Quelle Extend_Auts.c Sprache: C

/****************************************************************************
**
*A  Extend_Auts.c               ANUPQ source                   Eamonn O'Brien
**
*Y  Copyright 1995-2001,  Lehrstuhl D fuer Mathematik,  RWTH Aachen,  Germany
*Y  Copyright 1995-2001,  School of Mathematical Sciences, ANU,     Australia
**
*/

#include "pq_defs.h"
#include "pcp_vars.h"
#include "pq_functions.h"
#define SIZE 100
#define DEFAULT_SIZE 100000
#define DEBUG1

/* for each automorphism, compute its action on each of the generators;

   this code is a modified version of the code to be found in the file
   extend_automorphisms -- the modifications are introduced in order
   to store the automorphims much more efficiently than the 3-dimensional
   array used in that code; the efficiency is achieved by storing the
   description using two 1-dimensional arrays, head and list;

   these vectors are organised as follows --
   ptr = head[(alpha - 1) * lastg + i] is a pointer to action of
         automorphism alpha on generator i;
   length = list[ptr + 1] = length of generator-exponent string
         storing action;
   list[ptr + 2] ... list[ptr + 1 + length] contains the string */

void Extend_Auts(int **head, int **list, int start, struct pcp_vars *pcp)
{
   register int lastg = pcp->lastg;
   register int offset;
   register int alpha;
   int index = 0;
   int max_length;
   FILE *fp;
   int nmr_saved;
   int list_length;

   int saved_length;        /* total length of description saved to file */
   int restored_length = 0; /* amount of description restored from file */
   int new;                 /* new storage requirement */

   /* this used to be 5 * lastg + 4 -- April 1994 */
   if (is_space_exhausted(7 * lastg + 4, pcp))
      return;

   fp = TemporaryFile();

   save_auts(fp, *head, *list, pcp);

   fread(&nmr_saved, sizeof(int), 1, fp);
   fread(&saved_length, sizeof(int), 1, fp);

   max_length = MIN(SIZE, lastg) * lastg;
   list_length = max_length + DEFAULT_SIZE;

   if (pcp->cc != 1) {
      if ((*head)[0] < lastg)
         *head = reallocate_vector(
             *head, 1 + (*head)[0] * pcp->m, 1 + lastg * pcp->m, 0, FALSE);
      if ((*list)[0] < list_length)
         *list = reallocate_vector(
             *list, (*list)[0] + 1, list_length + 1, 0, FALSE);
      else
         list_length = (*list)[0];
   }

   for (alpha = 1; alpha <= pcp->m; ++alpha) {
      offset = (alpha - 1) * lastg;
      restored_length +=
          restore_auts(fp, offset, nmr_saved, start - 1, &index, *head, *list);
      Extend_Aut(
          start, max_length, &list_length, *head, list, offset, &index, pcp);
#ifdef DEBUG
      if (alpha != pcp->m) {
         printf("*** After automorphism %d, allocation is %d\n",
                alpha,
                list_length);
         printf("*** Value of index is now %d\n", index);
      }
#endif

      if ((new = saved_length - restored_length + index) > list_length) {
         *list = reallocate_vector(*list, list_length + 1, new + 1, 0, FALSE);
         list_length = new;
#ifdef DEBUG
         printf("*** Allocation is increased to %d\n", list_length);
#endif
      }
   }

   (*head)[0] = lastg;
   (*list)[0] = list_length;

   CloseFile(fp);

#ifdef DEBUG1
   printf("*** Final allocated space for automorphisms is %d\n", list_length);
   printf("*** Final amount used is %d\n", index);
#endif
}

/* list the action of each automorphism on each of the
   pcp generators, first .. last, by their image */

void List_Auts(int *head, int *list, int first, int last, struct pcp_vars *pcp)
{
   register int alpha, i, j, ptr, length;
   int offset = 0;
#include "access.h"

   for (alpha = 1; alpha <= pcp->m; ++alpha) {
      for (i = first; i <= MIN(last, pcp->lastg); ++i) {
         ptr = head[offset + i];
         length = list[ptr + 1];
         printf("%d --> ", i);
         for (j = ptr + 2; j <= ptr + length + 1; ++j) {
            printf("%d^%d ", FIELD2(list[j]), FIELD1(list[j]));
         }
         printf("\n");
      }
      offset += pcp->lastg;
   }
}

/* set up description of action of automorphisms on defining generators */

void Setup_Action(int **head,
                  int **list,
                  int ***auts,
                  int nmr_of_exponents,
                  struct pcp_vars *pcp)
{
   register int *y = y_address;

   register int i, generator;
   int position, max_length, exp, alpha, offset;
   int lastg = pcp->lastg;
   int fq_rank = y[pcp->clend + 1];
   int list_length;
   int index = 0;

