Fractals/mandelbrot-numerics

mandelbrot-numerics is a library for advanced numeric calculations related with Mandelbrot set by Claude Heiland-Allen.

This is unofficial wiki about it, containing most of maximus/book 'lib' and 'bin' (but no rendering)

See:
 * dictionary
 * notation
 * fork of the library
 * Libraries by Claude Heiland-Allen
 * exrtact various tools to manipulate EXR images compare with OpenExr
 * kf-extras programs for manipulating output from Kalles Fraktaler 2
 * mandelbrot-symbolics - symbolic algorithms related to the Mandelbrot set
 * mandelbrot-graphics - CPU-based visualisation of the Mandelbrot set
 * mandelbrot-text - 	parsing and pretty printing related to the Mandelbrot set

=Notation=

directories

 * bin for programs
 * lib for functions with 2 versions
 * d for double precision
 * r for arbitrary precision
 * include

precision
Precision here means precision of floating point numbers in the numerical computing.

It is a positive integer. It is a number of bits of binary number (

Function names ( notation)

 * double precision: m_d_*
 * functions taking pointers to double _Complex use them for output
 * arbitrary precision: m_r_*

how to find precision ?

 * start with minimal precision ( double = 53 bits )
 * compute new precision = 53 + f(parameter) where parameter is a feature of the procedure ( for example minimal size of the component for m-interior ). In other words new precision is grater then 53

=Installation = " you need to "make install" mandelbrot-symbolics lib before trying anything with mandelbrot-numerics"

dependencies

 * pkg-config
 * gmp (libgmp)
 * mpfr (libmpfr)
 * mpc (libmpc)
 * haskell - ghc
 * sndfile using : sudo apt-get install libsndfile1-dev

Gcc flags :

gcc -lmpc -lmpfr -lgmp -lm

From console :

sudo aptitude install libmpc-dev

clone
From directory in which you want to clone git repo:

git clone https://code.mathr.co.uk/mandelbrot-numerics.git

compile
and from the same directory :

make -C mandelbrot-numerics/c/lib prefix=${HOME}/opt install make -C mandelbrot-numerics/c/bin prefix=${HOME}/opt install

then to run do:

export LD_LIBRARY_PATH=${HOME}/opt/lib

check :

echo $LD_LIBRARY_PATH

result :

/home/a/opt/lib

or

export PATH=${HOME}/opt/bin:${PATH}

and check :

echo $PATH

To set it permanently change file .profile

sudo gedit ~/.profile

test
m-describe 53 20 100 0 0 4

LD_LIBRARY_PATH=${HOME}/opt/lib m-primary-separators

git
From console opened in the mandelbrot-numerics directory :

git pull

If you made some local changes you can undu them :

git checkout -f

then

git pull

Now install again

=structures=

=how to use it =
 * to include library c source should *only* have #include 
 * compile and link with pkg-config: see mandelbrot-numerics/c/bin/Makefile for an example
 * quickest way to get started is to just put your file in mandelbrot-numerics/c/bin and run make

Misiurewicz point

 * m-feature-database
 * m-misiurewicz

m-misiurewicz
usage:

m-misiurewicz precision guess-re guess-im preperiod period maxsteps

Description
 * uses preperiod of critical point

Examples using double precision

m-misiurewicz double -2.1 0 2 1 100 -2.0000000000000000e+00 0.0000000000000000e+00 m-misiurewicz double -1.6 0 3 1 100 -1.5436890126920764e+00 0.0000000000000000e+00

m-misiurewicz double -1.5 0 5 2 100 -1.4303576324513072e+00 0.0000000000000000e+00 m-misiurewicz double -1.4 0 9 4 100 -1.4074051181647020e+00 0.0000000000000000e+00

Using bits :

./m-misiurewicz 200 -1.4 0 9 4 100 -1.4074051181647020225078282291990509777838059260945479350908287e+00 0.0000000000000000000000000000000000000000000000000000000000000e+00

m-feature-database
m-feature-database

usage: ./m-feature-database preperiod period

m-feature-database 1 1

.1(0)	1/2	-2.00	0.00

m-describe
m-describe
 * official page
 * this procedure gives description of point c from the parameter plane.

