Fractals/Mathematics/Newton method

• from calculus book
• from numerical methods book
• other

types

Animation of Newton's method in case of real-valued function with one root

Newton method can be used for finding successively better approximations to one root (zero) x of function f :

$x:f(x)=0\,.$

If one wants to find another root must apply the method again with other initial point.

How to find all roots ?

Newton method can be applied to :

• a real-valued function
• a complex-valued function
• a system of equations

systems of nonlinear equations

One may also use Newton's method to solve systems of

• 2 (non-linear) equations
• with 2 complex variables : $x_{1},x_{2}$

${\begin{cases}F_{1}(x_{1},x_{2})=0\\F_{2}(x_{1},x_{2})=0\end{cases}}$

It can be expressed more synthetically using vector notation  :

$\mathbf {F} (\mathbf {x} )=0$

where $\mathbf {x}$ is a vector of 2 complex variables :

$\mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}$

and $\mathbf {F}$ is a vector of functions ( or is a vector-valued function of the vector variable x ) :

 $\mathbf {F} ={\begin{bmatrix}F_{1}\\F_{2}\end{bmatrix}}$

Solution can be find by iteration :

$\mathbf {x} ^{(k+1)}=\mathbf {x} ^{(k)}+\mathbf {s} ,\ k=0,1,2,\ldots .$

where s is an increment ( which is now a vector of 2 components ):

 $\mathbf {s} =\mathbf {x} ^{(k+1)}-\mathbf {x} ^{(k)}$
 

Increment can be computed by :

$\mathbf {s} =J^{-1}(\mathbf {x} ^{(k)})\mathbf {F} (\mathbf {x} ^{(k)})$

Here $J$ is Jacobian matrix of $\mathbf {F}$ at $\mathbf {x} ^{(k)}$ :

 $J_{\mathbf {F} }(\mathbf {x} ^{(k)})=(F'(\mathbf {x} ))_{ij}={\dfrac {\partial F_{i}}{\partial x_{j}}}(\mathbf {x} )={\begin{bmatrix}{\dfrac {\partial F_{1}}{\partial x_{1}}}&{\dfrac {\partial F_{1}}{\partial x_{2}}}\\[1em]{\dfrac {\partial F_{2}}{\partial x_{1}}}&{\dfrac {\partial F_{2}}{\partial x_{2}}}\end{bmatrix}}$

and $J^{-1}$ is inverse of the Jacobian matrix $J$.

In case of

• a small number of equations use the inverse matrix.
• a big number of equations : rather than actually computing the inverse of this matrix, one can save time by solving the system of linear equations $J_{F}(x_{n})(x_{n+1}-x_{n})=-F(x_{n})\,\!$ for the unknown x_n+1 − x_n.

Algorithm in pseudocode :

• start with an initial approximation ( guess) $\mathbf {x} ^{k};k=0$
• repeat until convergence ( iterate)
  • solve: $\mathbf {s} =J^{-1}(\mathbf {x} ^{(k)})\mathbf {F} (\mathbf {x} ^{(k)})$
  • compute : $\mathbf {x} ^{(k+1)}=\mathbf {x} ^{(k)}+\mathbf {s}$
  • $k=k+1$

Stop criteria :

• absolute error : $|s|<1.e-12$

What is needed ?

• function f ( a differentiable function )
• it's derivative f'
• starting point x0 ( which should be in the basin of root's attraction )
• stopping rule ( criteria to stop iteration )

stopping rule

$|x_{n+1}-x_{n}|<\epsilon \;$

Pitfalls or failure

Why method fails to find a root ?

• method has a cycle and never converge (
  • inflection point
  • local minimum
• method is diverging not converging
  • local minimum
• multiple root ( multiplicity of root > 1 , for example : f(x) = f'(x) = 0). Slow approximation, derivative tends to zero, trouble in division step ( do not divide by zero ). Use modified Newton's method
• slow convergence
  • bad value of the initial approximation ( an initial guess far from the root )
  • a function whose derivative vanishes near a root, or whose second derivative is unbounded near a root

• 
  Function has inflection point ( method has a cycle and never converge)
• 
  Local minimum

"For most problems, the application of Newton's method in double precision has 2-3 steps of global orientation and then 4-6 steps of quadratic convergence before the precision of the double floating point format is exceeded. Thus, one can be even bolder, if after 10 steps the iteration does not converge, the initial point was bad, be it that it leads to a periodic orbit or divergence to infinity. Most likely, close-by initial points will behave similarly, so make a non-trivial change in the initial point for the start of the next run of the iteration."LutzL

Pseudocode

The following is an example of using the Newton's Method to help find a root of a function f which has derivative fprime.

The initial guess will be $x_{0}=1$ and the function will be $f(x)=x^{2}-2$ so that $f'(x)=2x$.

Each new iterative of Newton's method will be denoted by x1. We will check during the computation whether the denominator (yprime) becomes too small (smaller than epsilon), which would be the case if $f'(x_{n})\approx 0$, since otherwise a large amount of error could be introduced.

%These choices depend on the problem being solved
x0 = 1                      %The initial value
f = @(x) x^2 - 2            %The function whose root we are trying to find
fprime = @(x) 2*x           %The derivative of f(x)
tolerance = 10^(-7)         %7 digit accuracy is desired
epsilon = 10^(-14)          %Don't want to divide by a number smaller than this

maxIterations = 20          %Don't allow the iterations to continue indefinitely
haveWeFoundSolution = false %Were we able to find the solution to the desired tolerance? not yet.

