A1. Single Function Iteration Algorithm (when only one INPUT FUNCTION is selected):

I. Call

• the selected INPUT FUNCTION by the name F(z)

• the (complex number) SEED VALUE by the name z₀

• the number of POINTS TO PLOT by the name N

• the number of SKIPS by the name S
   

II. Find a complex number z₁ such that F(z₁)=z₀ (this involves finding an inverse image under the map F(z) - if more than one inverse image exists, then one of these is randomly chosen to be z₁). Then, likewise find z₂ such that F(z₂)=z₁. Iterate this process N+S times to get a sequence of complex numbers z₀, z₁,..., z_N+S such that F(z_n)=z_n-1.

III. The set of points {z_S+1, z_S+2,..., z_N+S}, formed by skipping the first S points of the sequence, is called the (Approximate) Julia set of F(z).  It is this set which is created as an OUTPUT SET and labeled with the map which generated it.  For N and S large we get better approximations to the actual Julia set of F(z).
 

B1. Multiple Function (Semigroup) Algorithm (when multiple INPUT FUNCTIONS are selected):

We follow the above algorithm A1, but with a slight change.  Call the selected INPUT FUNCTIONS by the names G₁,..., G_k and change item II above as follows. Starting with z₀ we randomly choose a map F₁ from G₀, ..., G_k and then calculate z₁ such that F₁(z₁)=z₀.  Then randomly choose a map F₂ from G₁, ..., G₂ and then calculate z₂ such that F₂(z₂)=z₁ . Iterate this process N+S times to get a sequence of complex numbers z₀, z₁, ..., z_N+S.  The set of points {z_S+1, z_S+2,..., z_N+S}, formed by skipping the first S points of the sequence, is called the (Approximate) Julia set of the semigroup <G₁,...,G_k> (acting in parallel).  In the right window the OUTPUT SET is labeled the COMPOSITE JULIA SET.

A2. Single Function Iteration Algorithm (when only one INPUT FUNCTION is selected):

I. Call

• the selected INPUT FUNCTION by the name F(z)

• the (complex number) SEED VALUE by the name z₀

• the number of POINTS TO PLOT by the name N

• the number of SKIPS by the name S (this is not relevant when PROCESS TYPE = FULL)
   

II. Find F^-1({z₀}), the set of ALL complex numbers w such that F(w)=z₀ (this involves finding an inverse image under the map F(z)). Then find F^-2({z₀}) = F^-1(F^-1({z₀})), the set of ALL complex numbers w such that F²(w)=z₀. Continue this process until the set F^-j({z₀}) = F^-1(...(F^-1({z₀}))) has at least N points.  This algorithm will often produce a set with many more than N points.

Note that if F(z) has degree 1, then all sets of the form F^-j({z₀}) will have only one point.  In this case the program is set to simply terminate with j=N.

III. The set of points F^-j({z₀}) is called the (Approximate) Julia set of F(z).  It is this set which is created as an OUTPUT SET and labeled with the map which generated it.  For N large we get better approximations to the actual Julia set of F(z).
 

B2. Multiple Function (Semigroup) Algorithm (when multiple INPUT FUNCTIONS are selected):

We follow the above algorithm A2, but with a slight change.  Call the selected INPUT FUNCTIONS by the names G₁,..., G_k and change item II above as follows. Calculate the set J₁=G₁^-1({z₀})∪...∪G_k^-1({z₀}).  Then calculate the set J₂=G₁^-1({J₁})∪...∪G_k^-1({J₁}).  Continue this process until the set J_j=G₁^-1({J_j-1})∪...∪G_k^-1({J_j-1}) has at least N points. 

If multiple INPUT FUNCTIONS are selected, then the sets J_j will necessarily grow in size.  Although this is not mathematically true (e.g., if each of the multiple input functions is of degree one and fixes the seed value), the computer will regard the repeated points calculated as distinct points in the set and therefore the size of the sets J_j will grow.  

The set of points J_j is called the (Approximate) Julia set of the semigroup <G₁,...,G_k> (acting in parallel).  In the right window the OUTPUT SET is labeled the COMPOSITE JULIA SET.