Based on Unveiling Fractal Structure with Lagrangian Descriptors
/* partial pnorm
   input: z, c, p
   output ppn
*/

float ppnorm( vec2 z, vec2 c, float p){

vec3 s0,s1; // for 2 points on the Riemann sphere
float d; // denominator

// map from complex plane to riemann sphere
// z
d = z.x*z.x + z.y*z.y + 1.0;
s0 = vec3(2.0*z,(d-2.0))/d;
// c
d = c.x*c.x + c.y*c.y + 1.0;
s1 = vec3(2.0*c,(d-2.0))/d;
// sum
vec3 ss = pow(abs(s1 - s0),vec3(p));

return ss.x+ss.y+ss.z;
}

// DLD = Discret Lagrangian Descriptior
float lagrangian( vec2 z, vec2 c, float p ){

int i; // number of iteration
float d = 0.0; // DLD = sum

for (i=0; i<Iterations; ++i){
d += ppnorm(z, c, p); // sum z
z = cMul(z,z) +c; // complex iteration
if (cAbs(z) > 1e19 ) break; // exterior : upper limit of float type
}

d /= float(i); // averaging not summation

return d;
}



Modified and optimized for GLSL ( disclaimer: it may or may not be mathematically correct but it looks nice and it's fun to play with )



Linkback: https://fractalforums.org/code-snippets-fragments/74/lagrangian-descriptors-fragment-code/3612/

Here is my code. It shuld give similar Julia image in gray
For interior it gives similar result, but exterior not
How can I improve it?

/*

  Adam Majewski
  adammaj1 aaattt o2 dot pl  // o like oxygen not 0 like zero
 
 
 


 
  ==============================================
 
 
  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
 
  =======================
  # gnuplot "i.plt"
set terminal svg enhanced background rgb 'white'
set xlabel "re(z)"
set ylabel "DLD"
set title "Relation between z and DLD in the interior of Julia set for c = -1"
set output "interior.svg"
plot "i.txt" with lines

  ----------------------
  d0 - db  = 5.0000000000000000 - 4.5389870050569598 = 0.4610129949430402
 allways free memory (deallocate )  to avoid memory leaks
Numerical approximation of Julia set for fc(z)= z^2 + c
parameter c = ( -1.0000000000000000 ; 0.0000000000000000 )
Image Width = 4.000000 in world coordinate
PixelWidth = 0.004004
Maximal number of iterations = iterMax = 1000
ratio of image  = 1.000000 ; it should be 1.000 ...
gcc version: 7.5.0
 
 
*/

#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




// 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 = 5000; // 
// The size of array has to be a positive constant integer
static unsigned 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 = -1.8; //-0.05;
static const double ZxMax =  1.8; //0.75;
static const double ZyMin = -1.8; //-0.1;
static const double ZyMax =  1.8; //0.7;
static double PixelWidth; // =(ZxMax-ZxMin)/ixMax;
static double PixelHeight; // =(ZyMax-ZyMin)/iyMax;
static double ratio;


// complex numbers of parametr plane
//https://fractalforums.org/code-snippets-fragments/74/lagrangian-descriptors-fragment-code/3612/msg22426#msg22426
double complex c = -1.05204872; // parameter of function fc(z)=z^2 + c


// attracting period 2 cycle
double complex z21 = 0.049822582293598;
double complex z22 = -1.049822582293598;
double complex z1a  = -0.641073494565534; // alfa
complex double z1b = 1.641073494565534; // beta
double complex zb = 0.203208556149733  +0.377005347593583*I; // point on the boubndary of main componnet


int ARe = 60;
int ERe = 16;
double ER ;//= 1e60;
double AR ;//= 1e-16; // bigger values do not works


// DLD
const int N = 20; // fixed number : maximal number of iterations
double p  = 0.01444322;  //


// DLD colors
double me = 1.0;
double mi = 10.0;
double d21; // = lagrangian(z21, c, N, p);
double d22; //  = lagrangian(z22, c, N, p);
double db;// = lagrangian(z1a, c, N, p);
double dd;// = d1a-d21;



