REB_CayleyJulia

In addition to the the well-known Newton's method for determining roots of polynomials, three additional methods can be used: Householder's method, Halley's method and Schroder's Method. All three are described in Mathworld. This formula has seven additional methods which are hybrid methods.

One of the unique characteristics of this convergent formula is the presence of multiple julia sets similar to the classical julia sets. For this formula they are much smaller and hard to find. The julia sets can be found with the Newton, Mixed 1, Mixed 2, Mixed 3 and Mixed 4 convergence methods.

 class REB_CayleyJulia(common.ulb:ConvergentFormula) {
 ; Based upon a formula of Mark Townsend. <br>
 ; <p>
 ; In addition to the the well-known Newton's method for determining roots of
 ; polynomials, three additional methods can be used: Householder's method,
 ; Halley's method and Schroder's Method. All three are described in Mathworld.
 ; This formula has seven additional methods which are hybrid methods.
 ; <p>
 ; One of the unique characteristics of this convergent formula is the presence
 ; of multiple julia sets similar to the classical julia sets. For this formula
 ; they are much smaller and hard to find. The julia sets can be found with the
 ; Newton, Mixed 1, Mixed 2, Mixed 3 and Mixed 4 convergence methods.
 public:
   import "common.ulb"
 
   ; constructor
   func REB_CayleyJulia(Generic pparent)
      ConvergentFormula.ConvergentFormula(pparent)
    endfunc
 
   ; initialize the formula
   complex func Init(complex pz)
     ConvergentFormula.Init(pz)
     fz = 0
     fzp = 0
     fzp2 = 0
     oldz = 0
     m_iterations = 0
     return pz
   endfunc
 
   ; call for each iterated point
   complex func Iterate(complex pz)
     ConvergentFormula.Iterate(pz)
     complex seed = 0
     if @converge == 0 || @converge == 1 || @converge == 2 || @converge == 3 || \
        @converge == 4 || @converge == 5 || @converge == 6
       seed = @p10
     elseif @converge == 7
       seed = @p17
     elseif @converge == 8
       seed = @p18
     elseif @converge == 9
       seed = @p19
     elseif @converge == 10
       seed = @p110
     endif
     oldz = pz
     fz = pz^3 - seed*pz - seed + 1
     fzp = 3*pz^2 - seed
     fzp2 = 6*pz
     if @converge == 0                          ; Newton
       pz = pz - fz/fzp
     elseif @converge == 1                      ; Householder
       pz = pz - fz/fzp*(1 + fz*fzp2/(2*fzp^2))
     elseif @converge == 2                      ; Halley
       pz = pz - 2*fz*fzp/(2*fzp^2 - fz*fzp2)
     elseif @converge == 3                      ; Schroder
       pz = pz - fz*fzp/(fzp^2 - fz*fzp2)
     elseif @converge == 4                      ; Ho custom
       pz = pz - fz/fzp*(1 + fz*fzp2/(@custom*fzp^2))
     elseif @converge == 5                      ; Ha custom
       pz = pz - 2*fz*fzp/(@custom*fzp^2 - fz*fzp2)
     elseif @converge == 6                      ; H_S custom
       pz = pz - @custom*fz*fzp/(@custom*fzp^2 - fz*fzp2)
     elseif @converge == 7                      ; Mixed 1
       if  m_iterations % 2 == 0
         pz = pz - fz/fzp*(1 + fz*fzp2/(2*fzp^2))
       else
         pz = pz - fz/fzp
       endif
     elseif @converge == 8                      ; Mixed 2
       if  m_iterations % 2 == 0
         pz = pz - 2*fz*fzp/(2*fzp^2 - fz*fzp2)
       else
         pz = pz - fz/fzp
       endif
     elseif @converge == 9                      ; Mixed 3
       if  m_iterations % 2 == 0
         pz = pz - fz*fzp/(fzp^2 - fz*fzp2)
       else
         pz = pz - fz/fzp
       endif
     elseif @converge == 10                      ; Mixed 4
       if  m_iterations % 2 == 0
         pz = pz - @custom*fz*fzp/(@custom*fzp^2 - fz*fzp2)
       else
         pz = pz - fz/fzp
       endif
     endif
     return pz
   endfunc
 
 protected:  
    complex fz
    complex fzp
    complex fzp2
    complex oldz
 
 default:
   title = "Cayley Julia"
   int param v_cayleyjulia
     caption = "Version (Cayley Julia)"
     default = 101
     hint = "This version parameter is used to detect when a change has been made to the formula that is incompatible with the previous version. When that happens, this field will reflect the old version number to alert you to the fact that an alternate rendering is being used."
     visible = @v_cayleyjulia < 101
   endparam
   float param p_bailout
     caption = "Bailout value"
     default = 1e-12
   endparam
   param p10
     caption = "Julia seed"
     default = (0.360968017578125,0.00074462890625)
     visible = (@converge == 0 || @converge == 1 || @converge == 2 || @converge == 3 || \
        @converge == 4 || @converge == 5 || @converge == 6)
   endparam
   param p17
     caption = "Julia seed"
     default = (0.31216913789,1.38719537118)
     visible = @converge==7
   endparam
   param p18
     caption = "Julia seed"
     default = (0.0973120321799,1.15430065766)
     visible = @converge==8
   endparam
   param p19
     caption = "Julia seed"
     default = (0.127944,0.186804)
     visible = @converge==9
   endparam
   param p110
     caption = "Julia seed"
     default = (-0.009955549007286, 0.9678595903229)
     visible = @converge==10
   endparam
   heading
     caption = "Convergence Methods"
   endheading
   param converge
     caption = "Convergence Method"
     default = 0
     enum = "Newton" "Householder" "Halley" "Schroder" "Ho Custom" \
            "Ha Custom" "H_S Custom" "Mixed1" "Mixed2" "Mixed3" "Mixed4"
   endparam
   float param custom
     caption = "H_S Constant"
     default = 1.5
     visible = @converge==4 || @converge==5  || @converge==6 || @converge==10
   endparam
   param p_power
     caption = "Power"
     default = (3,0)
     visible = false
   endparam
 }

Method Summary
`complex`	`Init(complex pz)` initialize the formula
`complex`	`Iterate(complex pz)` call for each iterated point

Constructor Summary
`REB_CayleyJulia()`
`REB_CayleyJulia(Generic pparent)` constructor

reb Class REB_CayleyJulia

REB_CayleyJulia

REB_CayleyJulia

Init

Iterate

reb
Class REB_CayleyJulia