File:Julia set f(z)=1+z−3z2−3.75z3+1.5z4+2.25z5.png
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Julia set f(z)=1+z−3z2−3.75z3+1.5z4+2.25z5
Summary
[edit]DescriptionJulia set f(z)=1+z−3z2−3.75z3+1.5z4+2.25z5.png |
English: Julia set |
Date | |
Source | Own work |
Author | Adam majewski |
Licensing
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Summary
[edit]Input polynomial p(z)=(2.25+0i)*z^5+(1.5+0i)*z^4+(-3.75+0i)*z^3+(-3+0i)*z^2+(1+0i)*z^1+(1+0i) 4 critical points found cp#0: -0.81886248779132231324,-0.083366172059259224802 . It's critical orbit is bounded and enters cycle #0 length=3 and it's stability = |multiplier|=0.70431 =attractive cycle = { 0.051046483840772594043,-9.8813129168249308835e-324 ; 1.042741415337827604,-9.8813129168249308835e-324 ; -0.92377040934520726623,-2.9643938750474792651e-323 ; } cp#1: 0.1354031363439483493,0 . It's critical orbit is bounded and enters cycle #0 cp#2: -0.81886248779132408959,0.083366172059257226401 . It's critical orbit is bounded and enters cycle #0 cp#3: 0.96898850590536300675,-2.1898778728935287718e-27 . It's critical orbit is bounded and enters cycle #1 length=3 and it's stability = |multiplier|=0.99976 =attractive cycle = { -1.0000132826277488007,-1.4242477576956251569e-48 ; -1.3283554026521926517e-05,-1.4244464037113862021e-48 ; 0.99998671591662380287,-1.4245599311482756001e-48 ; }
c source code
[edit]/*
Adam Majewski
adammaj1 aaattt o2 dot pl // o like oxygen not 0 like zero
f(z)=1+z−3z2−3.75z3+1.5z4+2.25z5
- parabolic 3-cycle ( z=-1, z=0, z=1)
- an attracting 3-cycle :
cycle = { 0.051046483840772594043,-9.8813129168249308835e-324 ;
1.042741415337827604,-9.8813129168249308835e-324 ;
-0.92377040934520726623,-2.9643938750474792651e-323 ; }
==============================================
Structure of a program or how to analyze the program
============== Image X ========================
DrawImageOfX -> DrawPointOfX -> ComputeColorOfX
first 2 functions are identical for every X
check only last function = ComputeColorOfX
which computes color of one pixel !
==========================================
---------------------------------
indent d.c
default is gnu style
-------------------
c console progam
export OMP_DISPLAY_ENV="TRUE"
gcc d.c -lm -Wall -march=native -fopenmp
time ./a.out > b.txt
gcc d.c -lm -Wall -march=native -fopenmp
time ./a.out
time ./a.out >i.txt
time ./a.out >e.txt
convert -limit memory 1000mb -limit disk 1gb dd30010000_20_3_0.90.pgm -resize 2000x2000 10.png
*/
#include <stdio.h>
#include <stdlib.h> // malloc
#include <string.h> // strcat
#include <math.h> // M_PI; needs -lm also
#include <complex.h>
#include <omp.h> // OpenMP
#include <limits.h> // Maximum value for an unsigned long long int
// https://sourceforge.net/p/predef/wiki/Standards/
#if defined(__STDC__)
#define PREDEF_STANDARD_C_1989
#if defined(__STDC_VERSION__)
#if (__STDC_VERSION__ >= 199409L)
#define PREDEF_STANDARD_C_1994
#endif
#if (__STDC_VERSION__ >= 199901L)
#define PREDEF_STANDARD_C_1999
#endif
#endif
#endif
/* --------------------------------- global variables and consts ------------------------------------------------------------ */