for i = 1 : maxIterations

    y = f(x0)
    yprime = fprime(x0)

    if(abs(yprime) < epsilon)                         %Don't want to divide by too small of a number
        fprintf('WARNING: denominator is too small\n')
        break;                                        %Leave the loop
    end

    x1 = x0 - y/yprime                                %Do Newton's computation
    
    if(abs(x1 - x0)/abs(x1) < tolerance)              %If the result is within the desired tolerance
        haveWeFoundSolution = true
        break;                                        %Done, so leave the loop
    end

    x0 = x1                                           %Update x0 to start the process again                  
    
end

if (haveWeFoundSolution) % We found a solution within tolerance and maximum number of iterations
    fprintf('The root is: %f\n', x1);
else %If we weren't able to find a solution to within the desired tolerance
    fprintf('Warning: Not able to find solution to within the desired tolerance of %f\n', tolerance);
    fprintf('The last computed approximate root was %f\n', x1)
end

code

• c
• matlab : newtonm function
• Maxima CAS
• scilab
• Mathematica
• Animation engine for explanatory math videos in python

error analysis

• math.stackexchange question: what-is-the-equation-for-the-error-of-the-newton-raphson-method
• wolframalpha: Newton's method sin z = 4
• wolphram alpha : using Newton's method find (53)^(1/3)
• solve x^5-2 using newton method with x0=2 to 50 digits

applications of Newton's method in case of iterated complex quadratic polynomials

recurrence relations

A recurrence relation is an equation ( map) that recursively defines a sequence of points (for example orbit). To compute the sequence one needs:

• starting point
• map (equation = function = recurrence relation )

definitions

Define the iterated function $f^{n}$ by:

${\begin{aligned}f^{0}(z,c)&=z\\f^{1}(z,c)&=z^{2}+c\\f^{m+1}(z,c)&=f(f^{m}(z,c),c)\end{aligned}}$

Please note the other function:

 $F(z)=f(z)-z$

which is used for :

• finding periodic points

Pick arbitrary names for the iterated function and it's derivatives These derivatives can be computed by recurrence relations.

description	math notation	arbitrary names	Initial conditions	Recurrence steps	common names
iterated function	$f^{m}$	$A_{m}$	$A_{0}=z$	$A_{m+1}=A_{m}^{2}+c$	z or f
first derivative with respect to z	${\dfrac {\partial }{\partial z}}f^{m}$	$B_{m}$	$B_{0}=1$	$B_{m+1}=2A_{m}B_{m}$	d (from derivative) or p ( from prime) of f'
second derivative with respect to z	${\dfrac {\partial }{\partial z}}{\dfrac {\partial }{\partial z}}f^{m}$	$C_{m}$	$C_{0}=0$	$C_{m+1}=2(B_{m}^{2}+A_{m}C_{m})$
first derivative with respect to c	${\dfrac {\partial }{\partial c}}f^{m}$	$D_{m}$	$D_{0}=0$	$D_{m+1}=2A_{m}D_{m}+1$
second derivative with respect to c	${\dfrac {\partial }{\partial c}}{\dfrac {\partial }{\partial z}}f^{m}$	$E_{m}$	$E_{0}=0$	$E_{m+1}=2(A_{m}E_{m}+B_{m}D_{m})$

Derivation of recurrence relations

First basic rules:

• Iterated function $f^{m+1}(z,c)=f(f^{m}(z,c),c)$ is a function composition
• Applay chain rule for composite functions, so if $f(x)=h(g(x))$ then $f'(x)=h'(g(x))\cdot g'(x).$

$A_{0}\quad {\xrightarrow {definition\ of\ A}}\ f^{0}(z,c)\quad {\xrightarrow {definition\ of\ f}}\ \quad z$

$B_{0}\quad {\xrightarrow {definition\ of\ B}}\ {\dfrac {\partial }{\partial z}}f^{0}(z,c){\xrightarrow {definition\ of\ f}}\ {\dfrac {\partial }{\partial z}}z{\xrightarrow {derivative}}\ 1$

$C_{0}\quad {\xrightarrow {definition\ of\ C}}\ {\dfrac {\partial }{\partial z}}{\dfrac {\partial }{\partial z}}f^{0}(z,c){\xrightarrow {definition\ of\ f}}\ {\dfrac {\partial }{\partial z}}{\dfrac {\partial }{\partial z}}z{\xrightarrow {derivative}}{\dfrac {\partial }{\partial z}}1{\xrightarrow {derivative}}0$

$D_{0}\quad {\xrightarrow {definition\ of\ D}}{\dfrac {\partial }{\partial c}}f^{0}(z,c){\xrightarrow {definition\ of\ f}}{\dfrac {\partial }{\partial c}}z{\xrightarrow {derivative}}0$

$E_{0}\quad {\xrightarrow {definition\ of\ E}}{\dfrac {\partial }{\partial c}}{\dfrac {\partial }{\partial z}}f^{0}(z,c)\quad {\xrightarrow {definitionoff}}{\dfrac {\partial }{\partial c}}{\dfrac {\partial }{\partial z}}z\quad {\xrightarrow {derivative}}{\dfrac {\partial }{\partial c}}1\quad {\xrightarrow {derivative}}0$

$A_{m+1}\quad {\xrightarrow {definitionofA}}f^{m+1}(z,c)\quad {\xrightarrow {definition\ of\ f}}f(f^{m}(z,c),c)\quad {\xrightarrow {definition\ of\ A}}f(A_{m},c)\quad {\xrightarrow {definition\ of\ f}}A_{m}^{2}+c$

$B_{m+1}\quad {\xrightarrow {definition\ of\ B}}{\dfrac {\partial }{\partial z}}A_{m+1}\quad {\xrightarrow {definition\ of\ A}}{\dfrac {\partial }{\partial z}}(A_{m}^{2}+c)\quad {\xrightarrow {distributivity}}{\dfrac {\partial }{\partial z}}A_{m}^{2}+{\dfrac {\partial }{\partial z}}c\quad {\xrightarrow {constant\ derivative}}{\dfrac {\partial }{\partial z}}A_{m}^{2}+0\quad {\xrightarrow {zero}}{\dfrac {\partial }{\partial z}}A_{m}^{2}\quad {\xrightarrow {chain\ rule}}2A_{m}({\dfrac {\partial }{\partial z}}A_{m})\quad {\xrightarrow {definition\ of\ B}}2A_{m}B_{m}$

