December 2016 - Fractint - mailman.xmission.com

Fractal series with separate views.
by sciwise＠ihug.co.nz 22 Dec '16

22 Dec '16

With this version I've placed the image data and waveforms upon separate figures . Using a few values from the right of the main bulb , there are selections where the z series retains it's magnitude whilst the series for ed tends towards zero. I'm considering another approach to this , which I shall develop for the next version ; meanwhile try exploring. sciwise. """ plot2sz3.py General structure for examining fractal series . Intended for use with Linux type OS's and d1iMandelbrot formula with [x]fractint. (c) copyright 2016 sciwise(a)ihug.co.nz This version uses separate figures allowing for easier manual selection from the fractal and better discernability of the waveforms. """ # Import libraries import numpy as np import matplotlib.pyplot as plt import matplotlib.gridspec as gridspec # import matplotlib.cbook as cbook import matplotlib.cm as cm # # -------------------------------------------------------------------- # #####Event handler for button_press_event##### def enter_axes(event): print('enter_axes', event.inaxes) event.canvas.draw() def leave_axes(event): print('leave_axes', event.inaxes) event.canvas.draw() def onclick(event): ''' Event handler for button_press_event @param event MouseEvent ''' global ix, iy ix, iy = event.xdata, event.ydata for i, ax in enumerate([ax1, ax2 , ax3]): # For infomation, print which axes the click was in if ax == event.inaxes: # Check if the click was in ax1 if (i+1 == 1 ) : c = pt2c(ix,iy,xTL,yTL,xBR,yBR) d1imand(c) plt.figure(2) ax2.clear() ax2.plot(x2, y2r, 'r.-' ) ax2.plot(x2, y2i, 'b.-') ax3.clear() ax3.plot(x2, yed, 'm.-') plt.draw() plt.figure(1) #print " ",c.real," ",c.imag # Check if the click was in ax1 or ax2 if event.inaxes in [ax1, ax2,ax3]: return iy else: return # # .......................................................................... # # xTL x top left # yTL y top left # xBR x bottom right # yBE y bottom right # def pt2c(ix,iy,xTL,yTL,xBR,yBR): x=xTL+(xBR-xTL)*ix/800 y=yTL+(yBR-yTL)*iy/600 c=complex(x,y) return c def d1imand(c): """ Mandelbrot series = z first derivative = z1 second derivative = z2 Diff' equation = ed = diff(f(z),z) + 3*f(z)*z = sin(z) """ for i in range(n): y2r[i] = 0.0 y2i[i] = 0.0 yed[i] = 0.0 z = c z1 = 1.0 z2 = 0.0 itr=0 ed=0 while ( itr < n) and (abs(z2) < 1000000) and ( abs(z) > 0 ): z2 =2*z1*z1 +2*z2*z z1 = 2*z*z1 + 1 u = z z = z*z+c ed = 3*z*c + z1 - np.sin(u) y2r[itr] = z.real y2i[itr] = z.imag yed[itr] = abs(ed) itr = itr+1 for i in range(200): y2r[i] = 0.0 y2i[i] = 0.0 yed[i] = 0.0 return def fracCorners(file1) : f = open(file1, 'r') x = f.read() j=0 k=0 l=0 for i in range(0,len(x)-8): ns = x[i : i+8] if ( ns=='corners=' ) : l=int(l+8) break l=int(l+1) j=l for i in range(l,len(x)-1): ns = x[i : i+1] if ( ns== "/" ) : break j=int(j+1) s=x[l:j] xTL = np.float64(s) l=j+1 j=l for i in range(l,len(x)-1): ns = x[i : i+1] if ( ns== "/" ) : break j=int(j+1) s = x[l:j] xBR = np.float64(s) l=j+1 j=l for i in range(l,len(x)-1): ns = x[i : i+1] if ( ns== "/" ) : break j=int(j+1) s=x[l:j] yBR = np.float64(s) l=j +1 j=l for i in range(l,len(x)-6): ns = x[i : i+6] if ( ns== "float=" ) : break j=int(j+1) s=x[l:j] yTL = np.float64(s) l=j j=l for i in range(l,len(x)-8): ns = x[i : i+8] if ( ns== "maxiter=" ) : j=j+8 break j=int(j+1) # l=j j=l for i in range(l,len(x)-7): ns = x[i : i+7] if ( ns== "inside=" ) : break j=int(j+1) s = x[l:j] maxiter = np.int64(s) f.close() return [xTL,yTL,xBR,yBR,maxiter] # # --------------------------- Main ----------------------------------------- # # fractint gif file and associated par file. # filegif = 'fract008.gif' filepar = 'fract.par' # # ........................................................................... # fbase = '/usr/share/xfractint/pars/' file1= fbase + filepar xyc=fracCorners(file1) xTL = xyc[0] yTL = xyc[1] xBR =xyc[2] yBR =xyc[3] maxiter=xyc[4] # fbase = '/root/' file2= fbase + filegif imageFile = cbook.get_sample_data(file2) image = plt.imread(imageFile) # # --------------------------------------------------------------------------- # # Generate data # n=maxiter x2 = np.array((np.linspace(0,n,n))) y2r = np.array((np.linspace(0,n,n))) y2i = np.array((np.linspace(0,n,n))) yed = np.array((np.linspace(0,n,n))) # # initial value for c , this to be selected from image . # c=-0.25 + 0.3j #c=-0.47 + 0.018j #c=-0.75 + 0.0014j #c=-0.0099 + 0.005j #c=-0.108516194675302 + 0.9013927770462271j #c=-0.123991033848631 + 0.7429343365256118j # # # ........................................................................... # d1imand(c) # # ------------------------------------------------------------------- # # Plot on seperate figures fig1 = plt.figure() # large subplot ax1=fig1.add_subplot(1,1,1) plt.axis('off') plt.imshow(image, interpolation="none" ) fig1.tight_layout() fig2 = plt.figure() ax2=fig2.add_subplot(2,1,1) plt.plot(x2, y2r, 'r.-') plt.plot(x2, y2i, 'b.-') ax3=fig2.add_subplot(2,1,2) plt.plot(x2, yed, 'm.-') plt.ylabel('Magnitude') fig2.tight_layout() # # cid = fig1.canvas.mpl_connect('button_press_event', onclick) #fig.canvas.mpl_disconnect(cid) plt.show() exit(0) # # -------------------------------------------------------------------- # #fig.set_size_inches(w=11,h=7) #fig_name = 'plot.png' #fig.savefig(fig_name)