Use without parameters:

m-describe

gives usage description:

usage: m-describe precision maxperiod maxiters re im ncpus

Check:
 * point c = 0 = 0 +0*I= re+ im*I
 * with precision = 53 bits
 * max period= 20
 * max iters = 100

m-describe 53 20 100 0 0 4

output is :

the input point was +0e+00 + +0e+00 i the point didn't escape after 100 iterations nearby hyperbolic components to the input point:

- a period 1 cardioid with nucleus at +0e+00 + +0e+00 i the component has size 1.00000e+00 and is pointing west the atom domain has size 0.00000e+00 the atom domain coordinates of the input point are -nan + -nan i the atom domain coordinates in polar form are nan to the east the nucleus is 0.00000e+00 to the east of the input point the input point is interior to this component at radius 0.00000e+00 and angle 0.000000000000000000 (in turns) the multiplier is +0.00000e+00 + +0.00000e+00 i a point in the attractor is +0e+00 + +0e+00 i  external angles of this component are: .(0) .(1)

Another example:

m-describe 1000 1000 10000000 -1.749512 0.0 8 the input point was -1.74951199999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999e+00 + +0e+00 i the point didn't escape after 10000000 iterations nearby hyperbolic components to the input point:

- a period 1 cardioid with nucleus at +0e+00 + +0e+00 i the component has size 1.00000e+00 and is pointing west the atom domain has size 0.00000e+00 the atom domain coordinates of the input point are -nan + -nan i the atom domain coordinates in polar form are nan to the east the nucleus is 1.74951e+00 to the east of the input point the input point is exterior to this component at radius 1.82808e+00 and angle 0.500000000000000000 (in turns) the multiplier is -1.82808e+00 + +0.00000e+00 i a point in the attractor is -9.14032e-01 + +0e+00 i  external angles of this component are: .(0) .(1)

- a period 2 circle with nucleus at -1e+00 + +0e+00 i the component has size 5.00000e-01 and is pointing west the atom domain has size 1.00000e+00 the atom domain coordinates of the input point are -0.74951 + -0 i the atom domain coordinates in polar form are 0.74951 to the west the nucleus is 7.49512e-01 to the east of the input point the input point is exterior to this component at radius 2.99805e+00 and angle 0.500000000000000000 (in turns) the multiplier is -2.99805e+00 + +0.00000e+00 i a point in the attractor is -1.49976e+00 + +0e+00 i

- a period 3 cardioid with nucleus at -1.754878e+00 + +0e+00 i the component has size 1.90355e-02 and is pointing west the atom domain has size 2.34487e-01 the atom domain coordinates of the input point are -0.022921 + +0 i the atom domain coordinates in polar form are 0.022921 to the west the nucleus is 5.36567e-03 to the west of the input point the input point is exterior to this component at radius 1.24010e+00 and angle 0.000000000000000000 (in turns) the multiplier is +1.24010e+00 + +0.00000e+00 i a point in the attractor is -6.8613231e-02 + +0e+00 i  external angles of this component are: .(011) .(100)

- a period 237 cardioid with nucleus at -1.7495120000000000000000000000000116053893024686343718029506670414e+00 + +0e+00 i the component has size 5.10303e-60 and is pointing west the atom domain has size 1.34480e-32 the atom domain coordinates of the input point are -0.86149 + -0 i the atom domain coordinates in polar form are 0.86149 to the west the nucleus is 1.16054e-32 to the west of the input point the input point is exterior to this component at radius 5.87695e+33 and angle 0.500000000000000000 (in turns) the multiplier is -5.87695e+33 + +0.00000e+00 i a point in the attractor is +2.5892947214253619092848904520135803680047966975848204640792969525e-02 + +0e+00 i