$C_{m+1}\quad {\xrightarrow {definition\ of\ C}}{\dfrac {\partial }{\partial z}}B_{m+1}\quad {\xrightarrow {definition\ of\ B}}{\dfrac {\partial }{\partial z}}(2A_{m}B_{m})\quad {\xrightarrow {linearity}}2{\dfrac {\partial }{\partial z}}(A_{m}B_{m})\quad {\xrightarrow {product\ rule}}2(({\dfrac {\partial }{\partial z}}A_{m})B_{m}+A_{m}({\dfrac {\partial }{\partial z}}B_{m}))\quad {\xrightarrow {definition\ of\ B}}2(B_{m}B_{m}+A_{m}({\dfrac {\partial }{\partial z}}B_{m}))\quad {\xrightarrow {algebra}}2(B_{m}^{2}+A_{m}({\dfrac {\partial }{\partial z}}B_{m}))\quad {\xrightarrow {definition\ of\ C}}2(B_{m}^{2}+A_{m}C_{m})$

$D_{m+1}\quad {\xrightarrow {definition\ of\ D}}{\dfrac {\partial }{\partial c}}A_{m+1}\quad {\xrightarrow {definition\ of\ A}}{\dfrac {\partial }{\partial c}}(A_{m}^{2}+c)\quad {\xrightarrow {distributivity}}{\dfrac {\partial }{\partial c}}A_{m}^{2}+{\dfrac {\partial }{\partial c}}c\quad {\xrightarrow {derivative}}{\dfrac {\partial }{\partial c}}A_{m}^{2}+1\quad {\xrightarrow {chain\ rule}}2A_{m}({\dfrac {\partial }{\partial c}}A_{m})+1\quad {\xrightarrow {definition\ of\ D}}2A_{m}D_{m}+1$

$E_{m+1}\quad {\xrightarrow {definition\ of\ E}}{\dfrac {\partial }{\partial c}}B_{m+1}\quad {\xrightarrow {definition\ of\ B}}{\dfrac {\partial }{\partial c}}(2A_{m}B_{m})\quad {\xrightarrow {linearity}}2{\dfrac {\partial }{\partial c}}(A_{m}B_{m})\quad {\xrightarrow {product\ rule}}2(A_{m}({\dfrac {\partial }{\partial c}}B_{m})+({\dfrac {\partial }{\partial c}}A_{m})B_{m})\quad {\xrightarrow {definitionof\ E}}2(A_{m}E_{m}+({\dfrac {\partial }{\partial c}}A_{m})B_{m})\quad {\xrightarrow {definition\ of\ D}}2(A_{m}E_{m}+D_{m}B_{m})$

Computing

A = 0;
B = 1;
for (m = 0; m < mMax; m++){
  /* first compute B then A */
  B = 2*A*B; /* first derivative with respect to z */
  A = A*A + c; /* complex quadratic polynomial */
   }

or without complex numbers:

/* Maxima CAS */
(%i1) a:ax+ay*%i;
(%o1)                                                                                                          %i ay + ax
(%i2) b:bx+by*%i;
(%o2)                                                                                                          %i by + bx
(%i3) bn:2*a*b;
(%o3)                                                                                                 2 (%i ay + ax) (%i by + bx)
(%i4) realpart(bn);
(%o4)                                                                                                      2 (ax bx - ay by)
(%i5) imagpart(bn);
(%o5)                                                                                                      2 (ax by + ay bx)
(%i6) c:cx+cy*%i;
(%o6)                                                                                                          %i cy + cx
(%i7) an:a*a+c;
                                                                                                                                2
(%o7)                                                                                                  %i cy + cx + (%i ay + ax)
(%i8) realpart(an);
                                                                                                                    2     2
(%o8)                                                                                                        cx - ay  + ax
(%i9) imagpart(an);
(%o9)                                                                                                         cy + 2 ax ay

so :

• bx = 2 (ax bx - ay by)
• by = 2 (ax by + ay bx)
• ax = cx - ay^2 + ax^2
• ay = cy + 2 ax ay

where:

• a = ax + ay*i
• b = bx + by*i
• c = cx + cy*i

center

To use it in a GUI program :

• click on the menu item : nucleus
• using a mouse mark rectangle region around center of hyperbolic component

A center ( or nucleus ) $c=n$ of period $p$ satisfies :

$f^{p}(0,n)=0$

Applying Newton's method in one variable:

$n_{m+1}=N(n_{m})=n_{m}-{\frac {f^{p}(0,n_{m})}{{\dfrac {\partial }{\partial c}}f^{p}(0,n_{m})}}=n_{m}-{\frac {A_{p}(0,n_{m})}{D_{p}(0,n_{m})}}$

where initial estimate $n_{0}$ is arbitrary choosen ( abs(n) <2.0 )

stoppping rule : $A_{p}=0\Leftrightarrow n_{m+1}=n_{m}$

arbitrary names	Initial conditions	Recurrence steps
$A_{m}=f^{m}$	$A_{0}=z=0$	$A_{m+1}=A_{m}^{2}+c$
$D_{m}={\dfrac {\partial }{\partial c}}f^{m}$	$D_{0}=0$	$D_{m+1}=2A_{m}D_{m}+1$

/*

code from  unnamed c program ( book ) 
http://code.mathr.co.uk/book
see mandelbrot_nucleus.c

by Claude Heiland-Allen

COMPILE :

 gcc -std=c99 -Wall -Wextra -pedantic -O3 -ggdb  m.c -lm -lmpfr

usage:

./a.out bits cx cy period maxiters
    
output :  space separated complex nucleus on stdout
    
example

./a.out 53 -1.75 0 3 100
./a.out 53 -1.75 0 30 100
./a.out 53  0.471 -0.3541 14 100
./a.out 53 -0.12 -0.74 3 100
*/