// virtual 2D array and integer ( screen) coordinate
// Indexes of array starts from 0 not 1
//unsigned int ix, iy; // var
static unsigned int ixMin = 0; // Indexes of array starts from 0 not 1
static unsigned int ixMax; //
static unsigned int iWidth; // horizontal dimension of array
static unsigned int iyMin = 0; // Indexes of array starts from 0 not 1
static unsigned int iyMax; //
static unsigned int iHeight = 10000; //
// The size of array has to be a positive constant integer
static unsigned long long int iSize; // = iWidth*iHeight;
// memmory 1D array
unsigned char *data;
unsigned char *edge;
//unsigned char *edge2;
// unsigned int i; // var = index of 1D array
//static unsigned int iMin = 0; // Indexes of array starts from 0 not 1
static unsigned int iMax; // = i2Dsize-1 =
// The size of array has to be a positive constant integer
// unsigned int i1Dsize ; // = i2Dsize = (iMax -iMin + 1) = ; 1D array with the same size as 2D array
static const double ZxMin = -1.5; //-0.05;
static const double ZxMax = 1.5; //0.75;
static const double ZyMin = -0.75; //-0.1;
static const double ZyMax = 0.75; //0.7;
static double PixelWidth; // =(ZxMax-ZxMin)/ixMax;
static double PixelHeight; // =(ZyMax-ZyMin)/iyMax;
static double ratio;
/*
ER = pow(10,ERe);
AR = pow(10,-ARe);
*/
//int ARe ; // increase ARe until black ( unknown) points disapear
//int ERe ;
double ER;
double ER2; //= 1e60;
double AR; // bigger values do not works
double AR2;
double AR12;
int IterMax = 100000;
/* colors = shades of gray from 0 to 255
unsigned char colorArray[2][2]={{255,231}, {123,99}};
color = 245; exterior
*/
unsigned char iColorOfExterior = 245;
unsigned char iColorOfInterior1 = 99;
unsigned char iColorOfInterior2 = 183;
unsigned char iColorOfBoundary = 0;
unsigned char iColorOfUnknown = 5;
// pixel counters
unsigned long long int uUnknown = 0;
unsigned long long int uInterior = 0;
unsigned long long int uExterior = 0;
complex double zp[3]={-1,0,1};
complex double za[3]= {0.051046483840772594043, 1.042741415337827604, -0.92377040934520726623};
/* ------------------------------------------ functions -------------------------------------------------------------*/
//------------------complex numbers -----------------------------------------------------
// from screen to world coordinate ; linear mapping
// uses global cons
double
GiveZx (int ix)
{
return (ZxMin + ix * PixelWidth);
}
// uses globaal cons
double
GiveZy (int iy)
{
return (ZyMax - iy * PixelHeight);
} // reverse y axis
complex double
GiveZ (int ix, int iy)
{
double Zx = GiveZx (ix);
double Zy = GiveZy (iy);
return Zx + Zy * I;
}
double cabs2(complex double z){
return creal(z)*creal(z)+cimag(z)*cimag(z);
}
// =====================
int IsPointInsideTrap1(complex double z){
if ( cabs2(z+AR12)<AR2) {return 1;} // circle with prabolic point on it's boundary
return 0; // outside
}
// =====================
int IsPointInsideTrap2(complex double z){
if (cabs2(z - 0.051046483840772594043) <AR2) {return 1;} // circle around periodic point
return 0; // outside
}
// ****************** DYNAMICS = trap tests ( target sets) ****************************
