determine numerically the cycles
first find the critical points via classical Newton method using a set of equidistant starting points ( at a square the size 3 times the Lagrangian estimate for the location of the roots of f'(z) (usually a total of 1024 points).
Then I construct the orbit of those using double precision (or sometimes float128), but without rounding control, for usually 25 000 iterations (escaping is tested using an escape radius either manually set or computed via the Douady estimate).
At the end of the orbit computation, I check backwards whether the last iterate has already been visited before (squared Euclidean distance of two points below 10^-15) to get the cycle length.
One can use the code for polynomial cycles to also detect attracting/parabolic ones numerically in the rational case by making some changes:
- Load 2 polynomials f and g to denote the rational f/g.(routien "loadPolynom")
- Find the zeros of g (poles) using the "findeNullstellen" .routine.
- Construct symbolically the derivative as two polynomials (f'*g-g'f) and (gg) using "ableitenFA" for the polynomial derivative (I have a polynomial multiplication and addition routine here in case you haven't implemented those yourself)
- Find the zeros of (f'*g-g'*f) again using "findeNullstellen" (critical points)
- Construct numerically the orbits of all those values (critical points and poles) to get (attracting) cycles - this needs some work as it's best to implement this anew (from the routien "analysiereJulia"), implementing a check for inf/NaN as a numerical value. If needed, evaluating the symbolic derivative polynomials to get the multiplier (function Polynom::eval_arg_f)
- Max iteration and epsilon might need to be adjusted as the complex division constitutes another layer of numerical instability.
Program reads parameters from the input txt file and prints the output data (to the stdout or to the output txt file)
g++ main.cpp -lm -Wall
./a.out a.txt > a_out.txtor if one wants use all input files in the working directory use simply make :
makeExample result:
chmod +x m.sh
./m.sh
program compiled without errors
run the program for input txt file:
almost_failed_out.txt is output file = skipped
almost_failed.txt = input file almost_failed_out.txt = output file
cubic_p2_Bagula_out.txt is output file = skipped
cubic_p2_Bagula.txt = input file cubic_p2_Bagula_out.txt = output file
cubic_parab_out.txt is output file = skipped
cubic_parab.txt = input file cubic_parab_out.txt = output file
DancingRabbit_out.txt is output file = skipped
DancingRabbit.txt = input file DancingRabbit_out.txt = output file
dumbbell_out.txt is output file = skipped
dumbbell.txt = input file dumbbell_out.txt = output file
fractile3_out.txt is output file = skipped
fractile3.txt = input file fractile3_out.txt = output file
ijon_b012_out.txt is output file = skipped
ijon_b012.txt = input file ijon_b012_out.txt = output file
kawahira_sc_c3_out.txt is output file = skipped
kawahira_sc_c3.txt = input file kawahira_sc_c3_out.txt = output file
labirynth_out.txt is output file = skipped
labirynth.txt = input file labirynth_out.txt = output file
milnor_fig11_out.txt is output file = skipped
milnor_fig11.txt = input file milnor_fig11_out.txt = output file
notanewrecord_out.txt is output file = skipped
notanewrecord.txt = input file notanewrecord_out.txt = output file
p3near_out.txt is output file = skipped
p3near.txt = input file p3near_out.txt = output file
q_p2_1_out.txt is output file = skipped
q_p2_1.txt = input file q_p2_1_out.txt = output file
q_p2_2_out.txt is output file = skipped
q_p2_2.txt = input file q_p2_2_out.txt = output file
q_primitive3o7_out.txt is output file = skipped
q_primitive3o7.txt = input file q_primitive3o7_out.txt = output file
quadratic_out.txt is output file = skipped
quadratic_.txt = input file quadratic__out.txt = output file
rabbit_out.txt is output file = skipped
rabbit.txt = input file rabbit_out.txt = output file
spiral1_out.txt is output file = skipped
spiral1.txt = input file spiral1_out.txt = output file