- a period 756 cardioid with nucleus at -1.74951200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000437443253498601125336414441820009561885311706078736706424485680024027767522121183394207260374919267049e+00 + +0e+00 i the component has size 2.81459e-198 and is pointing west the atom domain has size 8.60400e-102 the atom domain coordinates of the input point are -0.51039 + +0 i the atom domain coordinates in polar form are 0.51039 to the west the nucleus is 4.37443e-102 to the west of the input point the input point is exterior to this component at radius 1.99523e+68 and angle 0.500000000000000000 (in turns) the multiplier is -1.99523e+68 + +0.00000e+00 i a point in the attractor is -1.32154302717027851153258720563184483110950266513429522776342823541752294814605666085090894143487158886716259121852162061940587472967643029955949630872182624723698360235112807868903937707485681031295828661e-02 + +0e+00 i

shape
Versions
 * m_d.shape.c
 * m_r_shape.c

Description

atom domain
Atom domain size estimate formula from http://ibiblio.org/e-notes/MSet/windows.htm <-- where I got the formula from

component size
Files:
 * m-size.c
 * m_d_size.c
 * output: complex double size
 * m_r_size.c
 * output : mpc_t size

// mandelbrot-numerics -- numerical algorithms related to the Mandelbrot set // Copyright (C) 2015-2018 Claude Heiland-Allen // License GPL3+ http://www.gnu.org/licenses/gpl.html

Use without parameters:

m-size

gives usage description :

usage: ./m-size precision nucleus-re nucleus-im period

Parameters:
 * precision
 * nucleus ( complex number c )
 * period
 * positive integer
 * string "double"

Results are 2 numbers is:
 * cabs(size) = size and carg(size) = an orientation estimate
 * mpc_abs(r, size, MPFR_RNDN); mpc_arg(t, size, MPFR_RNDN);

It gives the size as a complex number, the magnitude is the size relative to the period 1 continent and the phase / argument is the angle of rotation.

Code: $ m-size 53 -8.6915874972342078e-01 2.5565708568620021e-01 48 1.1864737161932281e-08 -5.9691937849660148e-01

my program splits it into magnitude and phase (in radians) for convenience, so the minibrot is 1.1865e-8 times the size of the period 1 continent, so multiply the view size of your initial view by this number to get the minibrot to appear about the same size.

Example usage:

m-size double 0 0 1

output :

1.0000000000000000e+00 0.0000000000000000e+00

In other words: the component has size 1.00000e+00 and is pointing west

Minibrot period 48

m-size 53 -8.6915874972342078e-01 2.5565708568620021e-01 48 1.1864737161932281e-08 -5.9691937849660148e-01

box-period
Computing period under complex quadratic polynomial

"But, the box period method detects the period of a *nucleus* within the box - if there is no nucleus there it will fail."

Find the period of the central minibrot. You can do this by iterating the 4 c corners of a box around the view, and seeing at which iteration count the 4 z points surround the origin. See http://www.mrob.com/pub/muency/period.html for a more detailed description of the method, I have an implementation which you can read here: http://code.mathr.co.uk/mandelbrot-numerics/blob/HEAD:/c/lib/m_d_box_period.c (there's also an arbitrary precision version in the same directory)

See:
 * http://www.mrob.com/pub/muency/period.html

binaries
m-box-period

The result :

usage: m-box-period precision center-re center-im radius maxperiod Example:

m-box-period 53 -0.8691524744 0.2556487868 1.25e-5 1000 48

code
This is one file program using code extracted from Claude's library.

m-interior
Interior coordinates in the Mandelbrot set

Steps: C is fixed throughout this process - it determines the pixel we want to colour.

The first step is to find a special z (maybe call it w) such that p iterations of z to z^2 + c starting from w, gets you back to w. This is the limit cycle attractor of c. This is done by m_d_attractor, using Newton's method to solve f_c^p(z) - z = 0.