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View fractal series
by sciwise＠ihug.co.nz 20 Dec '16

20 Dec '16

This python 2.7 code is intended for use with a Linux type system. The *.gif and *.par files are generated from xfractint and then read and displayed . The mouse is then used to select a pixel value from the image , this is then used to calculate z , z1 , z2 and ed . The real and imaginary components of the z series are then shown as blue and red waveforms. The differential equation results are shown in magenta. The objective is to have the magenta values go to zero whilst the red or blue values remain greater than zero. This may involve going back to fractint and selecting a another region , saving the *.gif and *.par files , then invoking the python code again ; with the new *.gif and *.par files. I use geany to edit and run python files. """ plot2sz2.py General structure for examining fractal series . Intended for use with Linux type OS's and d1iMandelbrot formula with fractint. (c) copyright 2016 sciwise(a)ihug.co.nz """ # Import libraries import numpy as np import matplotlib.pyplot as plt import matplotlib.gridspec as gridspec # import matplotlib.cbook as cbook import matplotlib.cm as cm # # -------------------------------------------------------------------- # #####Event handler for button_press_event##### def enter_axes(event): print('enter_axes', event.inaxes) event.canvas.draw() def leave_axes(event): print('leave_axes', event.inaxes) event.canvas.draw() def onclick(event): ''' Event handler for button_press_event @param event MouseEvent ''' global ix, iy ix, iy = event.xdata, event.ydata for i, ax in enumerate([ax1, ax2]): # For infomation, print which axes the click was in if ax == event.inaxes: # Check if the click was in ax1 if (i+1 == 1 ) : c = pt2c(ix,iy,xTL,yTL,xBR,yBR) d1imand(c) ax2.clear() ax2.plot(x2, y2r, 'r.-' ) ax2.plot(x2, y2i, 'b.-') ax2.plot(x2, yed, 'm.-') plt.draw() #print " ",c.real," ",c.imag # Check if the click was in ax1 or ax2 if event.inaxes in [ax1, ax2]: return iy else: return # # .......................................................................... # # xTL x top left # yTL y top left # xBR x bottom right # yBE y bottom right # def pt2c(ix,iy,xTL,yTL,xBR,yBR): x=xTL+(xBR-xTL)*ix/800 y=yTL+(yBR-yTL)*iy/600 c=complex(x,y) return c def d1imand(c): """ Mandelbrot series = z first derivative = z1 second derivative = z2 Diff' equation = ed = diff(f(z),z) + 3*f(z)*z = sin(z) """ for i in range(n): y2r[i] = 0.0 y2i[i] = 0.0 yed[i] = 0.0 z = c z1 = 1.0 z2 = 0.0 itr=0 ed=0 while ( itr < n) and (abs(z2) < 1000000) and ( abs(z) > 0 ): z2 =2*z1*z1 +2*z2*z z1 = 2*z*z1 + 1 u = z z = z*z+c ed = 3*z*c + z1 - np.sin(u) y2r[itr] = z.real y2i[itr] = z.imag yed[itr] = abs(ed) itr = itr+1 for i in range(200): y2r[i] = 0.0 y2i[i] = 0.0 yed[i] = 0.0 return def fracCorners(file1) : f = open(file1, 'r') x = f.read() j=0 k=0 l=0 for i in range(0,len(x)-8): ns = x[i : i+8] if ( ns=='corners=' ) : l=int(l+8) break l=int(l+1) j=l for i in range(l,len(x)-1): ns = x[i : i+1] if ( ns== "/" ) : break j=int(j+1) s=x[l:j] xTL = np.float64(s) l=j+1 j=l for i in range(l,len(x)-1): ns = x[i : i+1] if ( ns== "/" ) : break j=int(j+1) s = x[l:j] xBR = np.float64(s) l=j+1 j=l for i in range(l,len(x)-1): ns = x[i : i+1] if ( ns== "/" ) : break j=int(j+1) s=x[l:j] yBR = np.float64(s) l=j +1 j=l for i in range(l,len(x)-6): ns = x[i : i+6] if ( ns== "float=" ) : break j=int(j+1) s=x[l:j] yTL = np.float64(s) l=j j=l for i in range(l,len(x)-8): ns = x[i : i+8] if ( ns== "maxiter=" ) : j=j+8 break j=int(j+1) # l=j j=l for i in range(l,len(x)-7): ns = x[i : i+7] if ( ns== "inside=" ) : break j=int(j+1) s = x[l:j] maxiter = np.int64(s) f.close() return [xTL,yTL,xBR,yBR,maxiter] # # --------------------------- Main ----------------------------------------- # # fractint gif file and associated par file. # filegif = 'fract008.gif' filepar = 'fract.par' # # ........................................................................... # fbase = '/usr/share/xfractint/pars/' file1= fbase + filepar xyc=fracCorners(file1) xTL = xyc[0] yTL = xyc[1] xBR =xyc[2] yBR =xyc[3] maxiter=xyc[4] # fbase = '/root/' file2= fbase + filegif imageFile = cbook.get_sample_data(file2) image = plt.imread(imageFile) # # --------------------------------------------------------------------------- # # Generate data # n=maxiter x2 = np.array((np.linspace(0,n,n))) y2r = np.array((np.linspace(0,n,n))) y2i = np.array((np.linspace(0,n,n))) yed = np.array((np.linspace(0,n,n))) # # initial value for c , this to be selected from image . # c=-0.25 + 0.3j #c=-0.47 + 0.018j #c=-0.75 + 0.0014j #c=-0.0099 + 0.005j #c=-0.108516194675302 + 0.9013927770462271j #c=-0.123991033848631 + 0.7429343365256118j # # # ........................................................................... # d1imand(c) # # ------------------------------------------------------------------- # # Plot figure with subplots of different sizes fig = plt.figure(1) # set up subplot grid gridspec.GridSpec(6,6) # large subplot ax1 = plt.subplot2grid((6,6), (0,0), colspan=5, rowspan=5) plt.axis('off') plt.imshow(image, interpolation="none" ) # # # small subplot 3 ax2 = plt.subplot2grid((6,6), (5,0), colspan=6, rowspan=1) plt.locator_params(axis='x', nbins=6) plt.locator_params(axis='y', nbins=5) plt.plot(x2, y2r, 'r.-') plt.plot(x2, y2i, 'b.-') plt.plot(x2, yed, 'm.-') plt.ylabel('Magnitude') # fit subplots and save fig fig.tight_layout() cid = fig.canvas.mpl_connect('button_press_event', onclick) #fig.canvas.mpl_disconnect(cid) plt.show() exit(0) # # -------------------------------------------------------------------- # #fig.set_size_inches(w=11,h=7) #fig_name = 'plot.png' #fig.savefig(fig_name)

