source: git/Singular/walk.cc @ d471a7

/*****************************************
*  Computer Algebra System SINGULAR      *
*****************************************/
/* $Id$ */
/*
* ABSTRACT: Implementation of the Groebner walk
*/

// define if the Buchberger alg should be used
//   to compute a reduced GB of a omega-homogenoues ideal
// default: we use the hilbert driven algorithm.
#define BUCHBERGER_ALG  //we use the improved Buchberger alg.

//#define UPPER_BOUND //for the original "Tran" algorithm
//#define REPRESENTATION_OF_SIGMA //if one perturbs sigma in Tran

//#define TEST_OVERFLOW
//#define CHECK_IDEAL
//#define CHECK_IDEAL_MWALK

//#define NEXT_VECTORS_CC
//#define PRINT_VECTORS //to print vectors (sigma, tau, omega)

#define INVEPS_SMALL_IN_FRACTAL  //to choose the small invers of epsilon
#define INVEPS_SMALL_IN_MPERTVECTOR  //to choose the small invers of epsilon
#define INVEPS_SMALL_IN_TRAN  //to choose the small invers of epsilon

#define FIRST_STEP_FRACTAL // to define the first step of the fractal
//#define MSTDCC_FRACTAL // apply Buchberger alg to compute a red GB, if
//                          tau doesn't stay in the correct cone

//#define TIME_TEST // print the used time of each subroutine
//#define ENDWALKS //print the size of the last omega-homogenoues Groebner basis

/* includes */

#include <kernel/mod2.h>
#include <misc/intvec.h>
#include <Singular/cntrlc.h>
#include <misc/options.h>
#include <omalloc/omalloc.h>
#include <Singular/ipshell.h>
#include <Singular/ipconv.h>
#include <coeffs/coeffs.h>
#include <Singular/subexpr.h>

#include <polys/monomials/maps.h>

/* include Hilbert-function */
#include <kernel/combinatorics/stairc.h>

/** kstd2.cc */
#include <kernel/GBEngine/kutil.h>
#include <kernel/GBEngine/khstd.h>

#include <Singular/walk.h>
#include <kernel/polys.h>
#include <kernel/ideals.h>
#include <Singular/ipid.h>
#include <Singular/tok.h>
#include <coeffs/numbers.h>
#include <Singular/ipid.h>
#include <polys/monomials/ring.h>
#include <kernel/GBEngine/kstd1.h>
#include <polys/matpol.h>
#include <polys/weight.h>
#include <misc/intvec.h>
#include <kernel/GBEngine/syz.h>
#include <Singular/lists.h>
#include <polys/prCopy.h>
#include <polys/monomials/ring.h>
//#include <polys/ext_fields/longalg.h>
#include <polys/clapsing.h>

#include <coeffs/mpr_complex.h>

#include <stdio.h>
// === Zeit & System (Holger Croeni ===
#include <time.h>
#include <sys/time.h>
#include <math.h>
#include <sys/stat.h>
#include <unistd.h>
#include <float.h>
#include <misc/mylimits.h>
#include <sys/types.h>

int nstep;

extern BOOLEAN ErrorCheck();

extern BOOLEAN pSetm_error;

void Set_Error( BOOLEAN f) { pSetm_error=f; }

BOOLEAN Overflow_Error =  FALSE;

clock_t xtif, xtstd, xtlift, xtred, xtnw;
clock_t xftostd, xtextra, xftinput, to;

/****************************
 * utilities for TSet, LSet *
 ****************************/
inline static intset initec (int maxnr)
{
  return (intset)omAlloc(maxnr*sizeof(int));
}

inline static unsigned long* initsevS (int maxnr)
{
  return (unsigned long*)omAlloc0(maxnr*sizeof(unsigned long));
}
inline static int* initS_2_R (int maxnr)
{
  return (int*)omAlloc0(maxnr*sizeof(int));
}

/************************************
 * construct the set s from F u {P} *
 ************************************/
// unused
#if 0
static void initSSpecialCC (ideal F, ideal Q, ideal P,kStrategy strat)
{
  int   i,pos;

  if (Q!=NULL) i=((IDELEMS(Q)+(setmaxTinc-1))/setmaxTinc)*setmaxTinc;
  else i=setmaxT;

  strat->ecartS=initec(i);
  strat->sevS=initsevS(i);
  strat->S_2_R=initS_2_R(i);
  strat->fromQ=NULL;
  strat->Shdl=idInit(i,F->rank);
  strat->S=strat->Shdl->m;

  // - put polys into S -
  if (Q!=NULL)
  {
    strat->fromQ=initec(i);
    memset(strat->fromQ,0,i*sizeof(int));
    for (i=0; i<IDELEMS(Q); i++)
    {
      if (Q->m[i]!=NULL)
      {
        LObject h;
        h.p = pCopy(Q->m[i]);
        //if (TEST_OPT_INTSTRATEGY)
        //{
        //  //pContent(h.p);
        //  h.pCleardenom(); // also does a pContent
        //}
        //else
        //{
        //  h.pNorm();
        //}
        strat->initEcart(&h);
        if (rHasLocalOrMixedOrdering_currRing())
        {
          deleteHC(&h,strat);
        }
        if (h.p!=NULL)
        {
          if (strat->sl==-1)
            pos =0;
          else
          {
            pos = posInS(strat,strat->sl,h.p,h.ecart);
          }
          h.sev = pGetShortExpVector(h.p);
          h.SetpFDeg();
          strat->enterS(h,pos,strat, strat->tl+1);
          enterT(h, strat);
          strat->fromQ[pos]=1;
        }
      }
    }
  }
  //- put polys into S -
  for (i=0; i<IDELEMS(F); i++)
  {
    if (F->m[i]!=NULL)
    {
      LObject h;
      h.p = pCopy(F->m[i]);
      if (rHasGlobalOrdering(currRing))
      {
        //h.p=redtailBba(h.p,strat->sl,strat);
        h.p=redtailBba(h.p,strat->sl,strat);
      }
      else
      {
        deleteHC(&h,strat);
      }
      strat->initEcart(&h);
      if (h.p!=NULL)
      {
        if (strat->sl==-1)
          pos =0;
        else
          pos = posInS(strat,strat->sl,h.p,h.ecart);
        h.sev = pGetShortExpVector(h.p);
        strat->enterS(h,pos,strat, strat->tl+1);
        h.length = pLength(h.p);
        h.SetpFDeg();
        enterT(h,strat);
      }
    }
  }
#ifdef INITSSPECIAL
  for (i=0; i<IDELEMS(P); i++)
  {
    if (P->m[i]!=NULL)
    {
      LObject h;
      h.p=pCopy(P->m[i]);
      strat->initEcart(&h);
      h.length = pLength(h.p);
      if (TEST_OPT_INTSTRATEGY)
      {
        h.pCleardenom();
      }
      else
      {
        h.pNorm();
      }
      if(strat->sl>=0)
      {
        if (rHasGlobalOrdering(currRing))
        {
          h.p=redBba(h.p,strat->sl,strat);
          if (h.p!=NULL)
            h.p=redtailBba(h.p,strat->sl,strat);
        }
        else
        {
          h.p=redMora(h.p,strat->sl,strat);
          strat->initEcart(&h);
        }
        if(h.p!=NULL)
        {
          if (TEST_OPT_INTSTRATEGY)
          {
            h.pCleardenom();
          }
          else
          {
            h.is_normalized = 0;
            h.pNorm();
          }
          h.sev = pGetShortExpVector(h.p);
          h.SetpFDeg();
          pos = posInS(strat->S,strat->sl,h.p,h.ecart);
          enterpairsSpecial(h.p,strat->sl,h.ecart,pos,strat,strat->tl+1);
          strat->enterS(h,pos,strat, strat->tl+1);
          enterT(h,strat);
        }
      }
      else
      {
        h.sev = pGetShortExpVector(h.p);
        h.SetpFDeg();
        strat->enterS(h,0,strat, strat->tl+1);
        enterT(h,strat);
      }
    }
  }
#endif
}
#endif

/*****************
 *interreduce F  *
 *****************/
static ideal kInterRedCC(ideal F, ideal Q)
{
  int j;
  kStrategy strat = new skStrategy;

//  if (TEST_OPT_PROT)
//  {
//    writeTime("start InterRed:");
//    mflush();
//  }
  //strat->syzComp     = 0;
  strat->kHEdgeFound = (currRing->ppNoether) != NULL;
  strat->kNoether=pCopy((currRing->ppNoether));
  strat->ak = id_RankFreeModule(F, currRing);
  initBuchMoraCrit(strat);
  strat->NotUsedAxis = (BOOLEAN *)omAlloc((currRing->N+1)*sizeof(BOOLEAN));
  for(j=currRing->N; j>0; j--)
  {
    strat->NotUsedAxis[j] = TRUE;
  }
  strat->enterS      = enterSBba;
  strat->posInT      = posInT0;
  strat->initEcart   = initEcartNormal;
  strat->sl   = -1;
  strat->tl          = -1;
  strat->tmax        = setmaxT;
  strat->T           = initT();
  strat->R           = initR();
  strat->sevT        = initsevT();
  if(rHasLocalOrMixedOrdering_currRing())
  {
    strat->honey = TRUE;
  }

  //initSCC(F,Q,strat);
  initS(F,Q,strat);

  /*
  timetmp=clock();//22.01.02
  initSSpecialCC(F,Q,NULL,strat);
  tininitS=tininitS+clock()-timetmp;//22.01.02
  */
  if(TEST_OPT_REDSB)
  {
    strat->noTailReduction=FALSE;
  }
  updateS(TRUE,strat);

  if(TEST_OPT_REDSB && TEST_OPT_INTSTRATEGY)
  {
    completeReduce(strat);
  }
  pDelete(&strat->kHEdge);
  omFreeSize((ADDRESS)strat->T,strat->tmax*sizeof(TObject));
  omFreeSize((ADDRESS)strat->ecartS,IDELEMS(strat->Shdl)*sizeof(int));
  omFreeSize((ADDRESS)strat->sevS,IDELEMS(strat->Shdl)*sizeof(unsigned long));
  omFreeSize((ADDRESS)strat->NotUsedAxis,(currRing->N+1)*sizeof(BOOLEAN));
  omfree(strat->sevT);
  omfree(strat->S_2_R);
  omfree(strat->R);

  if(strat->fromQ)
  {
    for(j=0; j<IDELEMS(strat->Shdl); j++)
    {
      if(strat->fromQ[j])
      {
        pDelete(&strat->Shdl->m[j]);
      }
    }
    omFreeSize((ADDRESS)strat->fromQ,IDELEMS(strat->Shdl)*sizeof(int));
    strat->fromQ = NULL;
  }
//  if (TEST_OPT_PROT)
//  {
//    writeTime("end Interred:");
//    mflush();
//  }
  ideal shdl=strat->Shdl;
  idSkipZeroes(shdl);
  delete(strat);

  return shdl;
}

//unused
#if 0
static void TimeString(clock_t tinput, clock_t tostd, clock_t tif,clock_t tstd,
                       clock_t tlf,clock_t tred, clock_t tnw, int step)
{
  double totm = ((double) (clock() - tinput))/1000000;
  double ostd,mostd, mif, mstd, mlf, mred, mnw, mxif,mxstd,mxlf,mxred,mxnw,tot;
  // double mextra
  Print("\n// total time = %.2f sec", totm);
  Print("\n// tostd = %.2f sec = %.2f", ostd=((double) tostd)/1000000,
        mostd=((((double) tostd)/1000000)/totm)*100);
  Print("\n// tif   = %.2f sec = %.2f", ((double) tif)/1000000,
        mif=((((double) tif)/1000000)/totm)*100);
  Print("\n// std   = %.2f sec = %.2f", ((double) tstd)/1000000,
        mstd=((((double) tstd)/1000000)/totm)*100);
  Print("\n// lift  = %.2f sec = %.2f", ((double) tlf)/1000000,
        mlf=((((double) tlf)/1000000)/totm)*100);
  Print("\n// ired  = %.2f sec = %.2f", ((double) tred)/1000000,
        mred=((((double) tred)/1000000)/totm)*100);
  Print("\n// nextw = %.2f sec = %.2f", ((double) tnw)/1000000,
        mnw=((((double) tnw)/1000000)/totm)*100);
  PrintS("\n Time for the last step:");
  Print("\n// xinfo = %.2f sec = %.2f", ((double) xtif)/1000000,
        mxif=((((double) xtif)/1000000)/totm)*100);
  Print("\n// xstd  = %.2f sec = %.2f", ((double) xtstd)/1000000,
        mxstd=((((double) xtstd)/1000000)/totm)*100);
  Print("\n// xlift = %.2f sec = %.2f", ((double) xtlift)/1000000,
        mxlf=((((double) xtlift)/1000000)/totm)*100);
  Print("\n// xired = %.2f sec = %.2f", ((double) xtred)/1000000,
        mxred=((((double) xtred)/1000000)/totm)*100);
  Print("\n// xnextw= %.2f sec = %.2f", ((double) xtnw)/1000000,
        mxnw=((((double) xtnw)/1000000)/totm)*100);

  tot=mostd+mif+mstd+mlf+mred+mnw+mxif+mxstd+mxlf+mxred+mxnw;
  double res = (double) 100 - tot;
  Print("\n// &%d&%.2f&%.2f&%.2f&%.2f&%.2f&%.2f&%.2f&%.2f&%.2f&%.2f&%.2f&%.2f&%.2f&%.2f&%.2f(%.2f)\\ \\",
        step, ostd, totm, mostd,mif,mstd,mlf,mred,mnw,mxif,mxstd,mxlf,mxred,mxnw,tot,res,
        ((((double) xtextra)/1000000)/totm)*100);
}
#endif

//unused
#if 0
static void TimeStringFractal(clock_t tinput, clock_t tostd, clock_t tif,clock_t tstd,
                       clock_t textra, clock_t tlf,clock_t tred, clock_t tnw)
{

  double totm = ((double) (clock() - tinput))/1000000;
  double ostd, mostd, mif, mstd, mextra, mlf, mred, mnw, tot, res;
  Print("\n// total time = %.2f sec", totm);
  Print("\n// tostd = %.2f sec = %.2f", ostd=((double) tostd)/1000000,
        mostd=((((double) tostd)/1000000)/totm)*100);
  Print("\n// tif   = %.2f sec = %.2f", ((double) tif)/1000000,
        mif=((((double) tif)/1000000)/totm)*100);
  Print("\n// std   = %.2f sec = %.2f", ((double) tstd)/1000000,
        mstd=((((double) tstd)/1000000)/totm)*100);
  Print("\n// xstd  = %.2f sec = %.2f", ((double) textra)/1000000,
        mextra=((((double) textra)/1000000)/totm)*100);
  Print("\n// lift  = %.2f sec = %.2f", ((double) tlf)/1000000,
        mlf=((((double) tlf)/1000000)/totm)*100);
  Print("\n// ired  = %.2f sec = %.2f", ((double) tred)/1000000,
        mred=((((double) tred)/1000000)/totm)*100);
  Print("\n// nextw = %.2f sec = %.2f", ((double) tnw)/1000000,
        mnw=((((double) tnw)/1000000)/totm)*100);
  tot = mostd+mif+mstd+mextra+mlf+mred+mnw;
  res = (double) 100.00-tot;
  Print("\n// &%.2f &%.2f&%.2f &%.2f &%.2f &%.2f &%.2f &%.2f &%.2f&%.2f&%.2f\\ \\ ",
        ostd,totm,mostd,mif,mstd,mextra,mlf,mred,mnw,tot,res);
}
#endif

#ifdef CHECK_IDEAL_MWALK
static void idString(ideal L, const char* st)
{
  int i, nL = IDELEMS(L);

  Print("\n//  ideal %s =  ", st);
  for(i=0; i<nL-1; i++)
  {
    Print(" %s, ", pString(L->m[i]));
  }
  Print(" %s;", pString(L->m[nL-1]));
}
#endif

#if defined(CHECK_IDEAL_MWALK) || defined(ENDWALKS)
static void headidString(ideal L, char* st)
{
  int i, nL = IDELEMS(L);

  Print("\n//  ideal %s =  ", st);
  for(i=0; i<nL-1; i++)
  {
    Print(" %s, ", pString(pHead(L->m[i])));
  }
  Print(" %s;", pString(pHead(L->m[nL-1])));
}
#endif

#if defined(CHECK_IDEAL_MWALK) || defined(ENDWALKS)
static void idElements(ideal L, char* st)
{
  int i, nL = IDELEMS(L);
  int *K=(int *)omAlloc(nL*sizeof(int));

  Print("\n//  #monoms of %s =  ", st);
  for(i=0; i<nL; i++)
  {
    K[i] = pLength(L->m[i]);
  }
  int j, nsame;
  // int  nk=0;
  for(i=0; i<nL; i++)
  {
    if(K[i]!=0)
    {
      nsame = 1;
      for(j=i+1; j<nL; j++)
      {
        if(K[j]==K[i])
        {
          nsame ++;
          K[j]=0;
        }
      }
      if(nsame == 1)
      {
        Print("%d, ",K[i]);
      }
      else
      {
        Print("%d[%d], ", K[i], nsame);
      }
    }
  }
  omFree(K);
}
#endif


static void ivString(intvec* iv, const char* ch)
{
  int nV = iv->length()-1;
  Print("\n// intvec %s =  ", ch);

  for(int i=0; i<nV; i++)
  {
    Print("%d, ", (*iv)[i]);
  }
  Print("%d;", (*iv)[nV]);
}

//unused
#if 0
static void MivString(intvec* iva, intvec* ivb, intvec* ivc)
{
  int nV = iva->length()-1;
  int i;
  PrintS("\n//  (");
  for(i=0; i<nV; i++)
  {
    Print("%d, ", (*iva)[i]);
  }
  Print("%d) ==> (", (*iva)[nV]);
  for(i=0; i<nV; i++)
  {
    Print("%d, ", (*ivb)[i]);
  }
  Print("%d) := (", (*ivb)[nV]);

  for(i=0; i<nV; i++)
  {
    Print("%d, ", (*ivc)[i]);
  }
  Print("%d)", (*ivc)[nV]);
}
#endif

/********************************************************************
 * returns gcd of integers a and b                                  *
 ********************************************************************/
static inline long gcd(const long a, const long b)
{
  long r, p0 = a, p1 = b;
  //assume(p0 >= 0 && p1 >= 0);
  if(p0 < 0)
  {
    p0 = -p0;
  }
  if(p1 < 0)
  {
    p1 = -p1;
  }
  while(p1 != 0)
  {
    r = p0 % p1;
    p0 = p1;
    p1 = r;
  }
  return p0;
}

/*********************************************
 * cancel gcd of integers zaehler and nenner *
 *********************************************/
static void cancel(mpz_t zaehler, mpz_t nenner)
{
//  assume(zaehler >= 0 && nenner > 0);
  mpz_t g;
  mpz_init(g);
  mpz_gcd(g, zaehler, nenner);

  mpz_div(zaehler , zaehler, g);
  mpz_div(nenner ,  nenner, g);

  mpz_clear(g);
}

//unused
#if 0
static int isVectorNeg(intvec* omega)
{
  int i;

  for(i=omega->length(); i>=0; i--)
  {
    if((*omega)[i]<0)
    {
      return 1;
    }
  }
  return 0;
}
#endif