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <gmp.h> // arbitrary precision
#include <mpfr.h>

extern int mandelbrot_nucleus(mpfr_t cx, mpfr_t cy, const mpfr_t c0x, const mpfr_t c0y, int period, int maxiters) {
  int retval = 0;
  mpfr_t zx, zy, dx, dy, s, t, u, v;
  mpfr_inits2(mpfr_get_prec(c0x), zx, zy, dx, dy, s, t, u, v, (mpfr_ptr) 0);
  mpfr_set(cx, c0x, GMP_RNDN);
  mpfr_set(cy, c0y, GMP_RNDN);

  for (int i = 0; i < maxiters; ++i) {
    // z = 0
    mpfr_set_ui(zx, 0, GMP_RNDN);
    mpfr_set_ui(zy, 0, GMP_RNDN);
    // d = 0
    mpfr_set_ui(dx, 0, GMP_RNDN);
    mpfr_set_ui(dy, 0, GMP_RNDN);
    for (int p = 0; p < period; ++p) {
      // d = 2 * z * d + 1;
      mpfr_mul(u, zx, dx, GMP_RNDN);
      mpfr_mul(v, zy, dy, GMP_RNDN);
      mpfr_sub(u, u, v, GMP_RNDN);
      mpfr_mul_2ui(u, u, 1, GMP_RNDN);
      mpfr_mul(dx, dx, zy, GMP_RNDN);
      mpfr_mul(dy, dy, zx, GMP_RNDN);
      mpfr_add(dy, dx, dy, GMP_RNDN);
      mpfr_mul_2ui(dy, dy, 1, GMP_RNDN);
      mpfr_add_ui(dx, u, 1, GMP_RNDN);
      // z = z^2 + c;
      mpfr_sqr(u, zx, GMP_RNDN);
      mpfr_sqr(v, zy, GMP_RNDN);
      mpfr_mul(zy, zx, zy, GMP_RNDN);
      mpfr_sub(zx, u, v, GMP_RNDN);
      mpfr_mul_2ui(zy, zy, 1, GMP_RNDN);
      mpfr_add(zx, zx, cx, GMP_RNDN);
      mpfr_add(zy, zy, cy, GMP_RNDN);
    }
    // check d == 0
    if (mpfr_zero_p(dx) && mpfr_zero_p(dy)) {
      retval = 1;
      goto done;
    }

    
    // st = c - z / d
    mpfr_sqr(u, dx, GMP_RNDN);
    mpfr_sqr(v, dy, GMP_RNDN);
    mpfr_add(u, u, v, GMP_RNDN);
    mpfr_mul(s, zx, dx, GMP_RNDN);
    mpfr_mul(t, zy, dy, GMP_RNDN);
    mpfr_add(v, s, t, GMP_RNDN);
    mpfr_div(v, v, u, GMP_RNDN);
    mpfr_mul(s, zy, dx, GMP_RNDN);
    mpfr_mul(t, zx, dy, GMP_RNDN);
    mpfr_sub(zy, s, t, GMP_RNDN);
    mpfr_div(zy, zy, u, GMP_RNDN);
    mpfr_sub(s, cx, v, GMP_RNDN);
    mpfr_sub(t, cy, zy, GMP_RNDN);
    // uv = st - c
    mpfr_sub(u, s, cx, GMP_RNDN);
    mpfr_sub(v, t, cy, GMP_RNDN);
    // c = st
    mpfr_set(cx, s, GMP_RNDN);
    mpfr_set(cy, t, GMP_RNDN);
    // check uv = 0
    if (mpfr_zero_p(u) && mpfr_zero_p(v)) {
      retval = 2;
      goto done;
    }
  } // for (int i = 0; i < maxiters; ++i)

 done: mpfr_clears(zx, zy, dx, dy, s, t, u, v, (mpfr_ptr) 0);

 return retval;
}

void DescribeStop(int stop)
{
  switch( stop )
  {
    case 0:
    printf(" method stopped because i = maxiters\n");
    break;
    
    case 1:
    printf(" method stopped because derivative == 0\n");
    break;
    
    //...
    case 2:
    printf(" method stopped because uv = 0\n");
    break;
    
   }
}

void usage(const char *progname) {
  fprintf(stderr,
    "program finds one center ( nucleus) of hyperbolic component of Mandelbrot set using Newton method"
    "usage: %s bits cx cy period maxiters\n"
    "outputs space separated complex nucleus on stdout\n"
    "example %s 53 -1.75 0 3 100\n",
    progname, progname);
}

int main(int argc, char **argv) {
  
  // check the input 
  if (argc != 6) { usage(argv[0]); return 1; }
  
  // read the values 
  int bits = atoi(argv[1]);
  mpfr_t cx, cy, c0x, c0y;
  mpfr_inits2(bits, cx, cy, c0x, c0y, (mpfr_ptr) 0);
  mpfr_set_str(c0x, argv[2], 10, GMP_RNDN);
  mpfr_set_str(c0y, argv[3], 10, GMP_RNDN);
  int period = atoi(argv[4]);
  int maxiters = atoi(argv[5]);
  int stop;

  //
  stop = mandelbrot_nucleus(cx, cy, c0x, c0y, period, maxiters);

   
  //
  printf(" nucleus ( center) of component with period = %s near c = %s ; %s is : \n ", argv[4], argv[2], argv[3]);
  mpfr_out_str(0, 10, 0, cx, GMP_RNDN);
  putchar(' ');
  mpfr_out_str(0, 10, 0, cy, GMP_RNDN);
  putchar('\n');
  //
  DescribeStop(stop) ;

  // clear memeory 
  mpfr_clears(cx, cy, c0x, c0y, (mpfr_ptr) 0);
  
  // 
  return 0;
}

boundary

The boundary of the component with center $n$ of period $p$ can be parameterized by internal angles.

The boundary point $c=b$ at angle $t$ (measured in turns) satisfies system of 2 equations :

${\begin{cases}f^{p}(w,b)-w&=0\\{\dfrac {\partial }{\partial z}}f^{p}(w,b)-e^{2\pi it}&=0\end{cases}}$

Defining a function of two complex variables:

${\begin{aligned}g{\begin{pmatrix}z\\c\end{pmatrix}}&={\begin{pmatrix}f^{p}(z,c)-z\\{\dfrac {\partial }{\partial z}}f^{p}(z,c)-e^{2\pi it}\end{pmatrix}}\end{aligned}}$

and applying Newton's method in two variables:

${\begin{aligned}{\boldsymbol {v}}_{0}={\begin{pmatrix}n\\n\end{pmatrix}}\\J_{g}({\boldsymbol {v}}_{m})({\boldsymbol {v}}_{m+1}-{\boldsymbol {v}}_{m})=-g({\boldsymbol {v}}_{m})\end{aligned}}$

where

${\begin{aligned}J_{g}={\begin{pmatrix}{\dfrac {\partial }{\partial z}}(f^{p}(z,c)-z)&{\dfrac {\partial }{\partial c}}(f^{p}(z,c)-z)\\{\dfrac {\partial }{\partial z}}({\dfrac {\partial }{\partial z}}f^{p}(z,c)-e^{2\pi it})&{\dfrac {\partial }{\partial c}}({\dfrac {\partial }{\partial z}}f^{p}(z,c)-e^{2\pi it})\end{pmatrix}}\end{aligned}}$

This can be expressed using the recurrence relations as:

${\begin{aligned}{\begin{pmatrix}w_{0}\\b_{0}\end{pmatrix}}&={\begin{pmatrix}n\\n\end{pmatrix}}\\{\begin{pmatrix}B_{p}-1&D_{p}\\C_{p}&E_{p}\end{pmatrix}}{\begin{pmatrix}w_{m+1}-w_{m}\\b_{m+1}-b_{m}\end{pmatrix}}&=-{\begin{pmatrix}A_{p}-w_{m}\\B_{p}-e^{2\pi it}\end{pmatrix}}\end{aligned}}$

where $\{A,B,C,D,E\}_{p}$ are evaluated at $(w_{m},b_{m})$.

Example

"The process is for numerically finding one boundary point on one component at a time - you can't solve for the whole boundary of all the components at once. So pick and fix one particular nucleus n and one particular internal angle t " (Claude Heiland-Allen )

Lets try find point of boundary ( bond point )

$c=b$

of hyperbolic component of Mandelbrot set for :

• period p=3
• angle t=0
• center ( nucleus ) c = n = -0.12256116687665+0.74486176661974i

There is only one such point.

Initial estimates (first guess) will be center ( nucleus ) $c=n$ of this components. This center ( complex number) will be saved in a vector of complex numbers containing two copies of that nucleus:

${\begin{aligned}{\boldsymbol {v}}_{0}={\begin{pmatrix}n\\n\end{pmatrix}}={\begin{pmatrix}-0.12256116687665+0.74486176661974i\\-0.12256116687665+0.74486176661974i\end{pmatrix}}\end{aligned}}$

The boundary point at angle $t$ (measured in turns) satisfies system of 2 equations :

${\begin{cases}f^{3}(z,c)-z&=0\\{\dfrac {\partial }{\partial z}}f^{3}(z,c)-e^{2\pi i0}&=0\end{cases}}$

so :

${\begin{cases}z^{8}+4cz^{6}+6c^{2}z^{4}+2cz^{4}+4c^{3}z^{2}+4c^{2}z^{2}-z+c^{4}+2c^{3}+c^{2}+c=0\\8z^{7}+24cz^{5}+24c^{2}z^{3}+8cz^{3}+8c^{3}z+8c^{2}z-1=0\end{cases}}$

Then function g

${\begin{aligned}g({\boldsymbol {v}})={\begin{cases}z^{8}+4cz^{6}+6c^{2}z^{4}+2cz^{4}+4c^{3}z^{2}+4c^{2}z^{2}-z+c^{4}+2c^{3}+c^{2}+c\\8z^{7}+24cz^{5}+24c^{2}z^{3}+8cz^{3}+8c^{3}z+8c^{2}z-1\end{cases}}\end{aligned}}$

Jacobian matrix $J$ is

$J_{g}({\boldsymbol {v}})={\begin{pmatrix}8z^{7}+24cz^{5}+24c^{2}z^{3}+8cz^{3}+8c^{3}z+8c^{2}z-1&4z^{6}+12cz^{4}+2z^{4}+12c^{2}z^{2}+8cz^{2}+4c^{3}+6c^{2}+2c+1\\56z^{6}+120cz^{4}+72c^{2}z^{2}+24cz^{2}+8c^{3}+8c^{2}&24z^{5}+48cz^{3}+8z^{3}+24c^{2}z+16cz\end{pmatrix}}$

$J_{g}^{-1}({\boldsymbol {v}})={\begin{pmatrix}{\frac {24z^{5}+48cz^{3}+8z^{3}+24c^{2}z+16cz}{d}}&{\frac {-4*z^{6}-12*c*z^{4}-2*z^{4}-12*c^{2}*z^{2}-8*c*z^{2}-4*c^{3}-6*c^{2}-2*c-1}{d}}\\{\frac {-56*z^{6}-120*c*z^{4}-72*c^{2}*z^{2}-24*c*z^{2}-8*c^{3}-8*c^{2}}{d}}&{\frac {8*z^{7}+24*c*z^{5}+24*c^{2}*z^{3}+8*c*z^{3}+8*c^{3}*z+8*c^{2}*z-1}{d}}\end{pmatrix}}$

where denominator d is :

$d=(24z^{5}+48cz^{3}+8z^{3}+24c^{2}z+16cz)(8z^{7}+24cz^{5}+24c^{2}z^{3}+8cz^{3}+8c^{3}z+8c^{2}z-1)+(-56z^{6}-120cz^{4}-72c^{2}z^{2}-24cz^{2}-8c^{3}-8c^{2})(4z^{6}+12cz^{4}+2z^{4}+12c^{2}z^{2}+8cz^{2}+4c^{3}+6c^{2}+2c+1)$

Then find better aproximation of point c=b using iteration :

${\boldsymbol {v}}_{m+1}={\frac {{\boldsymbol {v}}_{m}-g({\boldsymbol {v}}_{m})}{J_{g}({\boldsymbol {v}}_{m})}}=({\boldsymbol {v}}_{m}-g({\boldsymbol {v}}_{m}))J_{g}^{-1}({\boldsymbol {v}}_{m})$

using ${\boldsymbol {v}}_{0}$ as an starting point :

${\boldsymbol {v}}_{0}$

${\boldsymbol {v}}_{1}=({\boldsymbol {v}}_{0}-g({\boldsymbol {v}}_{0}))J_{g}^{-1}({\boldsymbol {v}}_{0})$

