reb Class REB_CayleyMandel

Object common:Generic common:Formula common:ConvergentFormula reb:REB_CayleyMandel

class
ConvergentFormula:REB_CayleyMandel

Based upon a formula of Mark Townsend.

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.

Ultra Fractal Source
class REB_CayleyMandel(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_CayleyMandel(Generic pparent)
ConvergentFormula.ConvergentFormula(pparent)
endfunc

; initialize the formula
complex func Init(complex pz)
ConvergentFormula.Init(pz)
m_c = pz
fz = 0
fzp = 0
fzp2 = 0
oldz = 0
m_iterations = 0
pz = 0
return pz
endfunc

complex func Iterate(complex pz)
ConvergentFormula.Iterate(pz)
oldz = pz
fz = pz^3 - m_c*pz - m_c + 1
fzp = 3*pz^2 - m_c
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
complex m_c

default:
title = "Cayley Mandel"
int param v_cayleymandel
caption = "Version (Cayley Mandel)"
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_cayleymandel < 101
endparam
float param p_bailout
caption = "Bailout value"
default = 1e-12
endparam
caption = "Convergence Methods"
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
}

Constructor Summary
REB_CayleyMandel()

REB_CayleyMandel(Generic pparent)
constructor

Method Summary
complex Init(complex pz)
initialize the formula
complex Iterate(complex pz)
Produce the next value in the sequence

Methods inherited from class common:ConvergentFormula
GetLowerBailout, IsBailedOut

Methods inherited from class common:Formula
GetPrimaryExponent, GetUpperBailout

Methods inherited from class common:Generic
GetParent

Methods inherited from class Object

Constructor Detail

REB_CayleyMandel

public REB_CayleyMandel(Generic pparent)
constructor

REB_CayleyMandel

public REB_CayleyMandel()
Method Detail

Init

public complex Init(complex pz)
initialize the formula

Overrides:
Init in class ConvergentFormula
Parameters:
pz - seed value for the sequence; for a normal fractal formula, this will be #pixel
Returns:
first value in the sequence; this corresponds to #z in a fractal formula

Iterate

public complex Iterate(complex pz)
Description copied from class: ConvergentFormula
Produce the next value in the sequence

As long as the sequence has not bailed out, this function will be continually called to produce sequence values.

Overrides:
Iterate in class ConvergentFormula
Parameters:
pz - previous value in the sequence; corresponds to #z in a fractal formula. Note that you should always use this value for computing the next iteration, rather than a saved value, as the calling code may modify the returned value before passing it back to the next Iterate() call.
Returns:
the next value in the sequence