/* colors = shades of gray from 0 to 255 */
unsigned char iColorOfExterior = 250;
unsigned char iColorOfInterior = 200;
unsigned char iColorOfBoundary = 0;





/* ------------------------------------------ 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;




}




// ****************** 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;
}


// ***********************************************************************************************
// ********************** 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;
}







// ***************************************************************************************************************************
// ************************** DLD/J*****************************************
// ****************************************************************************************************************************



/* partial pnorm
   input: z , zn = f(z), p
   output ppn
   
   
*/
double ppnorm( complex double z, complex double zn, double p){

double s[2][3]; // array for 2 points on the Riemann sphere
int j;
double d; // denominator
double x;
double y;

double ds;
double ppn = 0.0;

// map from complex plane to riemann sphere
// z
x = creal(z);
y = cimag(z);
d = x*x + y*y + 1.0;

s[0][0] = (2.0*x)/d;
s[0][1] = (2.0*y)/d; 
s[0][2] = (d-2.0)/d; // (x^2 + y^2 - 1)/d

// zn
x = creal(zn);
y = cimag(zn);
d = x*x + y*y + 1.0;
s[1][0] = (2.0*x)/d;
s[1][1] = (2.0*y)/d; 
s[1][2] = (d-2.0)/d; // (x^2 + y^2 - 1)/d

// sum
for (j=0; j <3; ++j){
ds = fabs(s[1][j] - s[0][j]);
//  normal:  neither zero, subnormal, infinite, nor NaN
//if (fpclassify (ds) !=FP_INFINITE)
//if (isnormal(ds))
// it is solved by if (cabs(z) > 1e60 ) break; procedure in parent function
ppn += pow(ds,p); // |ds|^p
// else {ppn = 10000.0; printf("ds = infty\t");} //

}


return ppn;







}

// DLD = Discret Lagrangian Descriptior
double lagrangian( complex double z0, complex double c, int iMax, double p ){

int i; // number of iteration
double d = 0.0; // DLD = sum
double ppn; // partial pnorm
complex double z = z0;
complex double zn; // next z


for (i=0; i<iMax; ++i){




zn = z*z +c; // complex iteration
ppn = ppnorm(z, zn, p);
d += ppn; // sum
//
z = zn;

//if (! isnormal(d)) { return 0.0; } // not works
if (cabs(z) > ER ) break; // exterior : big values produces NAN error in ppnorm computing
if (cabs(z - z21) < AR || cabs(z -z22)< AR)
{ // interior
//return FP_ZERO;
d = -d;
break;

}


}





if (d<0.0) {// interior
// d(z1a) - d(z21) = -0.0804163521959989
d = - d;
d = (db - d) /dd ; // normalize, see test_interior

//if (d>1.0) {printf("d int > 1.0\n");d = d - (int)d;} // fractional part}
}
else {
d =  d/((double)i); // averaging not summation
d = d*me;} // exterior

return d;




}





unsigned char ComputeColorOfDLD(complex double z){

 
  //double cabsz;
  int iColor;
  double d;
 

  d = lagrangian(z,c, N,p);
  //if ( d==FP_ZERO)
  // {iColor = 255;}
  // else
  iColor = (int)(d*255)  % 255; // nMax or lower walues in denominator
 
 
  return (unsigned char) iColor;


}



// plots raster point (ix,iy)
int DrawPointOfDLD (unsigned char A[], int ix, int iy)
{
  int i; /* index of 1D array */
  unsigned char iColor;
  complex double z;


  i = Give_i (ix, iy); /* compute index of 1D array from indices of 2D array */
  z = GiveZ(ix,iy);
  iColor = ComputeColorOfDLD(z);
  A[i] = iColor ; // interior
 
  return 0;
}




// fill array
// uses global var :  ...
// scanning complex plane
int DrawImagerOfDLD (unsigned char A[])
{
  unsigned int ix, iy; // pixel coordinate

