January 2017 - Fractint - mailman.xmission.com

Lambert W
by sciwise＠ihug.co.nz 04 Feb '17

04 Feb '17

This is a late night , early morning , endeavour ; therefore not as well investigated as usual. There are fractals for the power tower however there didn't appear to be anything for the basic Lambert W function. LamW(XAXIS) {; Edward Montague (c) 2017 ; ; Checked for complex values with Maxima ; ; Lambert's W function via Halley's root finding ; algorithm , with addition imaginary constant . ; ; ; c = z = Pixel w=0.1 : wold=w w=w - (w*exp(w) - z)/((exp(w)*(w+1)-(w+2)*(w*exp(w)-z)/(2*w+2))+(0,0.000001)) .000001 < |w-wold| ; && |w|

2 1

Newman1
by sciwise＠ihug.co.nz 26 Jan '17

26 Jan '17

This formula was constructed after reading a pdf from Yu and Ben-Israel. Doesn't appear to generate what I expected , maybe I need to delve into this a bit more , any comments from the list might be useful. If you comment out the line p1=1 then the parameter p1 is selectable from the parameter box. Newman1(XAXIS) {; Edward Montague (c) 2017 ; ; Supposedly equivalent to the Mandelbrot Set. ; ; Ref , The Newton and Halley Methods for Complex Roots ; Lily . Yau and Adi Ben-Israel ; ; c = z = Pixel p1=1 : ed =exp(-(2*atan((2*z-1)/sqrt(4*c-1)))/sqrt(4*c-1)-(2*atan(1/sqrt(4*c-1)))/sqrt(4*c-1)) edp= -exp(-(2*atan((2*z-1)/sqrt(4*c-1)))/sqrt(4*c-1)-(2*atan(1/sqrt(4*c-1)))/sqrt(4*c-1))/(z^2-z+c) zold=z; z = z - p1*ed/edp .0001 < |z-zold| ; && |z|