/* ----------- array functions = drawing -------------- */
/* gives position of 2D point (ix,iy) in 1D array ; uses also global variable iWidth */
unsigned int
Give_i (unsigned int ix, unsigned int iy)
{
return ix + iy * iWidth;
}
// f(z)=1+z−3z2−3.75z3+1.5z4+2.25z5
unsigned char
ComputeColor_Fatou (complex double z, int IterMax)
{
complex double z2;
complex double z3;
double r2;
int i; // number of iteration
for (i = 0; i < IterMax; ++i)
{
z2 = z*z;
z3 = z*z2;
z = 1.0 +z -3.0*z2 -3.75*z3 +1.5*z2*z2 +2.25*z3*z2; // complex iteration z^6+A*z+c
r2 =cabs2(z);
//z = z*z;
if (r2 > ER2) // esaping = exterior
{
uExterior += 1;
return iColorOfExterior;
}
if ( IsPointInsideTrap1(z))
return iColorOfInterior1;
if (IsPointInsideTrap2(z))
return iColorOfInterior2;
}
uUnknown += 1;
return iColorOfUnknown;
}
// plots raster point (ix,iy)
int
DrawFatouPoint (unsigned char A[], int ix, int iy, int IterMax)
{
int i; /* index of 1D array */
unsigned char iColor = 0;
complex double z;
i = Give_i (ix, iy); /* compute index of 1D array from indices of 2D array */
z = GiveZ (ix, iy);
iColor = ComputeColor_Fatou (z, IterMax);
A[i] = iColor; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int
DrawFatouImage (unsigned char A[], int IterMax)
{
unsigned int ix, iy; // pixel coordinate
fprintf (stdout, "compute Fatou image \n");
// for all pixels of image
#pragma omp parallel for schedule(dynamic) private(ix,iy) shared(A, ixMax , iyMax, uUnknown, uInterior, uExterior)
for (iy = iyMin; iy <= iyMax; ++iy)
{
fprintf (stderr, " %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawFatouPoint (A, ix, iy, IterMax); //
}
return 0;
}
//=========
int IsInside (int x, int y, int xcenter, int ycenter, int r){
double dx = x- xcenter;
double dy = y - ycenter;
double d = sqrt(dx*dx+dy*dy);
if (d<r)
return 1;
return 0;
}
int PlotBigPoint(complex double z, unsigned char A[]){
unsigned int ix_seed = (creal(z)-ZxMin)/PixelWidth;
unsigned int iy_seed = (ZyMax - cimag(z))/PixelHeight;
unsigned int i;
/* mark seed point by big pixel */
int iSide =3.0*iWidth/2000.0 ; /* half of width or height of big pixel */
int iY;
int iX;
for(iY=iy_seed-iSide;iY<=iy_seed+iSide;++iY){
for(iX=ix_seed-iSide;iX<=ix_seed+iSide;++iX){
if (IsInside(iX, iY, ix_seed, iy_seed, iSide)) {
i= Give_i(iX,iY); /* index of _data array */
A[i]= 255-A[i];}}}
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int MarkAttractors (unsigned char A[])
{
int i;
fprintf (stderr, "mark attractors \n");
for (i = 0; i < 3; ++i)
{ PlotBigPoint(zp[i], A); // period 3 parabolic cycle
PlotBigPoint(za[i], A); // period 3 attracting cycle
}
return 0;
}
// =====================
int IsPointInsideTraps(unsigned int ix, unsigned int iy){
complex double z = GiveZ (ix, iy);
if ( IsPointInsideTrap1(z)) {return 1;} // circle with prabolic point on it's boundary
if (IsPointInsideTrap2(z)) {return 1;}
return 0; // outside
}
int MarkTraps(unsigned char A[]){
unsigned int ix, iy; // pixel coordinate
unsigned int i;
fprintf (stderr, "Mark traps \n");
// for all pixels of image
#pragma omp parallel for schedule(dynamic) private(ix,iy) shared(A, ixMax , iyMax, uUnknown, uInterior, uExterior)
