Newton_poly {
; coded by Anton Lorenz
; February 2004



init:

  complex x=#pixel
  complex delta_x
  bool continue = true  ; bailout
  complex a = 0
  complex b = 0
  float k = 0
  float l = 0
  float m = 0
  complex t = (0,0)
 
loop:
  px = (0,0)
  dpx = (0,0)

 if @f1 != 0.0
  a = @a1
  k = @k1
  px =  px + (x^k + a*x)*@f1
  dpx = dpx + (k*x^(k-1) + a)*@f1
 endif
 
if @f2 != 0.0
  a = @a2
  k = @k2
  px =  px + (x^k + a*x)*@f2
  dpx = dpx + (k*x^(k-1) + a)*@f2
endif 

if @f3 != 0.0
  a = @a3
  b = @b3
  px  = px + (b*x/(1+a)^2 + x^2/(2*(1+a)) - a*b*log(a*b+(1+a)*x)/(1+a)^3)*@f3
  dpx = dpx + (x/(a + x/(b+x)))*@f3
endif
if @f3a != 0.0
  a = @a3a
  b = @b3a
  k = @k3a
  l = @l3a
  m = @m3a
  px  = px + (x^k/(a + x^l/(b+x^m)))*@f3a
  dpx = dpx + (  (k*x^(k - 1))/(x^l/(x^m + b) + a) - \
  (x^k*((l*x^(l - 1))/(x^m + b) - (m*x^(l + m - 1))/ \
      (x^m + b)^2))/(x^l/(x^m + b) + a)^2  ) * @f3a
endif
if @f4 != 0.0
  k = @k4
  b = @b4
  t = b*log(x)
  px  = px + (((1+k)*x^(1+k)*cos(t)  + b*x^(1+k)*sin(t))/((1+k)^2-b^2))*@f4
  dpx = dpx + (x^k * Cos(b*Log(x)))*@f4
endif

if @f5 != 0.0
  a = @a5
  t = exp(a - x^2) 
  px  = px + ( t* x )*@f5
  dpx = dpx + ( -t/2 )*@f5
endif

if @f6 != 0.0
  k = @k6
  l = @l6
  a = @a6
  b = @b6
  t = x^l
  px  = px +  (  x^(1+k-l)/(1+k-l) + (a*x/(1-l) + b*x^2)/t )*@f6
  dpx = dpx + ( (a + b * x +  x^k)/t  )*@f6
endif

if @f7 != 0.0
  a = @a7
  b = @b7
  k = @k7
  l = @l7
  m = @m7
  px  = px + ( x^k/(a + x/(b + x)^m)^l )*@f7
  dpx = dpx + (  (k*x^(k - 1))/(x/(b + x)^m + a)^l \
            - l*x^k*((b + x)^(-m) - m*x*(b + x)^(-m - 1))*(x/(b + x)^m + a)^(-l - 1)  ) * @f7
endif
if @f8 != 0.0
  a = @a8
  b = @b8
  k = @k8
  l = @l8
  m = @m8
  px  = px + ( Sin(x^k/(a + x/(b + x)^l)^m)  )*@f8
  dpx = dpx + ( ((k*x^(k - 1))/(x/(b + x)^l + a)^m  - m*x^k*((b + x)^(-l) \
            - l*x*(b + x)^(-l - 1))*(x/(b + x)^l + a)^(-m - 1))* Cos(x^k/(x/(b + x)^l + a)^m)   ) * @f8
endif
if @f9 != 0.0
  a = @a9
  b = @b9
  k = @k9
  l = @l9
  m = @m9
  px  = px + ( x^l* b^(a + x)^m )*@f9
  dpx = dpx + ( b^(a + x)^m*l*x^(-1 + l)  \
            + b^(a + x)^m*m*x^l*(a + x)^(-1 + m)*Log(b)   ) * @f9
endif


  delta_x = (px + @q)  / dpx 
  x = x - delta_x
  
   if @xmode=="x"  

        z = delta_x
    elseif @xmode == "dx" 
        z = x
    elseif @xmode == "1/x"  
        z = 1/x 
    elseif @xmode == "1/dx"
        z = 1/delta_x
    elseif @xmode =="sum x" 
        z = z + x
    elseif @xmode == "sum dx"
        z = z + delta_x
    elseif @xmode == "sum 1/x"
        z = z + 1/x
    elseif @xmode == "sum 1/dx"
        z = z + 1/delta_x
    endif ; xmode 
    
  if |delta_x| < @delta
    continue = false
  endif 
bailout:
   continue
  
default:
  title = "x = x - P(x)/P'(x)" 
;  or      x = x - integral[P(x)])/P(x)
  periodicity = 0   ; turn off periodicity checking by default
  magn = 0.5       ; default zoom factor.
  
