NewtonRaphson method
Robert P. Munafo, 2010 Sep 12.
The NewtonRaphson method, also called Newton's method, is a simple iterative process often useful for solving equations with one free variable. If the variable is x, the equation to be solved is
f(x) = 0
where f(x) is some differentiable function of x. If f'(x) is the derivative of f(x) with respect to x, then the NewtonRaphson iteration involves calculating
x_{n+1} = x_{n}  f(x_{n}) / f'(x_{n})
which allows us to calculate a new approximation x_{n+1} from an existing approximation x_{n}.
Using complex math, Newton's method can be used to find the nucleus of a muatom of known period. In this case the equation being solved is L_{P}(C)=0 where P is the period of the muatom and L_{P} is the P^{th} Lemniscate function (that is, the function Z'=Z^{2}+C iterated P times with an initial value Z_{0}=0). For example, if the period is 3, the lemniscate function is
L_{3}(C) = ((0^{2}+C)^{2}+C)^{2}+C = C^{4} + 2 C^{3} + C^{2} + C
and its derivative is
(L_{3})'(C) = 4 C^{3} + 6 C^{2} + 2 C + 1
(NOTE: In general, it is not necessary to know the polynomial expansions of L_{P}(C) and L_{P}'(C) as shown here, because the values can be calculated iteratively. See below for details.)
Starting with an initial guess C_{0} we calculate L_{P}(C_{0}) and the derivative (L_{P})'(C_{0}). Then our new guess is
C_{1} = C_{0}  L_{3}(C_{0}) / (L_{3})'(C_{0})
As a concrete example, we can use the initial guess 0.12+0.76i and find the nucleus of R2.1/3a:

After just two steps of the iteration the approximation is accurate to 5 digits. The result after three steps, 0.12256116687638... + 0.74486176662039...i, agrees with the exact answer given in the R2.1/3a article (0.12256116628477... + 0.74486176791682...i) in the first 8 digits of both components.
Iterative Evaluation of L_{N} and its Derivative
In the example above, the lemnicate L_{3}(C) and its derivative L_{3}'(C) are shown as fully expanded polynomials. As the period increases, these polynomials rapidly become too big to manage (see Z_{5} and Z_{6} at the end of the lemniscates article). Fortunately, there is no need to know these polynomials when finding a muatom nucleus by the NewtonRaphson method, because L_{N}(C) and L_{N}'(C) can both be computed by iteration. This table shows how:

As you can see from the 1^{st} and 3^{rd} columns, each value of L(C) and L'(C) can be computed from the previous values with the formulas:
L_{N+1}(C) = (L_{N}(C))^{2} + C
L_{N+1}'(C) = 2 L_{N}(C) L_{N}'(C) + 1
The following shows the calculation of the nucleus of the period3 muatom R2.1/3a. This was done with maxima:
(%i1) l3(c):=c^4+2*c^3+c^2+c; 4 3 2 (%o1) l3(c) := c + 2 c + c + c (%i2) diff(l3(c),c); 3 2 (%o2) 4 c + 6 c + 2 c + 1 (%i3) d3(c):=4*c^3+6*c^2+2*c+1; 3 2 (%o3) d3(c) := 4 c + 6 c + 2 c + 1 (%i4) c0:0.12+0.76*%i; (%o4) 0.76 %i  0.12 (%i5) rectform(l3(c0)); (%o5) 0.0131404799999999  .02923264000000014 %i (%i6) rectform(d3(c0)); (%o6)  1.198976 %i  1.794368 (%i7) c1:rectform(c0l3(c0)/d3(c0)); (%o7) .7453543395553945 %i  .1224628812914806 (%i8) c2:rectform(c1l3(c1)/d3(c1)); (%o8) .7448623089253219 %i  .1225610213028558 (%i9) c3:rectform(c2l3(c2)/d3(c2)); (%o9) .7448617666203934 %i  .1225611668763843
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872017. Muency index
This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2017 Feb 02. s.11