for (iy = iyMin; iy <= iyMax; ++iy)
{
fprintf (stderr, " %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix){
if (IsPointInsideTraps(ix, iy)) {
i= Give_i(ix,iy); /* index of _data array */
A[i]= 255-A[i]; // inverse color
}}}
return 0;
}
int PlotPoint(complex double z, unsigned char A[]){
unsigned int ix = (creal(z)-ZxMin)/PixelWidth;
unsigned int iy = (ZyMax - cimag(z))/PixelHeight;
unsigned int i;
A[i]= 255-A[i]; // Mark point with inveres color
return 0;
}
// ***********************************************************************************************
// ********************** edge detection usung Sobel filter ***************************************
// ***************************************************************************************************
// from Source to Destination
int ComputeBoundaries(unsigned char S[], unsigned char D[])
{
unsigned int iX,iY; /* indices of 2D virtual array (image) = integer coordinate */
unsigned int i; /* index of 1D array */
/* sobel filter */
unsigned char G, Gh, Gv;
// boundaries are in D array ( global var )
// clear D array
memset(D, iColorOfExterior, iSize*sizeof(*D)); // for heap-allocated arrays, where N is the number of elements = FillArrayWithColor(D , iColorOfExterior);
// printf(" find boundaries in S array using Sobel filter\n");
#pragma omp parallel for schedule(dynamic) private(i,iY,iX,Gv,Gh,G) shared(iyMax,ixMax)
for(iY=1;iY<iyMax-1;++iY){
for(iX=1;iX<ixMax-1;++iX){
Gv= S[Give_i(iX-1,iY+1)] + 2*S[Give_i(iX,iY+1)] + S[Give_i(iX-1,iY+1)] - S[Give_i(iX-1,iY-1)] - 2*S[Give_i(iX-1,iY)] - S[Give_i(iX+1,iY-1)];
Gh= S[Give_i(iX+1,iY+1)] + 2*S[Give_i(iX+1,iY)] + S[Give_i(iX-1,iY-1)] - S[Give_i(iX+1,iY-1)] - 2*S[Give_i(iX-1,iY)] - S[Give_i(iX-1,iY-1)];
G = sqrt(Gh*Gh + Gv*Gv);
i= Give_i(iX,iY); /* compute index of 1D array from indices of 2D array */
if (G==0) {D[i]=255;} /* background */
else {D[i]=0;} /* boundary */
}
}
return 0;
}
// copy from Source to Destination
int CopyBoundaries(unsigned char S[], unsigned char D[])
{
unsigned int iX,iY; /* indices of 2D virtual array (image) = integer coordinate */
unsigned int i; /* index of 1D array */
//printf("copy boundaries from S array to D array \n");
for(iY=1;iY<iyMax-1;++iY)
for(iX=1;iX<ixMax-1;++iX)
{i= Give_i(iX,iY); if (S[i]==0) D[i]=0;}
return 0;
}
// *******************************************************************************************
// ********************************** save A array to pgm file ****************************
// *********************************************************************************************
int
SaveArray2PGMFile (unsigned char A[], int a, int b, int c, char *comment)
{
FILE *fp;
const unsigned int MaxColorComponentValue = 255; /* color component is coded from 0 to 255 ; it is 8 bit color file */
char name[100]; /* name of file */
snprintf (name, sizeof name, "%d_%d_%d", a, b, c ); /* */
char *filename = strcat (name, ".pgm");
char long_comment[200];
sprintf (long_comment, "fc(z)= 1+z−3z^2−3.75z^3+1.5z^4+2.25z^5 ; %s\tER = %e\tAR =%e", comment, ER, AR);
// save image array to the pgm file
fp = fopen (filename, "wb"); // create new file,give it a name and open it in binary mode