/********************************************************************
 * compute a weight degree of a monomial p w.r.t. a weight_vector   *
 ********************************************************************/
static inline int MLmWeightedDegree(const poly p, intvec* weight)
{
  /* 2147483647 is max. integer representation in SINGULAR */
  mpz_t sing_int;
  mpz_init_set_ui(sing_int,  2147483647);

  int i, wgrad;

  mpz_t zmul;
  mpz_init(zmul);
  mpz_t zvec;
  mpz_init(zvec);
  mpz_t zsum;
  mpz_init(zsum);

  for (i=currRing->N; i>0; i--)
  {
    mpz_set_si(zvec, (*weight)[i-1]);
    mpz_mul_ui(zmul, zvec, pGetExp(p, i));
    mpz_add(zsum, zsum, zmul);
  }

  wgrad = mpz_get_ui(zsum);

  if(mpz_cmp(zsum, sing_int)>0)
  {
    if(Overflow_Error ==  FALSE)
    {
      PrintLn();
      PrintS("\n// ** OVERFLOW in \"MwalkInitialForm\": ");
      mpz_out_str( stdout, 10, zsum);
      PrintS(" is greater than 2147483647 (max. integer representation)");
      Overflow_Error = TRUE;
    }
  }

  mpz_clear(zmul);
  mpz_clear(zvec);
  mpz_clear(zsum);
  mpz_clear(sing_int);

  return wgrad;
}

/********************************************************************
 * compute a weight degree of a polynomial p w.r.t. a weight_vector *
 ********************************************************************/
static inline int MwalkWeightDegree(poly p, intvec* weight_vector)
{
  assume(weight_vector->length() >= currRing->N);
  int max = 0, maxtemp;

  while(p != NULL)
  {
    maxtemp = MLmWeightedDegree(p, weight_vector);
    pIter(p);

    if (maxtemp > max)
    {
      max = maxtemp;
    }
  }
  return max;
}


/********************************************************************
 * compute a weight degree of a monomial p w.r.t. a weight_vector   *
 ********************************************************************/
static  void  MLmWeightedDegree_gmp(mpz_t result, const poly p, intvec* weight)
{
  /* 2147483647 is max. integer representation in SINGULAR */
  mpz_t sing_int;
  mpz_init_set_ui(sing_int,  2147483647);

  int i;

  mpz_t zmul;
  mpz_init(zmul);
  mpz_t zvec;
  mpz_init(zvec);
  mpz_t ztmp;
  mpz_init(ztmp);

  for (i=currRing->N; i>0; i--)
  {
    mpz_set_si(zvec, (*weight)[i-1]);
    mpz_mul_ui(zmul, zvec, pGetExp(p, i));
    mpz_add(ztmp, ztmp, zmul);
  }
  mpz_init_set(result, ztmp);
  mpz_clear(ztmp);
  mpz_clear(sing_int);
  mpz_clear(zvec);
  mpz_clear(zmul);
}


/*****************************************************************************
 * return an initial form of the polynom g w.r.t. a weight vector curr_weight *
 *****************************************************************************/
static poly MpolyInitialForm(poly g, intvec* curr_weight)
{
  if(g == NULL)
  {
    return NULL;
  }
  mpz_t max; mpz_init(max);
  mpz_t maxtmp; mpz_init(maxtmp);

  poly hg, in_w_g = NULL;

  while(g != NULL)
  {
    hg = g;
    pIter(g);
    MLmWeightedDegree_gmp(maxtmp, hg, curr_weight);

    if(mpz_cmp(maxtmp, max)>0)
    {
      mpz_init_set(max, maxtmp);
      pDelete(&in_w_g);
      in_w_g = pHead(hg);
    }
    else
    {
      if(mpz_cmp(maxtmp, max)==0)
      {
        in_w_g = pAdd(in_w_g, pHead(hg));
      }
    }
  }
  return in_w_g;
}

/************************************************************************
 * compute the initial form of an ideal <G> w.r.t. a weight vector iva  *
 ************************************************************************/
ideal MwalkInitialForm(ideal G, intvec* ivw)
{
  BOOLEAN nError =  Overflow_Error;
  Overflow_Error = FALSE;

  int i, nG = IDELEMS(G);
  ideal Gomega = idInit(nG, 1);

  for(i=nG-1; i>=0; i--)
  {
    Gomega->m[i] = MpolyInitialForm(G->m[i], ivw);
  }
  if(Overflow_Error == FALSE)
  {
    Overflow_Error = nError;
  }
  return Gomega;
}

/************************************************************************
 * test whether the weight vector iv is in the cone of the ideal G      *
 *     i.e. test whether in(in_w(g)) = in(g) for all g in G             *
 ************************************************************************/

static int test_w_in_ConeCC(ideal G, intvec* iv)
{
  if(G->m[0] == NULL)
  {
    PrintS("//** the result may be WRONG, i.e. 0!!\n");
    return 0;
  }

  BOOLEAN nError =  Overflow_Error;
  Overflow_Error = FALSE;

  int i, nG = IDELEMS(G);
  poly mi, gi;

  for(i=nG-1; i>=0; i--)
  {
    mi = MpolyInitialForm(G->m[i], iv);
    gi = G->m[i];

    if(mi == NULL)
    {
      pDelete(&mi);
      if(Overflow_Error == FALSE)
      {
        Overflow_Error = nError;
      }
      return 0;
    }
    if(!pLmEqual(mi, gi))
    {
      pDelete(&mi);
      if(Overflow_Error == FALSE)
      {
        Overflow_Error = nError;
      }
      return 0;
    }
    pDelete(&mi);
  }

  if(Overflow_Error == FALSE)
  {
    Overflow_Error = nError;
  }
  return 1;
}

/***************************************************
 * compute a least common multiple of two integers *
 ***************************************************/
static inline long Mlcm(long &i1, long &i2)
{
  long temp = gcd(i1, i2);
  return ((i1 / temp)* i2);
}


/***************************************************
 * return  the dot product of two intvecs a and b  *
 ***************************************************/
static inline long  MivDotProduct(intvec* a, intvec* b)
{
  assume( a->length() ==  b->length());
  int i, n = a->length();
  long result = 0;

  for(i=n-1; i>=0; i--)
    {
    result += (*a)[i] * (*b)[i];
    }
  return result;
}

/*****************************************************
 * Substract two given intvecs componentwise         *
 *****************************************************/
static intvec* MivSub(intvec* a, intvec* b)
{
  assume( a->length() ==  b->length());
  int i, n = a->length();
  intvec* result = new intvec(n);

  for(i=n-1; i>=0; i--)
  {
    (*result)[i] = (*a)[i] - (*b)[i];
  }
  return result;
}

/*****************************************************
 * return the "intvec" lead exponent of a polynomial *
 *****************************************************/
static intvec* MExpPol(poly f)
{
  int i, nR = currRing->N;
  intvec* result = new intvec(nR);

  for(i=nR-1; i>=0; i--)
  {
    (*result)[i] = pGetExp(f,i+1);
  }
  return result;
}

/*****************************************************
 * Compare two given intvecs and return 1, if they   *
 * are the same, otherwise 0                         *
 *****************************************************/
int MivSame(intvec* u , intvec* v)
{
  assume(u->length() == v->length());

  int i, niv = u->length();

  for (i=0; i<niv; i++)
  {
    if ((*u)[i] != (*v)[i])
    {
      return 0;
    }
  }
  return 1;
}

/******************************************************
 * Compare 3 given intvecs and return 0, if the first *
 * and the second are the same. Return 1, if the      *
 * the second and the third are the same, otherwise 2 *
 ******************************************************/
int M3ivSame(intvec* temp, intvec* u , intvec* v)
{
  assume(temp->length() == u->length() && u->length() == v->length());

  if((MivSame(temp, u)) == 1)
  {
    return 0;
  }
  if((MivSame(temp, v)) == 1)
  {
    return 1;
  }
  return 2;
}

/*****************************************************
 * compute a Groebner basis of an ideal              *
 *****************************************************/
static ideal MstdCC(ideal G)
{
  BITSET save1,save2;
  SI_SAVE_OPT(save1,save2);
  si_opt_1|=(Sy_bit(OPT_REDTAIL)|Sy_bit(OPT_REDSB));
  ideal G1 = kStd(G, NULL, testHomog, NULL);
  SI_RESTORE_OPT(save1,save2);

  idSkipZeroes(G1);
  return G1;
}

/*****************************************************
 * compute a Groebner basis of an homogeneous ideal  *
 *****************************************************/
static ideal MstdhomCC(ideal G)
{
  BITSET save1,save2;
  SI_SAVE_OPT(save1,save2);
  si_opt_1|=(Sy_bit(OPT_REDTAIL)|Sy_bit(OPT_REDSB));
  ideal G1 = kStd(G, NULL, isHomog, NULL);
  SI_RESTORE_OPT(save1,save2);

  idSkipZeroes(G1);
  return G1;
}


/*****************************************************************************
* create a weight matrix order as intvec of an extra weight vector (a(iv),lp)*
******************************************************************************/
intvec* MivMatrixOrder(intvec* iv)
{
  int i, nR = iv->length();

  intvec* ivm = new intvec(nR*nR);

  for(i=0; i<nR; i++)
  {
    (*ivm)[i] = (*iv)[i];
  }
  for(i=1; i<nR; i++)
  {
    (*ivm)[i*nR+i-1] = 1;
  }
  return ivm;
}

/*****************************************************************************
* create a weight matrix order as intvec of an extra weight vector (a(iv),lp)*
******************************************************************************/
intvec* MivMatrixOrderRefine(intvec* iv, intvec* iw)
{
  assume(iv->length() == iw->length());
  int i, nR = iv->length();

  intvec* ivm = new intvec(nR*nR);

  for(i=0; i<nR; i++)
  {
    (*ivm)[i] = (*iv)[i];
    (*ivm)[i+nR] = (*iw)[i];
  }
  for(i=2; i<nR; i++)
  {
    (*ivm)[i*nR+i-2] = 1;
  }
  return ivm;
}

/*******************************
 * return intvec = (1, ..., 1) *
 *******************************/
intvec* Mivdp(int nR)
{
  int i;
  intvec* ivm = new intvec(nR);

  for(i=nR-1; i>=0; i--)
  {
    (*ivm)[i] = 1;
  }
  return ivm;
}

/**********************************
 * return intvvec = (1,0, ..., 0) *
 **********************************/
intvec* Mivlp(int nR)
{
  intvec* ivm = new intvec(nR);
  (*ivm)[0] = 1;

  return ivm;
}

//unused
/*****************************************************************************
 * print the max total degree and the max coefficient of G                   *
 *****************************************************************************/
#if 0
static void checkComplexity(ideal G, char* cG)
{
  int nV = currRing->N;
  int nG = IDELEMS(G);
  intvec* ivUnit = Mivdp(nV);
  int i, tmpdeg, maxdeg=0;
  number tmpcoeff , maxcoeff=currRing->cf->nNULL;
  poly p;
  for(i=nG-1; i>=0; i--)
  {
    tmpdeg = MwalkWeightDegree(G->m[i], ivUnit);
    if(tmpdeg > maxdeg )
    {
      maxdeg = tmpdeg;
    }
  }

  for(i=nG-1; i>=0; i--)
  {
    p = pCopy(G->m[i]);
    while(p != NULL)
    {
      //tmpcoeff = pGetCoeff(pHead(p));
      tmpcoeff = pGetCoeff(p);
      if(nGreater(tmpcoeff,maxcoeff))
      {
         maxcoeff = nCopy(tmpcoeff);
      }
      pIter(p);
    }
    pDelete(&p);
  }
  p = pNSet(maxcoeff);
  char* pStr = pString(p);
  delete ivUnit;
  Print("// max total degree of %s = %d\n",cG, maxdeg);
  Print("// max coefficient of %s  = %s", cG, pStr);//ing(p));
  Print(" which consists of %d digits", (int)strlen(pStr));
  PrintLn();
}
#endif

/*****************************************************************************
* If target_ord = intmat(A1, ..., An) then calculate the perturbation        *
* vectors                                                                    *
*   tau_p_dep = inveps^(p_deg-1)*A1 + inveps^(p_deg-2)*A2 +... + A_p_deg     *
* where                                                                      *
*      inveps > totaldegree(G)*(max(A2)+...+max(A_p_deg))                    *
* intmat target_ord is an integer order matrix of the monomial ordering of   *
* basering.                                                                  *
* This programm computes a perturbated vector with a p_deg perturbation      *
* degree which smaller than the numbers of variables                         *
******************************************************************************/
intvec* MPertVectors(ideal G, intvec* ivtarget, int pdeg)
{
  // ivtarget is a matrix order of a degree reverse lex. order
  int nV = currRing->N;
  //assume(pdeg <= nV && pdeg >= 0);

  int i, j, nG = IDELEMS(G);
  intvec* v_null =  new intvec(nV);


  // Check that the perturbed degree is valid
  if(pdeg > nV || pdeg <= 0)
  {
    WerrorS("//** The perturbed degree is wrong!!");
    return v_null;
  }
  delete v_null;

  if(pdeg == 1)
  {
    return ivtarget;
  }
  mpz_t *pert_vector = (mpz_t*)omAlloc(nV*sizeof(mpz_t));
  //mpz_t *pert_vector1 = (mpz_t*)omAlloc(nV*sizeof(mpz_t));

  for(i=0; i<nV; i++)
  {
    mpz_init_set_si(pert_vector[i], (*ivtarget)[i]);
   // mpz_init_set_si(pert_vector1[i], (*ivtarget)[i]);
  }
  // Calculate max1 = Max(A2)+Max(A3)+...+Max(Apdeg),
  // where the Ai are the i-te rows of the matrix target_ord.
  int ntemp, maxAi, maxA=0;
  for(i=1; i<pdeg; i++)
  {
    maxAi = (*ivtarget)[i*nV];
    if(maxAi<0)
    {
      maxAi = -maxAi;
    }
    for(j=i*nV+1; j<(i+1)*nV; j++)
    {
      ntemp = (*ivtarget)[j];
      if(ntemp < 0)
      {
        ntemp = -ntemp;
      }
      if(ntemp > maxAi)
      {
        maxAi = ntemp;
      }
    }
    maxA += maxAi;
  }

  // Calculate inveps = 1/eps, where 1/eps > totaldeg(p)*max1 for all p in G.

  intvec* ivUnit = Mivdp(nV);

  mpz_t tot_deg; mpz_init(tot_deg);
  mpz_t maxdeg; mpz_init(maxdeg);
  mpz_t inveps; mpz_init(inveps);


  for(i=nG-1; i>=0; i--)
  {
     mpz_set_ui(maxdeg, MwalkWeightDegree(G->m[i], ivUnit));
     if (mpz_cmp(maxdeg,  tot_deg) > 0 )
     {
       mpz_set(tot_deg, maxdeg);
     }
  }

  delete ivUnit;
  mpz_mul_ui(inveps, tot_deg, maxA);
  mpz_add_ui(inveps, inveps, 1);


  // takes  "small" inveps
#ifdef INVEPS_SMALL_IN_MPERTVECTOR
  if(mpz_cmp_ui(inveps, pdeg)>0 && pdeg > 3)
  {
    //  Print("\n// choose the\"small\" inverse epsilon := %d / %d = ", mpz_get_si(inveps), pdeg);
    mpz_fdiv_q_ui(inveps, inveps, pdeg);
    // mpz_out_str(stdout, 10, inveps);
  }
#else
  // PrintS("\n// the \"big\" inverse epsilon: ");
  mpz_out_str(stdout, 10, inveps);
#endif

  // pert(A1) = inveps^(pdeg-1)*A1 + inveps^(pdeg-2)*A2+...+A_pdeg,
  // pert_vector := A1
  for( i=1; i < pdeg; i++ )
  {
    for(j=0; j<nV; j++)
    {
      mpz_mul(pert_vector[j], pert_vector[j], inveps);
      if((*ivtarget)[i*nV+j]<0)
      {
        mpz_sub_ui(pert_vector[j], pert_vector[j],-(*ivtarget)[i*nV+j]);
      }
      else
      {
        mpz_add_ui(pert_vector[j], pert_vector[j],(*ivtarget)[i*nV+j]);
      }
    }
  }
  mpz_t ztemp;
  mpz_init(ztemp);
  mpz_set(ztemp, pert_vector[0]);
  for(i=1; i<nV; i++)
  {
    mpz_gcd(ztemp, ztemp, pert_vector[i]);
    if(mpz_cmp_si(ztemp, 1)  == 0)
    {
      break;
    }
  }
  if(mpz_cmp_si(ztemp, 1) != 0)
  {
    for(i=0; i<nV; i++)
    {
      mpz_divexact(pert_vector[i], pert_vector[i], ztemp);
    }
  }

  intvec *pert_vector1= new intvec(nV);
  j = 0;
  for(i=0; i<nV; i++)
  {
    (* pert_vector1)[i] = mpz_get_si(pert_vector[i]);
    (* pert_vector1)[i] = 0.1*(* pert_vector1)[i];
    (* pert_vector1)[i] = floor((* pert_vector1)[i] + 0.5);
    if((* pert_vector1)[i] == 0)
    {
      j++;
    }
  }
  if(j > nV - 1)
  {
    // Print("\n//  MPertVectors: geaenderter vector gleich Null! \n");
    delete pert_vector1;
    goto CHECK_OVERFLOW;
  }

// check that the perturbed weight vector lies in the Groebner cone
  if(test_w_in_ConeCC(G,pert_vector1) != 0)
  {
    // Print("\n//  MPertVectors: geaenderter vector liegt in Groebnerkegel! \n");
    for(i=0; i<nV; i++)
    {
      mpz_set_si(pert_vector[i], (*pert_vector1)[i]);
    }
  }
  else
  {
    //Print("\n// MpertVectors: geaenderter vector liegt nicht in Groebnerkegel! \n");
  }
  delete pert_vector1;

  CHECK_OVERFLOW:
  intvec* result = new intvec(nV);

  /* 2147483647 is max. integer representation in SINGULAR */
  mpz_t sing_int;
  mpz_init_set_ui(sing_int,  2147483647);

  int ntrue=0;
  for(i=0; i<nV; i++)
  {
    (*result)[i] = mpz_get_si(pert_vector[i]);
    if(mpz_cmp(pert_vector[i], sing_int)>=0)
    {
      ntrue++;
      if(Overflow_Error == FALSE)
      {
        Overflow_Error = TRUE;
        PrintS("\n// ** OVERFLOW in \"MPertvectors\": ");
        mpz_out_str( stdout, 10, pert_vector[i]);
        PrintS(" is greater than 2147483647 (max. integer representation)");
        Print("\n//  So vector[%d] := %d is wrong!!", i+1, (*result)[i]);
      }
    }
  }

  if(Overflow_Error == TRUE)
  {
    ivString(result, "pert_vector");
    Print("\n// %d element(s) of it is overflow!!", ntrue);
  }

  mpz_clear(ztemp);
  mpz_clear(sing_int);
  omFree(pert_vector);
  //omFree(pert_vector1);
  mpz_clear(tot_deg);
  mpz_clear(maxdeg);
  mpz_clear(inveps);

  rComplete(currRing);
  for(j=0; j<IDELEMS(G); j++)
  {
    poly p=G->m[j];
    while(p!=NULL)
    {
      p_Setm(p,currRing); pIter(p);
    }
  }
  return result;
}

/*****************************************************************************
 * The following procedure returns                                           *
 *     Pert(A1) = 1/eps^(pdeg-1)*A_1 + 1/eps^(pdeg-2)*A_2+...+A_pdeg,        *
 * where the A_i are the i-th rows of the matrix target_ord and              *
 *     1/eps > deg(p)*(max(A_2) + max(A_3)+...+max(A_pdeg))                  *
 *****************************************************************************/
intvec* MPertVectorslp(ideal G, intvec* ivtarget, int pdeg)
{
  // ivtarget is a matrix order of the lex. order
  int nV = currRing->N;
  //assume(pdeg <= nV && pdeg >= 0);

  int i, j, nG = IDELEMS(G);
  intvec* pert_vector =  new intvec(nV);

  //Checking that the perturbated degree is valid
  if(pdeg > nV || pdeg <= 0)
  {
    WerrorS("//** The perturbed degree is wrong!!");
    return pert_vector;
  }
  for(i=0; i<nV; i++)
  {
    (*pert_vector)[i]=(*ivtarget)[i];
  }
  if(pdeg == 1)
  {
    return pert_vector;
  }
  // Calculate max1 = Max(A2)+Max(A3)+...+Max(Apdeg),
  // where the Ai are the i-te rows of the matrix target_ord.
  int ntemp, maxAi, maxA=0;
  for(i=1; i<pdeg; i++)
  {
    maxAi = (*ivtarget)[i*nV];
    for(j=i*nV+1; j<(i+1)*nV; j++)
    {
      ntemp = (*ivtarget)[j];
      if(ntemp > maxAi)
      {
        maxAi = ntemp;
      }
    }
    maxA += maxAi;
  }

  // Calculate inveps := 1/eps, where 1/eps > deg(p)*max1 for all p in G.
  int inveps, tot_deg = 0, maxdeg;

  intvec* ivUnit = Mivdp(nV);//19.02
  for(i=nG-1; i>=0; i--)
  {
    // maxdeg = pTotaldegree(G->m[i], currRing); //it's wrong for ex1,2,rose
    maxdeg = MwalkWeightDegree(G->m[i], ivUnit);
    if (maxdeg > tot_deg )
    {
      tot_deg = maxdeg;
    }
  }
  delete ivUnit;

  inveps = (tot_deg * maxA) + 1;