2 1

NewtDiffmaxc
by sciwise＠ihug.co.nz 21 Jan '17

21 Jan '17

Maxima is mostly an algebraic system , however using various simplifications numerical calculations are also possible ; even those that have a tendency to cause numerical overflow. Yes numerical overflow is possible in maxima unless various precautions are taken. This is the maxima equivalent to the fractint formula d2jbMandelbrot that I posted recently . You may not need to compile() the function , I tend to do this as I'm always assuming that a speed increase will result. The file iterdiff2c.wxm /* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/ /* [ Created with wxMaxima version 11.08.0 ] */ /* [wxMaxima: comment start ] . iterdiff2c.wxm (c) Copyright 2017 , sciwise(a)ihug.co.nz Methodology for expressing differential equations in terms of an iterated function. In particular a first order differential equation and the derivative thereof . Note that f(z,c) might be any iterated function . . [wxMaxima: comment end ] */ /* [wxMaxima: comment start ] ' Differential equation we're examining . diff(f(z),z) + 3*f(z)*z = sin(z) for an iterated function z = f(z) , g(z) =f(f(z)) diff(g(z),z) + 3*g(z)*f(z) = sin(f(z)) . [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ kill(all)$ eq1:diff(g(z,c),z) + 3*g(z,c)*f(z,c) - sin(f(z,c))$ expand(eq1); expand(diff(eq1,z)) ; /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] For the iterated Mandelbrot fractal function . [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ f(z,c):=z*z+c$ g(z,c):=f(f(z,c),c)$ f(z,c); g(z,c); /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ eq1:diff(g(z,c),z) + 3*g(z,c)*f(z,c) - sin(f(z,c))$ ed:expand(eq1)$ edp:expand(diff(eq1,z))$ /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ ed; edp; /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] This may well simplify the construction of fractint formulas for this type of equation . [wxMaxima: comment end ] */ /* [wxMaxima: comment start ] z=Pixel c=Pixel : ed = -sin(z^2+c)+3*z^6+9*c*z^4+4*z^3+9*c^2*z^2+3*c*z^2+4*c*z+3*c^3+3*c^2 edp = -2*z*cos(z^2+c)+18*z^5+36*c*z^3+12*z^2+18*c^2*z+6*c*z+4*c [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ kill(all)$ /* [wxMaxima: input end ] */ /* [wxMaxima: input start ] */ ratprint : false$ /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] Iterated function , Differential and Newton's method. [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ iterdiffn(c,n):=block([z,ed,edp,mag,j,i], mag:0.001, j:0, z:c, ed:1, for i:0 while( i < n) and ( abs(ed) > mag ) do( z:expand(z), z:radcan(z), z:realpart(z) + %i*imagpart(z), z:ev(z,numer), ed : -sin(z^2+c)+3*z^6+9*c*z^4+4*z^3+9*c^2*z^2+3*c*z^2+4*c*z+3*c^3+3*c^2, ed:expand(ed), ed:radcan(ed), ed:realpart(ed) + %i*imagpart(ed), ed:ev(ed,numer), edp : -2*z*cos(z^2+c)+18*z^5+36*c*z^3+12*z^2+18*c^2*z+6*c*z+4*c, edp:expand(edp), edp:radcan(edp), edp:realpart(edp) + %i*imagpart(edp), edp:ev(edp,numer), z : z - (ed/edp), j : j+1 ), realpart(ed) + %i*imagpart(ed) )$ /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ compile(iterdiffn)$ /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] This returns the value of the differential equation for the intial value of c. If this is close to zero then the last value of z might be a complex root of the Differential Equation . [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ n:6$ c:0.01 + %i*0.1$ iterdiffn(c,n); /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] [wxMaxima: comment end ] */ /* Maxima can't load/batch files which end with a comment! */ "Created with wxMaxima"$ Also the file iterdiff2c.mac /* . iterdiff2c.mac (c) Copyright 2017 , sciwise(a)ihug.co.nz Methodology for expressing differential equations in terms of an iterated function. In particular a first order differential equation and the derivative thereof . Note that f(z,c) might be any iterated function . . */ /* ' Differential equation we're examining . diff(f(z),z) + 3*f(z)*z = sin(z) for an iterated function z = f(z) , g(z) =f(f(z)) diff(g(z),z) + 3*g(z)*f(z) = sin(f(z)) . */ kill(all)$ eq1:diff(g(z,c),z) + 3*g(z,c)*f(z,c) - sin(f(z,c))$ expand(eq1); expand(diff(eq1,z)) ; /* For the iterated Mandelbrot fractal function . */ f(z,c):=z*z+c$ g(z,c):=f(f(z,c),c)$ f(z,c); g(z,c); /* */ eq1:diff(g(z,c),z) + 3*g(z,c)*f(z,c) - sin(f(z,c))$ ed:expand(eq1)$ edp:expand(diff(eq1,z))$ /* */ ed; edp; /* This may well simplify the construction of fractint formulas for this type of equation . */ /* z=Pixel c=Pixel : ed = -sin(z^2+c)+3*z^6+9*c*z^4+4*z^3+9*c^2*z^2+3*c*z^2+4*c*z+3*c^3+3*c^2 edp = -2*z*cos(z^2+c)+18*z^5+36*c*z^3+12*z^2+18*c^2*z+6*c*z+4*c */ kill(all)$ ratprint : false$ /* Iterated function , Differential and Newton's method. */ iterdiffn(c,n):=block([z,ed,edp,mag,j,i], mag:0.001, j:0, z:c, ed:1, for i:0 while( i < n) and ( abs(ed) > mag ) do( z:expand(z), z:radcan(z), z:realpart(z) + %i*imagpart(z), z:ev(z,numer), ed : -sin(z^2+c)+3*z^6+9*c*z^4+4*z^3+9*c^2*z^2+3*c*z^2+4*c*z+3*c^3+3*c^2, ed:expand(ed), ed:radcan(ed), ed:realpart(ed) + %i*imagpart(ed), ed:ev(ed,numer), edp : -2*z*cos(z^2+c)+18*z^5+36*c*z^3+12*z^2+18*c^2*z+6*c*z+4*c, edp:expand(edp), edp:radcan(edp), edp:realpart(edp) + %i*imagpart(edp), edp:ev(edp,numer), z : z - (ed/edp), j : j+1 ), realpart(ed) + %i*imagpart(ed) )$ /* */ compile(iterdiffn)$ /* This returns the value of the differential equation for the intial value of c. If this is close to zero then the last value of z might be a complex root of the Differential Equation . */ n:6$ c:0.01 + %i*0.1$ iterdiffn(c,n); /* */