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Re: [Fractint] Another diff equation in Fractint
by studiomusician0 11 Dec '16

11 Dec '16

Hi, do you know of any good software that works with window 10? I wish they'd bring XP back. Thanks Tony . Sent from my Verizon, Samsung Galaxy smartphone -------- Original message --------From: sciwise(a)ihug.co.nz Date: 12/11/16 8:18 AM (GMT-05:00) To: Fractint and General Fractals Discussion <fractint(a)mailman.xmission.com> Subject: [Fractint] Another diff equation in Fractint This uses the Mandelbrot series as the base series . I used Maxima CAS to check that this iteration code was the same as that obtained via symbolic differentiation at each iteration ; they are. Two features that I noticed , firstly the 'channels' around the bulbs are filling up with 'silt' ; secondly there may now be some extra minibrots to the right of the main bulb. Overall this remains bland . Using Python I shall look at the series in more detail , however I'm not expecting much. d1iMandelbrot(XAXIS) {; Edward Montague (c) 2016 ; ; Mandelbrot series = z ; First Derivative of Mandelbrot Series = z1 ; Second Derivative of Mandelbrot Series = z2 ; ; Differential Equation is , diff(f(z),z) + 3*f(z)*z = sin(z) ; Actually w.r.t c , where c == Pixel. ; z = Pixel z1=1 z2 = 0 u = 0 ed = 0: z2 = 2*z*z2+2*z1^2 z1 = 2*z*z1+1 u = z z = z*z + Pixel ed = 3*z*Pixel + z1 - sin(u) |ed| < 10000 } _______________________________________________ Fractint mailing list Fractint(a)mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint

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Another diff equation in Fractint
by sciwise＠ihug.co.nz 11 Dec '16

11 Dec '16

This uses the Mandelbrot series as the base series . I used Maxima CAS to check that this iteration code was the same as that obtained via symbolic differentiation at each iteration ; they are. Two features that I noticed , firstly the 'channels' around the bulbs are filling up with 'silt' ; secondly there may now be some extra minibrots to the right of the main bulb. Overall this remains bland . Using Python I shall look at the series in more detail , however I'm not expecting much. d1iMandelbrot(XAXIS) {; Edward Montague (c) 2016 ; ; Mandelbrot series = z ; First Derivative of Mandelbrot Series = z1 ; Second Derivative of Mandelbrot Series = z2 ; ; Differential Equation is , diff(f(z),z) + 3*f(z)*z = sin(z) ; Actually w.r.t c , where c == Pixel. ; z = Pixel z1=1 z2 = 0 u = 0 ed = 0: z2 = 2*z*z2+2*z1^2 z1 = 2*z*z1+1 u = z z = z*z + Pixel ed = 3*z*Pixel + z1 - sin(u) |ed| < 10000 }

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Re: [Fractint] Fractint Digest, Vol 166, Issue 1
by Jovan Trujillo 08 Dec '16