  complex param q
    caption="q"
    hint="The coefficient q in the function P(x) := f(x) + q"
    default=(-3,0.5)
  endparam
;---------------------------  
  heading   
    caption = "f1-------------"
  endheading
  float param f1
    caption = "function 1"
    default = 1.0
  endparam
  
  float param k1
    visible = @f1 != 0.0
    default = -5.0
  endparam
  
  complex param a1
    visible = @f1 != 0.0
    default =(5,0.20)
  endparam
;------------------------------
  heading   
    caption = "f2-------------"
  endheading

  float param f2
    caption = "function 2"
    default = 0.0
  endparam
  
  float param k2
    visible = @f2 != 0.0
    default = -2
  endparam 
  complex param a2
    visible = @f2 != 0.0
    default =(1,0)
  endparam

;--------------------------------
  heading   
    caption = "f3-------------"
  endheading
  float param f3
    caption = "function 3"
    default = 0.0
  endparam

  complex param a3
    visible = @f3 != 0.0
    default =(1,0)
  endparam

  complex param b3
    visible = @f3 != 0.0
    default =(1,0)
  endparam
;-------------------------------- 
  heading   
    caption = "f3A-------------"
  endheading
  float param f3a
    caption = "function 3a"
    default = 0.0
  endparam
  float param k3a
    visible = @f3a != 0.0
    default = 2.0
  endparam
  float param l3a
    visible = @f3a != 0.0
    default = 1.0
  endparam
  float param m3a
    visible = @f3a != 0.0
    default = 1.0
  endparam
  complex param a3a
    visible = @f3a != 0.0
    default =(1,0)
  endparam

  complex param b3a
    visible = @f3a != 0.0
    default =(1,0)
  endparam
;-------------------------------- 
  heading   
    caption = "f4-------------"
  endheading
  float param f4
    caption = "function 4"
    default = 0.0
  endparam

  float param k4
    visible = @f4 != 0.0
    default = 2
  endparam

  complex param b4
    visible = @f4 != 0.0
    default =(1,0)
  endparam
;-------------------------------- 
  heading   
    caption = "f5-------------"
  endheading
  float param f5
    caption = "function 5"
    default = 0.0
  endparam

  complex param a5
    visible = @f5 != 0.0
    default =(1,0)
  endparam

;-------------------------------- 

  heading   
    caption = "f6-------------"
  endheading
  float param f6
    caption = "function 6"
    default = 1.0
  endparam

  float param k6
    visible = @f6 != 0.0
    default = -1.0
  endparam
  float param l6
    visible = @f6 != 0.0
    default = 3  
    endparam
  complex param a6
    visible = @f6 != 0.0
    default =(-2,0)
  endparam
  complex param b6
    visible = @f6 != 0.0
    default =(1,0)
  endparam
;-------------------------------- 
   heading   
    caption = "f7-------------"
  endheading
  float param f7
    caption = "function 7"
    default = 0.0
  endparam
  float param k7
    visible = @f7 != 0.0
    default = 1.0
  endparam
  float param l7
    visible = @f7 != 0.0
    default = -3.0
  endparam
  float param m7
    visible = @f7 != 0.0
    default = 5.0
  endparam
  complex param a7
    visible = @f7 != 0.0
    default =(2,0)
  endparam

  complex param b7
    visible = @f7 != 0.0
    default =(2,0)
  endparam
;-------------------------------- 
    heading   
    caption = "f8-------------"
  endheading
  float param f8
    caption = "function 8"
    default = 0.0
  endparam
  float param k8
    visible = @f8 != 0.0
    default = 1.0
  endparam
  float param l8
    visible = @f8 != 0.0
    default = -2.0
  endparam
  float param m8
    visible = @f8 != 0.0
    default = -3.0
  endparam
  complex param a8
    visible = @f8 != 0.0
    default =(1,0)
  endparam

  complex param b8
    visible = @f8 != 0.0
    default =(1,0)
  endparam
;-------------------------------- 
     heading   
    caption = "f9-------------"
  endheading
  float param f9
    caption = "function 9"
    default = 0.0
  endparam
  float param k9
    visible = @f9 != 0.0
    default = 1.0
  endparam
  float param l9
    visible = @f9 != 0.0
    default = -2.0
  endparam
  float param m9
    visible = @f9 != 0.0
    default = -3.0
  endparam
  complex param a9
    visible = @f9 != 0.0
    default =(1,0)
  endparam

  complex param b9
    visible = @f9 != 0.0
    default =(1,0)
  endparam
;-------------------------------- 
 
  heading   
    caption = "Bailout"
  endheading
  
  float param delta
    caption="Bailout delta"
    default=1e-4
    min=0
  endparam
  
  heading 
    caption = "Colouring"
  endheading

  param xmode
    caption = "final z"
    enum = "x" "dx" "1/x" "1/dx" "sum x" "sum dx" "sum 1/x" "sum 1/dx"
    default = 1
  endparam
  
}