1 0

DiffNewtMaxb
by sciwise＠ihug.co.nz 21 Jan '17

21 Jan '17

Assuming that the fractal function is made inherent within the equations ed and edp the equation z=z*z+c might be removed. Using this was deflecting Newton's method from finding the basin of attraction. The new formula is somewhat faster and more intricate ; also this might provide an accurate location for the roots , I may only know once I evaluate the characteristic equation for this simple differential equation. The characteristic equation approach doesn't appear to be applicable to more complicated Differential equations. d2jbMandelbrot(XAXIS) {; Edward Montague (c) 2017 c = Pixel z = c : ed = -sin(z^2+c)+3*z^6+9*c*z^4+4*z^3+9*c^2*z^2+3*c*z^2+4*c*z+3*c^3+3*c^2 edp = -2*z*cos(z^2+c)+18*z^5+36*c*z^3+12*z^2+18*c^2*z+6*c*z+4*c z = z - ed/edp .0001 < |ed| } test { reset=2004 type=formula formulafile=fractint.frm formulaname=d2jbMandelbrot corners=-2/2/-1.5/1.5 float=y maxiter=1024 inside=bof60 colors=(a)default.map }

1 0

DiffNewtMax
by sciwise＠ihug.co.nz 20 Jan '17

20 Jan '17

So that you might comprehend how I derived the formula for d2jaMandelbrot I'm including the Maxima Cas code that I used . file iterdiff2a.wxm . /* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/ /* [ Created with wxMaxima version 11.08.0 ] */ /* [wxMaxima: comment start ] . (c) Copyright 2017 , sciwise(a)ihug.co.nz Methodology for expressing differential equations in terms of an iterated function. In particular a first order differential equation and the derivative thereof . Note that f(z,c) might be any iterated function . . [wxMaxima: comment end ] */ /* [wxMaxima: comment start ] ' Differential equation we're examining . diff(f(z),z) + 3*f(z)*z = sin(z) for an iterated function z = f(z) , g(z) =f(f(z)) diff(g(z),z) + 3*g(z)*f(z) = sin(f(z)) . [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ kill(all)$ eq1:diff(g(z,c),z) + 3*g(z,c)*f(z,c) - sin(f(z,c))$ expand(eq1); expand(diff(eq1,z)) ; /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] For the iterated Mandelbrot fractal function . [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ f(z,c):=z*z+c$ g(z,c):=f(f(z,c),c)$ f(z,c); g(z,c); /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ eq1:diff(g(z,c),z) + 3*g(z,c)*f(z,c) - sin(f(z,c))$ ed:expand(eq1)$ edp:expand(diff(eq1,z))$ /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] [wxMaxima: comment end ] */ /* [wxMaxima: input start ] */ ed; edp; /* [wxMaxima: input end ] */ /* [wxMaxima: comment start ] This may well simplify the construction of fractint formulas for this type of equation . [wxMaxima: comment end ] */ /* [wxMaxima: comment start ] z=Pixel c=Pixel : z=z*z+c ed = -sin(z^2+c)+3*z^6+9*c*z^4+4*z^3+9*c^2*z^2+3*c*z^2+4*c*z+3*c^3+3*c^2 edp = -2*z*cos(z^2+c)+18*z^5+36*c*z^3+12*z^2+18*c^2*z+6*c*z+4*c [wxMaxima: comment end ] */ /* Maxima can't load/batch files which end with a comment! */ "Created with wxMaxima"$ And file iterdiff2a.mac . /* . (c) Copyright 2017 , sciwise(a)ihug.co.nz Methodology for expressing differential equations in terms of an iterated function. In particular a first order differential equation and the derivative thereof . Note that f(z,c) might be any iterated function . . */ /* ' Differential equation we're examining . diff(f(z),z) + 3*f(z)*z = sin(z) for an iterated function z = f(z) , g(z) =f(f(z)) diff(g(z),z) + 3*g(z)*f(z) = sin(f(z)) . */ kill(all)$ eq1:diff(g(z,c),z) + 3*g(z,c)*f(z,c) - sin(f(z,c))$ expand(eq1); expand(diff(eq1,z)) ; /* For the iterated Mandelbrot fractal function . */ f(z,c):=z*z+c$ g(z,c):=f(f(z,c),c)$ f(z,c); g(z,c); /* */ eq1:diff(g(z,c),z) + 3*g(z,c)*f(z,c) - sin(f(z,c))$ ed:expand(eq1)$ edp:expand(diff(eq1,z))$ /* */ ed; edp; /* This may well simplify the construction of fractint formulas for this type of equation . */ /* z=Pixel c=Pixel : z=z*z+c ed = -sin(z^2+c)+3*z^6+9*c*z^4+4*z^3+9*c^2*z^2+3*c*z^2+4*c*z+3*c^3+3*c^2 edp = -2*z*cos(z^2+c)+18*z^5+36*c*z^3+12*z^2+18*c^2*z+6*c*z+4*c */