#ifdef INVEPS_SMALL_IN_FRACTAL
  //  Print("\n// choose the\"small\" inverse epsilon := %d / %d = ", inveps, pdeg);
  if(inveps > pdeg && pdeg > 3)
  {
    inveps = inveps / pdeg;
  }
  // Print(" %d", inveps);
#else
  PrintS("\n// the \"big\" inverse epsilon %d", inveps);
#endif

  // Pert(A1) = inveps^(pdeg-1)*A1 + inveps^(pdeg-2)*A2+...+A_pdeg
  for ( i=1; i < pdeg; i++ )
  {
    for(j=0; j<nV; j++)
    {
      (*pert_vector)[j] = inveps*((*pert_vector)[j]) + (*ivtarget)[i*nV+j];
    }
  }

  int temp = (*pert_vector)[0];
  for(i=1; i<nV; i++)
  {
    temp = gcd(temp, (*pert_vector)[i]);
    if(temp == 1)
    {
      break;
    }
  }
  if(temp != 1)
  {
    for(i=0; i<nV; i++)
    {
      (*pert_vector)[i] = (*pert_vector)[i] / temp;
    }
  }

  intvec* result = pert_vector;
  delete pert_vector;
  return result;
}

/*****************************************************************************
 * define a lexicographic order matrix as intvec                             *
 *****************************************************************************/
intvec* MivMatrixOrderlp(int nV)
{
  int i;
  intvec* ivM = new intvec(nV*nV);

  for(i=0; i<nV; i++)
  {
    (*ivM)[i*nV + i] = 1;
  }
  return(ivM);
}


/*****************************************************************************
 * define a reverse lexicographic order (dp) matrix as intvec                *
 *****************************************************************************/
intvec* MivMatrixOrderdp(int nV)
{
  int i;
  intvec* ivM = new intvec(nV*nV);

  for(i=0; i<nV; i++)
  {
    (*ivM)[i] = 1;
  }
  for(i=1; i<nV; i++)
  {
    (*ivM)[(i+1)*nV - i] = -1;
  }
  return(ivM);
}

/*****************************************************************************
 * creates an intvec of the monomial order Wp(ivstart)                       *
 *****************************************************************************/
intvec* MivWeightOrderlp(intvec* ivstart)
{
  int i;
  int nV = ivstart->length();
  intvec* ivM = new intvec(nV*nV);

  for(i=0; i<nV; i++)
  {
    (*ivM)[i] = (*ivstart)[i];
  }
  for(i=1; i<nV; i++)
  {
    (*ivM)[i*nV + i-1] = 1;
  }
  return(ivM);
}

/*****************************************************************************
 * creates an intvec of the monomial order dp(ivstart)                       *
 *****************************************************************************/
intvec* MivWeightOrderdp(intvec* ivstart)
{
  int i;
  int nV = ivstart->length();
  intvec* ivM = new intvec(nV*nV);

  for(i=0; i<nV; i++)
  {
    (*ivM)[i] = (*ivstart)[i];
  }
  for(i=0; i<nV; i++)
  {
    (*ivM)[nV+i] = 1;
  }
  for(i=2; i<nV; i++)
  {
    (*ivM)[(i+1)*nV - i] = -1;
  }
  return(ivM);
}

//unused
#if 0
static intvec* MatrixOrderdp(int nV)
{
  int i;
  intvec* ivM = new intvec(nV*nV);

  for(i=0; i<nV; i++)
  {
    (*ivM)[i] = 1;
  }
  for(i=1; i<nV; i++)
  {
    (*ivM)[(i+1)*nV - i] = -1;
  }
  return(ivM);
}
#endif

intvec* MivUnit(int nV)
{
  int i;
  intvec* ivM = new intvec(nV);
  for(i=nV-1; i>=0; i--)
  {
    (*ivM)[i] = 1;
  }
  return(ivM);
}


/************************************************************************
*  compute a perturbed weight vector of a matrix order w.r.t. an ideal  *
*************************************************************************/
int Xnlev;
intvec* Mfpertvector(ideal G, intvec* ivtarget)
{
  int i, j, nG = IDELEMS(G);
  int nV = currRing->N;
  int niv = nV*nV;


  // Calculate maxA = Max(A2) + Max(A3) + ... + Max(AnV),
  // where the Ai are the i-te rows of the matrix 'targer_ord'.
  int ntemp, maxAi, maxA=0;
  for(i=1; i<nV; i++)
  {
    maxAi = (*ivtarget)[i*nV];
    if(maxAi<0)
    {
      maxAi = -maxAi;
    }
    for(j=i*nV+1; j<(i+1)*nV; j++)
    {
      ntemp = (*ivtarget)[j];
      if(ntemp < 0)
      {
        ntemp = -ntemp;
      }
      if(ntemp > maxAi)
      {
        maxAi = ntemp;
      }
    }
    maxA = maxA + maxAi;
  }
  intvec* ivUnit = Mivdp(nV);

  // Calculate inveps = 1/eps, where 1/eps > deg(p)*maxA for all p in G.
  mpz_t tot_deg; mpz_init(tot_deg);
  mpz_t maxdeg; mpz_init(maxdeg);
  mpz_t inveps; mpz_init(inveps);


  for(i=nG-1; i>=0; i--)
  {
    mpz_set_ui(maxdeg, MwalkWeightDegree(G->m[i], ivUnit));
    if (mpz_cmp(maxdeg,  tot_deg) > 0 )
    {
      mpz_set(tot_deg, maxdeg);
    }
  }

  delete ivUnit;
  //inveps = (tot_deg * maxA) + 1;
  mpz_mul_ui(inveps, tot_deg, maxA);
  mpz_add_ui(inveps, inveps, 1);

  // takes  "small" inveps
#ifdef INVEPS_SMALL_IN_FRACTAL
  if(mpz_cmp_ui(inveps, nV)>0 && nV > 3)
  {
    mpz_cdiv_q_ui(inveps, inveps, nV);
  }
  //PrintS("\n// choose the \"small\" inverse epsilon!");
#endif

  // PrintLn();  mpz_out_str(stdout, 10, inveps);

  // Calculate the perturbed target orders:
  mpz_t *ivtemp=(mpz_t *)omAlloc(nV*sizeof(mpz_t));
  mpz_t *pert_vector=(mpz_t *)omAlloc(niv*sizeof(mpz_t));

  for(i=0; i < nV; i++)
  {
    mpz_init_set_si(ivtemp[i], (*ivtarget)[i]);
    mpz_init_set_si(pert_vector[i], (*ivtarget)[i]);
  }

  mpz_t ztmp; mpz_init(ztmp);
  // BOOLEAN isneg = FALSE;

  for(i=1; i<nV; i++)
  {
    for(j=0; j<nV; j++)
    {
      mpz_mul(ztmp, inveps, ivtemp[j]);
      if((*ivtarget)[i*nV+j]<0)
      {
        mpz_sub_ui(ivtemp[j], ztmp, -(*ivtarget)[i*nV+j]);
      }
      else
      {
        mpz_add_ui(ivtemp[j], ztmp,(*ivtarget)[i*nV+j]);
      }
    }

    for(j=0; j<nV; j++)
      {
      mpz_init_set(pert_vector[i*nV+j],ivtemp[j]);
      }
  }

  /* 2147483647 is max. integer representation in SINGULAR */
  mpz_t sing_int;
  mpz_init_set_ui(sing_int,  2147483647);

  intvec* result = new intvec(niv);
  intvec* result1 = new intvec(niv);
  BOOLEAN nflow = FALSE;

  // computes gcd
  mpz_set(ztmp, pert_vector[0]);
  for(i=0; i<niv; i++)
  {
    mpz_gcd(ztmp, ztmp, pert_vector[i]);
    if(mpz_cmp_si(ztmp, 1)==0)
    {
      break;
    }
  }

  for(i=0; i<niv; i++)
  {
    mpz_divexact(pert_vector[i], pert_vector[i], ztmp);
    (* result)[i] = mpz_get_si(pert_vector[i]);
  }

  j = 0;
  for(i=0; i<nV; i++)
  {
    (* result1)[i] = mpz_get_si(pert_vector[i]);
    (* result1)[i] = 0.1*(* result1)[i];
    (* result1)[i] = floor((* result1)[i] + 0.5);
    if((* result1)[i] == 0)
    {
      j++;
    }
  }
  if(j > nV - 1)
  {
    // Print("\n//  MfPertwalk: geaenderter vector gleich Null! \n");
    delete result1;
    goto CHECK_OVERFLOW;
  }

// check that the perturbed weight vector lies in the Groebner cone
  if(test_w_in_ConeCC(G,result1) != 0)
  {
    // Print("\n//  MfPertwalk: geaenderter vector liegt in Groebnerkegel! \n");
    delete result;
    result = result1;
    for(i=0; i<nV; i++)
    {
      mpz_set_si(pert_vector[i], (*result1)[i]);
    }
  }
  else
  {
    delete result1;
    // Print("\n// Mfpertwalk: geaenderter vector liegt nicht in Groebnerkegel! \n");
  }

  CHECK_OVERFLOW:

  for(i=0; i<niv; i++)
  {
    if(mpz_cmp(pert_vector[i], sing_int)>0)
    {
      if(nflow == FALSE)
      {
        Xnlev = i / nV;
        nflow = TRUE;
        Overflow_Error = TRUE;
        Print("\n// Xlev = %d and the %d-th element is", Xnlev,  i+1);
        PrintS("\n// ** OVERFLOW in \"Mfpertvector\": ");
        mpz_out_str( stdout, 10, pert_vector[i]);
        PrintS(" is greater than 2147483647 (max. integer representation)");
        Print("\n//  So vector[%d] := %d is wrong!!", i+1, (*result)[i]);
      }
    }
  }
  if(Overflow_Error == TRUE)
  {
    ivString(result, "new_vector");
  }
  omFree(pert_vector);
  omFree(ivtemp);
  mpz_clear(ztmp);
  mpz_clear(tot_deg);
  mpz_clear(maxdeg);
  mpz_clear(inveps);
  mpz_clear(sing_int);

  rComplete(currRing);
  for(j=0; j<IDELEMS(G); j++)
  {
    poly p=G->m[j];
    while(p!=NULL)
    {
      p_Setm(p,currRing); pIter(p);
    }
  }
  return result;
}

/****************************************************************
 * Multiplication of two ideals element by element              *
 * i.e. Let be A := (a_i) and B := (b_i), return C := (a_i*b_i) *
 *  destroy A, keeps B                                          *
 ****************************************************************/
static ideal MidMult(ideal A, ideal B)
{
  int mA = IDELEMS(A), mB = IDELEMS(B);

  if(A==NULL || B==NULL)
  {
    return NULL;
  }
  if(mB < mA)
  {
    mA = mB;
  }
  ideal result = idInit(mA, 1);

  int i, k=0;
  for(i=0; i<mA; i++)
    {
      result->m[k] = pMult(A->m[i], pCopy(B->m[i]));
      A->m[i]=NULL;
      if (result->m[k]!=NULL)
      {
        k++;
      }
    }

  idDelete(&A);
  idSkipZeroes(result);
  return result;
}

/*********************************************************************
 * G is a red. Groebner basis w.r.t. <_1                             *
 * Gomega is an initial form ideal of <G> w.r.t. a weight vector w   *
 * M is a subideal of <Gomega> and M selft is a red. Groebner basis  *
 *    of the ideal <Gomega> w.r.t. <_w                               *
 * Let m_i = h1.gw1 + ... + hs.gws for each m_i in M; gwi in Gomega  *
 * return F with n(F) = n(M) and f_i = h1.g1 + ... + hs.gs for each i*
 ********************************************************************/
static ideal MLifttwoIdeal(ideal Gw, ideal M, ideal G)
{
  ideal Mtmp = idLift(Gw, M, NULL, FALSE, TRUE, TRUE, NULL);

  // If Gw is a GB, then isSB = TRUE, otherwise FALSE
  // So, it is better, if one tests whether Gw is a GB
  // in ideals.cc:
  // idLift (ideal mod, ideal submod,ideal * rest, BOOLEAN goodShape,
  //           BOOLEAN isSB,BOOLEAN divide,matrix * unit)

  // Let be Mtmp = {m1,...,ms}, where mi=sum hij.in_gj, for all i=1,...,s
  // We compute F = {f1,...,fs}, where fi=sum hij.gj
  int i, j, nM = IDELEMS(Mtmp);
  ideal idpol, idLG;
  ideal F = idInit(nM, 1);

  for(i=0; i<nM; i++)
  {
     idpol = idVec2Ideal(Mtmp->m[i]);
     idLG = MidMult(idpol, G);
     idpol = NULL;
     F->m[i] = NULL;
     for(j=IDELEMS(idLG)-1; j>=0; j--)
     {
       F->m[i] = pAdd(F->m[i], idLG->m[j]);
       idLG->m[j]=NULL;
     }
     idDelete(&idLG);
  }
  idDelete(&Mtmp);
  return F;
}

//unused
#if 0
static void checkidealCC(ideal G, char* Ch)
{
  int i,nmon=0,ntmp;
  int nG = IDELEMS(G);
  int n = nG-1;
  Print("\n//** Ideal %s besteht aus %d Polynomen mit ", Ch, nG);

  for(i=0; i<nG; i++)
  {
    ntmp =  pLength(G->m[i]);
    nmon += ntmp;

    if(i != n)
    {
      Print("%d, ", ntmp);
    }
    else
    {
      Print(" bzw. %d ", ntmp);
    }
  }
  PrintS(" Monomen.\n");
  Print("//** %s besitzt %d Monome.", Ch, nmon);
  PrintLn();
}
#endif

//unused
#if 0
static void HeadidString(ideal L, char* st)
{
  int i, nL = IDELEMS(L)-1;

  Print("//  The head terms of the ideal %s = ", st);
  for(i=0; i<nL; i++)
  {
    Print(" %s, ", pString(pHead(L->m[i])));
  }
  Print(" %s;\n", pString(pHead(L->m[nL])));
}
#endif

static inline int MivComp(intvec* iva, intvec* ivb)
{
  assume(iva->length() == ivb->length());
  int i;
  for(i=iva->length()-1; i>=0; i--)
  {
    if((*iva)[i] - (*ivb)[i] != 0)
    {
      return 0;
    }
  }
  return 1;
}

/**********************************************
 * Look for the smallest absolut value in vec *
 **********************************************/
static int MivAbsMax(intvec* vec)
{
  int i,k;
  if((*vec)[0] < 0)
  {
    k = -(*vec)[0];
  }
  else
  {
    k = (*vec)[0];
  }
  for(i=1; i < (vec->length()); i++)
  {
    if((*vec)[i] < 0)
    {
      if(-(*vec)[i] > k)
      {
        k = -(*vec)[i];
      }
    }
    else
    {
      if((*vec)[i] > k)
      {
        k = (*vec)[i];
      }
    }
  }
  return k;
}

/**********************************************************************
 * Compute a next weight vector between curr_weight and target_weight *
 * with respect to an ideal <G>.                                      *
**********************************************************************/
static intvec* MwalkNextWeightCC(intvec* curr_weight, intvec* target_weight,
                                 ideal G)
{
  BOOLEAN nError = Overflow_Error;
  Overflow_Error = FALSE;

  assume(currRing != NULL && curr_weight != NULL &&
         target_weight != NULL && G != NULL);

  int nRing = currRing->N;
  int checkRed, j, kkk, nG = IDELEMS(G);
  intvec* ivtemp;

  mpz_t t_zaehler, t_nenner;
  mpz_init(t_zaehler);
  mpz_init(t_nenner);

  mpz_t s_zaehler, s_nenner, temp, MwWd;
  mpz_init(s_zaehler);
  mpz_init(s_nenner);
  mpz_init(temp);
  mpz_init(MwWd);

  mpz_t sing_int;
  mpz_init(sing_int);
  mpz_set_si(sing_int,  2147483647);

  mpz_t sing_int_half;
  mpz_init(sing_int_half);
  mpz_set_si(sing_int_half,  3*(1073741824/2));

  mpz_t deg_w0_p1, deg_d0_p1;
  mpz_init(deg_w0_p1);
  mpz_init(deg_d0_p1);

  mpz_t sztn, sntz;
  mpz_init(sztn);
  mpz_init(sntz);

  mpz_t t_null;
  mpz_init(t_null);

  mpz_t ggt;
  mpz_init(ggt);

  mpz_t dcw;
  mpz_init(dcw);

  //int tn0, tn1, tz1, ncmp, gcd_tmp, ntmp;
  int gcd_tmp;
  intvec* diff_weight = MivSub(target_weight, curr_weight);

  intvec* diff_weight1 = MivSub(target_weight, curr_weight);
  poly g;
  //poly g, gw;
  for (j=0; j<nG; j++)
  {
    g = G->m[j];
    if (g != NULL)
    {
      ivtemp = MExpPol(g);
      mpz_set_si(deg_w0_p1, MivDotProduct(ivtemp, curr_weight));
      mpz_set_si(deg_d0_p1, MivDotProduct(ivtemp, diff_weight));
      delete ivtemp;

      pIter(g);
      while (g != NULL)
      {
        ivtemp = MExpPol(g);
        mpz_set_si(MwWd, MivDotProduct(ivtemp, curr_weight));
        mpz_sub(s_zaehler, deg_w0_p1, MwWd);

        if(mpz_cmp(s_zaehler, t_null) != 0)
        {
          mpz_set_si(MwWd, MivDotProduct(ivtemp, diff_weight));
          mpz_sub(s_nenner, MwWd, deg_d0_p1);

          // check for 0 < s <= 1
          if( (mpz_cmp(s_zaehler,t_null) > 0 &&
               mpz_cmp(s_nenner, s_zaehler)>=0) ||
              (mpz_cmp(s_zaehler, t_null) < 0 &&
               mpz_cmp(s_nenner, s_zaehler)<=0))
          {
            // make both positive
            if (mpz_cmp(s_zaehler, t_null) < 0)
            {
              mpz_neg(s_zaehler, s_zaehler);
              mpz_neg(s_nenner, s_nenner);
            }

            //compute a simple fraction of s
            cancel(s_zaehler, s_nenner);

            if(mpz_cmp(t_nenner, t_null) != 0)
            {
              mpz_mul(sztn, s_zaehler, t_nenner);
              mpz_mul(sntz, s_nenner, t_zaehler);

              if(mpz_cmp(sztn,sntz) < 0)
              {
                mpz_add(t_nenner, t_null, s_nenner);
                mpz_add(t_zaehler,t_null, s_zaehler);
              }
            }
            else
            {
              mpz_add(t_nenner, t_null, s_nenner);
              mpz_add(t_zaehler,t_null, s_zaehler);
            }
          }
        }
        pIter(g);
        delete ivtemp;
      }
    }
  }
//Print("\n// Alloc Size = %d \n", nRing*sizeof(mpz_t));
  mpz_t *vec=(mpz_t*)omAlloc(nRing*sizeof(mpz_t));


  // there is no 0<t<1 and define the next weight vector that is equal to the current weight vector
  if(mpz_cmp(t_nenner, t_null) == 0)
  {
    #ifndef SING_NDEBUG
    Print("\n//MwalkNextWeightCC: t_nenner ist Null!");
    #endif
    delete diff_weight;
    diff_weight = ivCopy(curr_weight);//take memory
    goto FINISH;
  }

  // define the target vector as the next weight vector, if t = 1
  if(mpz_cmp_si(t_nenner, 1)==0 && mpz_cmp_si(t_zaehler,1)==0)
  {
    delete diff_weight;
    diff_weight = ivCopy(target_weight); //this takes memory
    goto FINISH;
  }

   checkRed = 0;

  SIMPLIFY_GCD:

