The example Use Newtonâ€™s method to draw a fractal in C# shows how to use Newton’s method to draw fractals by finding the roots of equations of the form F(z) = z^{C} – 1 for complex numbers z and some constant C.

You can also use Newton’s method to find the roots of other equations. This example find roots for equations of the form F(z) = z^{C} – C^{z} for constants C.

The previous example picks a point’s color depending on the root to which it is drawn by Newton’s method. It’s not as easy to find the roots of this equation analytically, so this example uses a different method for coloring points. It colors a point based on how many iterations it takes for Newton’s method to settle down to a particular value for the point.

The following code shows how this program colors its points.

// Calculate the values. Complex x0 = new Complex(Wxmin, 0); for (int i = 0; i < wid; i++) { x0.Im = Wymin; for (int j = 0; j < hgt; j++) { int clr = 0; // Default to black. Complex x = x0; Complex epsilon; const int max_iter = 400; int iter = 0; do { if (++iter > max_iter) break; epsilon = -(F(x) / dFdx(x)); x += epsilon; } while (epsilon.MagnitudeSquared() > cutoff); // Set the color. if (iter <= max_iter) { clr = iter % (Colors.GetUpperBound(0) + 1); } // Set the pixel's color. Bm.SetPixel(i, j, Colors[clr]); // Move to the next point. x0.Im += dy; } // For j x0.Re += dx; // Let the user know we're not dead. if (i % 10 == 0) picCanvas.Refresh(); } // For i

See the code and the previous example for additional details.

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