#include "access.h"

   *head = allocate_vector(fq_rank * pcp->m + 1, 0, FALSE);
   max_length = MIN(SIZE, lastg) * lastg;
   list_length = max_length + DEFAULT_SIZE;
   *list = allocate_vector(list_length + 1, 0, FALSE);

   for (alpha = 1; alpha <= pcp->m; ++alpha) {
      offset = (alpha - 1) * fq_rank;
      for (generator = 1; generator <= fq_rank; ++generator) {
         position = (*head)[offset + generator] = index;
         (*list)[++position] = 0;
         ++index;
         for (i = 1; i <= nmr_of_exponents; ++i) {
            if ((exp = auts[alpha][generator][i]) != 0) {
               ++(*list)[position];
               (*list)[++index] = PACK2(exp, i);
            }
         }
      }
   }

   (*head)[0] = fq_rank;
   (*list)[0] = list_length;
}

/* extend the automorphism whose action on the defining generators
   of the group is described in the two 1-dimensional arrays, head
   and list, to act on the generators of the group; the first
   generator whose image is computed is start */

void Extend_Aut(int start,
                int max_length,
                int *list_length,
                int *head,
                int **list,
                int offset,
                int *index,
                struct pcp_vars *pcp)
{
   register int *y = y_address;

   register int i, generator;
   register int lastg = pcp->lastg;
   register int structure = pcp->structure;
   int cp1 = pcp->submlg - lastg - 2;
   int cp2 = cp1 - lastg;
   int result = cp2 - lastg;
   register int value;
   int u, v;
   int exp;
   int position, new;

#include "access.h"

   /* update submlg because of possible call to power */
   pcp->submlg -= (3 * lastg + 2);

   /* for each specified generator, compute its image under
      the action of the automorphism */

   for (generator = start; generator <= lastg; ++generator) {

#ifdef DEBUG
      if (generator % 100 == 0)
         printf("Processed generator %d\n", generator);
#endif

      /* check if there is sufficient space allocated */
      if (generator % SIZE == 1 && (new = *index + max_length) > *list_length) {
         *list = reallocate_vector(*list, *list_length + 1, new + 1, 0, FALSE);
         *list_length = new;
      }

      /* examine the definition of generator */
      value = y[structure + generator];

      if (value <= 0) {
         evaluate_image(head, *list, offset, -value, result, pcp);
      } else {

         u = PART2(value);
         v = PART3(value);

         if (v == 0)
            Extend_Pow(cp1, cp2, u, offset, head, *list, pcp);
         else
            Extend_Comm(cp1, cp2, u, v, offset, head, *list, pcp);

#if defined(GROUP)
         /* solve the appropriate equation, storing the image
            of generator under the action of alpha at result;
            in the Lie Program, Extend_Comm has already
            set up the answer at location result */

         solve_equation(cp1, cp2, result, pcp);

#endif
      }

      /* now copy the result to list */

      position = head[offset + generator] = *index;
      (*list)[++position] = 0;
      ++*index;

      for (i = 1; i <= lastg; ++i) {
         if ((exp = y[result + i]) != 0) {
            ++(*list)[position];
            (*list)[++*index] = PACK2(exp, i);
         }
      }
   }

   /* reset value of submlg */
   pcp->submlg += (3 * lastg + 2);
}

void evaluate_image(
    int *head, int *list, int offset, int ptr, int cp, struct pcp_vars *pcp)
{
   register int *y = y_address;

   int lastg = pcp->lastg;
   int i, j, start, u;
   int pointer;
   int exp;
   int image_length, relation_length;
   int next_gen, next_exp;
   int p = pcp->p;
#include "access.h"

   for (i = 1; i <= lastg; ++i)
      y[cp + i] = 0;

   if (ptr == 0)
      return;

   /* length of redundant relation */
   relation_length = y[ptr + 1];

   /* first generator in redundant relation */
   u = FIELD2(y[ptr + 1 + 1]);

   /* its exponent */
   exp = FIELD1(y[ptr + 1 + 1]);

   /* set up exp power of the image of u under alpha as exponent vector at cp */
   traverse_list(exp, head[offset + u], list, cp, pcp);

   /* now reduce the entries mod p */
   for (i = 1; i <= lastg; ++i)
      y[cp + i] %= p;

   /* now set up image of second generator as word with
      base address pointer */

   pointer = pcp->lused + 1;
   for (i = 2; i <= relation_length; ++i) {
      next_gen = FIELD2(y[ptr + 1 + i]);
      next_exp = FIELD1(y[ptr + 1 + i]);
      start = head[offset + next_gen];
      image_length = list[++start];
      y[pointer + 1] = image_length;
      for (j = 1; j <= image_length; ++j)
         y[pointer + 1 + j] = list[start + j];
      for (; next_exp > 0; --next_exp)
         collect(-pointer, cp, pcp);
   }
}