  // simplify the vectors curr_weight and diff_weight (C-int)
  gcd_tmp = (*curr_weight)[0];

  for (j=1; j<nRing; j++)
  {
    gcd_tmp = gcd(gcd_tmp, (*curr_weight)[j]);
    if(gcd_tmp == 1)
    {
      break;
    }
  }
  if(gcd_tmp != 1)
  {
    for (j=0; j<nRing; j++)
    {
      gcd_tmp = gcd(gcd_tmp, (*diff_weight)[j]);
      if(gcd_tmp == 1)
      {
        break;
      }
    }
  }
  if(gcd_tmp != 1)
  {
    for (j=0; j<nRing; j++)
    {
      (*curr_weight)[j] =  (*curr_weight)[j]/gcd_tmp;
      (*diff_weight)[j] =  (*diff_weight)[j]/gcd_tmp;
    }
  }
  if(checkRed > 0)
  {
    for (j=0; j<nRing; j++)
    {
      mpz_set_si(vec[j], (*diff_weight)[j]);
    }
    goto TEST_OVERFLOW;
  }

#ifdef  NEXT_VECTORS_CC
  Print("\n// gcd of the weight vectors (current and target) = %d", gcd_tmp);
  ivString(curr_weight, "new cw");
  ivString(diff_weight, "new dw");

  PrintS("\n// t_zaehler: ");  mpz_out_str( stdout, 10, t_zaehler);
  PrintS(", t_nenner: ");  mpz_out_str( stdout, 10, t_nenner);
#endif

  //  BOOLEAN isdwpos;

  // construct a new weight vector
  for (j=0; j<nRing; j++)
  {
    mpz_set_si(dcw, (*curr_weight)[j]);
    mpz_mul(s_nenner, t_nenner, dcw);

    if( (*diff_weight)[j]>0)
    {
      mpz_mul_ui(s_zaehler, t_zaehler, (*diff_weight)[j]);
    }
    else
    {
      mpz_mul_ui(s_zaehler, t_zaehler, -(*diff_weight)[j]);
      mpz_neg(s_zaehler, s_zaehler);
    }
    mpz_add(sntz, s_nenner, s_zaehler);
    mpz_init_set(vec[j], sntz);

#ifdef NEXT_VECTORS_CC
    Print("\n//   j = %d ==> ", j);
    PrintS("(");
    mpz_out_str( stdout, 10, t_nenner);
    Print(" * %d)", (*curr_weight)[j]);
    Print(" + ("); mpz_out_str( stdout, 10, t_zaehler);
    Print(" * %d) =  ",  (*diff_weight)[j]);
    mpz_out_str( stdout, 10, s_nenner);
    PrintS(" + ");
    mpz_out_str( stdout, 10, s_zaehler);
    PrintS(" = "); mpz_out_str( stdout, 10, sntz);
    Print(" ==> vector[%d]: ", j); mpz_out_str(stdout, 10, vec[j]);
#endif

    if(j==0)
    {
      mpz_set(ggt, sntz);
    }
    else
    {
      if(mpz_cmp_si(ggt,1) != 0)
      {
        mpz_gcd(ggt, ggt, sntz);
      }
    }
  }

#ifdef  NEXT_VECTORS_CC
  PrintS("\n// gcd of elements of the vector: ");
  mpz_out_str( stdout, 10, ggt);
#endif

/**********************************************************************
* construct a new weight vector and check whether vec[j] is overflow, *
* i.e. vec[j] > 2^31.                                                 *
* If vec[j] doesn't overflow, define a weight vector. Otherwise,      *
* report that overflow appears. In the second case, test whether the  *
* the correctness of the new vector plays an important role           *
**********************************************************************/
  kkk=0;
  for(j=0; j<nRing; j++)
    {
    if(mpz_cmp(vec[j], sing_int_half) >= 0)
      {
      goto REDUCTION;
      }
    }
  checkRed = 1;
  for (j=0; j<nRing; j++)
    {
      (*diff_weight)[j] = mpz_get_si(vec[j]);
    }
  goto SIMPLIFY_GCD;

  REDUCTION:
  for (j=0; j<nRing; j++)
  {
    (*diff_weight)[j] = mpz_get_si(vec[j]);
  }
  while(MivAbsMax(diff_weight) >= 5)
  {
    for (j=0; j<nRing; j++)
    {
      if(mpz_cmp_si(ggt,1)==0)
      {
        (*diff_weight1)[j] = floor(0.1*(*diff_weight)[j] + 0.5);
        // Print("\n//  vector[%d] = %d \n",j+1, (*diff_weight1)[j]);
      }
      else
      {
        mpz_divexact(vec[j], vec[j], ggt);
        (*diff_weight1)[j] = floor(0.1*(*diff_weight)[j] + 0.5);
        // Print("\n//  vector[%d] = %d \n",j+1, (*diff_weight1)[j]);
      }
/*
      if((*diff_weight1)[j] == 0)
      {
        kkk = kkk + 1;
      }
*/
    }


/*
    if(kkk > nRing - 1)
    {
      // diff_weight was reduced to zero
      // Print("\n // MwalkNextWeightCC: geaenderter Vector gleich Null! \n");
      goto TEST_OVERFLOW;
    }
*/

    if(test_w_in_ConeCC(G,diff_weight1) != 0)
    {
      Print("\n// MwalkNextWeightCC: geaenderter vector liegt in Groebnerkegel! \n");
      for (j=0; j<nRing; j++)
      {
        (*diff_weight)[j] = (*diff_weight1)[j];
      }
      if(MivAbsMax(diff_weight) < 5)
      {
        checkRed = 1;
        goto SIMPLIFY_GCD;
      }
    }
    else
    {
      // Print("\n// MwalkNextWeightCC: geaenderter vector liegt nicht in Groebnerkegel! \n");
      break;
    }
  }

 TEST_OVERFLOW:

  for (j=0; j<nRing; j++)
  {
    if(mpz_cmp(vec[j], sing_int)>=0)
    {
      if(Overflow_Error == FALSE)
      {
        Overflow_Error = TRUE;
        PrintS("\n// ** OVERFLOW in \"MwalkNextWeightCC\": ");
        mpz_out_str( stdout, 10, vec[j]);
        PrintS(" is greater than 2147483647 (max. integer representation)\n");
        //Print("//  So vector[%d] := %d is wrong!!\n",j+1, vec[j]);// vec[j] is mpz_t
      }
    }
  }

 FINISH:
   delete diff_weight1;
   mpz_clear(t_zaehler);
   mpz_clear(t_nenner);
   mpz_clear(s_zaehler);
   mpz_clear(s_nenner);
   mpz_clear(sntz);
   mpz_clear(sztn);
   mpz_clear(temp);
   mpz_clear(MwWd);
   mpz_clear(deg_w0_p1);
   mpz_clear(deg_d0_p1);
   mpz_clear(ggt);
   omFree(vec);
   mpz_clear(sing_int_half);
   mpz_clear(sing_int);
   mpz_clear(dcw);
   mpz_clear(t_null);



  if(Overflow_Error == FALSE)
  {
    Overflow_Error = nError;
  }
 rComplete(currRing);
  for(kkk=0; kkk<IDELEMS(G);kkk++)
  {
    poly p=G->m[kkk];
    while(p!=NULL)
    {
      p_Setm(p,currRing);
      pIter(p);
    }
  }
return diff_weight;
}

/**********************************************************************
* Compute an intermediate weight vector from iva to ivb w.r.t.        *
* the reduced Groebner basis G.                                       *
* Return NULL, if it is equal to iva or iva = avb.                    *
**********************************************************************/
intvec* MkInterRedNextWeight(intvec* iva, intvec* ivb, ideal G)
{
  intvec* tmp = new intvec(iva->length());
  intvec* result;

  if(G == NULL)
  {
    return tmp;
  }
  if(MivComp(iva, ivb) == 1)
  {
    return tmp;
  }
  result = MwalkNextWeightCC(iva, ivb, G);

  if(MivComp(result, iva) == 1)
  {
    delete result;
    return tmp;
  }

  delete tmp;
  return result;
}

/**************************************************************
 * define and execute a new ring which order is (a(vb),a(va),lp,C)  *
 * ************************************************************/
#if 0
// unused
static void VMrHomogeneous(intvec* va, intvec* vb)
{

  if ((currRing->ppNoether)!=NULL)
  {
    pDelete(&(currRing->ppNoether));
  }
  if (((sLastPrinted.rtyp>BEGIN_RING) && (sLastPrinted.rtyp<END_RING)) ||
      ((sLastPrinted.rtyp==LIST_CMD)&&(lRingDependend((lists)sLastPrinted.data))))
  {
    sLastPrinted.CleanUp();
  }

  ring r = (ring) omAlloc0Bin(sip_sring_bin);
  int i, nv = currRing->N;

  r->cf  = currRing->cf;
  r->N   = currRing->N;
  int nb = 4;


  //names
  char* Q; // In order to avoid the corrupted memory, do not change.
  r->names = (char **) omAlloc0(nv * sizeof(char_ptr));
  for(i=0; i<nv; i++)
  {
    Q = currRing->names[i];
    r->names[i]  = omStrDup(Q);
  }

  //weights: entries for 3 blocks: NULL Made:???
  r->wvhdl = (int **)omAlloc0(nb * sizeof(int_ptr));
  r->wvhdl[0] = (int*) omAlloc(nv*sizeof(int));
  r->wvhdl[1] = (int*) omAlloc((nv-1)*sizeof(int));

  for(i=0; i<nv-1; i++)
  {
    r->wvhdl[1][i] = (*vb)[i];
    r->wvhdl[0][i] = (*va)[i];
  }
  r->wvhdl[0][nv] = (*va)[nv];

  // order: (1..1),a,lp,C
  r->order = (int *) omAlloc(nb * sizeof(int *));
  r->block0 = (int *)omAlloc0(nb * sizeof(int *));
  r->block1 = (int *)omAlloc0(nb * sizeof(int *));

  // ringorder a for the first block: var 1..nv
  r->order[0]  = ringorder_a;
  r->block0[0] = 1;
  r->block1[0] = nv;

 // ringorder a for the second block: var 2..nv
  r->order[1]  = ringorder_a;
  r->block0[1] = 2;
  r->block1[1] = nv;

  // ringorder lp for the third block: var 2..nv
  r->order[2]  = ringorder_lp;
  r->block0[2] = 2;
  r->block1[2] = nv;

  // ringorder C for the 4th block
  // it is very important within "idLift",
  // especially, by ring syz_ring=rCurrRingAssure_SyzComp();
  // therefore, nb  must be (nBlocks(currRing)  + 1)
  r->order[3]  = ringorder_C;

  // polynomial ring
  r->OrdSgn    = 1;

  // complete ring intializations

  rComplete(r);

  rChangeCurrRing(r);
}
#endif

/**************************************************************
 * define and execute a new ring which order is (a(va),lp,C)  *
 * ************************************************************/
static ring VMrDefault(intvec* va)
{

  if ((currRing->ppNoether)!=NULL)
  {
    pDelete(&(currRing->ppNoether));
  }
  if (((sLastPrinted.rtyp>BEGIN_RING) && (sLastPrinted.rtyp<END_RING)) ||
      ((sLastPrinted.rtyp==LIST_CMD)&&(lRingDependend((lists)sLastPrinted.data))))
  {
    sLastPrinted.CleanUp();
  }

  ring r = (ring) omAlloc0Bin(sip_sring_bin);
  int i, nv = currRing->N;

  r->cf  = currRing->cf;
  r->N   = currRing->N;

  int nb = 4;

  //names
  char* Q; // In order to avoid the corrupted memory, do not change.
  r->names = (char **) omAlloc0(nv * sizeof(char_ptr));
  for(i=0; i<nv; i++)
  {
    Q = currRing->names[i];
    r->names[i]  = omStrDup(Q);
  }

  /*weights: entries for 3 blocks: NULL Made:???*/
  r->wvhdl = (int **)omAlloc0(nb * sizeof(int_ptr));
  r->wvhdl[0] = (int*) omAlloc(nv*sizeof(int));
  for(i=0; i<nv; i++)
    r->wvhdl[0][i] = (*va)[i];

  /* order: a,lp,C,0 */
  r->order = (int *) omAlloc(nb * sizeof(int *));
  r->block0 = (int *)omAlloc0(nb * sizeof(int *));
  r->block1 = (int *)omAlloc0(nb * sizeof(int *));

  // ringorder a for the first block: var 1..nv
  r->order[0]  = ringorder_a;
  r->block0[0] = 1;
  r->block1[0] = nv;

  // ringorder lp for the second block: var 1..nv
  r->order[1]  = ringorder_lp;
  r->block0[1] = 1;
  r->block1[1] = nv;

  // ringorder C for the third block
  // it is very important within "idLift",
  // especially, by ring syz_ring=rCurrRingAssure_SyzComp();
  // therefore, nb  must be (nBlocks(currRing)  + 1)
  r->order[2]  = ringorder_C;

  // the last block: everything is 0
  r->order[3]  = 0;

  // polynomial ring
  r->OrdSgn    = 1;

  // complete ring intializations

  rComplete(r);
  return r;
  //rChangeCurrRing(r);
}

static ring VMrDefault1(intvec* va)
{

  if ((currRing->ppNoether)!=NULL)
  {
    pDelete(&(currRing->ppNoether));
  }
  if (((sLastPrinted.rtyp>BEGIN_RING) && (sLastPrinted.rtyp<END_RING)) ||
      ((sLastPrinted.rtyp==LIST_CMD)&&(lRingDependend((lists)sLastPrinted.data))))
  {
    sLastPrinted.CleanUp();
  }

  ring r = (ring) omAlloc0Bin(sip_sring_bin);
  int i, nv = currRing->N;

  r->cf  = currRing->cf;
  r->N   = currRing->N;

  int nb = 4;

  //names
  char* Q; // In order to avoid the corrupted memory, do not change.
  r->names = (char **) omAlloc0(nv * sizeof(char_ptr));
  for(i=0; i<nv; i++)
  {
    Q = currRing->names[i];
    r->names[i]  = omStrDup(Q);
  }

  /*weights: entries for 3 blocks: NULL Made:???*/
  r->wvhdl = (int **)omAlloc0(nb * sizeof(int_ptr));
  r->wvhdl[0] = (int*) omAlloc(nv*sizeof(int));
  for(i=0; i<nv; i++)
    r->wvhdl[0][i] = (*va)[i];

  /* order: a,lp,C,0 */
  r->order = (int *) omAlloc(nb * sizeof(int *));
  r->block0 = (int *)omAlloc0(nb * sizeof(int *));
  r->block1 = (int *)omAlloc0(nb * sizeof(int *));

  // ringorder a for the first block: var 1..nv
  r->order[0]  = ringorder_a;
  r->block0[0] = 1;
  r->block1[0] = nv;

  // ringorder lp for the second block: var 1..nv
  r->order[1]  = ringorder_lp;
  r->block0[1] = 1;
  r->block1[1] = nv;

  // ringorder C for the third block
  // it is very important within "idLift",
  // especially, by ring syz_ring=rCurrRingAssure_SyzComp();
  // therefore, nb  must be (nBlocks(currRing)  + 1)
  r->order[2]  = ringorder_C;

  // the last block: everything is 0
  r->order[3]  = 0;

  // polynomial ring
  r->OrdSgn    = 1;

  // complete ring intializations

  rComplete(r);

  //rChangeCurrRing(r);
  return r;
}

/****************************************************************
 * define and execute a new ring with ordering (a(va),Wp(vb),C) *
 * **************************************************************/

static ring VMrRefine(intvec* va, intvec* vb)
{

  if ((currRing->ppNoether)!=NULL)
  {
    pDelete(&(currRing->ppNoether));
  }
  if (((sLastPrinted.rtyp>BEGIN_RING) && (sLastPrinted.rtyp<END_RING)) ||
      ((sLastPrinted.rtyp==LIST_CMD)&&(lRingDependend((lists)sLastPrinted.data))))
  {
    sLastPrinted.CleanUp();
  }

  ring r = (ring) omAlloc0Bin(sip_sring_bin);
  int i, nv = currRing->N;

  r->cf  = currRing->cf;
  r->N   = currRing->N;
  //int nb = nBlocks(currRing)  + 1;
  int nb = 4;

  //names
  char* Q; // In order to avoid the corrupted memory, do not change.
  r->names = (char **) omAlloc0(nv * sizeof(char_ptr));
  for(i=0; i<nv; i++)
  {
    Q = currRing->names[i];
    r->names[i]  = omStrDup(Q);
  }

  //weights: entries for 3 blocks: NULL Made:???
  r->wvhdl = (int **)omAlloc0(nb * sizeof(int_ptr));
  r->wvhdl[0] = (int*) omAlloc(nv*sizeof(int));
  r->wvhdl[1] = (int*) omAlloc(nv*sizeof(int));

  for(i=0; i<nv; i++)
  {
    r->wvhdl[0][i] = (*va)[i];
    r->wvhdl[1][i] = (*vb)[i];
  }
  r->wvhdl[2]=NULL;
  r->wvhdl[3]=NULL;

  // order: (1..1),a,lp,C
  r->order = (int *) omAlloc(nb * sizeof(int *));
  r->block0 = (int *)omAlloc0(nb * sizeof(int *));
  r->block1 = (int *)omAlloc0(nb * sizeof(int *));

  // ringorder a for the first block: var 1..nv
  r->order[0]  = ringorder_a;
  r->block0[0] = 1;
  r->block1[0] = nv;

 // ringorder Wp for the second block: var 1..nv
  r->order[1]  = ringorder_Wp;
  r->block0[1] = 1;
  r->block1[1] = nv;

  // ringorder lp for the third block: var 1..nv
  r->order[2]  = ringorder_C;
  r->block0[2] = 1;
  r->block1[2] = nv;

  // ringorder C for the 4th block
  // it is very important within "idLift",
  // especially, by ring syz_ring=rCurrRingAssure_SyzComp();
  // therefore, nb  must be (nBlocks(currRing)  + 1)
  r->order[3]  = 0;

  // polynomial ring
  r->OrdSgn    = 1;

  // complete ring intializations

  rComplete(r);

  //rChangeCurrRing(r);
  return r;
}

/*****************************************************
 * define and execute a new ring with ordering (M,C) *
 *****************************************************/
static ring VMatrDefault(intvec* va)
{

  if ((currRing->ppNoether)!=NULL)
  {
    pDelete(&(currRing->ppNoether));
  }
  if (((sLastPrinted.rtyp>BEGIN_RING) && (sLastPrinted.rtyp<END_RING)) ||
      ((sLastPrinted.rtyp==LIST_CMD)&&(lRingDependend((lists)sLastPrinted.data))))
  {
    sLastPrinted.CleanUp();
  }

  ring r = (ring) omAlloc0Bin(sip_sring_bin);
  int i, nv = currRing->N;

  r->cf  = currRing->cf;
  r->N   = currRing->N;

  int nb = 4;

  //names
  char* Q; // In order to avoid the corrupted memory, do not change.
  r->names = (char **) omAlloc0(nv * sizeof(char_ptr));
  for(i=0; i<nv; i++)
  {
    Q = currRing->names[i];
    r->names[i]  = omStrDup(Q);
  }

  /*weights: entries for 3 blocks: NULL Made:???*/
  r->wvhdl = (int **)omAlloc0(nb * sizeof(int_ptr));
  r->wvhdl[0] = (int*) omAlloc(nv*nv*sizeof(int));
  r->wvhdl[1] =NULL; // (int*) omAlloc(nv*sizeof(int));
  r->wvhdl[2]=NULL;
  r->wvhdl[3]=NULL;
  for(i=0; i<nv*nv; i++)
    r->wvhdl[0][i] = (*va)[i];

  /* order: a,lp,C,0 */
  r->order = (int *) omAlloc(nb * sizeof(int *));
  r->block0 = (int *)omAlloc0(nb * sizeof(int *));
  r->block1 = (int *)omAlloc0(nb * sizeof(int *));

  // ringorder a for the first block: var 1..nv
  r->order[0]  = ringorder_M;
  r->block0[0] = 1;
  r->block1[0] = nv;

  // ringorder C for the second block
  r->order[1]  = ringorder_C;
  r->block0[1] = 1;
  r->block1[1] = nv;


// ringorder C for the third block: var 1..nv
  r->order[2]  = ringorder_C;
  r->block0[2] = 1;
  r->block1[2] = nv;


  // the last block: everything is 0
  r->order[3]  = 0;

  // polynomial ring
  r->OrdSgn    = 1;

  // complete ring intializations

  rComplete(r);

