File:Periodic points of f(z) = z*z-0.75 for period =3 as intersections of 2 implicit curves.png
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[edit]DescriptionPeriodic points of f(z) = z*z-0.75 for period =3 as intersections of 2 implicit curves.png |
English: Periodic points of f(z) = z*z-0.75 for period =3 as intersections of 2 implicit curves "(which are related by the Cauchy-Riemann equations) separately. Their intersections give the complex roots of the original function. "[1] |
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Source | Own work |
Author | Adam majewski |
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[edit]I, the copyright holder of this work, hereby publish it under the following license:
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Maxima CAS src code
[edit]/* find periodic points of f^n(z,c) zn = z0 A useful way to visualize the roots of a complex function is to plot the 0 contours of the real and imaginary parts. That is, compute z = Dm(...) on a reasonably dense grid, and then use matplotlib's contour function to plot the contours where z.real is 0 and where z.imag is zero. The roots of the function are the points where these contours intersect. Warren Weckesser https://stackoverflow.com/questions/24419164/storing-roots-of-a-complex-function-in-an-array-in-scipy/24421779#24421779 */ kill(all); remvalue(all); ratprint:false; numer:true$ display2d:false$ declare (z, complex)$ declare ([x,y], real)$ z:x+y*%i; /* -------------------functions --------------------------------------*/ f(z):= z*z+c$ /* complex quadratic polynomial */ /* iterated function */ fn(n, z) := if n=0 then z else (if n=1 then f(z) else f(fn(n-1, z)) )$ /* for periodic points {z: zp=z0 }*/ Fn(n,z) := fn(n, z) - z$ /* converts complex number z = x*y*%i to the list in a draw format: [x,y] */ dr(z):=[float(realpart(z)), float(imagpart(z))]$ ToPoints(myList):= points(map(dr,myList))$ compile(all)$ /* constants */ period :5$ c:-3/4$ /* ------------------ computations ---------------------------------------*/ zp: Fn(period, z)$ e1: realpart(zp )=0$ e2: imagpart(zp )=0$ /* find periodic points using numerical method */ polyfactor:false$ if ( period < 6) /* allroots fails for period >5 */ then sol: allroots(%i*Fn(period, w)) else ( /* increase precision of numerical computations */ print("bfloat"), fpprec : 32, /*Default value: 16, it is the number of significant digits for arithmetic on bigfloat numbers */ float2bf : true, sol: bfallroots(%i*Fn(period, bfloat(w) )) )$ sol: map(rhs,sol)$ intersections:ToPoints(sol)$ dSize : 2$ /* image size in world coordinate = x, -dSize,dSize, y, -dSize,dSize), */ path:"~/Dokumenty/newton/2/"$ /* pwd, if empty then file is in a home dir , path should end with "/" */ /* draw it using draw package ( Maxima-Gnuplot interface) by Mario Rodríguez Riotorto */ draw2d( file_name = sconcat(path, string(period)), terminal = pngcairo, dimensions = [1000,1000], /* the text */ color = black, font = "Courier", font_size = 15, title = sconcat("Periodic points f(z) = z*z-3/4 period = ", string(period), " as intersections of 2 implicit curves"), user_preamble = "set key box opaque ", /* legend ovelaps the graph */ /* */ grid = false, xaxis = false, yaxis = false, xaxis_type = solid, yaxis_type = solid, xaxis_color = black, yaxis_color = black, proportional_axes = xy, /* implicit curves */ ip_grid = [200, 200], /* precision and time of computations for implicit curves */ line_width = 1.7, line_type = solid, /* first curve */ key = "re", color = blue, implicit(e1, x, -dSize,dSize, y, -dSize,dSize), /* second curve */ color = red, key = "im", implicit(e2, x, -dSize,dSize, y, -dSize,dSize), /* points */ point_type= filled_circle, point_size = 1.5, color= black, key = "periodic", intersections ) $
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current | 11:58, 27 December 2020 | 1,000 × 1,000 (109 KB) | Soul windsurfer (talk | contribs) | Uploaded own work with UploadWizard |
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