08 Dec '16

Well I now have new motivation to read Differential Equations on Fractals. Thanks sciwise! I'll have to test out this code next week when finals are over and I finally graduate. Sent from my iPhone > On Dec 5, 2016, at 12:00 PM, fractint-request(a)mailman.xmission.com wrote: > > Send Fractint mailing list submissions to > fractint(a)mailman.xmission.com > > To subscribe or unsubscribe via the World Wide Web, visit > https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint > or, via email, send a message with subject or body 'help' to > fractint-request(a)mailman.xmission.com > > You can reach the person managing the list at > fractint-owner(a)mailman.xmission.com > > When replying, please edit your Subject line so it is more specific > than "Re: Contents of Fractint digest..." > > > Today's Topics: > > 1. Maxima cas for Fractint (sciwise(a)ihug.co.nz) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Mon, 05 Dec 2016 18:22:18 +1300 > From: sciwise(a)ihug.co.nz > To: Fractint and General Fractals Discussion > <fractint(a)mailman.xmission.com> > Subject: [Fractint] Maxima cas for Fractint > Message-ID: <de06bfcfb027b2792ff64c0109b1b4ac(a)ihug.co.nz> > Content-Type: text/plain; charset=UTF-8 > > > > This is code that I've written in Maxima CAS . > > This finds the > first and second derivative of an iterated function , g(z,c) . > > To > construct a fractal substitute z:g(z,c) with the equation for g(z,c) , > > > rather than calling a function. > > Similarly for z1:g1(z,z1,c) and > z2:g2(z,z1,z2,c) . > > I've provided an example of what this should look > like. > > The validity of this approach is tested by symbolically > calculating the second > > derivative at each iteration and comparing this > with the second derivative from the > > above method. This is done for > three iterations and , if correct , will display TRUE. > > The file > difffrac1h.mac > > /* > . > > (c) 2016 , sciwise(a)ihug.co.nz . > > A general > method for defining the first and second derivatives of an > iterated > fractal function , g(z,c) . > > Some care is required to ensure that > g(0,c) is defined . > > . > */ > > /* > Example Fractint code . > > d2zmand(XAXIS){; > Edward Montague > ; Mandelbrot series = z > ; 1st Derivative of > Mandelbrot series = z1. > ; 2nd Derivative of Mandelbrot series = z2. > > z > = Pixel > z1 = 1 > z2 = 0: > z2 = 2*z1*z1 +2*z2*z > z1 = 2*z*z1 + 1 > z = > z*z+Pixel > |z2| z1 > */ > > eq:diff(g(z,c),z,1)*z1 + diff(g(0,c),c); > > define(g1(z,z1,c),eq)$ > > /* > Second Derivative --> > z2 > */ > > eq:diff(g(z,c),z,2)*z1*z1 + diff(g(z,c),z)*z2 + diff(g(0,c),c,2); > > define(g2(z,z1,z2,c),eq)$ > > /* > Initial values > */ > > z : c; > z1 : > diff(z,c); > z2 : diff(z,c,2); > > /* > > First iteration. > */ > > z2 : > g2(z,z1,z2,c); > z1 : g1(z,z1,c)$ > z : > g(z,c); > eq:diff(z,c,2); > is(equal(eq,z2)); > > /* > Second iteration. > */ > > z2 > : g2(z,z1,z2,c); > z1 : g1(z,z1,c)$ > z : > g(z,c); > eq:diff(z,c,2)$ > is(equal(eq,z2)); > > /* > Third iteration. > */ > > z2 : > g2(z,z1,z2,c); > z1 : g1(z,z1,c)$ > z : > g(z,c)$ > eq:diff(z,c,2)$ > is(equal(eq,z2)); > > /* > > */ > > And the file > difffrac1h.wxm > > /* [wxMaxima batch file version 1] [ DO NOT EDIT BY > HAND! ]*/ > /* [ Created with wxMaxima version 11.08.0 ] */ > > /* [wxMaxima: > comment start ] > . > > (c) 2016 , sciwise(a)ihug.co.nz . > > A general method > for defining the first and second derivatives of an > iterated fractal > function , g(z,c) . > > Some care is required to ensure that g(0,c) is > defined . > > . > [wxMaxima: comment end ] */ > > /* [wxMaxima: comment start > ] > Example Fractint code . > > d2zmand(XAXIS){; Edward Montague > ; > Mandelbrot series = z > ; 1st Derivative of Mandelbrot series = z1. > ; > 2nd Derivative of Mandelbrot series = z2. > > z = Pixel > z1 = 1 > z2 = 0: > > z2 = 2*z1*z1 +2*z2*z > z1 = 2*z*z1 + 1 > z = z*z+Pixel > |z2| z1 > > [wxMaxima: comment end ] */ > > /* [wxMaxima: input start ] > */ > eq:diff(g(z,c),z,1)*z1 + diff(g(0,c),c); > define(g1(z,z1,c),eq)$ > /* > [wxMaxima: input end ] */ > > /* [wxMaxima: comment start ] > Second > Derivative --> z2 > [wxMaxima: comment end ] */ > > /* [wxMaxima: input > start ] */ > eq:diff(g(z,c),z,2)*z1*z1 + diff(g(z,c),z)*z2 + > diff(g(0,c),c,2); > define(g2(z,z1,z2,c),eq)$ > /* [wxMaxima: input end ] > */ > > /* [wxMaxima: comment start ] > Initial values > [wxMaxima: comment > end ] */ > > /* [wxMaxima: input start ] */ > z : c; > z1 : diff(z,c); > z2 : > diff(z,c,2); > /* [wxMaxima: input end ] */ > > /* [wxMaxima: comment start > ] > > First iteration. > [wxMaxima: comment end ] */ > > /* [wxMaxima: input > start ] */ > > z2 : g2(z,z1,z2,c); > z1 : g1(z,z1,c)$ > z : > g(z,c); > eq:diff(z,c,2); > is(equal(eq,z2)); > /* [wxMaxima: input end ] > */ > > /* [wxMaxima: comment start ] > Second iteration. > [wxMaxima: comment > end ] */ > > /* [wxMaxima: input start ] */ > z2 : g2(z,z1,z2,c); > z1 : > g1(z,z1,c)$ > z : g(z,c); > eq:diff(z,c,2)$ > is(equal(eq,z2)); > /* [wxMaxima: > input end ] */ > > /* [wxMaxima: comment start ] > Third iteration. > > [wxMaxima: comment end ] */ > > /* [wxMaxima: input start ] */ > z2 : > g2(z,z1,z2,c); > z1 : g1(z,z1,c)$ > z : > g(z,c)$ > eq:diff(z,c,2)$ > is(equal(eq,z2)); > /* [wxMaxima: input end ] > */ > > /* [wxMaxima: comment start ] > > [wxMaxima: comment end ] */ > > /* > Maxima can't load/batch files which end with a comment! */ > "Created with > wxMaxima"$ > > > > ------------------------------ > > Subject: Digest Footer > > _______________________________________________ > Fractint mailing list > Fractint(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint > > > ------------------------------ > > End of Fractint Digest, Vol 166, Issue 1 > ****************************************