${\boldsymbol {v}}_{2}=({\boldsymbol {v}}_{1}-g({\boldsymbol {v}}_{1}))J_{g}^{-1}({\boldsymbol {v}}_{1})$

$...$

${\boldsymbol {v}}_{m+1}=({\boldsymbol {v}}_{m}-g({\boldsymbol {v}}_{m}))J_{g}^{-1}({\boldsymbol {v}}_{m})$

Maxima CAS code

Boundaries of Components of Mandelbrot set by Newton method. Image and Maxima CAS code

C++ code

/*
cpp code by Claude Heiland-Allen 
http://mathr.co.uk/blog/
licensed GPLv3+

g++ -std=c++98 -Wall -pedantic -Wextra -O3 -o bonds bonds.cc

./bonds period nucleus_re nucleus_im internal_angle
./bonds 1 0 0 0
./bonds 1 0 0 0.5
./bonds 1 0 0 0.333333333333333
./bonds 2 -1 0 0
./bonds 2 -1 0 0.5
./bonds 2 -1 0 0.333333333333333
./bonds 3 -0.12256116687665 0.74486176661974 0
./bonds 3 -0.12256116687665 0.74486176661974 0.5
./bonds 3 -0.12256116687665 0.74486176661974 0.333333333333333

for b in $(seq -w 0 999)
do
  ./bonds 1 0 0 0.$b 2>/dev/null
done > period-1.txt
gnuplot
plot "./period-1.txt" with lines

-------------
for b in $(seq -w 0 999)
do
  ./bonds 3 -0.12256116687665 0.74486176661974 0.$b 2>/dev/null
done > period-3a.txt

gnuplot
plot "./period-3a.txt" with lines
*/
#include <complex>
#include <stdio.h>
#include <stdlib.h>

typedef std::complex<double> complex;

const double pi = 3.141592653589793;
const double eps = 1.0e-12;

struct param {
  int period;
  complex nucleus, bond;
};

struct recurrence {
  complex A, B, C, D, E;
};

void recurrence_init(recurrence &r, const complex &z) {
  r.A = z;
  r.B = 1;
  r.C = 0;
  r.D = 0;
  r.E = 0;
}

/*
void recurrence_step(recurrence &rr, const recurrence &r, const complex &c) {
  rr.A = r.A * r.A + c;
  rr.B = 2.0 * r.A * r.B;
  rr.C = 2.0 * (r.B * r.B + r.A * r.C);
  rr.D = 2.0 * r.A * r.D + 1.0;
  rr.E = 2.0 * (r.A * r.E + r.B * r.D);
}
*/

void recurrence_step_inplace(recurrence &r, const complex &c) {
  // this works because no component of r is read after it is written
  r.E = 2.0 * (r.A * r.E + r.B * r.D);
  r.D = 2.0 * r.A * r.D + 1.0;
  r.C = 2.0 * (r.B * r.B + r.A * r.C);
  r.B = 2.0 * r.A * r.B;
  r.A = r.A * r.A + c;
}

struct matrix {
  complex a, b, c, d;
};

struct vector {
  complex u, v;
};

vector operator+(const vector &u, const vector &v) {
  vector vv = { u.u + v.u, u.v + v.v };
  return vv;
}

vector operator*(const matrix &m, const vector &v) {
  vector vv = { m.a * v.u + m.b * v.v, m.c * v.u + m.d * v.v };
  return vv;
}

matrix operator*(const complex &s, const matrix &m) {
  matrix mm = { s * m.a, s * m.b, s * m.c, s * m.d };
  return mm;
}

complex det(const matrix &m) {
  return m.a * m.d - m.b * m.c;
}

matrix inv(const matrix &m) {
  matrix mm = { m.d, -m.b, -m.c, m.a };
  return (1.0/det(m)) * mm;
}

// newton stores <z, c> into vector as <u, v>

vector newton_init(const param &h) {
  vector vv = { h.nucleus, h.nucleus };
  return vv;
}

void print_equation(const matrix &m, const vector &v, const vector &k) {
  fprintf(stderr, "/  % 20.16f%+20.16fi  % 20.16f%+20.16fi  \\  /  z  -  % 20.16f%+20.16fi  \\  =  /  % 20.16f%+20.16fi  \\\n", real(m.a), imag(m.a), real(m.b), imag(m.b), real(v.u), imag(v.u), real(k.u), imag(k.u));
  fprintf(stderr, "\\  % 20.16f%+20.16fi  % 20.16f%+20.16fi  /  \\  c  -  % 20.16f%+20.16fi  /     \\  % 20.16f%+20.16fi  /\n", real(m.c), imag(m.c), real(m.d), imag(m.d), real(v.v), imag(v.v), real(k.v), imag(k.v));
}

vector newton_step(const vector &v, const param &h) {
  recurrence r;
  recurrence_init(r, v.u);
  for (int i = 0; i < h.period; ++i) {
    recurrence_step_inplace(r, v.v);
  }
  // matrix equation: J * (vv - v) = -g
  // solved by: vv = inv(J) * -g + v
  // note: the C++ variable g has the value of semantic variable -g
  matrix J = { r.B - 1.0, r.D, r.C, r.E };
  vector g = { v.u - r.A, h.bond - r.B };
  print_equation(J, v, g);
  return inv(J) * g + v;
}