fprintf (fp, "P5\n # %s\n %u %u\n %u\n", long_comment, iWidth, iHeight, MaxColorComponentValue); // write header to the file
fwrite (A, iSize, 1, fp); // write array with image data bytes to the file in one step
fclose (fp);
// info
printf ("File %s saved ", filename);
if (long_comment == NULL || strlen (long_comment) == 0)
printf ("\n");
else
printf (". Comment = %s \n", long_comment);
return 0;
}
int
PrintCInfo ()
{
printf ("gcc version: %d.%d.%d\n", __GNUC__, __GNUC_MINOR__, __GNUC_PATCHLEVEL__); // https://stackoverflow.com/questions/20389193/how-do-i-check-my-gcc-c-compiler-version-for-my-eclipse
// OpenMP version is displayed in the console : export OMP_DISPLAY_ENV="TRUE"
printf ("__STDC__ = %d\n", __STDC__);
printf ("__STDC_VERSION__ = %ld\n", __STDC_VERSION__);
printf ("c dialect = ");
switch (__STDC_VERSION__)
{ // the format YYYYMM
case 199409L:
printf ("C94\n");
break;
case 199901L:
printf ("C99\n");
break;
case 201112L:
printf ("C11\n");
break;
case 201710L:
printf ("C18\n");
break;
//default : /* Optional */
}
return 0;
}
int
PrintProgramInfo ()
{
// display info messages
printf ("Numerical approximation of Julia set for fc(z)= 1+z−3z^2−3.75z^3+1.5z^4+2.25z^5 \n");
//printf ("iPeriodParent = %d \n", iPeriodParent);
//printf ("iPeriodOfChild = %d \n", iPeriodChild);
//printf ("parameter c = ( %.16f ; %.16f ) \n", creal (c), cimag (c));
printf ("Image Width = %f in world coordinate\n", ZxMax - ZxMin);
printf ("PixelWidth = %.16f \n", PixelWidth);
printf ("AR = %.16f = %f *PixelWidth\n", AR, AR / PixelWidth);
printf("pixel counters\n");
printf ("uUnknown = %llu\n", uUnknown);
printf ("uExterior = %llu\n", uExterior);
printf ("uInterior = %llu\n", uInterior);
printf ("Sum of pixels = %llu\n", uInterior+uExterior + uUnknown);
printf ("all pixels of the array = iSize = %llu\n", iSize);
// image corners in world coordinate
// center and radius
// center and zoom
// GradientRepetition
printf ("Maximal number of iterations = iterMax = %d \n", IterMax);
printf ("ratio of image = %f ; it should be 1.000 ...\n", ratio);
//
return 0;
}
// *****************************************************************************
//;;;;;;;;;;;;;;;;;;;;;; setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
// **************************************************************************************
int
setup ()
{
fprintf (stderr, "setup start\n");
/* 2D array ranges */
iWidth = iHeight*2;
iSize = iWidth * iHeight; // size = number of points in array
// iy
iyMax = iHeight - 1; // Indexes of array starts from 0 not 1 so the highest elements of an array is = array_name[size-1].
//ix
ixMax = iWidth - 1;
/* 1D array ranges */
// i1Dsize = i2Dsize; // 1D array with the same size as 2D array
iMax = iSize - 1; // Indexes of array starts from 0 not 1 so the highest elements of an array is = array_name[size-1].
/* Pixel sizes */
PixelWidth = (ZxMax - ZxMin) / ixMax; // ixMax = (iWidth-1) step between pixels in world coordinate
PixelHeight = (ZyMax - ZyMin) / iyMax;
ratio = ((ZxMax - ZxMin) / (ZyMax - ZyMin)) / ((double) iWidth / (double) iHeight); // it should be 1.000 ...