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Re: [Fractint] Python for fractint
by sciwise＠ihug.co.nz 08 Dec '16

08 Dec '16

This is a work in progress. Unlike mainstream mathematics this is more like an experiment than a typical mathematical construct. I'm attempting to use the Mandelbrot set and the first and second derivatives thereof in the construction and solution of differential equations. Just as the Maxwell equations have a general solution in terms of a complex exponential so might other differential equations have a solution in terms of the Mandelbrot set. This is a big if and a differential equation with a known result is required ; that's my next step. This is a numerical approach that utilizes the power of Fractint to find the regions of stability ; the black regions when inside colour is numb. That's interesting about P3 , the main difference between P2 and P3 that I found is how the print statement is used. Python is unexpectedly powerful , the 3d graphics via Mayavi is something that I've been wanting for a long time. I think that the Mandelbulber program was written in Python. Sciwise On 08/12/2016 17:29, LLOYD GARRICK wrote: > Hey thanx for that! > > Cool graphs, but I am not sure exactly what they mean...... > > I am just learning Python too, the first thing I try is fractals. But it seems all the points are plotted in memory first and then the entire graph is shown. I am trying to figure how to plot points in real time so you can see it drawing. > > Your code runs in both P2 and P3 (Linux); but P3 seems much faster. > >> On December 7, 2016 at 3:00 AM sciwise(a)ihug.co.nz wrote: >> >> Maxima CAS has a few limitations , hence I now program in Python >> too. >> >> This code calculates the second derivative of the Mandelbrot >> series for an >> >> initial complex value of c . The real and imaginary >> components are then graphed. >> >> The first 80 values have been set to a >> constant so that the stable condition might >> >> be observed. >> >> File >> mltplotz.py >> >> --- code --- >> >> """ >> (c) copyright 2016 >> sciwise(a)ihug.co.nz >> >> Simple demo with multiple subplots. >> >> Derivatives >> of Mandelbrot series . >> >> """ >> import numpy as np >> import matplotlib.pyplot >> as plt >> >> n=128 >> >> x2 = np.array((np.linspace(0,n,n))) >> y2r = >> np.array((np.linspace(0,n,n))) >> y2i = np.array((np.linspace(0,n,n))) >> >> for >> i in range(n): >> y2r[i] = 0.0 >> y2i[i] = 0.0 >> >> ''' >> >> Fractal calculations >> . >> >> ''' >> >> c=-0.5 + 0.3j >> z = c >> z1 = 1.0 >> z2 = 0.0 >> >> itr=0 >> while ( itr < n) >> and (abs(z2) < 100000): >> z2 =2*z1*z1 +2*z2*z >> z1 = 2*z*z1 + 1 >> z = >> z*z+c >> y2r[itr] = z2.real >> y2i[itr] = z2.imag >> itr=itr+1 >> >> """ >> >> --------------------------------------------------------------------- >> >> Values after 60 iterations are of interest . >> >> """ >> >> for i in range(80): >> >> y2r[i] = 0.29 >> y2i[i] = 0.186 >> >> plt.subplot(2, 1, 1) >> plt.plot(x2, y2r, >> 'r.-') >> plt.title('Second Derivative of Mandelbrot >> series.') >> plt.xlabel('index') >> plt.ylabel('Real z2') >> >> plt.subplot(2, 1, >> 2) >> plt.plot(x2, y2i, 'b.-') >> plt.xlabel('index') >> plt.ylabel('Imaginary >> z2') >> >> plt.show() >> >> exit(0) >> >> _______________________________________________ >> Fractint mailing list >> Fractint(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint

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Python for fractint
by sciwise＠ihug.co.nz 07 Dec '16

07 Dec '16

Maxima CAS has a few limitations , hence I now program in Python too. This code calculates the second derivative of the Mandelbrot series for an initial complex value of c . The real and imaginary components are then graphed. The first 80 values have been set to a constant so that the stable condition might be observed. File mltplotz.py --- code --- """ (c) copyright 2016 sciwise(a)ihug.co.nz Simple demo with multiple subplots. Derivatives of Mandelbrot series . """ import numpy as np import matplotlib.pyplot as plt n=128 x2 = np.array((np.linspace(0,n,n))) y2r = np.array((np.linspace(0,n,n))) y2i = np.array((np.linspace(0,n,n))) for i in range(n): y2r[i] = 0.0 y2i[i] = 0.0 ''' Fractal calculations . ''' c=-0.5 + 0.3j z = c z1 = 1.0 z2 = 0.0 itr=0 while ( itr < n) and (abs(z2) < 100000): z2 =2*z1*z1 +2*z2*z z1 = 2*z*z1 + 1 z = z*z+c y2r[itr] = z2.real y2i[itr] = z2.imag itr=itr+1 """ --------------------------------------------------------------------- Values after 60 iterations are of interest . """ for i in range(80): y2r[i] = 0.29 y2i[i] = 0.186 plt.subplot(2, 1, 1) plt.plot(x2, y2r, 'r.-') plt.title('Second Derivative of Mandelbrot series.') plt.xlabel('index') plt.ylabel('Real z2') plt.subplot(2, 1, 2) plt.plot(x2, y2i, 'b.-') plt.xlabel('index') plt.ylabel('Imaginary z2') plt.show() exit(0)

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Maxima cas for Fractint
by sciwise＠ihug.co.nz 05 Dec '16