Then to find the interior coordinate for c, take your w, and iterate it p times, calculating the derivative w.r.t. z:

z = w; dz = 1; for (int i = 0; i < p; ++i) dz = 2 * z * dz z = z * z + c

then the interior coordinate is simply dz after the loop. if |dz| < 1 then c is interior to a component of period p

you can colour pixels based on dz (eg angle -> hue, radius -> saturation)

for highlighting the edge you can optionally use the w to find the interior distance estimate (m_d_interior_de).

Description

 * from math.stackexchange
 * blog
 * "traced boundaries using Newton's method in two variables"
 * theory

an algorithm to find points on the boundary of the Mandelbrot set, given a particular hyperbolic component and the desired internal angle. It involves Newton's method in two complex variables to solve system of 2 equations:

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

where:
 * F is function ( complex quadratic polynomial)
 * $$ F^p$$ is iterated polynomial
 * p is the period of the target component
 * $$ b= re^{2\pi i \theta} $$with |r|≤1.
 * θ is the desired internal angle.
 * First equation describes periodic point
 * second equation describes attractive periodic point ( magnitude r of multiplier b is not grater then 1 )

The resulting c is the coordinates of the point on the boundary. It can also be modified to find points in the interior, simply set b=re2πiθ with |r|≤1.

The algorithm for finding numerically interior coordinate b by Claude:
 * When c is outside the Mandelbrot set, give up now;
 * For each period p, starting from 1 and increasing:
 * Find z0 such that Fp(z0,c)=z0 using Newton's method in one complex variable;
 * Find b by evaluating first derivative with respect to z at z0;
 * If |b|≤1 then return b, otherwise continue with the next p.

usage
complex double z = 0; complex double c = 0; m_d_interior(&z, &c, zre + I * zim, cre + I * cim, ir * cexp(I * twopi * it), period, maxsteps); printf("z = %.16e%+.16e\t c = %.16e%+.16e\n", creal(z), cimag(z), creal(c), cimag(c));

Use without parameters:

m-interior

gives usage

usage: ./m-interior precision z-guess-re z-guess-im c-guess-re c-guess-im interior-r interior-t period maxsteps

z-guess = z-guess-re+z-guess-im*I c-guess = c-guess-re + c-guess=im*I

Output:
 * point c with given internal coordinate from hyperbolic componenet described by period and nucleus
 * periodic point z (wucleus)

Example :

./a.out double 0 0 0 0 1 0 1 100 5.0000000000000000e-01 0.0000000000000000e+00 2.5000000000000000e-01 0.0000000000000000e+00

How to read it :
 * z = 0.5 ( fixed point for c = 1/4)
 * c = 0.25

General algorithm:
 * choose period
 * compute / choose center c from period = initial aproximation
 * compute size of component with center c. Compute precision from size
 * choose t = p/q and r
 * compute

code
Program m-interior from bin directory

Source code :
 * mandelbrot-numerics/c/bin/ m-interior.c
 * mandelbrot-numerics/c/lib/m_d_interior.c
 * mandelbrot-numerics/c/lib/m_r_interior.c

Functions:
 * m_d_interior(&z, &c, zre + I * zim, cre + I * cim, ir * cexp(I * twopi * it), period, maxsteps);
 * m_r_interior(z, c, z, c, interior, period, maxsteps);

m-interior-de
description

m-attractor
to find a special z (maybe call it w) such that p iterations of z to z^2 + c starting from w, gets you back to w. This is the limit cycle attractor of c. It is done using Newton's method to solve f_c^p(z) - z = 0.

usage: %s precision z-guess-re z-guess-im c-re c-im period maxsteps

Finds one periodic z point ( point on dynamical plane ) near z-guess
 * for :
 * period
 * c= c-re + c-im* i
 * using Newton method with :
 * z-guess = z-guess-re + z-guess-im* i ( initial aproximation of the root)
 * maxima number of steps = maxsteps
 * using number with width of significant = precision. Here one can use :
 * double or 53
 * positive integer = number of bits for MPFR