/* given generator t of the p-multiplicator, whose definition is
   u^p; hence, we have the equation

                      u^p = W * t

   where W is a word (possibly trivial) in the generators of the group;
   find the image of t under alpha by setting up (W)alpha at cp1,
   ((u)alpha)^p at cp2, and then call solve_equation */

void Extend_Pow(int cp1,
                int cp2,
                int u,
                int offset,
                int *head,
                int *list,
                struct pcp_vars *pcp)
{
   register int *y = y_address;

   register int i;
   register int value;
   register int lastg = pcp->lastg;

   for (i = 1; i <= lastg; ++i)
      y[cp1 + i] = y[cp2 + i] = 0;

   /* set up the image of u under alpha at cp2 */
   traverse_list(1, head[offset + u], list, cp2, pcp);

   /* raise the image of u under alpha to its pth power */
   power(pcp->p, cp2, pcp);

   /* set up image of W under alpha at cp1 */
   if ((value = y[pcp->ppower + u]) < 0)
      Collect_Image_Of_Str(-value, cp1, offset, head, list, pcp);
}

#if defined(GROUP)

/* given generator t of the p-multiplicator, whose definition is
   [u, v]; hence, we have the equation

   [u, v] = W * t, or equivalently, u * v = v * u * W * t

   where W is a word (possibly trivial) in the generators of the group;
   find the image of t under alpha by setting up
   (v)alpha * (u)alpha * (W)alpha at cp1, (u)alpha * (v)alpha at cp2
   and then call solve_equation */

void Extend_Comm(int cp1,
                 int cp2,
                 int u,
                 int v,
                 int offset,
                 int *head,
                 int *list,
                 struct pcp_vars *pcp)
{
   register int *y = y_address;

   register int i;
   register int pointer, value;
   register int lastg = pcp->lastg;

   for (i = 1; i <= lastg; ++i)
      y[cp1 + i] = y[cp2 + i] = 0;

   /* set up the image of u under alpha at cp2 */
   traverse_list(1, head[offset + u], list, cp2, pcp);

   /* collect image of v under alpha at cp2 */
   Collect_Image_Of_Gen(cp2, head[offset + v], list, pcp);

   /* set up image of v under alpha at cp1 */
   traverse_list(1, head[offset + v], list, cp1, pcp);

   /* collect image of u under alpha at cp1 */
   Collect_Image_Of_Gen(cp1, head[offset + u], list, pcp);

   /* collect image of W under alpha at cp1 */
   pointer = y[pcp->ppcomm + u];
   if ((value = y[pointer + v]) < 0)
      Collect_Image_Of_Str(-value, cp1, offset, head, list, pcp);
}

#endif

/* there may be a case where each of the exponent and p is large
   to use the power routine to compute the exp power of the
   image of generator under automorphism -- it does not seem to
   be worthwhile where p = 5 -- needs further investigation */

void Pq_Collect_Image_Of_Gen(
    int exp, int cp, int head, int *list, struct pcp_vars *pcp)
{
   register int *y = y_address;

   register int lused = pcp->lused;
   int str = lused + pcp->lastg;
   register int i;

   for (i = 1; i <= pcp->lastg; ++i)
      y[lused + i] = 0;

   traverse_list(1, head, list, lused, pcp);
   power(exp, lused, pcp);
   vector_to_string(lused, str, pcp);

   collect(-str, cp, pcp);
}

/* collect image of a generator under the
   action of an automorphism and store the result at cp */

void Collect_Image_Of_Gen(int cp, int head, int *list, struct pcp_vars *pcp)
{
   register int *y = y_address;

   register int lused = pcp->lused;
   register int length = list[++head];
   register int i;

   y[lused + 1] = length;

   for (i = 1; i <= length; ++i)
      y[lused + 1 + i] = list[head + i];

   collect(-lused, cp, pcp);
}

/* collect image of supplied string under the action of
   supplied automorphism and store the result at cp */

void Collect_Image_Of_Str(
    int string, int cp, int offset, int *head, int *list, struct pcp_vars *pcp)
{
   register int *y = y_address;

   register int i;
   register int generator, exp;
   register int value;
   register int length = y[string + 1] - 1; /* last element of string
                                               is in p-multiplicator */
#include "access.h"

   /* process the string generator by generator, collecting exp
      copies of the image of generator under action of automorphism
      -- should power routine be used? */

   for (i = 1; i <= length; ++i) {
      value = y[string + 1 + i];
      generator = FIELD2(value);
      exp = FIELD1(value);
      while (exp > 0) {
         Collect_Image_Of_Gen(cp, head[offset + generator], list, pcp);
         --exp;
      }
   }
}

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