  //rChangeCurrRing(r);
  return r;
}

/***********************************************************
 * define and execute a new ring with ordering (a(vb),M,C) *
 ***********************************************************/
static ring VMatrRefine(intvec* va, intvec* vb)
{

  if ((currRing->ppNoether)!=NULL)
  {
    pDelete(&(currRing->ppNoether));
  }
  if (((sLastPrinted.rtyp>BEGIN_RING) && (sLastPrinted.rtyp<END_RING)) ||
      ((sLastPrinted.rtyp==LIST_CMD)&&(lRingDependend((lists)sLastPrinted.data))))
  {
    sLastPrinted.CleanUp();
  }

  ring r = (ring) omAlloc0Bin(sip_sring_bin);
  int i, nv = currRing->N;
  int nvs = nv*nv;
  r->cf  = currRing->cf;
  r->N   = currRing->N;

  int nb = 4;

  //names
  char* Q; // In order to avoid the corrupted memory, do not change.
  r->names = (char **) omAlloc0(nv * sizeof(char_ptr));
  for(i=0; i<nv; i++)
  {
    Q = currRing->names[i];
    r->names[i]  = omStrDup(Q);
  }

  /*weights: entries for 3 blocks: NULL Made:???*/
  r->wvhdl = (int **)omAlloc0(nb * sizeof(int_ptr));
  r->wvhdl[0] = (int*) omAlloc(nv*sizeof(int));
  r->wvhdl[1] = (int*) omAlloc(nvs*sizeof(int));
  r->wvhdl[2]=NULL;
  r->wvhdl[3]=NULL;
  for(i=0; i<nvs; i++)
  {
    r->wvhdl[1][i] = (*va)[i];
  }
  for(i=0; i<nv; i++)
  {
    r->wvhdl[0][i] = (*vb)[i];
  }
  /* order: a,lp,C,0 */
  r->order = (int *) omAlloc(nb * sizeof(int *));
  r->block0 = (int *)omAlloc0(nb * sizeof(int *));
  r->block1 = (int *)omAlloc0(nb * sizeof(int *));

  // ringorder a for the first block: var 1..nv
  r->order[0]  = ringorder_a;
  r->block0[0] = 1;
  r->block1[0] = nv;

  // ringorder M for the second block: var 1..nv
  r->order[1]  = ringorder_M;
  r->block0[1] = 1;
  r->block1[1] = nv;

  // ringorder C for the third block: var 1..nv
  r->order[2]  = ringorder_C;
  r->block0[2] = 1;
  r->block1[2] = nv;


  // the last block: everything is 0
  r->order[3]  = 0;

  // polynomial ring
  r->OrdSgn    = 1;

  // complete ring intializations

  rComplete(r);

  //rChangeCurrRing(r);
  return r;
}

/**********************************************************************
* define and execute a new ring which order is  a lexicographic order *
***********************************************************************/
static void VMrDefaultlp(void)
{

  if ((currRing->ppNoether)!=NULL)
  {
    pDelete(&(currRing->ppNoether));
  }
  if (((sLastPrinted.rtyp>BEGIN_RING) && (sLastPrinted.rtyp<END_RING)) ||
      ((sLastPrinted.rtyp==LIST_CMD)&&(lRingDependend((lists)sLastPrinted.data))))

  {
    sLastPrinted.CleanUp();
  }

  ring r = (ring) omAlloc0Bin(sip_sring_bin);
  int i, nv = currRing->N;

  r->cf  = currRing->cf;
  r->N   = currRing->N;
  int nb = rBlocks(currRing) + 1;

  // names
  char* Q; // to avoid the corrupted memory, do not change!!
  r->names = (char **) omAlloc0(nv * sizeof(char_ptr));
  for(i=0; i<nv; i++)
  {
    Q = currRing->names[i];
    r->names[i]  = omStrDup(Q);
  }

  /*weights: entries for 3 blocks: NULL Made:???*/

  r->wvhdl = (int **)omAlloc0(nb * sizeof(int_ptr));

  /* order: lp,C,0 */
  r->order = (int *) omAlloc(nb * sizeof(int *));
  r->block0 = (int *)omAlloc0(nb * sizeof(int *));
  r->block1 = (int *)omAlloc0(nb * sizeof(int *));

  /* ringorder lp for the first block: var 1..nv */
  r->order[0]  = ringorder_lp;
  r->block0[0] = 1;
  r->block1[0] = nv;

  /* ringorder C for the second block */
  r->order[1]  = ringorder_C;

  /* the last block: everything is 0 */
  r->order[2]  = 0;

  /*polynomial ring*/
  r->OrdSgn    = 1;

  /* complete ring intializations */

  rComplete(r);

  rChangeCurrRing(r);
}


/* define a ring with parameters und change to it */
/* DefRingPar and DefRingParlp corrupt still memory */
static void DefRingPar(intvec* va)
{
  int i, nv = currRing->N;
  int nb = rBlocks(currRing) + 1;

  ring res=(ring)omAllocBin(sip_sring_bin);

  memcpy(res,currRing,sizeof(ip_sring));

  res->VarOffset = NULL;
  res->ref=0;

  res->cf = currRing->cf; currRing->cf->ref++;


  /*weights: entries for 3 blocks: NULL Made:???*/
  res->wvhdl = (int **)omAlloc0(nb * sizeof(int_ptr));
  res->wvhdl[0] = (int*) omAlloc(nv*sizeof(int));
  for(i=0; i<nv; i++)
    res->wvhdl[0][i] = (*va)[i];

  /* order: a,lp,C,0 */

  res->order = (int *) omAlloc(nb * sizeof(int *));
  res->block0 = (int *)omAlloc0(nb * sizeof(int *));
  res->block1 = (int *)omAlloc0(nb * sizeof(int *));

  // ringorder a for the first block: var 1..nv
  res->order[0]  = ringorder_a;
  res->block0[0] = 1;
  res->block1[0] = nv;

  // ringorder lp for the second block: var 1..nv
  res->order[1]  = ringorder_lp;
  res->block0[1] = 1;
  res->block1[1] = nv;

  // ringorder C for the third block
  // it is very important within "idLift",
  // especially, by ring syz_ring=rCurrRingAssure_SyzComp();
  // therefore, nb  must be (nBlocks(currRing)  + 1)
  res->order[2]  = ringorder_C;

  // the last block: everything is 0
  res->order[3]  = 0;

  // polynomial ring
  res->OrdSgn    = 1;


  res->names   = (char **)omAlloc0(nv * sizeof(char_ptr));
  for (i=nv-1; i>=0; i--)
  {
    res->names[i] = omStrDup(currRing->names[i]);
  }
  // complete ring intializations
  rComplete(res);

  // clean up history
  if (sLastPrinted.RingDependend())
  {
    sLastPrinted.CleanUp();
  }


  // execute the created ring
  rChangeCurrRing(res);
}


static void DefRingParlp(void)
{
  int i, nv = currRing->N;

  ring r=(ring)omAllocBin(sip_sring_bin);

  memcpy(r,currRing,sizeof(ip_sring));

  r->VarOffset = NULL;
  r->ref=0;

  r->cf = currRing->cf; currRing->cf->ref++;


  r->cf  = currRing->cf;
  r->N   = currRing->N;
  int nb = rBlocks(currRing) + 1;

  // names
  char* Q;
  r->names = (char **) omAlloc0(nv * sizeof(char_ptr));
  for(i=nv-1; i>=0; i--)
  {
    Q = currRing->names[i];
    r->names[i]  = omStrDup(Q);
  }

  /*weights: entries for 3 blocks: NULL Made:???*/

  r->wvhdl = (int **)omAlloc0(nb * sizeof(int_ptr));

  /* order: lp,C,0 */
  r->order = (int *) omAlloc(nb * sizeof(int *));
  r->block0 = (int *)omAlloc0(nb * sizeof(int *));
  r->block1 = (int *)omAlloc0(nb * sizeof(int *));

  /* ringorder lp for the first block: var 1..nv */
  r->order[0]  = ringorder_lp;
  r->block0[0] = 1;
  r->block1[0] = nv;

  /* ringorder C for the second block */
  r->order[1]  = ringorder_C;

  /* the last block: everything is 0 */
  r->order[2]  = 0;

  /*polynomial ring*/
  r->OrdSgn    = 1;


//   if (rParameter(currRing)!=NULL)
//   {
//     r->cf->extRing->qideal->m[0]=p_Copy(currRing->cf->extRing->qideal->m[0], currRing->cf->extRing);
//     int l=rPar(currRing);
//     r->cf->extRing->names=(char **)omAlloc(l*sizeof(char_ptr));
//
//     for(i=l-1;i>=0;i--)
//     {
//       rParameter(r)[i]=omStrDup(rParameter(currRing)[i]);
//     }
//   }

  // complete ring intializations

  rComplete(r);

  // clean up history
  if (sLastPrinted.RingDependend())
  {
    sLastPrinted.CleanUp();
  }

  // execute the created ring
  rChangeCurrRing(r);
}

//unused
/**************************************************************
 * check wheather one or more components of a vector are zero *
 **************************************************************/
#if 0
static int isNolVector(intvec* hilb)
{
  int i;
  for(i=hilb->length()-1; i>=0; i--)
  {
    if((* hilb)[i]==0)
    {
      return 1;
    }
  }
  return 0;
}
#endif

/******************************  Februar 2002  ****************************
 * G is a Groebner basis w.r.t. (a(curr_weight),lp) and                   *
 * we compute a GB of <G> w.r.t. the lex. order by the perturbation walk  *
 * its perturbation degree is tp_deg                                      *
 * We call the following subfunction LastGB, if                           *
 * the computed intermediate weight vector or                             *
 * if the perturbed target weight vector does NOT lie n the correct cone  *
 **************************************************************************/

static ideal LastGB(ideal G, intvec* curr_weight,int tp_deg)
{
  BOOLEAN nError = Overflow_Error;
  Overflow_Error = FALSE;

  int i, nV = currRing->N;
  int nwalk=0, endwalks=0, nnwinC=1;
  int nlast = 0;
  ideal Gomega, M, F, Gomega1, Gomega2, M1,F1,result,ssG;
  ring newRing, oldRing, TargetRing;
  intvec* iv_M_lp;
  intvec* target_weight;
  intvec* iv_lp = Mivlp(nV); //define (1,0,...,0)
  intvec* pert_target_vector;
  intvec* ivNull = new intvec(nV);
  intvec* extra_curr_weight = new intvec(nV);
  intvec* next_weight;

#ifndef  BUCHBERGER_ALG
  intvec* hilb_func;
#endif

  // to avoid (1,0,...,0) as the target vector
  intvec* last_omega = new intvec(nV);
  for(i=nV-1; i>0; i--)
  {
    (*last_omega)[i] = 1;
  }
  (*last_omega)[0] = 10000;

  ring EXXRing = currRing;

  // compute a pertubed weight vector of the target weight vector
  if(tp_deg > 1 && tp_deg <= nV)
  {
    //..25.03.03 VMrDefaultlp();//    VMrDefault(target_weight);
    if (rParameter (currRing) != NULL)
    {
      DefRingParlp();
    }
    else
    {
      VMrDefaultlp();
    }
    TargetRing = currRing;
    ssG = idrMoveR(G,EXXRing,currRing);
    iv_M_lp = MivMatrixOrderlp(nV);
    //target_weight = MPertVectorslp(ssG, iv_M_lp, tp_deg);
    target_weight = MPertVectors(ssG, iv_M_lp, tp_deg);
    delete iv_M_lp;
    pert_target_vector = target_weight;

    rChangeCurrRing(EXXRing);
    G = idrMoveR(ssG, TargetRing,currRing);
  }
  else
  {
    target_weight = Mivlp(nV);
  }
  //Print("\n// ring r%d_%d = %s;\n", tp_deg, nwalk, rString(currRing));

  while(1)
  {
    nwalk++;
    nstep++;
    to=clock();
   // compute a next weight vector
    next_weight = MkInterRedNextWeight(curr_weight,target_weight, G);
    xtnw=xtnw+clock()-to;

#ifdef PRINT_VECTORS
    MivString(curr_weight, target_weight, next_weight);
#endif

    if(Overflow_Error == TRUE)
    {
      newRing = currRing;
      nnwinC = 0;
      if(tp_deg == 1)
      {
        nlast = 1;
      }
      delete next_weight;

      //idElements(G, "G");
      //Print("\n// ring r%d_%d = %s;\n", tp_deg, nwalk, rString(currRing));

      break;
    }

    if(MivComp(next_weight, ivNull) == 1)
    {
      //Print("\n// ring r%d_%d = %s;\n", tp_deg, nwalk, rString(currRing));
      newRing = currRing;
      delete next_weight;
      break;
    }

    if(MivComp(next_weight, target_weight) == 1)
      endwalks = 1;

    for(i=nV-1; i>=0; i--)
    {
      (*extra_curr_weight)[i] = (*curr_weight)[i];
    }
    /* 06.11.01 NOT Changed */
    for(i=nV-1; i>=0; i--)
    {
      (*curr_weight)[i] = (*next_weight)[i];
    }
    oldRing = currRing;
    to=clock();
    // compute an initial form ideal of <G> w.r.t. "curr_vector"
    Gomega = MwalkInitialForm(G, curr_weight);
    xtif=xtif+clock()-to;

#ifdef ENDWALKS
    if(endwalks == 1)
    {
      Print("\n// ring r%d_%d = %s;\n", tp_deg, nwalk, rString(currRing));
      idElements(Gomega, "Gw");
      headidString(Gomega, "Gw");
    }
#endif

#ifndef  BUCHBERGER_ALG
    if(isNolVector(curr_weight) == 0)
    {
      hilb_func = hFirstSeries(Gomega,NULL,NULL,curr_weight,currRing);
    }
    else
    {
      hilb_func = hFirstSeries(Gomega,NULL,NULL,last_omega,currRing);
    }
#endif // BUCHBERGER_ALG

    /* define a new ring that its ordering is "(a(curr_weight),lp) */
    //..25.03.03 VMrDefault(curr_weight);
    if (rParameter (currRing) != NULL)
    {
      DefRingPar(curr_weight);
    }
    else
    {
      rChangeCurrRing(VMrDefault(curr_weight)); //Aenderung
    }
    newRing = currRing;
    Gomega1 = idrMoveR(Gomega, oldRing,currRing);

    to=clock();
    /* compute a reduced Groebner basis of <Gomega> w.r.t. "newRing" */
#ifdef  BUCHBERGER_ALG
    M = MstdhomCC(Gomega1);
#else
    M=kStd(Gomega1,NULL,isHomog,NULL,hilb_func,0,NULL,curr_weight);
    delete hilb_func;
#endif // BUCHBERGER_ALG
    xtstd=xtstd+clock()-to;
    /* change the ring to oldRing */
    rChangeCurrRing(oldRing);
    M1 =  idrMoveR(M, newRing,currRing);
    Gomega2 =  idrMoveR(Gomega1, newRing,currRing);

    to=clock();
    /* compute a reduced Groebner basis of <G> w.r.t. "newRing" */
    F = MLifttwoIdeal(Gomega2, M1, G);
    xtlift=xtlift+clock()-to;

    idDelete(&M1);
    idDelete(&G);

    /* change the ring to newRing */
    rChangeCurrRing(newRing);
    F1 = idrMoveR(F, oldRing,currRing);

    to=clock();
    /* reduce the Groebner basis <G> w.r.t. new ring */
    G = kInterRedCC(F1, NULL);
    xtred=xtred+clock()-to;
    idDelete(&F1);

    if(endwalks == 1)
    {
      //Print("\n// ring r%d_%d = %s;\n", tp_deg, nwalk, rString(currRing));
      break;
    }

    delete next_weight;
  }//while

  delete ivNull;

  if(tp_deg != 1)
  {
    //..25.03.03 VMrDefaultlp();//define and execute the ring "lp"
    if (rParameter (currRing) != NULL)
    {
      DefRingParlp();
    }
    else
    {
      VMrDefaultlp();
    }
    F1 = idrMoveR(G, newRing,currRing);

    if(nnwinC == 0 || test_w_in_ConeCC(F1, pert_target_vector) != 1)
    {
      oldRing = currRing;
      rChangeCurrRing(newRing);
      G = idrMoveR(F1, oldRing,currRing);
      Print("\n// takes %d steps and calls the recursion of level %d:",
             nwalk, tp_deg-1);

      F1 = LastGB(G,curr_weight, tp_deg-1);
    }

    TargetRing = currRing;
    rChangeCurrRing(EXXRing);
    result = idrMoveR(F1, TargetRing,currRing);
  }
  else
  {
    if(nlast == 1)
    {
      //OMEGA_OVERFLOW_LASTGB:
      /*
      if(MivSame(curr_weight, iv_lp) == 1)
        if (rParameter(currRing) != NULL)
          DefRingParlp();
        else
          VMrDefaultlp();
      else
        if (rParameter(currRing) != NULL)
          DefRingPar(curr_weight);
        else
          VMrDefault(curr_weight);
      */

        //..25.03.03 VMrDefaultlp();//define and execute the ring "lp"
        if (rParameter (currRing) != NULL)
        {
          DefRingParlp();
        }
        else
        {
          VMrDefaultlp();
        }

      F1 = idrMoveR(G, newRing,currRing);
      //Print("\n// Apply \"std\" in ring r%d_%d = %s;\n", tp_deg, nwalk, rString(currRing));

      G = MstdCC(F1);
      idDelete(&F1);
      newRing = currRing;
    }

    rChangeCurrRing(EXXRing);
    result = idrMoveR(G, newRing,currRing);
  }
  delete target_weight;
  delete last_omega;
  delete iv_lp;

  if(Overflow_Error == FALSE)
  {
    Overflow_Error = nError;
  }
  return(result);
}

/**********************************************************
 * check whether a polynomial of G has least 3 monomials  *
 **********************************************************/
static int lengthpoly(ideal G)
{
  int i;
  for(i=IDELEMS(G)-1; i>=0; i--)
  {
#if 0
    if(pLength(G->m[i])>2)
    {
      return 1;
    }
#else
    if((G->m[i]!=NULL) /* len >=0 */
       && (G->m[i]->next!=NULL) /* len >=1 */
       && (G->m[i]->next->next!=NULL) /* len >=2 */
       && (G->m[i]->next->next->next!=NULL) /* len >=3 */
      //&& (G->m[i]->next->next->next->next!=NULL) /* len >=4 */
       )
    {
    return 1;
    }
#endif
  }
  return 0;
}

/*********************************************************
 * check whether a polynomial of G has least 2 monomials *
**********************************************************/
static int islengthpoly2(ideal G)
{
  int i;
  for(i=IDELEMS(G)-1; i>=0; i--)
  {
    if((G->m[i]!=NULL) /* len >=0 */
       && (G->m[i]->next!=NULL) /* len >=1 */
       && (G->m[i]->next->next!=NULL)) /* len >=2 */
    {
      return 1;
    }
  }
  return 0;
}



/* Implementation of the improved Groebner walk algorithm which is written
   by Quoc-Nam Tran (2000).
   One perturbs the original target weight vector, only if
   the next intermediate weight vector is equal to the current target weight
   vector. This must be repeated until the wanted reduced Groebner basis
   to reach.
   If the numbers of variables is big enough, the representation of the origin
   weight vector may be very big. Therefore, it is possible the intermediate
   weight vector doesn't stay in the correct Groebner cone.
   In this case we have just a reduced Groebner basis of the given ideal
   with respect to another monomial order. Then we have to compute
   a wanted reduced Groebner basis of it with respect to the given order.
   At the following subroutine we use the improved Buchberger algorithm or
   the changed perturbation walk algorithm with a decrased degree.
 */

/***************************************
 * return the initial term of an ideal *
 ***************************************/
static ideal idHeadCC(ideal h)
{
  int i, nH =IDELEMS(h);

  ideal m = idInit(nH,h->rank);

  for (i=nH-1;i>=0; i--)
  {
    if (h->m[i]!=NULL)
    {
      m->m[i]=pHead(h->m[i]);
    }
  }
  return m;
}

/**********************************************
 * check whether two head-ideals are the same *
 **********************************************/
static inline int test_G_GB_walk(ideal H0, ideal H1)
{
  int i, nG = IDELEMS(H0);

  if(nG != IDELEMS(H1))
  {
    return 0;
  }
  for(i=nG-1; i>=0; i--)
  {
#if 0
    poly t;
    if((t=pSub(pCopy(H0->m[i]), pCopy(H1->m[i]))) != NULL)
    {
      pDelete(&t);
      return 0;
    }
    pDelete(&t);
#else
    if(!pEqualPolys(H0->m[i],H1->m[i]))
    {
      return 0;
    }
#endif
  }
  return 1;
}