./m-attractor double  0 0 -0.965 0.085 2 100

or ./m-attractor 53 0 0 -0.965 0.085 2 100

result :

-2.7669310528133408e-02 -8.9979332165608591e-02

The precision is set manually in m-attractor to let user experiment.

m-exray-in
Parameter external ray:
 * traced inward ( from infinity to the boundary )
 * using Newton method

When tracing inwards, one peels off the most-significant bit (aka angle doubling) each time the ray crosses a dwell band (integer part of normalized iteration count increases by 1).

binaries
m-exray-in usage: m-exray-in precision angle sharpness maxsteps

m-exray-in double 1/3 4 200

m-exray-in double 0 4 200 2.5405429038323935e-01 0.0000000000000000e+00

Code
Source code:
 * c/bin/m-exray-in.c = program showing usage of the library function
 * c/lib/m_d_exray_in.c = double ( machine double precision ) version of procedure
 * c/lib/m_r_exray_in.c = mpfr version of procedure = (dynamically changed as necessary) precision

variables :
 * native ( if true then use double= native precision)
 * precision (can be double, see arg_precision )

double version :

mpfr version :

m-exray-out
The trick when tracing outwards is to prepend bits when crossing dwell bands, depending if the outer cell was entered from its left or right inner cell. A picture may make it clearer:

Parameter external ray:
 * traced outward ( from point c ( from exterior of Mandelbrot set ) to the infinity )
 * using Newton method

"there are m_r_exray_out_* functions that can be used to get the external angle by tracing an external ray. but it is O(dwell^2), too slow to be practical beyond a few 1000, and it does not yet have adaptive precision so it cam get stuck in tight gaps between spirals..." checkin what ray
 * crosses given c point
 * land on that c point

arguments:
 * precision: input string is converted to integer number, in bits "replace double with 1000 to use 1000 bits instead of native 53". It can be "double" string = native precision.
 * c-re = real part of complex c
 * c-im = imaginary part of complex c
 * int sharpness ( replace 8 with 4 to be less sharp, but don't go too low or it may break )
 * int maxdwell
 * used only for formatting output (" if you have a desired (pre)period, use that to get the angle formatted nicely" )
 * preperiod ( int)
 * period (int)

usage :

m-exray-out precision c-re c-im sharpness maxdwell preperiod period

Example

m-exray-out double -0.1 0.651 8 10000 100000 1 | tail -n 1

output ( only binary expansion of externala angle):

.00100100100100100100100100100100100100100100100100100100100100100100100100

or full output:

m-exray-out double -1.7904997268969142e-01 1.0856197533070304e+00 4  20 100 1 -1.7904997268969142e-01 1.0856197533070304e+00 -1.7935443511056054e-01 1.0854961235734493e+00 -1.7969015435633140e-01 1.0853555826931407e+00 -1.8051917357961708e-01 1.0850119514784207e+00 1 -1.8269681168554189e-01 1.0840986080730519e+00 -1.8930978756749961e-01 1.0809967205891908e+00 0 -1.9360134634853554e-01 1.0737664210007076e+00 1 -2.0801776265667046e-01 1.0704315884719302e+00 0 -2.4370145596832851e-01 1.0627226337492655e+00 1 -2.7278498707172455e-01 1.0400463032627758e+00 1 -3.6142246174842857e-01 9.5627327660306660e-01 -4.0875942404913329e-01 9.4318016425218687e-01 -4.5809426255241681e-01 9.9407738931502432e-01 1 -6.0223759082842288e-01 9.3674887784190841e-01 -8.0664626534435835e-01 8.8455840042031386e-01 0 -8.3743553612789834e-01 9.3674304615806725e-01 -8.7287981833135664e-01 9.9952074265609137e-01 -9.1381642842183941e-01 1.0750773818725967e+00 -9.6138941844723436e-01 1.1665050307228815e+00 0 -1.0172147464760770e+00 1.2782611560409487e+00 -1.0836516873794186e+00 1.4168346080502183e+00 -1.1642543320399381e+00 1.5917680433570660e+00 -1.2645279518271537e+00 1.8173272090846146e+00 0 -1.3932089082214374e+00 2.1153654197070240e+00 -1.5644798765083927e+00 2.5204286897782238e+00 -1.8019543103478257e+00 3.0891768094384502e+00 -2.1462162170666716e+00 3.9184694463323830e+00 1 -2.6699272827383060e+00 5.1817748128576184e+00 -3.5099957053533952e+00 7.2066815445348764e+00 -4.9407140183321250e+00 1.0650884836446950e+01 -7.5526171722332034e+00 1.6932133257514206e+01 0 -1.2727799761990893e+01 2.9370322256332429e+01 -2.4030360674257757e+01 5.6528056606204473e+01 -5.1753810887543764e+01 1.2313571132413860e+02 -1.2986129881788275e+02 3.1079020626913712e+02 1 -3.8951035577778663e+02 9.3459826885312373e+02 -1.4412515103402823e+03 3.4614095793299703e+03 -6.8365543576278487e+03 1.6423640211807055e+04 -4.3547867632624519e+04 1.0462387704049867e+05 0 -3.9380510085478675e+05 9.4611788566395245e+05 -5.4017847803638447e+06 1.2977803446720989e+07 -1.2161023621086851e+08 2.9216894171553040e+08

.01010001110101

Input : ./ray_in ".(0101101011010110101101011010110101100)" 1000

Example:

m-exray-out 100 -0.7432918908524301 0.1312405523087976 8 1000 24 4

result :

.010101010101010101010100(1010)

which is

.01010101010101010101010(01)

external ray tracing implementation, that switches between: * perturbation methods (including acceleration with bilinear approximation) * arbitrary precision iterations depending on how close neighbouring points on the ray are. Trying to be correct in all cases but faster than just using arbitrary precision for the whole thing.

m-feigenbaum
"Estimate location of period 2P atom using size of period P island, use Newton's method to find its nucleus and size more precisely, repeat"

computes c centers of hyperbolic components for period doubling cascade on the real axis ( so imaginary part is 0):
 * center for period 1 : c = 0 ( omitted)
 * center for period 2 : c = -1
 * center for period 2^2= 4 : c = -1.310702641336833008
 * center for period 2^n
 * center for period 2^n

Small modification:

It seems that double precision is not enough

m-feigenbaum3
"estimate location of tip of antenna (renormalized from -2+0i) of period P island using size estimate, from there find nucleus of period 3P island using Newton's method, repeat"

usage:

m-feigenbaum3 re(location) im(location) periodFactor

m-feigenbaum3 -2 0 3 C = -1.786440255563804813e+00,0.000000000000000000e+00 Z = 5.659992713573920042e+01,0.000000000000000000e+00 T = 44 C = -1.786440255563804813e+00,0.000000000000000000e+00 Z = 5.492879663528381684e+01,0.000000000000000000e+00 T = 43 C = -1.786440255563804813e+00,0.000000000000000000e+00 Z = 5.526412200825988208e+01,0.000000000000000000e+00 T = 44 C = -1.786440255563804813e+00,0.000000000000000000e+00 Z = 5.524573293871185342e+01,0.000000000000000000e+00 T = 44 C = -1.786440255563804813e+00,0.000000000000000000e+00 Z = 5.524447443629173904e+01,0.000000000000000000e+00 T = 44 C = -1.786440255563804813e+00,0.000000000000000000e+00 Z = 5.510191709584042741e+01,0.000000000000000000e+00 T = 44 C = -1.786440255563804813e+00,0.000000000000000000e+00 Z = 4.825684575389948350e+01,0.000000000000000000e+00 T = 42 C = -1.786440255563804813e+00,0.000000000000000000e+00 Z = 6.901913875598086001e+00,0.000000000000000000e+00 T = 21 C = -1.786440255563804813e+00,0.000000000000000000e+00 Z = inf,-nan T = -2147483648 C = -1.786440255563804813e+00,0.000000000000000000e+00 Z = -nan,-nan T = -2147483648 C = -1.786440255563804813e+00,0.000000000000000000e+00 Z = -nan,-nan T = -2147483648 C = -1.786440255563804813e+00,0.000000000000000000e+00 Z = -nan,-nan T = -2147483648
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 * 2) preset
 * 1) endpreset

