File:Backward Iteration.svg
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Summary
[edit]DescriptionBackward Iteration.svg |
English: Backward iteration of complex quadratic polynomial with proper chose of the preimage |
Date | |
Source | Own work |
Author | Adam majewski |
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[edit]-
Preimages of the circle under map f(z) = z*z+0.25
Summary
[edit]"In the dynamic plane, external rays can be drawn by backwards iteration. It is most effective for a periodic or preperiodic angle.
You must keep track of points on the finite collection of rays with angles
Say corresponds to a radius and the angle .
Then maps to .
This point, which was constructed before, has two preimages under .
The one that is closer to is the correct one. This criterion was proved by Thierry Bousch. The ray will look better when you introduce intermediate points." Wolf Jung
Maxima CAS src code
[edit]/* batch file for Maxima CAS It shows how to choose one from 2 preimages under complex quadratic polynomial comments are from Wolf Jung program Mandel http://www.mndynamics.com/indexp.html */ kill(all); remvalue(all); /* Say z_{l, j } corresponds to: - a radius = R^{(1/2) ^l } - the angle = t*(2^j) Then fc(z) maps z_{l,j} to z_{(l-1),(j+1)}. Inverse function maps z_{l,j} to z_{(l+1),(j-1)} */ p(radius, angle, l,j):= radius^((1/2)^l) * %e^(%i*angle*(2^j)*2*%pi); /* circle D={w:abs(w)=1 } where w=l(t,r) t is angle in turns ; 1 turn = 360 degree = 2*Pi radians */ tMax:100; /* exponential for of complex number with angle in turns */ GiveCirclePoint(t):=R*%e^(%i*t*2*%pi)$ /* gives point of unit circle for angle t in turns */ /*-------------- unit circle ------------*/ R:1; circle_angles:makelist(t/tMax,t,0,tMax-1)$ CirclePoints:map(GiveCirclePoint,circle_angles)$ /* initial points on the rays t and 2*t */ R:20; t:1/3; /* initial angle */ /* initial points on periodic rays " You must keep track of points on the finite collection of rays with angles t, 2t, 4t... Say z_{l, j } corresponds to: - a radius = R^{(1/2) ^l } - the angle = t*(2^j) " */ z00:p(R,t,0,0); /* on ray 1/3 */ z01:p(R,t,0,1); /* on ray 2*1/3=2/3 */ z1m1:p(R,t,1,-1); /* preimages of z00 */ pre1:[z1m1,-z1m1]; if (z1m1.z01>0) then z11:z1m1 else z11:-z1m1; ray1:[z01,z11]; load(draw); /* Mario Rodriguez Riotorto http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html */ draw2d(file_name = "iteration6", pic_width=1000, pic_height= 1000, terminal = 'svg, title = "Backward iteration with proper chose of preimage ", user_preamble = "set angles degrees; set grid polar 30; set xtics 5; set mxtics 5; set size square;set key out;set key top right", yrange = [-20,20], xrange = [-20,20], points_joined =true, color = black, point_type = filled_circle, point_size = 0.1, points(map(realpart, CirclePoints),map(imagpart, CirclePoints)), points_joined =true, point_size = 0.9, line_width = 3, color = black, key = "ray 2/3", points(map(realpart,ray1),map(imagpart,ray1)), color = red, point_size = 1.0, points_joined =false, key = "z0 on ray 1/3", label(["z00",realpart(z00)+2,imagpart(z00)+2]), points([realpart(z00)],[imagpart(z00)]), color = green, key = "z0 on ray 2/3", label(["z01",realpart(z01)+2,imagpart(z01)-2]), points([realpart(z01)],[imagpart(z01)]), color = blue, key = "good preimage of z0 from ray 1/3", points([realpart(z11)],[imagpart(z11)]), color = black, key = "bad preimage of z0 from ray 1/3", points([realpart(-z11)],[imagpart(-z11)]) );
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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue |
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 21:00, 17 March 2018 | 1,000 × 1,000 (21 KB) | Soul windsurfer (talk | contribs) | optimised with svgomg | |
15:18, 26 September 2012 | 1,000 × 1,000 (34 KB) | Soul windsurfer (talk | contribs) | {{Information |Description ={{en|1=Backward iteration of complex quadratic polynomial with proper chose of the preimage}} |Source ={{own}} |Author =Adam majewski |Date =26.09.2012 |Permission = |o... |
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