tried the formula z^p + c / z^q (with only one critical point). Here is an example with p = q = 2 :

I've also been playing around with fractals with multiple critical points lately. I think the "most official" way to display these fractals is to color a point black only if it's in the set for every critical point. That's because the resulting set of points is the connectedness map for the corresponding julia sets, which is a key property of the original mandelbrot set. I don't think there's a clear-cut best algorithm for coloring these things; the method I use is to iterate all of the critical points simultaneously and use the point with the greatest distance from the origin for coloring after bailout.

The problem with the above method is that the results can be kinda ugly, or at least nowhere near as graceful as the classic M-set. The resultant sets often don't seem to be truly fractal in some spots, although you can find interesting fractal detail in other spots.

So the other method I've experimented with is coloring each point black if it's in the set for ANY critical point. It comes a lot closer to the grace and symmetry of the M-set, but it it results in clearly overlapping and intersecting features, which make me feel it's somehow not ideal, either. Something I'm  reeally curious about: if the set of points on the parameter plane where all the critical points escape under iteration is the connectedness map of the corresponding Julia sets, then does the set of points where ANY critical points escape have some special topological meaning as well? I doubt anyone would know, but it's interesting to ponder.

Anyway, I read on Wikipedia that the behavior of cubic and higher degree polynomial maps is much more complicated than that of quadratic maps, so connectedness maps for cubic functions aren't really considered useful. I guess a simple, binary measure of behavior can't cut it for cubic fractals.

So, that brings me back to what you're doing. I think you're on the right track by combining the images from each critical point. But I wish I understood all this complex analysis and topology stuff so I could figure out what's really going on behind the scenes.