m-feigenbaum3 -2 0 3 period = 3 	    C = -1.786440255563804813e+00 +0.000000000000000000e+00	Z = 5.659992713573920042e+01 +0.000000000000000000e+00	T = 44 	 size = 1.000000000000000000e+00 period = 9 	    C = -1.786440255563804813e+00 +0.000000000000000000e+00	Z = 5.492879663528381684e+01 +0.000000000000000000e+00	T = 43 	 size = 1.903551591313245098e-02 period = 27 	    C = -1.786440255563804813e+00 +0.000000000000000000e+00	Z = 5.526412200825988208e+01 +0.000000000000000000e+00	T = 44 	 size = 3.464641937910663597e-04 period = 81 	    C = -1.786440255563804813e+00 +0.000000000000000000e+00	Z = 5.524573293871185342e+01 +0.000000000000000000e+00	T = 44 	 size = 6.269877899222824167e-06 period = 243 	    C = -1.786440255563804813e+00 +0.000000000000000000e+00	Z = 5.524447443629173904e+01 +0.000000000000000000e+00	T = 44 	 size = 1.134900148763894982e-07 period = 729 	    C = -1.786440255563804813e+00 +0.000000000000000000e+00	Z = 5.510191709584042741e+01 +0.000000000000000000e+00	T = 44 	 size = 2.054226124560898465e-09 period = 2187 	    C = -1.786440255563804813e+00 +0.000000000000000000e+00	Z = 4.825684575389948350e+01 +0.000000000000000000e+00	T = 42 	 size = 3.718249148787694584e-11 period = 6561 	    C = -1.786440255563804813e+00 +0.000000000000000000e+00	Z = 6.901913875598086001e+00 +0.000000000000000000e+00	T = 21 	 size = 6.727899706332398247e-13 period = 19683 	    C = -1.786440255563804813e+00 +0.000000000000000000e+00	Z = inf -nan	                                        T = -2147483648  size = 1.225497697338603077e-14

m-nearest-roots
prints:
 * period + 1
 * distance

nucleus
Example

m-nucleus 53 -0.8691524744 0.2556487868 48 64 -8.6915874972342078e-01 2.5565708568620021e-01

Using mandelbrot-numerics code to find a period 20000 minibrot near c=i:

time ( m-size 50000 `m-nucleus 50000 0 1 20000 32 | tee /dev/stderr` 20000 ) result:

Real: -7.94371635020719854891738160412783355655571655934220979864154437186665266381839752694827412839798647961500393288206001718243574947969041001571452764573813270386773043890657915447590440682262982847984850747954651397530112905749405475656322952683099658907368560442258160650020607848404019574360015928310732321162892248353767660248370197290525936147027400641162839284358719120232798279335946667807358105438875167793741593547325266735069327302557778608644106179622519866122935283445393696343557633856896321836603613442558461094153199875350130482163704063056830338236495563673439039804535280785508071799728050867261772866764321466004183218020127059471832891869155131777148793623783744855003978556166275945987592382854914346001636616560970935690194642474350799683472931444242279303532082494681724726730215793068524121651255539648642881438793908961503336534572650294138320256272266401511874110343350075804826433466557434970496377256776889864232244369405531324605647258859564801875394658149346182974446044225587075779288942180746778730140063112382965935757129792815213763215220282228648802900473854773094929308599411389335234593128655705961816766820656646416251957119598501598919583996552648916958372557223670483862248293618337708488536783821892414834079173630777856887040339159451454451413820320985031133861211418216189764359098268780128103113152388389279428861457957252585570673517148583636601768074742304119073329884963616469528208780469933279651675925059743420699231452511254055112982107353820023514444751111958838715373753570192253249668373295851727004150649290597195686694158355363059076051950150675515327178060489599587207233515407720451367112670080914348577789947679519710429658805332761281607579456713410248426316697399348421904216287458230574853218767593126985409292256088816256083677149976337938821371162520666596848675233205259982355595191500395698228043440242483460453282606623505067669594412249210020781347809878342083995720267516303410640761285854328048228996451219086494380791861046262232250868350267279732070531069335456058075810861738389728075822784952247649213082437825670400017964695116284920963804738640435281128826388500084996503976174713121619465115431089540477322147326041925187615904292563389974241734512135917137265582720029921886674069030548689883618376890512670679090079403961656868087951068688380648550051080896212339507469642787298243494620219194528289122384879485919998382046000898894730797410798833415546961249865583490298587694425666197438155027638440977015278683283223237592285196828837817724864498950232968868903951567024882703973711396957106959906822062720685131843478689858753554377359860973443661102201460984546699116768724225074667262627728443966879920239803931804122526209629516006944876489894140461251413591077416433107817245402694455983274699016792738982236483908761297057937386400485705370954170438363459688523396294500025677001897775779648467905371046183931068799398357323949041524949463987272313849137872497458230017382893116457271038101978878775155394583543192631042855892679379794207412977468749003748893885495396328384416499711295357789312684393969978236394635812530246566920668407291430144255645534655305448512127921870136609665653341022339956029962840058531526194761671357631375481810866487416153177391152464571345907563692209886121824231068882728708949417669512506874667148077590404245131137769192663052846934827860000123183553702030005106499274369377990905739319653275146021741942478690314508498885788246072923605336506056596266334519288571982849118618955182528667685108492718112716744662153390074944257995892663303680374948085233220720153557023638725611501821394872889498558529148351911516472149535805779927657689235894521400922077748496598398591526894024900490146467804508553126342120419075974877882112231259535823874740555586322514596968577784108668373885354661780761108872715009186020735991100087828283498286019313596173574889070066927305748826842911520575560834348435316554497942304212184963480634409578595209299971155537132232724598938882225997145003158857582262512263839071008673739469491592103578094090661720264483653057268796903859030849653572938688839592414951182283574611329461089531972357016051084879329881399143836366382053366229072545212113372153549597813714998389405754608829232525356889838962795771410261837258032500259413668985519953173813432956024328548806809131054171714034788475725642201515019456412574072563407137580820711779059935656841767991738364945040633030627084938839175336944176880176190382309416787606626640570119659108002315396155092035685244233937354309136838434087989304238873837337071083152968515375527737951745716998725756754653076444863665684778600168820965083068993504957459413897743314655089063431674416519026715839254840077615901260018754086987760375481730005373105146271036737191067238754775650117874385944218918812933763418368173879599988846201728111862257411032737719753998571746301789152093746301249446548094347476639986415400892494510447473437939127386845304100653416668115765038751440800413766355884454192477208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5084450912408790993841951019053768851872131163025230275e-7527 Imag: 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Size: 2.395769e-15051

m-render-wakes
Draw wakes filled with solid color, uses mpq type from gmp library

m-render-wakes > m-render-wakes.ppm

=Images=

=Problems=

error while loading shared libraries
cd ~/mandelbrot-numerics/c/bin

./m-interior

./m-interior: error while loading shared libraries: libmandelbrot-numerics.so: cannot open shared object file: No such file or directory

export LD_LIBRARY_PATH=${HOME}/opt/lib

export PATH=${HOME}/opt/bin:${PATH}

./m-interior

usage: ./m-interior precision z-guess-re z-guess-im c-guess-re c-guess-im interior-r interior-t period maxsteps

=References=