M-set Derivatives






First derivative of the Mandelbrot function Fc with respect to z :

using power and sum rule in differentiation one gets:

Z' = dFc / dZ = F'c(Z)= 2*Z

because Z and Z' are complex numbers so :

Re( Z' ) = Z'.X = 2 * Z.X
Im (Z' ) = Z'. Y = 2 * Z.Y

in Maxima:
(%i3)P(n):=if n=0 then z else P(n-1)^2+c;
(%o3) P(n):=if n=0 then z else P(n-1)^2+c
(%i4) diff(P(1), z);
(%o4) 2*z

Applications:

First derivative of the Mandelbrot map with respect to c:

It can be stated in any of several equivalent forms:
dFc(n)(Z0)/dZ =
dFM(Z0,c,n)/dZ=
F'M(Z0,c,n)=

definition of function in pseudocode:
FM(Z0=0,c,n):=
 begin
  Zi= Z0
  For i=1 to n do
     Zi+1=Fc(Zi):=(Zi*Zi) + c
  return Zi
 end;

-------------------------------
Z'n+1=dZn+1/dc = 2 * Zn * Z'n + 1
---------------------------------

It goes like that :

Z0= 0
Z'0= 0

Z1= Z02+ c = c
Z'1= 2 * Z0* Z'0+ 1 = 1

Z2= Z12+ c = c2+ c
Z'2= 2 * Z1* Z'1+ 1 = 2c + 1

Z3= Z22+ c = c4 + 2 c3 + c2 + c
Z'3= 2 * Z2* Z'2+ 1 = 4 c3 + 6 c2 + 2 c + 1
...
ZN+1= ZN2+c
Z'N+1= 2 * ZN * Z'N + 1

and so on ...


-----------------------------

How it was computed ?

First some notes:

1.folding a function:
Fn(c)= F( ...F(F(c)) // n-times
example:
F2(c)= F(F(c)) // it is NOT power of function (F(c))2=SQR(F(c))

Here one may find usage of words : itereation, recursion
in the same meaning while in the programming theory thay have diffrent meaning.

2.it is a recursive definition of function

Zn+1=Fn+1(c):=(Fn(c))2 +c

3.Lets write definition of map
Zn+1=Fc(Zn):=(Zn*Zn) + c
in a diffrent form :
Zn+1:= Fn+1(c)= G(c) + c :=(Fn(c))2 + c
Here G(c) is a composite function
G(c):=E(I(c))
which consisist of external function E(u) and internal function I(c):

E(u)=u2
u:=Fn-1(c):=Zn
so:
E':=dE/du:=2*u:=2*Zn

I(c):= Fn(c) := Z n
I'(c):=dI/dc := Z'n
(Here one do not compute Z'n because it must be computed by recursion)

To compute derivative of composite function G(c) one must apply chain rule
dG/dc:=dE/du * dI/dc
so:
G'(c)=dG/dc = E'(u) * I'(c) = (2*Zn)*Z'n

Tu compute derivative of map:
Zn+1:= Fn+1(c)= G(c) + H(c)
which is a sum of 2 functions use sum rule in differentiation:
F'(c)= F(n+1)(c)= G'(c) + H'(c)
here H(c)=c
It is a monic polynomial of degree=1 ( (:-))) = linear function
To compute derivative of H(c) use The power rule:
H'(c)=dH/dc= 1


so
Z'n+1=dZn+1/dc = 2 * Zn * Z'n + 1

--------------------------------------------
Z and Z' are complex numbers:
Z = ZX + ZY *i
Z' = ZX' + ZY'*i

so using rules for Addition and Multiplication of complex numbers  one gets:

ZYn+1=2 * ZXn*ZYn + CY;
ZXn+1=(ZXn * ZXn) - ( ZYn * ZYn ) + CX

ZX'n+1=2*(ZXn*ZXn' - ZYn*ZYn') + 1
ZY'n+1=2*(Zyn*ZXn' + Zxn*ZYn')

------------------------------------------------

-------------------------------------------------------

Applications of first derivative:

Second derivative of the Mandelbrot map (iteration of function) with respect to c:
Z''n+1 = 2 * (Zn * Z''n + Z'n2)

See:


Main page


Autor: Adam Majewski adammaj1-at-o2-dot--pl
Feel free to e-mail me. (:-))
About