  //printf("compute image \n");
  // for all pixels of image
#pragma omp parallel for schedule(dynamic) private(ix,iy) shared(A, ixMax , iyMax)
  for (iy = iyMin; iy <= iyMax; ++iy){
    printf (" %d from %d \r", iy, iyMax); //info
    for (ix = ixMin; ix <= ixMax; ++ix)
      DrawPointOfDLD(A, ix, iy); // 
  }

  return 0;
}





// test how values changes to tune color
// from z21 to zb
int test_interior(){

complex double z = z21;
complex double dz = cabs(z21 - zb)/15.0;
double r = cabs(zb);


printf("# re(z) \t d\n"); // gnuplot
while (creal(z) < r){ // from z21 to z1a

double d = lagrangian(z, c, N, p);
//int iColor = ComputeColorOfDLD(z);

// printf(" z = %.16f d = %.16f color = %d \n",creal(z), d, iColor);
printf(" %.16f %.16f \n",creal(z), d); // gnuplot
z += dz;
}

//
d21 = lagrangian(z21, c, N, p);
db = lagrangian(zb, c, N, p);
d22 = lagrangian(z22, c, N, p);
dd = db - d21;
printf("d(z21) = %.16f \n",d21);
printf("d(z22) = %.16f \n",d22);
printf("d(zb) = %.16f \n",db);
printf("d(zb) - d(z21) = %.16f\n",dd);;



return 0;


}
 
 
 

// test how values changes to tune color
int test_exterior(){

complex double z;
complex double z0 = 1.62;
complex double z1 = 3.0;
complex double dz = 0.001;


z = z0;
printf("# z d\n"); // gnuplot
while (creal(z) < creal(z1)){

double d = lagrangian(z, c, N, p);
//int iColor = ComputeColorOfDLD(z);

// printf(" z = %.16f d = %.16f color = %d \n",creal(z), d, iColor);
printf(" %.16f %.16f \n",creal(z), d); // gnuplot
z += dz;
}

//
double d0 = lagrangian(z0, c, N, p);
double d1 = lagrangian(z1, c, N, p);
double dd = d0 - d1;
printf("d0 - d1  = %.16f - %.16f = %.16f\n",d0, d1, dd);

return 0;


}
 
 








// *******************************************************************************************
// ********************************** save A array to pgm file ****************************
// *********************************************************************************************

int SaveArray2PGMFile( unsigned char A[], int a, int b, int c, int d,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_%d", a,b,c,d); /*  */
  char *filename =strcat(name,".pgm");
  char long_comment[200];
  sprintf (long_comment, "%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)= z^2 + c \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 = %f \n", PixelWidth);
 
 
  // image corners in world coordinate
  // center and radius
  // center and zoom
  // GradientRepetition
  printf ("Maximal number of iterations = iterMax = %d \n", N);
  printf ("ratio of image  = %f ; it should be 1.000 ...\n", ratio);
  //
 
 
 
 
  return 0;
}






// *****************************************************************************
//;;;;;;;;;;;;;;;;;;;;;;  setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
// **************************************************************************************

int setup ()
{

  printf ("setup start\n");
   
 
 
 
 

  /* 2D array ranges */
 
  iWidth = iHeight;
  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 = pow(10,ARe);
   AR = pow(10,-ERe);

 
   
  /* create dynamic 1D arrays for colors ( shades of gray ) */
  data = malloc (iSize * sizeof (unsigned char));
   
 
  if (data == NULL  ){
    fprintf (stderr, " Could not allocate memory");
    return 1;
  }

 
 
 
 
  test_interior();
 
 
  printf (" end of setup \n");

  return 0;

} // ;;;;;;;;;;;;;;;;;;;;;;;;; end of the setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;




int end(){


  printf (" allways free memory (deallocate )  to avoid memory leaks \n"); // https://en.wikipedia.org/wiki/C_dynamic_memory_allocation
  free (data);
 
 
  PrintProgramInfo();
  PrintCInfo();
  return 0;

}

// ********************************************************************************************************************
/* -----------------------------------------  main   -------------------------------------------------------------*/
// ********************************************************************************************************************

int main () {
  setup ();
 
 
 
  DrawImagerOfDLD(data);
  SaveArray2PGMFile (data, iWidth,N,ERe, ARe,"DLD/J , name = iWidth_N_ER_AR");


  //test_exterior();
 
  end();

  return 0;
}


Thanks you for sharing, Dick! 