//unused
/*****************************************************
 * find the maximal total degree of polynomials in G *
 *****************************************************/
#if 0
static int Trandegreebound(ideal G)
{
  int i, nG = IDELEMS(G);
  // int np=1;
  int nV = currRing->N;
  int degtmp, result = 0;
  intvec* ivUnit = Mivdp(nV);

  for(i=nG-1; i>=0; i--)
  {
    // find the maximal total degree of the polynomial G[i]
    degtmp = MwalkWeightDegree(G->m[i], ivUnit);
    if(degtmp > result)
    {
      result = degtmp;
    }
  }
  delete ivUnit;
  return result;
}
#endif

//unused
/************************************************************************
 * perturb the weight vector iva w.r.t. the ideal G.                    *
 *  the monomial order of the current ring is the w_1 weight lex. order *
 *  define w := d^(n-1)w_1+ d^(n-2)w_2, ...+ dw_(n-1)+ w_n              *
 *  where d := 1 + max{totdeg(g):g in G}*m, or                          *
 *  d := (2*maxdeg*maxdeg + (nV+1)*maxdeg)*m;                           *
 ************************************************************************/
#if 0
static intvec* TranPertVector(ideal G, intvec* iva)
{
  BOOLEAN nError = Overflow_Error;
  Overflow_Error = FALSE;

  int i, j;
  // int nG = IDELEMS(G);
  int nV = currRing->N;

  // define the sequence which expresses the current monomial ordering
  // w_1 = iva; w_2 = (1,0,..,0); w_n = (0,...,0,1,0)
  intvec* ivMat = MivMatrixOrder(iva);

  int  mtmp, m=(*iva)[0];

  for(i=ivMat->length(); i>=0; i--)
  {
    mtmp = (*ivMat)[i];
    if(mtmp <0)
    {
      mtmp = -mtmp;
    }
    if(mtmp > m)
    {
      m = mtmp;
    }
  }

  // define the maximal total degree of polynomials of G
  mpz_t ndeg;
  mpz_init(ndeg);

 // 12 Juli 03
#ifndef UPPER_BOUND
  mpz_set_si(ndeg, Trandegreebound(G)+1);
#else
  mpz_t ztmp;
  mpz_init(ztmp);

  mpz_t maxdeg;
  mpz_init_set_si(maxdeg, Trandegreebound(G));

  //ndeg = (2*maxdeg*maxdeg + (nV+1)*maxdeg)*m;//Kalkbrenner (1999)
  mpz_pow_ui(ztmp, maxdeg, 2);
  mpz_mul_ui(ztmp, ztmp, 2);
  mpz_mul_ui(maxdeg, maxdeg, nV+1);
  mpz_add(ndeg, ztmp, maxdeg);
  mpz_mul_ui(ndeg, ndeg, m);

  mpz_clear(ztmp);

  //PrintS("\n// with the new upper degree bound (2d^2+(n+1)d)*m ");
  //Print("\n//         where d = %d, n = %d and bound = %d", maxdeg, nV, ndeg);
#endif //UPPER_BOUND

#ifdef INVEPS_SMALL_IN_TRAN
  if(mpz_cmp_ui(ndeg, nV)>0 && nV > 3)
  {
    mpz_cdiv_q_ui(ndeg, ndeg, nV);
  }
 //PrintS("\n// choose the \"small\" inverse epsilon:");
 //mpz_out_str(stdout, 10, ndeg);
#endif
  mpz_t deg_tmp;
  mpz_init_set(deg_tmp, ndeg);

  mpz_t *ivres=( mpz_t *) omAlloc(nV*sizeof(mpz_t));
  mpz_init_set_si(ivres[nV-1],1);

  for(i=nV-2; i>=0; i--)
  {
    mpz_init_set(ivres[i], deg_tmp);
    mpz_mul(deg_tmp, deg_tmp, ndeg);
  }

  mpz_t *ivtmp=(mpz_t *)omAlloc(nV*sizeof(mpz_t));
  for(i=0; i<nV; i++)
  {
    mpz_init(ivtmp[i]);
  }
  mpz_t sing_int;
  mpz_init_set_ui(sing_int,  2147483647);

  intvec* repr_vector = new intvec(nV);

  // define ivtmp := ndeg^(n-1).w_1 + ndeg^(n-2).w_2 + ... + w_n
  for(i=0; i<nV; i++)
  {
    for(j=0; j<nV; j++)
    {
      if( (*ivMat)[i*nV+j] >= 0 )
      {
        mpz_mul_ui(ivres[i], ivres[i], (*ivMat)[i*nV+j]);
      }
      else
      {
        mpz_mul_ui(ivres[i], ivres[i], -(*ivMat)[i*nV+j]);
        mpz_neg(ivres[i], ivres[i]);
      }
      mpz_add(ivtmp[j], ivtmp[j], ivres[i]);
    }
  }
  delete ivMat;

  int ntrue=0;
  for(i=0; i<nV; i++)
  {
    (*repr_vector)[i] = mpz_get_si(ivtmp[i]);
    if(mpz_cmp(ivtmp[i], sing_int)>=0)
    {
      ntrue++;
      if(Overflow_Error == FALSE)
      {
        Overflow_Error = TRUE;

        PrintS("\n// ** OVERFLOW in \"Repr.Vector\": ");
        mpz_out_str( stdout, 10, ivtmp[i]);
        PrintS(" is greater than 2147483647 (max. integer representation)");
        Print("\n//  So vector[%d] := %d is wrong!!\n",i+1,(*repr_vector)[i]);
      }
    }
  }
  if(Overflow_Error == TRUE)
  {
    ivString(repr_vector, "repvector");
    Print("\n// %d element(s) of it are overflow!!", ntrue);
  }

  if(Overflow_Error == FALSE)
    Overflow_Error=nError;

  omFree(ivres);
  omFree(ivtmp);

  mpz_clear(sing_int);
  mpz_clear(deg_tmp);
  mpz_clear(ndeg);

  return repr_vector;
}
#endif

//unused
#if 0
static intvec* TranPertVector_lp(ideal G)
{
  BOOLEAN nError = Overflow_Error;
  Overflow_Error = FALSE;
  // int j, nG = IDELEMS(G);
  int i;
  int nV = currRing->N;

  // define the maximal total degree of polynomials of G
  mpz_t ndeg;
  mpz_init(ndeg);

 // 12 Juli 03
#ifndef UPPER_BOUND
  mpz_set_si(ndeg, Trandegreebound(G)+1);
#else
  mpz_t ztmp;
  mpz_init(ztmp);

  mpz_t maxdeg;
  mpz_init_set_si(maxdeg, Trandegreebound(G));

  //ndeg = (2*maxdeg*maxdeg + (nV+1)*maxdeg);//Kalkbrenner (1999)
  mpz_pow_ui(ztmp, maxdeg, 2);
  mpz_mul_ui(ztmp, ztmp, 2);
  mpz_mul_ui(maxdeg, maxdeg, nV+1);
  mpz_add(ndeg, ztmp, maxdeg);
  // PrintS("\n// with the new upper degree bound (2d^2+(n+1)d)*m ");
  // Print("\n//         where d = %d, n = %d and bound = %d",
  // mpz_get_si(maxdeg), nV, mpz_get_si(ndeg));

 mpz_clear(ztmp);

#endif

#ifdef INVEPS_SMALL_IN_TRAN
 if(mpz_cmp_ui(ndeg, nV)>0 && nV > 3)
    mpz_cdiv_q_ui(ndeg, ndeg, nV);

 //PrintS("\n// choose the \"small\" inverse epsilon:");
 // mpz_out_str(stdout, 10, ndeg);
#endif

  mpz_t deg_tmp;
  mpz_init_set(deg_tmp, ndeg);

  mpz_t *ivres=(mpz_t *)omAlloc(nV*sizeof(mpz_t));
  mpz_init_set_si(ivres[nV-1], 1);

  for(i=nV-2; i>=0; i--)
  {
    mpz_init_set(ivres[i], deg_tmp);
    mpz_mul(deg_tmp, deg_tmp, ndeg);
  }

  mpz_t sing_int;
  mpz_init_set_ui(sing_int,  2147483647);

  intvec* repr_vector = new intvec(nV);
  int ntrue=0;
  for(i=0; i<nV; i++)
  {
    (*repr_vector)[i] = mpz_get_si(ivres[i]);

    if(mpz_cmp(ivres[i], sing_int)>=0)
    {
      ntrue++;
      if(Overflow_Error == FALSE)
      {
        Overflow_Error = TRUE;
        PrintS("\n// ** OVERFLOW in \"Repr.Vector\": ");
        mpz_out_str( stdout, 10, ivres[i]);
        PrintS(" is greater than 2147483647 (max. integer representation)");
        Print("\n//  So vector[%d] := %d is wrong!!\n",i+1,(*repr_vector)[i]);
      }
    }
  }
  if(Overflow_Error == TRUE)
  {
    ivString(repr_vector, "repvector");
    Print("\n// %d element(s) of it are overflow!!", ntrue);
  }
  if(Overflow_Error == FALSE)
    Overflow_Error = nError;

  omFree(ivres);

  mpz_clear(ndeg);
  mpz_clear(sing_int);

  return repr_vector;
}
#endif

//unused
#if 0
static intvec* RepresentationMatrix_Dp(ideal G, intvec* M)
{
  BOOLEAN nError = Overflow_Error;
  Overflow_Error = FALSE;

  int i, j;
  int nV = currRing->N;

  intvec* ivUnit = Mivdp(nV);
  int degtmp, maxdeg = 0;

  for(i=IDELEMS(G)-1; i>=0; i--)
  {
    // find the maximal total degree of the polynomial G[i]
    degtmp = MwalkWeightDegree(G->m[i], ivUnit);
    if(degtmp > maxdeg)
      maxdeg = degtmp;
  }

  mpz_t ztmp;
  mpz_init_set_si(ztmp, maxdeg);
  mpz_t *ivres=(mpz_t *)omAlloc(nV*sizeof(mpz_t));
  mpz_init_set_si(ivres[nV-1], 1); // (*ivres)[nV-1] = 1;

  for(i=nV-2; i>=0; i--)
  {
    mpz_init_set(ivres[i], ztmp); //(*ivres)[i] = ztmp;
    mpz_mul_ui(ztmp, ztmp, maxdeg); //ztmp *=maxdeg;
  }

  mpz_t *ivtmp=(mpz_t*)omAlloc(nV*sizeof(mpz_t));
  for(i=0; i<nV; i++)
    mpz_init(ivtmp[i]);

  // define ivtmp := ndeg^(n-1).w_1 + ndeg^(n-2).w_2 + ... + w_n
  for(i=0; i<nV; i++)
    for(j=0; j<nV; j++)
    {
      if((*M)[i*nV+j] < 0)
      {
        mpz_mul_ui(ztmp, ivres[i], -(*M)[i*nV+j]);
        mpz_neg(ztmp, ztmp);
      }
      else
        mpz_mul_ui(ztmp, ivres[i], (*M)[i*nV+j]);

      mpz_add(ivtmp[j], ivtmp[j], ztmp);
    }
  delete ivres;
  mpz_t sing_int;
  mpz_init_set_ui(sing_int,  2147483647);

  int ntrue=0;
  intvec* repvector = new intvec(nV);
  for(i=0; i<nV; i++)
  {
    (*repvector)[i] = mpz_get_si(ivtmp[i]);
    if(mpz_cmp(ivtmp[i], sing_int)>0)
    {
      ntrue++;
      if(Overflow_Error == FALSE)
      {
        Overflow_Error = TRUE;
        PrintS("\n// ** OVERFLOW in \"Repr.Matrix\": ");
        mpz_out_str( stdout, 10, ivtmp[i]);
        PrintS(" is greater than 2147483647 (max. integer representation)");
        Print("\n//  So vector[%d] := %d is wrong!!\n",i+1,(*repvector)[i]);
      }
    }
  }
  if(Overflow_Error == TRUE)
  {
    ivString(repvector, "repvector");
    Print("\n// %d element(s) of it are overflow!!", ntrue);
  }

  if(Overflow_Error == FALSE)
    Overflow_Error = nError;

  mpz_clear(sing_int);
  mpz_clear(ztmp);
  omFree(ivtmp);
  omFree(ivres);
  return repvector;
}
#endif

/*****************************************************************************
 * The following subroutine is the implementation of our first improved      *
 * Groebner walk algorithm, i.e. the first altervative algorithm.            *
 * First we use the Grobner walk algorithm and then we call the changed      *
 * perturbation walk algorithm with decreased degree, if an intermediate     *
 * weight vector is equal to the current target weight vector.               *
 * This call will be only repeated until we get the wanted reduced Groebner  *
 * basis or n times, where n is the numbers of variables.                    *
 *****************************************************************************/

//unused
#if 0
static int testnegintvec(intvec* v)
{
  int n = v->length();
  int i;
  for(i=0; i<n; i++)
  {
    if((*v)[i]<0)
    {
      return(1);
    }
  }
  return(0);
}
#endif

// npwinc = 0, if curr_weight doesn't stay in the correct Groebner cone
static ideal Rec_LastGB(ideal G, intvec* curr_weight,
                        intvec* orig_target_weight, int tp_deg, int npwinc)
{
  BOOLEAN nError = Overflow_Error;
  Overflow_Error = FALSE;
  // BOOLEAN nOverflow_Error = FALSE;

  clock_t tproc=0;
  clock_t tinput = clock();

  int i,  nV = currRing->N;
  int nwalk=0, endwalks=0, nnwinC=1;
  int nlast = 0;
  ideal Gomega, M, F, Gomega1, Gomega2, M1,F1,result,ssG;
  ring newRing, oldRing, TargetRing;
  intvec* iv_M_lp;
  intvec* target_weight;
  intvec* ivNull = new intvec(nV); //define (0,...,0)
  ring EXXRing = currRing;
  //int NEG=0; //19 juni 03
  intvec* next_weight;
#ifndef  BUCHBERGER_ALG
  //08 Juli 03
  intvec* hilb_func;
#endif
  // to avoid (1,0,...,0) as the target vector
  intvec* last_omega = new intvec(nV);
  for(i=nV-1; i>0; i--)
    (*last_omega)[i] = 1;
  (*last_omega)[0] = 10000;

  BOOLEAN isGB = FALSE;

  // compute a pertubed weight vector of the target weight vector
  if(tp_deg > 1 && tp_deg <= nV)
  {
    ideal H0 = idHeadCC(G);

    if (rParameter (currRing) != NULL)
    {
      DefRingParlp();
    }
    else
    {
      VMrDefaultlp();
    }
    TargetRing = currRing;
    ssG = idrMoveR(G,EXXRing,currRing);

    ideal H0_tmp = idrMoveR(H0,EXXRing,currRing);
    ideal H1 = idHeadCC(ssG);

    // Apply Lemma 2.2 in Collart et. al (1997) to check whether cone(k-1) is equal to cone(k)
    if(test_G_GB_walk(H0_tmp,H1)==1)
    {
      idDelete(&H0_tmp);
      idDelete(&H1);
      G = ssG;
      ssG = NULL;
      newRing = currRing;
      delete ivNull;

      if(npwinc != 0)
      {
        goto LastGB_Finish;
      }
      else
      {
        isGB = TRUE;
        goto KSTD_Finish;
      }
    }
    idDelete(&H0_tmp);
    idDelete(&H1);

    iv_M_lp = MivMatrixOrderlp(nV);
    target_weight  = MPertVectors(ssG, iv_M_lp, tp_deg);
    delete iv_M_lp;
    //PrintS("\n// Input is not GB!!");
    rChangeCurrRing(EXXRing);
    G = idrMoveR(ssG, TargetRing,currRing);

    if(Overflow_Error == TRUE)
    {
      //nOverflow_Error = Overflow_Error;
      //NEG = 1;
      newRing = currRing;
      goto JUNI_STD;
    }
  }

  while(1)
  {
    nwalk ++;
    nstep++;

    if(nwalk==1)
    {
      goto FIRST_STEP;
    }
    to=clock();
    // compute an initial form ideal of <G> w.r.t. "curr_vector"
    Gomega = MwalkInitialForm(G, curr_weight);
    xtif=xtif+clock()-to;

#ifndef  BUCHBERGER_ALG
    if(isNolVector(curr_weight) == 0)
    {
      hilb_func = hFirstSeries(Gomega,NULL,NULL,curr_weight,currRing);
    }
    else
    {
      hilb_func = hFirstSeries(Gomega,NULL,NULL,last_omega,currRing);
    }
#endif // BUCHBERGER_ALG

    oldRing = currRing;

    // defiNe a new ring that its ordering is "(a(curr_weight),lp)
    if (rParameter(currRing) != NULL)
    {
      DefRingPar(curr_weight);
    }
    else
    {
      rChangeCurrRing(VMrDefault(curr_weight)); //Aenderung
    }
    newRing = currRing;
    Gomega1 = idrMoveR(Gomega, oldRing,currRing);
    to=clock();
    // compute a reduced Groebner basis of <Gomega> w.r.t. "newRing"
#ifdef  BUCHBERGER_ALG
    M = MstdhomCC(Gomega1);
#else
    M=kStd(Gomega1,NULL,isHomog,NULL,hilb_func,0,NULL,curr_weight);
    delete hilb_func;
#endif // BUCHBERGER_ALG
    xtstd=xtstd+clock()-to;
    // change the ring to oldRing
    rChangeCurrRing(oldRing);
    M1 =  idrMoveR(M, newRing,currRing);
    Gomega2 =  idrMoveR(Gomega1, newRing,currRing);

     to=clock();
    // compute a reduced Groebner basis of <G> w.r.t. "newRing" by the lifting process
    F = MLifttwoIdeal(Gomega2, M1, G);
    xtlift=xtlift+clock()-to;
    idDelete(&M1);
    idDelete(&Gomega2);
    idDelete(&G);

    // change the ring to newRing
    rChangeCurrRing(newRing);
    F1 = idrMoveR(F, oldRing,currRing);

    to=clock();
    // reduce the Groebner basis <G> w.r.t. new ring
    G = kInterRedCC(F1, NULL);
    xtred=xtred+clock()-to;
    idDelete(&F1);

    if(endwalks == 1)
    {
      break;
    }
  FIRST_STEP:
    to=clock();
    Overflow_Error = FALSE;
    // compute a next weight vector
    next_weight = MkInterRedNextWeight(curr_weight,target_weight, G);
    xtnw=xtnw+clock()-to;
#ifdef PRINT_VECTORS
    MivString(curr_weight, target_weight, next_weight);
#endif

    if(Overflow_Error == TRUE)
    {
      //PrintS("\n// ** The next vector does NOT stay in Cone!!\n");
#ifdef TEST_OVERFLOW
      goto  LastGB_Finish;
#endif

      nnwinC = 0;
      if(tp_deg == nV)
      {
        nlast = 1;
      }
      delete next_weight;
      break;
    }

    if(MivComp(next_weight, ivNull) == 1)
    {
      //newRing = currRing;
      delete next_weight;
      break;
    }

    if(MivComp(next_weight, target_weight) == 1)
    {
      if(tp_deg == nV)
      {
        endwalks = 1;
      }
      else
      {
        // REC_LAST_GB_ALT2:
        //nOverflow_Error = Overflow_Error;
        tproc=tproc+clock()-tinput;
        /*
          Print("\n// takes %d steps and calls \"Rec_LastGB\" (%d):",
          nwalk, tp_deg+1);
        */
        G = Rec_LastGB(G,curr_weight, orig_target_weight, tp_deg+1,nnwinC);
        newRing = currRing;
        delete next_weight;
        break;
      }
    }

    for(i=nV-1; i>=0; i--)
    {
      (*curr_weight)[i] = (*next_weight)[i];
    }
    delete next_weight;
  }//while

  delete ivNull;

  if(tp_deg != nV)
  {
    newRing = currRing;

    if (rParameter(currRing) != NULL)
    {
      DefRingParlp();
    }
    else
    {
      VMrDefaultlp();
    }
    F1 = idrMoveR(G, newRing,currRing);

    if(nnwinC == 0 || test_w_in_ConeCC(F1, target_weight) != 1 )
    {
      // nOverflow_Error = Overflow_Error;
      //Print("\n//  takes %d steps and calls \"Rec_LastGB (%d):", tp_deg+1);
      tproc=tproc+clock()-tinput;
      F1 = Rec_LastGB(F1,curr_weight, orig_target_weight, tp_deg+1,nnwinC);
    }
    delete target_weight;

