# Newton flowers

The »Newton fractal« (flower) for a given complex function is constructed by coloring each complex number (or rather each point on a finite grid). We iteratively apply Newton’s method to construct the sequence . If we detect that the sequence converges after terms (that is, if the difference between -th and previous term is smaller than a fixed convergence threshold constant), we will paint the pixel corresponding to according to , and . If the sequence does not converge after a fixed number of iterations, we deem it divergent and paint the pixel black. Note that if the sequence does converge, it is to some root of function .

The renderer implements two generalizations of »Newton fractals«.

### Nova fractals

The Newton method step in the above is substituted by a generalized

for some complex constants .

### Repetitions

A new layer of complexity can be implanted into the fractal in the following way. Repeated a fixed amount of times, when we detect convergence at the term , we simply restart the sequence by setting and continue with (restarting simultaneously the counter of the maximal number of iterations). The effects can be seen below, for repetitions set to 1 (usual Newton fractal), 2 and 5 respectively.

#### Function and its derivative

##### Must be a JavaScript expression. Available complex functions: .add(), .sub(), .mul(), .div(), .pow(), .sin(), .cos(), .abs(), .tan(); Examples:

##### Complex number must be entered as “new Complex(real, imag)”.

#### Parameters for the Newton’s method

##### Sets parameters a, c for the modified Newton’s method. Increase repetitions for added complexity.

#### Color scheme

##### Sets the color for a point as a function of n, the number of iterations until convergence. Needs to be a JavaScrip expression.

#### Display window and resolution

##### Sets the part of complex plane to be displayed and the number of pixels. More pixels will cause slower rendering.

##### Controls: left-click on canvas to *zoom*, right-click to *unzoom*. Click on the below arrows for *shift*.