spiral6_out.txt is output file = skipped
spiral6.txt = input file spiral6_out.txt = output file
spiral_marc_10_out.txt is output file = skipped
spiral_marc_10.txt = input file spiral_marc_10_out.txt = output file
spiral_marc_out.txt is output file = skipped
spiral_marc.txt = input file spiral_marc_out.txt = output file
square_spiral_out.txt is output file = skipped
square_spiral.txt = input file square_spiral_out.txt = output file
thomasson85center_out.txt is output file = skipped
thomasson85center.txt = input file thomasson85center_out.txt = output file
thomasson85_out.txt is output file = skipped
thomasson85.txt = input file thomasson85_out.txt = output file
validationFailed_out2.txt = input file validationFailed_out2_out.txt = output file
validationFailed_out.txt is output file = skipped
validationFailed.txt = input file validationFailed_out.txt = output file
end- see files in txt directory:
- input files: *.txt
- output files : *_out.txt
- cubic Julia set - location by Roger Lee Bagula
- Critical orbit f(z) = zz+-0.749413589136570+0.015312826507689i
_%3D_z*z%2Bc_and_c%3D-0.749413589136570%2B0.015312826507689*i.png){kind=link}
In case of multiple roots the numerical algorithm can show 2 roots close to each other. See output of the cubic_p2_Bagula.txt:
coefficients read from input file cubic_p2_Bagula.txt
degree 3 coefficient = ( +1.0000000000000000 +0.0000000000000000*i)
degree 2 coefficient = ( +0.0000000000000000 +0.0000000000000000*i)
degree 1 coefficient = ( +0.0000000000000000 +0.0000000000000000*i)
degree 0 coefficient = ( -0.0400000000000000 -0.7800000000000000*i)
Input polynomial p(z)=(1+0i)*z^3+(-0.040000000000000035527-0.78000000000000002665i)
derivative dp/dz = (3+0i)*z^2
2 critical points found
cp#0: -2.2351741790771484375e-08,-2.2351741790771484375e-08 . It's critical orbit is bounded and enters cycle #0 length=2
and it's stability = |multiplier|=0.95704 =attractive
internal angle = 0.045777099465243623055
cycle = { 0.12845610612214106161,-0.42699144182978676643 ; -0.108141353107358687,-0.7232875185669475071 ; }
cp#1: -2.2351741790771484375e-08,9.4296410679817199707e-09 . It's critical orbit is bounded and enters cycle #0one can check using Maxima CAS that there is only one critical point z=0
%i1) f:z^3+c;
3
(%o1) z + c
(%i2) diff(f,z,1);
2
(%o2) 3 z
(%i3) solve(3*z^2=0);
(%o3) [z = 0]
Some cases need a longer iteration. I recommend you change the following two lines in the code:
int32_t maxit=25000;to some other value - 50000 is working here.
If you want to use a higher maxit than 2^16 you also need to change the memory allocation constant:
const int32_t MAXFIXORBITLEN=(1 << 16); // max length of a total orbit to some higher value than maxit.- Periodic points of complex quadratic polynomial using Newton method in c
- program describe from mandelbrot-numerics library by Claude Heiland-Allen
- prm - Find polynomial roots with multiplicities using mpmath by Bob Boehm
- basic stuff is the (modified) universal set of Newton starting points due to [Sutherland] The initial points selected for the Newton iterations are based on a modification/simplification for practical purposes of the universal set of starting points by Sutherland. Sutherland et al use a set of circles that guarantee at least one converging starting point for every basin of attraction, the square used here (out of practical considerations) has no such guarantee.
- marcm200 ( most of the code)
MIT License
Copyright (c) 2020
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
git init
git add README.md
git commit -m "first commit"
git branch -M main
git remote add origin git@github.com:adammaj1/Describe-iterated-map-.git
git push -u origin main
Subdirectory:
mkdir txt
git add *.txt
git mv *.txt ./txt
git commit -m "move"
git push -u origin main
to overwrite
git mv -f
to remove file from remote repo
git rm --cached file1.txt
git commit -m "remove"
git push -u origin main
local repo : ~Dokumenty/periodic
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