int main(int argc, char **argv) {
  if (argc < 5) {
    fprintf(stderr, "usage: %s period nucleus_re nucleus_im internal_angle\n", argv[0]);
    return 1;
  }
  param h;
  h.period  = atoi(argv[1]);
  h.nucleus = complex(atof(argv[2]), atof(argv[3]));
  double angle = atof(argv[4]);
  if (angle == 0 && h.period != 1) {
    fprintf(stderr, "warning: possibly wrong results due to convergence into parent!\n");
  }
  h.bond = std::polar(1.0, 2.0 * pi * angle);
  fprintf(stderr, "initial parameters:\n");
  fprintf(stderr, "period = %d\n", h.period);
  fprintf(stderr, "nucleus = % 20.16f%+20.16fi\n", real(h.nucleus), imag(h.nucleus));
  fprintf(stderr, "bond = % 20.16f%+20.16fi\n", real(h.bond), imag(h.bond));
  vector v = newton_init(h);
  fprintf(stderr, "z =% 20.16f%+20.16fi\n", real(v.u), imag(v.u));
  fprintf(stderr, "c =% 20.16f%+20.16fi\n", real(v.v), imag(v.v));
  fprintf(stderr, "\n");
  fprintf(stderr, "newton iteration:\n");
  vector vv;
  do {
    vv = v;
    v = newton_step(vv, h);
    fprintf(stderr, "z =% 20.16f%+20.16fi\n", real(v.u), imag(v.u));
    fprintf(stderr, "c =% 20.16f%+20.16fi\n", real(v.v), imag(v.v));
    fprintf(stderr, "\n");
  } while (abs(v.u - vv.u) + abs(v.v - vv.v) > eps);
  fprintf(stderr, "bond location:\n");
  fprintf(stderr, "c =% 20.16f%+20.16fi\n", real(v.v), imag(v.v));
  // suitable for gnuplot
  printf("%.16f %.16f\n", real(v.v), imag(v.v));
  return 0;
}

size

Newton's method (providing the initial estimate is sufficiently good and the function is well-behaved) provides successively more accurate approximations. How to tell when the approximation is good enough? One way might be to compare the center with points on its boundary, and continue to increase precision until this distance is accurate to enough bits.

Algorithm:

1. given a center location estimate $n_{m}$ accurate to $P$ bits of precision
2. compute a boundary location estimate $b_{m}$ accurate to the same precision, using center $n_{m}$ as starting value
3. compute $|n_{m}-b_{m}|$ to find an estimate of the size of the component
   1. if it isn't zero, and if it is accurate enough (to a few 10s of bits, perhaps) then finish
   2. otherwise increase precision, refine center estimate to new precision, try again from the top

Measuring effective accuracy of the distance between two very close points might be done by comparing floating point exponents with floating point mantissa precision.

internal coordinate

Mandelbrot set - multiplier map

Internal coordinate of hyperbolic component of Mandelbrot set

• https://mathr.co.uk/blog/2013-04-01_interior_coordinates_in_the_mandelbrot_set.html

It is possible to map hyperbolic component $\mathrm {H}$ to closed unit disk centered at origin ${\bar {D}}_{1}$:

$\lambda _{p}:\mathrm {H} _{p}\to {\bar {D}}_{1}$

$\lambda _{p}(c)=b$

${\bar {D}}_{1}=\{b\in \mathbb {C} :\left|b\right|\leq 1\}$.

This relation is described by system of 2 equations :

${\begin{cases}F^{p}(z,c)=z\\{\frac {\partial }{\partial z}}F^{p}(z,c)=b\end{cases}}$

where

• p is the period of the target hyperbolic component on parameter plane
• $\theta$ the desired internal angle of points c and b
• $r\leq 1$ is an internal radius
• $b=r*e^{2\pi i\theta }$ is a point of unit circle = internal coordinate
• c is a point of parameter plane
• z is a point of dynamic plane
• $(F^{0}(z,c)=z$ and $F^{q+1}(z,c)=F^{q}(F(z,c)^{2}+c)$

The algorithm by Claude Heiland-Allen for finding internal coordinate b from c :

• choose c
• check c ( bailout test on dynamical plane). When c is outside the Mandelbrot set, give up now ( or compute external coordinate );
• Start with period one : $p=1$
• while $p<p_{max}$
  • Find periodic point $z_{0}$ such that $F^{p}(z_{0},c)=z_{0}$using Newton's method in one complex variable;
  • Find b by evaluating ${\frac {\partial }{\partial z}}F^{p}(z,c)$ at $z_{0}$
  • If $\left|b\right|\leq 1$ then return b
  • otherwise continue with the next p  : $p=p+1$

To solve such system of equations one can use Newton method.

internal ray

Green internal and red external rays

/*
http://code.mathr.co.uk/mandelbrot-graphics/blob/HEAD:/c/bin/m-subwake-diagram-a.c
by Claude Heiland-Allen
*/

#include <mandelbrot-graphics.h>
#include <mandelbrot-numerics.h>
#include <mandelbrot-symbolics.h>
#include <cairo.h>

const double twopi = 6.283185307179586;

void draw_internal_ray(m_image *image, m_d_transform *transform, int period, double _Complex nucleus, const char *angle, double pt, m_pixel_t colour) {
  int steps = 128;
  mpq_t theta;
  mpq_init(theta);
  mpq_set_str(theta, angle, 10);
  mpq_canonicalize(theta);
  double a = twopi * mpq_get_d(theta);
  mpq_clear(theta);
  double _Complex interior = cos(a) + I * sin(a);

  double _Complex cl = 0, cl2 = 0;
  double _Complex c = nucleus;
  double _Complex z = c;
  cairo_surface_t *surface = m_image_surface(image);
  cairo_t *cr = cairo_create(surface);
  cairo_set_source_rgba(cr, m_pixel_red(colour), m_pixel_green(colour), m_pixel_blue(colour), m_pixel_alpha(colour));
  for (int i = 0; i < steps; ++i) {
    if (2 * i == steps) {
      cl = c;
    }
    if (2 * i == steps + 2) {
      cl2 = c;
    }
    double radius = (i + 0.5) / steps;
    m_d_interior(&z, &c, z, c, radius * interior, period, 64);
    double _Complex pc = c;
    double _Complex pdc = 1;
    m_d_transform_reverse(transform, &pc, &pdc);
    if (i == 0) {
      cairo_move_to(cr, creal(pc), cimag(pc));
    } else {
      cairo_line_to(cr, creal(pc), cimag(pc));
    }
  }
  cairo_stroke(cr);
  if (a != 0) {
    double t = carg(cl2 - cl);
    cairo_save(cr);
    double _Complex dcl = 1;
    m_d_transform_reverse(transform, &cl, &dcl);
    cairo_translate(cr, creal(cl), cimag(cl));
    cairo_rotate(cr, -t);
    cairo_translate(cr, 0, -pt/3);
    cairo_select_font_face(cr, "LMSans10", CAIRO_FONT_SLANT_NORMAL, CAIRO_FONT_WEIGHT_NORMAL);
    cairo_set_font_size(cr, pt);
    cairo_text_path(cr, angle);
    cairo_fill(cr);
    cairo_restore(cr);
  }
  cairo_destroy(cr);
}

external rays

Dynamic ray

//  qmnplane.cpp by Wolf Jung (C) 2007-2014.
// part of Mandel 5.11 http://mndynamics.com/indexp.html
// the GNU General   Public License