ER = 2.0; //pow (10, ERe);
ER2 = ER*ER;
AR = PixelWidth*10.0;
AR2 = AR * AR;
AR12 = AR/2.0;
/* create dynamic 1D arrays for colors ( shades of gray ) */
data = malloc (iSize * sizeof (unsigned char));
edge = malloc (iSize * sizeof (unsigned char));
if (data == NULL || edge == NULL)
{
fprintf (stderr, " Could not allocate memory");
return 1;
}
fprintf (stderr, " end of setup \n");
return 0;
} // ;;;;;;;;;;;;;;;;;;;;;;;;; end of the setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
int
end ()
{
fprintf (stderr, " allways free memory (deallocate ) to avoid memory leaks \n"); // https://en.wikipedia.org/wiki/C_dynamic_memory_allocation
free (data);
free(edge);
PrintProgramInfo ();
PrintCInfo ();
return 0;
}
// ********************************************************************************************************************
/* ----------------------------------------- main -------------------------------------------------------------*/
// ********************************************************************************************************************
int
main ()
{
setup ();
DrawFatouImage (data, IterMax); // first find Fatou
SaveArray2PGMFile (data, iWidth, IterMax, 0, "Fatou, name = iWidth_IterMax_n");
ComputeBoundaries(data,edge);
SaveArray2PGMFile (edge, iWidth, IterMax, 1, "Boundaries of Fatou; name = iWidth_IterMax_n");
CopyBoundaries(edge,data);
SaveArray2PGMFile (data, iWidth, IterMax, 2, "Fatou with boundaries; name = iWidth_IterMax_n");
//MarkAttractors(data);
MarkTraps(data);
SaveArray2PGMFile (data, iWidth, IterMax, 4, "Fatou with boundaries and traps; name = iWidth_IterMax_n");
end ();
return 0;
}
text output
[edit]PENMP DISPLAY ENVIRONMENT BEGIN _OPENMP = '201511' OMP_DYNAMIC = 'FALSE' OMP_NESTED = 'FALSE' OMP_NUM_THREADS = '8' OMP_SCHEDULE = 'DYNAMIC' OMP_PROC_BIND = 'FALSE' OMP_PLACES = '' OMP_STACKSIZE = '0' OMP_WAIT_POLICY = 'PASSIVE' OMP_THREAD_LIMIT = '4294967295' OMP_MAX_ACTIVE_LEVELS = '2147483647' OMP_CANCELLATION = 'FALSE' OMP_DEFAULT_DEVICE = '0' OMP_MAX_TASK_PRIORITY = '0' OMP_DISPLAY_AFFINITY = 'FALSE' OMP_AFFINITY_FORMAT = 'level %L thread %i affinity %A' OPENMP DISPLAY ENVIRONMENT END compute Fatou image File 20000_100000_0.pgm saved . Comment = fc(z)= 1+z−3z^2−3.75z^3+1.5z^4+2.25z^5 ; Fatou, name = iWidth_IterMax_n ER = 2.000000e+00 AR =1.500075e-03 File 20000_100000_1.pgm saved . Comment = fc(z)= 1+z−3z^2−3.75z^3+1.5z^4+2.25z^5 ; Boundaries of Fatou; name = iWidth_IterMax_n ER = 2.000000e+00 AR =1.500075e-03 File 20000_100000_2.pgm saved . Comment = fc(z)= 1+z−3z^2−3.75z^3+1.5z^4+2.25z^5 ; Fatou with boundaries; name = iWidth_IterMax_n ER = 2.000000e+00 AR =1.500075e-03 Mark traps File 20000_100000_4.pgm saved . Comment = fc(z)= 1+z−3z^2−3.75z^3+1.5z^4+2.25z^5 ; Fatou with boundaries and traps; name = iWidth_IterMax_n ER = 2.000000e+00 AR =1.500075e-03 Numerical approximation of Julia set for fc(z)= 1+z−3z^2−3.75z^3+1.5z^4+2.25z^5 Image Width = 3.000000 in world coordinate PixelWidth = 0.0001500075003750 AR = 0.0015000750037502 = 10.000000 *PixelWidth pixel counters uUnknown = 0 uExterior = 95170413 uInterior = 0 Sum of pixels = 95170413 all pixels of the array = iSize = 200000000 Maximal number of iterations = iterMax = 100000 ratio of image = 1.000000 ; it should be 1.000 ... gcc version: 9.3.0 __STDC__ = 1 __STDC_VERSION__ = 201710 c dialect = C18
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