05 Dec '16

This is code that I've written in Maxima CAS . This finds the first and second derivative of an iterated function , g(z,c) . To construct a fractal substitute z:g(z,c) with the equation for g(z,c) , rather than calling a function. Similarly for z1:g1(z,z1,c) and z2:g2(z,z1,z2,c) . I've provided an example of what this should look like. The validity of this approach is tested by symbolically calculating the second derivative at each iteration and comparing this with the second derivative from the above method. This is done for three iterations and , if correct , will display TRUE. The file difffrac1h.mac /* . (c) 2016 , sciwise(a)ihug.co.nz . A general method for defining the first and second derivatives of an iterated fractal function , g(z,c) . Some care is required to ensure that g(0,c) is defined . . */ /* Example Fractint code . d2zmand(XAXIS){; Edward Montague ; Mandelbrot series = z ; 1st Derivative of Mandelbrot series = z1. ; 2nd Derivative of Mandelbrot series = z2. z = Pixel z1 = 1 z2 = 0: z2 = 2*z1*z1 +2*z2*z z1 = 2*z*z1 + 1 z = z*z+Pixel |z2| z1 */ eq:diff(g(z,c),z,1)*z1 + diff(g(0,c),c); define(g1(z,z1,c),eq)$ /* Second Derivative --> z2 */ eq:diff(g(z,c),z,2)*z1*z1 + diff(g(z,c),z)*z2 + diff(g(0,c),c,2); define(g2(z,z1,z2,c),eq)$ /* Initial values */ z : c; z1 : diff(z,c); z2 : diff(z,c,2); /* First iteration. */ z2 : g2(z,z1,z2,c); z1 : g1(z,z1,c)$ z : g(z,c); eq:diff(z,c,2); is(equal(eq,z2)); /* Second iteration. */ z2 : g2(z,z1,z2,c); z1 : g1(z,z1,c)$ z : g(z,c); eq:diff(z,c,2)$ is(equal(eq,z2)); /* Third iteration. */ z2 : g2(z,z1,z2,c); z1 : g1(z,z1,c)$ z : g(z,c)$ eq:diff(z,c,2)$ is(equal(eq,z2)); /* */ And the file difffrac1h.wxm /* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/ /* [ Created with wxMaxima version 11.08.0 ] */ /* [wxMaxima: comment start ] . (c) 2016 , sciwise(a)ihug.co.nz . A general method for defining the first and second derivatives of an iterated fractal function , g(z,c) . Some care is required to ensure that g(0,c) is defined . . [wxMaxima: comment end ] */ /* [wxMaxima: comment start ] Example Fractint code . d2zmand(XAXIS){; Edward Montague ; Mandelbrot series = z ; 1st Derivative of Mandelbrot series = z1. ; 2nd Derivative of Mandelbrot series = z2. z = Pixel z1 = 1 z2 = 0: z2 = 2*z1*z1 +2*z2*z z1 = 2*z*z1 + 1 z = z*z+Pixel |z2| z1 [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ eq:diff(g(z,c),z,1)*z1 + diff(g(0,c),c); define(g1(z,z1,c),eq)$ /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] Second Derivative --> z2 [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ eq:diff(g(z,c),z,2)*z1*z1 + diff(g(z,c),z)*z2 + diff(g(0,c),c,2); define(g2(z,z1,z2,c),eq)$ /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] Initial values [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ z : c; z1 : diff(z,c); z2 : diff(z,c,2); /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] First iteration. [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ z2 : g2(z,z1,z2,c); z1 : g1(z,z1,c)$ z : g(z,c); eq:diff(z,c,2); is(equal(eq,z2)); /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] Second iteration. [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ z2 : g2(z,z1,z2,c); z1 : g1(z,z1,c)$ z : g(z,c); eq:diff(z,c,2)$ is(equal(eq,z2)); /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] Third iteration. [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ z2 : g2(z,z1,z2,c); z1 : g1(z,z1,c)$ z : g(z,c)$ eq:diff(z,c,2)$ is(equal(eq,z2)); /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] [wxMaxima: comment end ] */ /* Maxima can't load/batch files which end with a comment! */ "Created with wxMaxima"$

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