//Time ~ nmax^2 , therefore limited  nmax .
int QmnPlane::newtonRay(int signtype, qulonglong N, qulonglong D,
   double &a, double &b, int q, QColor color, int mode) //5, white, 1
{  uint n; int j, i, k, i0, k0; mndAngle t; t.setAngle(N, D, j);
   double logR = 14.0, x, y, u;
   u = exp(0.5*logR); y = t.radians();
   x = u*cos(y); y = u*sin(y);
   if (pointToPos(x, y, i0, k0) > 1) i0 = -10;
   stop(); QPainter *p = new QPainter(buffer); p->setPen(color);
   for (n = 1; n <= (nmax > 5000u ? 5000u : nmax + 2); n++)
   {  t.twice();
      for (j = 1; j <= q; j++)
      {  if (mode & 4 ? tricornNewton(signtype, n, a, b, x, y,
                exp(-j*0.69315/q)*logR, t.radians())
           : rayNewton(signtype, n, a, b, x, y,
                exp(-j*0.69315/q)*logR, t.radians()) )
         { n = (n <= 20 ? 65020u : 65010u); break; }
         if (pointToPos(x, y, i, k) > 1) i = -10;
         if (i0 > -10)
         {  if (i > -10) { if (!(mode & 8)) p->drawLine(i0, k0, i, k); }
            else { n = 65020u; break; }
         }
         i0 = i; k0 = k;
      }
   }
   //if rayNewton fails after > 20 iterations, endpoint may be accepted
   delete p; update(); if (n >= 65020u) return 2;
   if (mode & 2) { a = x; b = y; }
   if (n >= 65010u) return 1; else return 0;
} //newtonRay

int QmnPlane::rayNewton(int signtype, uint n, double a, double b,
   double &x, double &y, double rlog, double ilog)
{  uint k, l; double fx, fy, px, py, u, v = 0.0, d = 1.0 + x*x + y*y, t0, t1;
   for (k = 1; k <= 60; k++)
   {  if (signtype > 0) { a = x; b = y; }
      fx = cos(ilog); fy = sin(ilog);
      t0 = exp(rlog)*fx - 0.5*exp(-rlog)*(a*fx + b*fy);
      t1 = exp(rlog)*fy + 0.5*exp(-rlog)*(a*fy - b*fx);
      fx = x; fy = y; px = 1.0; py = 0.0;
      for (l = 1; l <= n; l++)
      {  u = 2.0*(fx*px - fy*py); py = 2.0*(fx*py + fy*px);
         px = u; if (signtype > 0) px++;
         u = fx*fx; v = fy*fy; fy = 2.0*fx*fy + b; fx = u - v + a;
         u += v; v = px*px + py*py; if (u + v > 1.0e100) return 1;
      }
      fx -= t0; fy -= t1; if (v < 1.0e-50) return 2;
      u = (fx*px + fy*py)/v; v = (fx*py - fy*px)/v;
      px = u*u + v*v; if (px > 9.0*d) return -1;
      x -= u; y += v; d = px; if (px < 1.0e-28 && k >= 5) break;
   }
   return 0;
} //rayNewton

Parameter ray

External parameter rays

equipotential line

Boundaries of potential or escape time level sets ( equipotential curves) are polynomial lemniscates. Computing from symbolic equations gets exponentially more complicated as boundary gest closer to Mandelbrot or Julia set.

Is there a more efficient way of computing for the Mandelbrot lemniscates?

Complex iteration of $z\to z^{2}+c$ gives a degree $2^{n-1}$ polynomial in $c$ in $O(n)$ work.

Suppose

$c=x+iy$

$f_{c}(z)=z^{2}+c$

$f_{c}^{\circ (n+1)}(z)=f_{c}^{\circ n}(f_{c}(z))$

Then

${\frac {d}{dc}}f_{c}^{\circ (n+1)}(z)=2f_{c}^{\circ n}(z){\frac {d}{dc}}f_{c}^{\circ n}(z)+1$

These can be calculated together in one inner loop (being careful not to overwrite old values that are still needed).

Now you can use Newton's root finding method to solve the implicit form $f_{c}^{\circ n}(0)=re^{2\pi i\theta }=t$ by

$c_{m+1}=c_{m}-{\frac {f_{c_{m}}^{\circ n}(0)-t}{{\frac {d}{dc}}f_{c_{m}}^{\circ n}(0)}}$

Use the previous $c$ as initial guess for next $\theta$. The increment in $\theta$ needs to be smaller than about ${\frac {1}{4}}$ for the algorithm to be stable. Note that $\theta$ (measured in turns) wraps around the unit circle $2^{n-1}$ times before $c$ gets back to its starting point.

This approach is inspired by An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira

Periodic points

• of complex quadratic polynomial
• general numerical method for complex quadratic polynomial

See also

• The Newton-Raphson fractal
• Calculus : Newton Method
• Newton-Hines fractals
• Fractalzoomer - root finding methods

References

• NEWTON’S METHOD IN PRACTICE: FINDING ALL ROOTS OF POLYNOMIALS OF DEGREE ONE MILLION EFFICIENTLY by DIERK SCHLEICHER AND ROBIN STOLL
• NEWTON’S METHOD IN PRACTICE II: THE ITERATED REFINEMENT NEWTON METHOD AND NEAR-OPTIMAL COMPLEXITY FOR FINDING ALL ROOTS OF SOME POLYNOMIALS OF VERY LARGE DEGREES by DIERK SCHLEICHER AND ROBIN STOLL
• Finding polynomial roots by dynamical systems -- a case studyby Sergey Shemyakov, Roman Chernov, Dzmitry Rumiantsau, Dierk Schleicher, Simon Schmitt, Anton Shemyakov