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) 19872013. Muency index