    TargetRing = currRing;
    rChangeCurrRing(EXXRing);
    result = idrMoveR(F1, TargetRing,currRing);
  }
  else
  {
    if(nlast == 1)
    {
      JUNI_STD:

      newRing = currRing;
      if (rParameter(currRing) != NULL)
      {
        DefRingParlp();
      }
      else
      {
        VMrDefaultlp();
      }
      KSTD_Finish:
      if(isGB == FALSE)
      {
        F1 = idrMoveR(G, newRing,currRing);
      }
      else
      {
        F1 = G;
      }
      to=clock();
      // Print("\n// apply the Buchberger's alg in ring = %s",rString(currRing));
      // idElements(F1, "F1");
      G = MstdCC(F1);
      xtextra=xtextra+clock()-to;


      idDelete(&F1);
      newRing = currRing;
    }

    LastGB_Finish:
    rChangeCurrRing(EXXRing);
    result = idrMoveR(G, newRing,currRing);
  }

  if(Overflow_Error == FALSE)
    {
    Overflow_Error=nError;
    }
// Print("\n// \"Rec_LastGB\" (%d) took %d steps and %.2f sec.Overflow_Error (%d)", tp_deg, nwalk, ((double) tproc)/1000000, nOverflow_Error);
  return(result);
}

/* The following subroutine is the implementation of our second improved
   Groebner walk algorithm, i.e. the second altervative algorithm.
   First we use the Grobner walk algorithm and then we call the changed
   perturbation walk algorithm with increased degree, if an intermediate
   weight vector is equal to the current target weight vector.
   This call will be only repeated until we get the wanted reduced Groebner
   basis or n times, where n is the numbers of variables.
*/

/******************************
 * walk + recursive LastGB    *
 ******************************/
ideal MAltwalk2(ideal Go, intvec* curr_weight, intvec* target_weight)
{
  Set_Error(FALSE);
  Overflow_Error = FALSE;
  //BOOLEAN nOverflow_Error = FALSE;
  //Print("// pSetm_Error = (%d)", ErrorCheck());

  xtif=0; xtstd=0; xtlift=0; xtred=0; xtnw=0; xtextra=0;
  xftinput = clock();
  clock_t tostd, tproc;

  nstep = 0;
  int i, nV = currRing->N;
  int nwalk=0, endwalks=0;
  // int nhilb = 1;
  ideal Gomega, M, F, Gomega1, Gomega2, M1, F1, G;
  //ideal  G1;
  //ring endRing;
  ring newRing, oldRing;
  intvec* ivNull = new intvec(nV);
  intvec* next_weight;
#if 0
  intvec* extra_curr_weight = new intvec(nV);
#endif
  //intvec* hilb_func;
  intvec* exivlp = Mivlp(nV);

  ring XXRing = currRing;

  //Print("\n// ring r_input = %s;", rString(currRing));
  to = clock();
  /* compute the reduced Groebner basis of the given ideal w.r.t.
     a "fast" monomial order, e.g. degree reverse lex. order (dp) */
  G = MstdCC(Go);
  tostd=clock()-to;

  /*
  Print("\n// Computation of the first std took = %.2f sec",
        ((double) tostd)/1000000);
  */
  if(currRing->order[0] == ringorder_a)
  {
    goto NEXT_VECTOR;
  }
  while(1)
  {
    nwalk ++;
    nstep ++;
    to = clock();
    /* compute an initial form ideal of <G> w.r.t. "curr_vector" */
    Gomega = MwalkInitialForm(G, curr_weight);
    xtif=xtif+clock()-to;
#if 0
    if(Overflow_Error == TRUE)
    {
      for(i=nV-1; i>=0; i--)
        (*curr_weight)[i] = (*extra_curr_weight)[i];
      delete extra_curr_weight;
      goto LAST_GB_ALT2;
    }
#endif
    oldRing = currRing;

    /* define a new ring that its ordering is "(a(curr_weight),lp) */
    if (rParameter(currRing) != NULL)
    {
      DefRingPar(curr_weight);
    }
    else
    {
      rChangeCurrRing(VMrDefault(curr_weight)); // Aenderung
    }
    newRing = currRing;
    Gomega1 = idrMoveR(Gomega, oldRing,currRing);
    to = clock();
    /* compute a reduced Groebner basis of <Gomega> w.r.t. "newRing" */
    M = MstdhomCC(Gomega1);
    xtstd=xtstd+clock()-to;
    /* change the ring to oldRing */
    rChangeCurrRing(oldRing);
    M1 =  idrMoveR(M, newRing,currRing);
    Gomega2 =  idrMoveR(Gomega1, newRing,currRing);

    to = clock();
    /* compute the reduced Groebner basis of <G> w.r.t. "newRing"
       by the liftig process */
    F = MLifttwoIdeal(Gomega2, M1, G);
    xtlift=xtlift+clock()-to;
    idDelete(&M1);
    idDelete(&Gomega2);
    idDelete(&G);

    /* change the ring to newRing */
    rChangeCurrRing(newRing);
    F1 = idrMoveR(F, oldRing,currRing);

    to = clock();
    /* reduce the Groebner basis <G> w.r.t. newRing */
    G = kInterRedCC(F1, NULL);
    xtred=xtred+clock()-to;
    idDelete(&F1);

    if(endwalks == 1)
      break;

  NEXT_VECTOR:
    to = clock();
    /* compute a next weight vector */
    next_weight = MkInterRedNextWeight(curr_weight,target_weight, G);
    xtnw=xtnw+clock()-to;
#ifdef PRINT_VECTORS
    MivString(curr_weight, target_weight, next_weight);
#endif

    if(Overflow_Error == TRUE)
    {
      /*
        ivString(next_weight, "omega");
        PrintS("\n// ** The weight vector does NOT stay in Cone!!\n");
      */
#ifdef TEST_OVERFLOW
      goto  TEST_OVERFLOW_OI;
#endif

      newRing = currRing;
      if (rParameter(currRing) != NULL)
      {
        DefRingPar(target_weight);
      }
      else
      {
        rChangeCurrRing(VMrDefault(target_weight)); // Aenderung
      }
      F1 = idrMoveR(G, newRing,currRing);
      G = MstdCC(F1);
      idDelete(&F1);
      newRing = currRing;
      break;
    }

    if(MivComp(next_weight, ivNull) == 1)
    {
      newRing = currRing;
      delete next_weight;
      break;
    }

    if(MivComp(next_weight, target_weight) == 1)
    {
      if(MivSame(target_weight, exivlp)==1)
      {
     // LAST_GB_ALT2:
        //nOverflow_Error = Overflow_Error;
        tproc = clock()-xftinput;
        //Print("\n// takes %d steps and calls the recursion of level 2:",  nwalk);
        /* call the changed perturbation walk algorithm with degree 2 */
        G = Rec_LastGB(G, curr_weight, target_weight, 2,1);
        newRing = currRing;
        delete next_weight;
        break;
      }
      endwalks = 1;
    }

    for(i=nV-1; i>=0; i--)
    {
      //(*extra_curr_weight)[i] = (*curr_weight)[i];
      (*curr_weight)[i] = (*next_weight)[i];
    }
    delete next_weight;
  }
#ifdef TEST_OVERFLOW
 TEST_OVERFLOW_OI:
#endif
  rChangeCurrRing(XXRing);
  G = idrMoveR(G, newRing,currRing);
  delete ivNull;
  delete exivlp;

#ifdef TIME_TEST
 // Print("\n// \"Main procedure\"  took %d steps dnd %.2f sec. Overflow_Error (%d)", nwalk, ((double) tproc)/1000000, nOverflow_Error);

  TimeStringFractal(xftinput, tostd, xtif, xtstd, xtextra,xtlift, xtred,xtnw);

  Print("\n// pSetm_Error = (%d)", ErrorCheck());
  //Print("\n// Overflow_Error? (%d)", nOverflow_Error);
  Print("\n// Awalk2 took %d steps!!", nstep);
#endif

  return(G);
}


/**************************************
 * perturb the matrix order of  "lex" *
 **************************************/
static intvec* NewVectorlp(ideal I)
{
  int nV = currRing->N;
  intvec* iv_wlp =  MivMatrixOrderlp(nV);
  intvec* result = Mfpertvector(I, iv_wlp);
  delete iv_wlp;
  return result;
}

int ngleich;
intvec* Xsigma;
intvec* Xtau;
int xn;
intvec* Xivinput;
intvec* Xivlp;

#if 0
/********************************
 * compute a next weight vector *
 ********************************/
static intvec* MWalkRandomNextWeight(ideal G, intvec* curr_weight, intvec* target_weight, int weight_rad, int pert_deg)
{
  int i, weight_norm;
  int nV = currRing->N;
  intvec* next_weight2;
  intvec* next_weight22 = new intvec(nV);
  intvec* next_weight = MwalkNextWeightCC(curr_weight,target_weight, G);
  if(MivComp(next_weight, target_weight) == 1)
  {
    return(next_weight);
  }
  else
  {
    //compute a perturbed next weight vector "next_weight1"
    intvec* next_weight1 = MkInterRedNextWeight(MPertVectors(G, MivMatrixOrder(curr_weight), pert_deg), target_weight, G);
    //Print("\n // size of next_weight1 = %d", sizeof((*next_weight1)));

    //compute a random next weight vector "next_weight2"
    while(1)
    {
      weight_norm = 0;
      while(weight_norm == 0)
      {
        for(i=0; i<nV; i++)
        {
          //Print("\n//  next_weight[%d]  = %d", i, (*next_weight)[i]);
          (*next_weight22)[i] = rand() % 60000 - 30000;
          weight_norm = weight_norm + (*next_weight22)[i]*(*next_weight22)[i];
        }
        weight_norm = 1 + floor(sqrt(weight_norm));
      }

      for(i=nV-1; i>=0; i--)
      {
        if((*next_weight22)[i] < 0)
        {
          (*next_weight22)[i] = 1 + (*curr_weight)[i] + floor(weight_rad*(*next_weight22)[i]/weight_norm);
        }
        else
        {
          (*next_weight22)[i] = (*curr_weight)[i] + floor(weight_rad*(*next_weight22)[i]/weight_norm);
        }
      //Print("\n//  next_weight22[%d]  = %d", i, (*next_weight22)[i]);
      }

      if(test_w_in_ConeCC(G, next_weight22) == 1)
      {
        //Print("\n//MWalkRandomNextWeight: next_weight2 im Kegel\n");
        next_weight2 = MkInterRedNextWeight(next_weight22, target_weight, G);
        delete next_weight22;
        break;
      }
    }
    intvec* result = new intvec(nV);
    ideal G_test = MwalkInitialForm(G, next_weight);
    ideal G_test1 = MwalkInitialForm(G, next_weight1);
    ideal G_test2 = MwalkInitialForm(G, next_weight2);

    // compare next_weights
    if(IDELEMS(G_test1) < IDELEMS(G_test))
    {
      if(IDELEMS(G_test2) <= IDELEMS(G_test1)) // |G_test2| <= |G_test1| < |G_test|
      {
        for(i=0; i<nV; i++)
        {
          (*result)[i] = (*next_weight2)[i];
        }
      }
      else // |G_test1| < |G_test|, |G_test1| < |G_test2|
      {
        for(i=0; i<nV; i++)
        {
          (*result)[i] = (*next_weight1)[i];
        }
      }
    }
    else
    {
      if(IDELEMS(G_test2) <= IDELEMS(G_test)) // |G_test2| <= |G_test| <= |G_test1|
      {
        for(i=0; i<nV; i++)
        {
          (*result)[i] = (*next_weight2)[i];
        }
      }
      else // |G_test| <= |G_test1|, |G_test| < |G_test2|
      {
        for(i=0; i<nV; i++)
        {
          (*result)[i] = (*next_weight)[i];
        }
      }
    }
    delete next_weight;
    delete next_weight1;
    idDelete(&G_test);
    idDelete(&G_test1);
    idDelete(&G_test2);
    if(test_w_in_ConeCC(G, result) == 1)
    {
      delete next_weight2;
      return result;
    }
    else
    {
      delete result;
      return next_weight2;
    }
  }
}
#endif

/********************************
 * compute a next weight vector *
 ********************************/
static intvec* MWalkRandomNextWeight(ideal G, intvec* curr_weight, intvec* target_weight, int weight_rad, int pert_deg)
{
  int i, weight_norm;
  //int randCount=0;
  int nV = currRing->N;
  intvec* next_weight2;
  intvec* next_weight22 = new intvec(nV);
  intvec* result = new intvec(nV);

  //compute a perturbed next weight vector "next_weight1"
  //intvec* next_weight1 = MkInterRedNextWeight(MPertVectors(G,MivMatrixOrderRefine(curr_weight,target_weight),pert_deg),target_weight,G);
  intvec* next_weight1 =MkInterRedNextWeight(curr_weight,target_weight,G);
  //compute a random next weight vector "next_weight2"
  while(1)
  {
    weight_norm = 0;
    while(weight_norm == 0)
    {
      for(i=0; i<nV; i++)
      {
        (*next_weight22)[i] = rand() % 60000 - 30000;
        weight_norm = weight_norm + (*next_weight22)[i]*(*next_weight22)[i];
      }
      weight_norm = 1 + floor(sqrt(weight_norm));
    }
    for(i=0; i<nV; i++)
    {
      if((*next_weight22)[i] < 0)
      {
        (*next_weight22)[i] = 1 + (*curr_weight)[i] + floor(weight_rad*(*next_weight22)[i]/weight_norm);
      }
      else
      {
        (*next_weight22)[i] = (*curr_weight)[i] + floor(weight_rad*(*next_weight22)[i]/weight_norm);
      }
    }
    if(test_w_in_ConeCC(G, next_weight22) == 1)
    {
      next_weight2 = MkInterRedNextWeight(next_weight22,target_weight,G);
      delete next_weight22;
      break;
    }
  }
  // compute "usual" next weight vector
  intvec* next_weight = MwalkNextWeightCC(curr_weight,target_weight, G);
  ideal G_test = MwalkInitialForm(G, next_weight);
  ideal G_test2 = MwalkInitialForm(G, next_weight2);

  // compare next weights
  if(Overflow_Error == FALSE)
  {
    ideal G_test1 = MwalkInitialForm(G, next_weight1);
    if(IDELEMS(G_test1) < IDELEMS(G_test))
    {
      if(IDELEMS(G_test2) < IDELEMS(G_test1))
      {
        // |G_test2| < |G_test1| < |G_test|
        for(i=0; i<nV; i++)
        {
          (*result)[i] = (*next_weight2)[i];
        }
      }
      else
      {
        // |G_test1| < |G_test|, |G_test1| <= |G_test2|
        for(i=0; i<nV; i++)
        {
          (*result)[i] = (*next_weight1)[i];
        }
      }
    }
    else
    {
      if(IDELEMS(G_test2) < IDELEMS(G_test)) // |G_test2| < |G_test| <= |G_test1|
      {
        for(i=0; i<nV; i++)
        {
          (*result)[i] = (*next_weight2)[i];
        }
      }
      else
      {
        // |G_test| < |G_test1|, |G_test| <= |G_test2|
        for(i=0; i<nV; i++)
        {
          (*result)[i] = (*next_weight)[i];
        }
      }
    }
    idDelete(&G_test1);
  }
  else
  {
    Overflow_Error = FALSE;
    if(IDELEMS(G_test2) < IDELEMS(G_test))
    {
      for(i=1; i<nV; i++)
      {
        (*result)[i] = (*next_weight2)[i];
      }
    }
    else
    {
      for(i=0; i<nV; i++)
      {
        (*result)[i] = (*next_weight)[i];
      }
    }
  }
  idDelete(&G_test);
  idDelete(&G_test2);
  if(test_w_in_ConeCC(G, result) == 1)
  {
    delete next_weight2;
    delete next_weight;
    delete next_weight1;
    return result;
  }
  else
  {
    delete result;
    delete next_weight2;
    delete next_weight1;
    return next_weight;
  }
}


/***************************************************************************
 * The procedur REC_GB_Mwalk computes a GB for <G> w.r.t. the weight order *
 * otw, where G is a reduced GB w.r.t. the weight order cw.                *
 * The new procedur Mwalk calls REC_GB.                                    *
 ***************************************************************************/
static ideal REC_GB_Mwalk(ideal G, intvec* curr_weight, intvec* orig_target_weight,
                          int tp_deg, int npwinc)
{
  BOOLEAN nError = Overflow_Error;
  Overflow_Error = FALSE;

  int i,  nV = currRing->N;
  int nwalk=0, endwalks=0, nnwinC=1, nlast = 0;
  ideal Gomega, M, F, Gomega1, Gomega2, M1,F1,result,ssG;
  ring newRing, oldRing, TargetRing;
  intvec* target_weight;
  intvec* ivNull = new intvec(nV);
#ifndef BUCHBERGER_ALG
  intvec* hilb_func;
  // to avoid (1,0,...,0) as the target vector
  intvec* last_omega = new intvec(nV);
  for(i=nV-1; i>0; i--)
  {
    (*last_omega)[i] = 1;
  }
  (*last_omega)[0] = 10000;
#endif
  BOOLEAN isGB = FALSE;

  ring EXXRing = currRing;

  // compute a pertubed weight vector of the target weight vector
  if(tp_deg > 1 && tp_deg <= nV)
  {
    ideal H0 = idHeadCC(G);
    if (rParameter(currRing) != NULL)
    {
      DefRingPar(orig_target_weight);
    }
    else
    {
      rChangeCurrRing(VMrDefault(orig_target_weight)); // Aenderung
    }
    TargetRing = currRing;
    ssG = idrMoveR(G,EXXRing,currRing);

    ideal H0_tmp = idrMoveR(H0,EXXRing,currRing);
    ideal H1 = idHeadCC(ssG);
    id_Delete(&H0,EXXRing);

    if(test_G_GB_walk(H0_tmp,H1)==1)
    {
      //Print("\n//REC_GB_Mwalk: input in %d-th recursive is a GB!\n",tp_deg);
      idDelete(&H0_tmp);
      idDelete(&H1);
      G = ssG;
      ssG = NULL;
      newRing = currRing;
      delete ivNull;
      if(npwinc == 0)
      {
        isGB = TRUE;
        goto KSTD_Finish;
      }
      else
      {
        goto LastGB_Finish;
      }
    }
    idDelete(&H0_tmp);
    idDelete(&H1);

    target_weight  = MPertVectors(ssG, MivMatrixOrder(orig_target_weight), tp_deg);

    rChangeCurrRing(EXXRing);
    G = idrMoveR(ssG, TargetRing,currRing);
  }

  while(1)
  {
    nwalk ++;
    nstep++;
    if(nwalk == 1)
    {
      goto NEXT_STEP;
    }
    //Print("\n//REC_GB_Mwalk: Entering the %d-th step in the %d-th recursive:\n",nwalk,tp_deg);
    to = clock();
    // compute an initial form ideal of <G> w.r.t. "curr_vector"
    Gomega = MwalkInitialForm(G, curr_weight);
    xtif = xtif + clock()-to;

#ifndef  BUCHBERGER_ALG
    if(isNolVector(curr_weight) == 0)
    {
      hilb_func = hFirstSeries(Gomega,NULL,NULL,curr_weight,currRing);
    }
    else
    {
      hilb_func = hFirstSeries(Gomega,NULL,NULL,last_omega,currRing);
    }
#endif

    oldRing = currRing;

    // define a new ring with ordering "(a(curr_weight),lp)
    if (rParameter(currRing) != NULL)
    {
      DefRingPar(curr_weight);
    }
    else
    {
      rChangeCurrRing(VMrDefault(curr_weight)); // Aenderung
    }
    newRing = currRing;
    Gomega1 = idrMoveR(Gomega, oldRing,currRing);

    to = clock();
    // compute a reduced Groebner basis of <Gomega> w.r.t. "newRing"
#ifdef  BUCHBERGER_ALG
    M = MstdhomCC(Gomega1);
#else
    M=kStd(Gomega1,NULL,isHomog,NULL,hilb_func,0,NULL,curr_weight);
    delete hilb_func;
#endif
    xtstd = xtstd + clock() - to;