I didn't explicitly give permission to post the fragment code but its done now. Its ok as long as you reference FractalForums as the source and me as the author of that code based on your code.

I would like to remove the wikibooks fragmentarium stuff as its out of date and wiki books doesn't give me the freedom I need to maintain it properly... they complain when I post a screenshot and tell me that I'm not allowed to post copyrighted material...FragM is opensource and I have contributed more than half of the code, so wikibooks is not the place I want people going for proper and up to date documentation.

I have a wiki on the github site that is up to date and would like the fragmentarium wikibooks page to link there instead.

Feel free to incorporate any improvements or optimizations Ive made into your own code.

oh no,  leave it there, everything I write for FragM is for the community, as I said, "Its ok as long as you reference FractalForums as the source and me as the author of that code based on your code."

Wikibooks has a purpose and provides great information but as soon as I publish something there it's no longer mine and it seems I need someone else's permission or review or whatever before it's actually put on the page. Wikibooks is imho not a place for program documentation, it is a place for a user to write about things they've done or learned or maybe a tutorial/how-to but the actual program  documentation must remain attached to the program and must not subject to approval by someone who may know nothing about the program or programming.

The material that you've gathered on wikibooks is great, it's really nice to have that as a reference.  your efforts are greatly appreciated.

Now... if you want to discuss your C program please do it in the original thread, this board/thread is about FragM and it's fragment code.

I've found that with the GLSL the second lower/interior test is not required and it renders  the same image, I've also optimized it a little further, the main function is just the standard mandelbrot so moved all that code into the vec3 color(vec2 c) function and the float ppnorm( vec2 z, vec2 c, float p) function is only 6 lines of actual code that does the work, 8 with decls...
float ppnorm( vec2 z, vec2 c, float p){

vec3 s0,s1; // for 2 points on the Riemann sphere
float d; // denominator

// map from complex plane to riemann sphere
// z
d = z.x*z.x + z.y*z.y + 1.0;
s0 = vec3(z,(d-2.0))/d;
// c
d = c.x*c.x + c.y*c.y + 1.0;
s1 = vec3(c,(d-2.0))/d;
// sum
vec3 ss = pow(abs(s1 - s0),vec3(p));

return ss.x+ss.y+ss.z;
}

// DLD = Discret Lagrangian Descriptior
vec3 color(vec2 c) {
vec2 z = Julia ? c : vec2(0.,0.);
int i; // number of iterations
float d = 0.0; // DLD = sum

if(Julia) c = JuliaXY;

for (i=0; i<Iterations; ++i){
d += ppnorm(z, c, p); // sum z
z = cMul(z,z) +c; // complex iteration
if (length(z) > 1e+19 ) break; // upper limit of float type 3.402823466e+38
}

float co = d / float(i); // averaging not summation
return .5+.5*cos(6.2831*co+RGB);
}
... it runs at max fps... not sure of the exact number but well beyond 60fps on my GPU moving the ppnorm code into the main color function should speed it up a bit more but , depending on the gfx driver this may be redundant after the glsl compiler optimizes things.

your second image looks exactly like mine just no color, the only fragment code that gets used from the included files for calculating values is the complex multiply function cMul() and it is just the standard complex multiply so I don't think any of the included code accounts for difference, (I don't see the difference)

and the fragment code is all 32bit float type while your C proggie is using 64bit double type, that may account for a small difference.