2 2

NewtDiffa
by sciwise＠ihug.co.nz 19 Jan '17

19 Jan '17

In this instance I've used Maxima Cas to directly calculate the formula for the differential equation , using the Mandelbrot Set as the function. Again I'm using Newton's method in an attempt to find the roots of the Differential equation ; if successful then I might be able to extend this to more complicated differential equations. Parameter file test { reset=2004 type=formula formulafile=fractint.frm formulaname=d2jaMandelbrot corners=-2/2/-1.5/1.5 float=y maxiter=1024 inside=bof60 outside=0 logmap=yes colors=(a)chroma.map } Fractal formula . d2jaMandelbrot(XAXIS) {; Edward Montague (c) 2017 c = Pixel z = c : z=z*z+c ed = -sin(z^2+c)+3*z^6+9*c*z^4+4*z^3+9*c^2*z^2+3*c*z^2+4*c*z+3*c^3+3*c^2 edp = -2*z*cos(z^2+c)+18*z^5+36*c*z^3+12*z^2+18*c^2*z+6*c*z+4*c z = z - ed/edp .0001 < |ed| }

1 0

DiffNewt
by sciwise＠ihug.co.nz 17 Jan '17

17 Jan '17

A fractal that's a merge between my fractal differential equation and Newton's method ; yet to be checked via Maxima CAS. I call this Garden at night. test { reset=2004 type=formula formulafile=fractint.frm formulaname=d1jMandelbrot corners=-0.15769712/0.42302879/-0.77360787/-0.33806344 float=y maxiter=1024 inside=bof60 outside=0 colors=00000000d00f00h00i00k00m0ew0fx0hx0iy0ky0mznwzpwzrxzzzz7zz6zz4zz3zz2zz0zz07z06z04z03z02z00z002000000 } d1jMandelbrot(XAXIS) {; Edward Montague (c) 2017 ; ; 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 edp = 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) edp = z2 + 3*Pixel*z1 + 3*z - cos(u) 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) edp = z2 + 3*Pixel*z1 + 3*z - cos(u) : 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) edp = z2 + 3*Pixel*z1 + 3*z - cos(u) z = z - ed/edp .0001 < |ed| }

2 1

D1iMz
by sciwise＠ihug.co.nz 08 Jan '17

08 Jan '17

To download the file FracD1M.bas , go to http://www.freebasic.net , Projects , Fractal map , FracD1M.bas

1 0

Diff 1 2 & eq via Mandz
by sciwise＠ihug.co.nz 08 Jan '17

08 Jan '17

This is the parameter and formula file for generating the mandelbrot series , first and second derivatives thereof and an associated first order differential equation . For possible use with FreeBASIC , FracD1iM.bas code . To do so convert the *.gif file produced to fract003.bmp and place in the same directory as FracD1iM.bas . The FreeBASIC code should run in Windows , Linux and possibly DOS , you'll require the appropriate FreeBASIC compiler. file fract03.par test { reset=2004 type=formula formulafile=fractint.frm formulaname=d1iMandelbrot corners=-1.599499/1.008761/-0.9269799/1.029215 float=y maxiter=1024 inside=bof60 colors=(a)blues.map } formula , append this to fractint.frm 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 && |z| > 0 }

1 0