    // change the ring to oldRing
    rChangeCurrRing(oldRing);

    M1 =  idrMoveR(M, newRing,currRing);
    Gomega2 =  idrMoveR(Gomega1, newRing,currRing);

    to = clock();
    F = MLifttwoIdeal(Gomega2, M1, G);
    xtlift = xtlift + clock() -to;

    idDelete(&M1);
    idDelete(&Gomega2);
    idDelete(&G);


    // change the ring to newRing
    rChangeCurrRing(newRing);
    F1 = idrMoveR(F, oldRing,currRing);

    to = clock();
    // reduce the Groebner basis <G> w.r.t. new ring
    G = kInterRedCC(F1, NULL);
    xtred = xtred + clock() -to;

    idDelete(&F1);

    if(endwalks == 1)
    {
      break;
    }
  NEXT_STEP:
    to = clock();
    // compute a next weight vector
    intvec* next_weight = MkInterRedNextWeight(curr_weight,target_weight, G);


    xtnw = xtnw + clock() - to;

#ifdef PRINT_VECTORS
    MivString(curr_weight, target_weight, next_weight);
#endif

    if(Overflow_Error == TRUE)
    {
      //PrintS("\n//REC_GB_Mwalk: The computed vector does NOT stay in the correct cone!!\n");
      nnwinC = 0;
      if(tp_deg == nV)
      {
        nlast = 1;
      }
      delete next_weight;
      break;
    }
    if(MivComp(next_weight, ivNull) == 1)
    {
      newRing = currRing;
      delete next_weight;
      break;
    }

    if(MivComp(next_weight, target_weight) == 1)
    {
      if(tp_deg == nV)
      {
        endwalks = 1;
      }
      else
      {
        G = REC_GB_Mwalk(G,curr_weight, orig_target_weight, tp_deg+1,nnwinC);
        newRing = currRing;
        delete next_weight;
        break;
      }
    }

    for(i=nV-1; i>=0; i--)
    {
      (*curr_weight)[i] = (*next_weight)[i];
    }
    delete next_weight;
  }

  delete ivNull;

  if(tp_deg != nV)
  {
    newRing = currRing;

    if (rParameter(currRing) != NULL)
    {
      DefRingPar(orig_target_weight);
    }
    else
    {
      rChangeCurrRing(VMrDefault(orig_target_weight)); // Aenderung
    }
    F1 = idrMoveR(G, newRing,currRing);

    if(nnwinC == 0)
    {
      F1 = REC_GB_Mwalk(F1,curr_weight, orig_target_weight, tp_deg+1,nnwinC);
    }
    else
    {
      if(test_w_in_ConeCC(F1, target_weight) != 1)
      {
        F1 = REC_GB_Mwalk(F1,curr_weight, orig_target_weight,tp_deg+1,nnwinC);
      }
    }
    delete target_weight;

    TargetRing = currRing;
    rChangeCurrRing(EXXRing);
    result = idrMoveR(F1, TargetRing,currRing);
  }
  else
  {
    if(nlast == 1)
    {
      if (rParameter(currRing) != NULL)
      {
        DefRingPar(orig_target_weight);
      }
      else
      {
        rChangeCurrRing(VMrDefault(orig_target_weight)); // Aenderung
      }
    KSTD_Finish:
      if(isGB == FALSE)
      {
        F1 = idrMoveR(G, newRing,currRing);
      }
      else
      {
        F1 = G;
      }
      to=clock();
      // apply Buchberger alg to compute a red. GB of F1
      G = MstdCC(F1);
      xtextra=clock()-to;
      idDelete(&F1);
      newRing = currRing;
    }

  LastGB_Finish:
    rChangeCurrRing(EXXRing);
    result = idrMoveR(G, newRing,currRing);
  }

  if(Overflow_Error == FALSE)
    {
    Overflow_Error = nError;
    }
#ifndef BUCHBERGER_ALG
  delete last_omega;
#endif
  return(result);
}


// THE NEW GROEBNER WALK ALGORITHM
// Groebnerwalk with a recursive "second" alternative GW, called REC_GB_Mwalk that only computes the last reduced GB
ideal MwalkAlt(ideal Go, intvec* curr_weight, intvec* target_weight)
{
  Set_Error(FALSE);
  Overflow_Error = FALSE;
  //Print("// pSetm_Error = (%d)", ErrorCheck());

  clock_t tinput, tostd, tif=0, tstd=0, tlift=0, tred=0, tnw=0;
  xtif=0; xtstd=0; xtlift=0; xtred=0; xtnw=0;
  tinput = clock();
  clock_t tim;
  nstep=0;
  int i;
  int nV = currRing->N;
  int nwalk=0;
  int endwalks=0;

  ideal Gomega, M, F, Gomega1, Gomega2, M1, F1, G;
  //ideal G1;
  //ring endRing;
  ring newRing, oldRing;
  intvec* ivNull = new intvec(nV);
  intvec* exivlp = Mivlp(nV);
#ifndef BUCHBERGER_ALG
  intvec* hilb_func;
#endif
  intvec* tmp_weight = new intvec(nV);
  for(i=nV-1; i>=0; i--)
    (*tmp_weight)[i] = (*curr_weight)[i];

   // to avoid (1,0,...,0) as the target vector
  intvec* last_omega = new intvec(nV);
  for(i=nV-1; i>0; i--)
    (*last_omega)[i] = 1;
  (*last_omega)[0] = 10000;

  ring XXRing = currRing;

  to = clock();
  // the monomial ordering of this current ring would be "dp"
  G = MstdCC(Go);
  tostd = clock()-to;

  if(currRing->order[0] == ringorder_a)
    goto NEXT_VECTOR;

  while(1)
  {
    nwalk ++;
    nstep ++;
    to = clock();
    // compute an initial form ideal of <G> w.r.t. "curr_vector"
    Gomega = MwalkInitialForm(G, curr_weight);
    tif = tif + clock()-to;
    oldRing = currRing;

    if(endwalks == 1)
    {
      /* compute a reduced Groebner basis of Gomega w.r.t. >>_cw by
         the recursive changed perturbation walk alg. */
      tim = clock();
      /*
        Print("\n// **** Grï¿œbnerwalk took %d steps and ", nwalk);
        PrintS("\n// **** call the rec. Pert. Walk to compute a red GB of:");
        idElements(Gomega, "G_omega");
      */

      if(MivSame(exivlp, target_weight)==1)
        M = REC_GB_Mwalk(idCopy(Gomega), tmp_weight, curr_weight, 2,1);
      else
        goto NORMAL_GW;
      /*
        Print("\n//  time for the last std(Gw)  = %.2f sec",
        ((double) (clock()-tim)/1000000));
        PrintS("\n// ***************************************************\n");
      */
#ifdef CHECK_IDEAL_MWALK
      idElements(Gomega, "G_omega");
      headidString(Gomega, "Gw");
      idElements(M, "M");
      //headidString(M, "M");
#endif
      to = clock();
      F = MLifttwoIdeal(Gomega, M, G);
      xtlift = xtlift + clock() - to;

      idDelete(&Gomega);
      idDelete(&M);
      idDelete(&G);

      oldRing = currRing;

      /* create a new ring newRing */
       if (rParameter(currRing) != NULL)
       {
         DefRingPar(curr_weight);
       }
       else
       {
         rChangeCurrRing(VMrDefault(curr_weight)); // Aenderung
       }
      newRing = currRing;
      F1 = idrMoveR(F, oldRing,currRing);
    }
    else
    {
    NORMAL_GW:
#ifndef  BUCHBERGER_ALG
      if(isNolVector(curr_weight) == 0)
      {
        hilb_func = hFirstSeries(Gomega,NULL,NULL,curr_weight,currRing);
      }
      else
      {
        hilb_func = hFirstSeries(Gomega,NULL,NULL,last_omega,currRing);
      }
#endif // BUCHBERGER_ALG

      // define a new ring that its ordering is "(a(curr_weight),lp)
      if (rParameter(currRing) != NULL)
      {
        DefRingPar(curr_weight);
      }
      else
      {
        rChangeCurrRing(VMrDefault(curr_weight));  //Aenderung
      }
      newRing = currRing;
      Gomega1 = idrMoveR(Gomega, oldRing,currRing);

      to = clock();
      // compute a reduced Groebner basis of <Gomega> w.r.t. "newRing"
#ifdef  BUCHBERGER_ALG
      M = MstdhomCC(Gomega1);
#else
      M=kStd(Gomega1,NULL,isHomog,NULL,hilb_func,0,NULL,curr_weight);
      delete hilb_func;
#endif // BUCHBERGER_ALG
      tstd = tstd + clock() - to;

      // change the ring to oldRing
      rChangeCurrRing(oldRing);
      M1 =  idrMoveR(M, newRing,currRing);
      Gomega2 =  idrMoveR(Gomega1, newRing,currRing);

      to = clock();
      // compute a representation of the generators of submod (M) with respect to those of mod (Gomega). Gomega is a reduced Groebner basis w.r.t. the current ring.
      F = MLifttwoIdeal(Gomega2, M1, G);
      tlift = tlift + clock() - to;

      idDelete(&M1);
      idDelete(&Gomega2);
      idDelete(&G);

      // change the ring to newRing
      rChangeCurrRing(newRing);
      F1 = idrMoveR(F, oldRing,currRing);
    }

    to = clock();
    // reduce the Groebner basis <G> w.r.t. new ring
    G = kInterRedCC(F1, NULL);
    if(endwalks != 1)
    {
      tred = tred + clock() - to;
    }
    else
    {
      xtred = xtred + clock() - to;
    }
    idDelete(&F1);
    if(endwalks == 1)
    {
      break;
    }
  NEXT_VECTOR:
    to = clock();
    // compute a next weight vector
    intvec* next_weight = MkInterRedNextWeight(curr_weight,target_weight,G);
    tnw = tnw + clock() - to;
#ifdef PRINT_VECTORS
    MivString(curr_weight, target_weight, next_weight);
#endif

    //if(test_w_in_ConeCC(G, next_weight) != 1)
    if(Overflow_Error == TRUE)
    {
      newRing = currRing;
      PrintS("\n// ** The computed vector does NOT stay in Cone!!\n");

      if (rParameter(currRing) != NULL)
      {
        DefRingPar(target_weight);
      }
      else
      {
        rChangeCurrRing(VMrDefault(target_weight)); // Aenderung
      }
      F1 = idrMoveR(G, newRing,currRing);
      G = MstdCC(F1);
      idDelete(&F1);

      newRing = currRing;
      break;
    }

    if(MivComp(next_weight, ivNull) == 1)
    {
      newRing = currRing;
      delete next_weight;
      break;
    }
    if(MivComp(next_weight, target_weight) == 1)
    {
      endwalks = 1;
    }
    for(i=nV-1; i>=0; i--)
    {
      (*tmp_weight)[i] = (*curr_weight)[i];
      (*curr_weight)[i] = (*next_weight)[i];
    }
    delete next_weight;
  }
  rChangeCurrRing(XXRing);
  G = idrMoveR(G, newRing,currRing);

  delete tmp_weight;
  delete ivNull;
  delete exivlp;

#ifdef TIME_TEST
  TimeString(tinput, tostd, tif, tstd, tlift, tred, tnw, nstep);

  Print("\n// pSetm_Error = (%d)", ErrorCheck());
  Print("\n// Overflow_Error? (%d)\n", Overflow_Error);
#endif
  return(G);
}


/*******************************
 * THE GROEBNER WALK ALGORITHM *
 *******************************/
ideal Mwalk(ideal Go, intvec* orig_M, intvec* target_M, ring baseRing)
{
  BITSET save1 = si_opt_1; // save current options
  //si_opt_1 &= (~Sy_bit(OPT_REDSB)); // no reduced Groebner basis
  //si_opt_1 &= (~Sy_bit(OPT_REDTAIL)); // not tail reductions
  //si_opt_1|=(Sy_bit(OPT_REDTAIL)|Sy_bit(OPT_REDSB));
  Set_Error(FALSE);
  Overflow_Error = FALSE;
#ifdef TIME_TEST
  clock_t tinput, tostd, tif=0, tstd=0, tlift=0, tred=0, tnw=0;
  xtif=0; xtstd=0; xtlift=0; xtred=0; xtnw=0;
  tinput = clock();
  clock_t tim;
#endif
  nstep=0;
  int i,nwalk,endwalks = 0;
  int nV = baseRing->N;

  ideal Gomega, M, F, Gomega1, Gomega2, M1; //, F1;
  ring newRing;
  ring XXRing = baseRing;
  intvec* ivNull = new intvec(nV);
  intvec* curr_weight = new intvec(nV);
  intvec* target_weight = new intvec(nV);
  intvec* exivlp = Mivlp(nV);
  intvec* tmp_weight = new intvec(nV);
  for(i=0; i<nV; i++)
  {
    (*tmp_weight)[i] = (*target_M)[i];
  }
  for(i=0; i<nV; i++)
  {
    (*curr_weight)[i] = (*orig_M)[i];
    (*target_weight)[i] = (*target_M)[i];
  }
#ifndef BUCHBERGER_ALG
  intvec* hilb_func;
   // to avoid (1,0,...,0) as the target vector
  intvec* last_omega = new intvec(nV);
  for(i=nV-1; i>0; i--)
  {
    (*last_omega)[i] = 1;
  }
  (*last_omega)[0] = 10000;
#endif
  rComplete(currRing);
#ifdef CHECK_IDEAL_MWALK
    idString(Go,"Go");
#endif
#ifdef TIME_TEST
  to = clock();
#endif
     if(orig_M->length() == nV)
      {
        newRing = VMrDefault(curr_weight); // define a new ring with ordering "(a(curr_weight),lp)
      }
      else
      {
        newRing = VMatrDefault(orig_M);
      }
  rChangeCurrRing(newRing);
  ideal G = MstdCC(idrMoveR(Go,baseRing,currRing));
  baseRing = currRing;
#ifdef TIME_TEST
  tostd = clock()-to;
#endif

  nwalk = 0;
  while(1)
  {
    nwalk ++;
    nstep ++;
#ifdef TIME_TEST
    to = clock();
#endif
#ifdef CHECK_IDEAL_MWALK
    idString(G,"G");
#endif
    Gomega = MwalkInitialForm(G, curr_weight); // compute an initial form ideal of <G> w.r.t. "curr_vector"
#ifdef TIME_TEST
    tif = tif + clock()-to; //time for computing initial form ideal
#endif
#ifdef CHECK_IDEAL_MWALK
    idString(Gomega,"Gomega");
#endif
#ifndef  BUCHBERGER_ALG
    if(isNolVector(curr_weight) == 0)
    {
      hilb_func = hFirstSeries(Gomega,NULL,NULL,curr_weight,currRing);
    }
    else
    {
      hilb_func = hFirstSeries(Gomega,NULL,NULL,last_omega,currRing);
    }
#endif
    if(nwalk == 1)
    {
      if(orig_M->length() == nV)
      {
        newRing = VMrDefault(curr_weight); // define a new ring with ordering "(a(curr_weight),lp)
      }
      else
      {
        newRing = VMatrDefault(orig_M);
      }
    }
    else
    {
     if(target_M->length() == nV)
     {
       newRing = VMrRefine(curr_weight,target_weight); //define a new ring with ordering "(a(curr_weight),Wp(target_weight))"
     }
     else
     {
       newRing = VMatrRefine(target_M,curr_weight);
     }
    }
    rChangeCurrRing(newRing);
    Gomega1 = idrMoveR(Gomega, baseRing,currRing);
    idDelete(&Gomega);
    // compute a reduced Groebner basis of <Gomega> w.r.t. "newRing"
#ifdef TIME_TEST
    to = clock();
#endif
#ifndef  BUCHBERGER_ALG
    M=kStd(Gomega1,NULL,isHomog,NULL,hilb_func,0,NULL,curr_weight);
    delete hilb_func;
#else
    M = kStd(Gomega1,NULL,testHomog,NULL,NULL,0,0,NULL);
#endif
#ifdef TIME_TEST
    tstd = tstd + clock() - to;
#endif
    idSkipZeroes(M);
#ifdef CHECK_IDEAL_MWALK
    PrintS("\n//** Mwalk: computed M.\n");
    idString(M, "M");
#endif
    //change the ring to baseRing
    rChangeCurrRing(baseRing);
    M1 =  idrMoveR(M, newRing,currRing);
    idDelete(&M);
    Gomega2 = idrMoveR(Gomega1, newRing,currRing);
    idDelete(&Gomega1);
#ifdef TIME_TEST
    to = clock();
#endif
    // compute a representation of the generators of submod (M) with respect to those of mod (Gomega), where Gomega is a reduced Groebner basis w.r.t. the current ring
    F = MLifttwoIdeal(Gomega2, M1, G);
#ifdef TIME_TEST
    tlift = tlift + clock() - to;
#endif
#ifdef CHECK_IDEAL_MWALK
    idString(F, "F");
#endif
    idDelete(&Gomega2);
    idDelete(&M1);
    rChangeCurrRing(newRing); // change the ring to newRing
    G = idrMoveR(F,baseRing,currRing);
    idDelete(&F);
    baseRing = currRing;
#ifdef TIME_TEST
    to = clock();
#endif
    //G = kStd(F1,NULL,testHomog,NULL,NULL,0,0,NULL);
#ifdef TIME_TEST
    tstd = tstd + clock() - to;
#endif
    idSkipZeroes(G);
#ifdef CHECK_IDEAL_MWALK
    idString(G, "G");
#endif
#ifdef TIME_TEST
    to = clock();
#endif
    intvec* next_weight = MwalkNextWeightCC(curr_weight,target_weight,G);
#ifdef TIME_TEST
    tnw = tnw + clock() - to;
#endif
#ifdef PRINT_VECTORS
    MivString(curr_weight, target_weight, next_weight);
#endif
    if(MivComp(next_weight, ivNull) == 1 || MivComp(target_weight,curr_weight) == 1)// || test_w_in_ConeCC(G, target_weight) == 1 || MivComp(next_weight,curr_weight) == 1)
    {
#ifdef CHECK_IDEAL_MWALK
      PrintS("\n//** Mwalk: entering last cone.\n");
#endif
      Gomega = MwalkInitialForm(G, curr_weight); // compute an initial form ideal of <G> w.r.t. "curr_vector"
      if(target_M->length() == nV)
      {
        newRing = VMrDefault(target_weight); // define a new ring with ordering "(a(curr_weight),lp)
      }
      else
      {
        newRing = VMatrDefault(target_M);
      }
      rChangeCurrRing(newRing);
      Gomega1 = idrMoveR(Gomega, baseRing,currRing);
      idDelete(&Gomega);
#ifdef CHECK_IDEAL_MWALK
      idString(Gomega1, "Gomega");
#endif
      M = kStd(Gomega1,NULL,testHomog,NULL,NULL,0,0,NULL);
#ifdef CHECK_IDEAL_MWALK
      idString(M,"M");
#endif
      rChangeCurrRing(baseRing);
      M1 =  idrMoveR(M, newRing,currRing);
      idDelete(&M);
      Gomega2 = idrMoveR(Gomega1, newRing,currRing);
      idDelete(&Gomega1);
      F = MLifttwoIdeal(Gomega2, M1, G);
#ifdef CHECK_IDEAL_MWALK
      idString(F,"F");
#endif
      idDelete(&Gomega2);
      idDelete(&M1);
      rChangeCurrRing(newRing); // change the ring to newRing
      G = idrMoveR(F,baseRing,currRing);
      idDelete(&F);
      baseRing = currRing;
      si_opt_1 = save1; //set original options, e. g. option(RedSB)
      idSkipZeroes(G);
#ifdef TIME_TEST
      to = clock();
#endif
 //     if(si_opt_1 == (Sy_bit(OPT_REDSB)))
  //    {
        G = kInterRedCC(G,NULL); //reduce the Groebner basis <G> w.r.t. currRing, if option(redSB) is set
  //    }
#ifdef TIME_TEST
      tred = tred + clock() - to;
#endif
      idSkipZeroes(G);
      delete next_weight;
      break;
#ifdef CHECK_IDEAL_MWALK
      PrintS("\n//** Mwalk: last cone.\n");
#endif
    }
#ifdef CHECK_IDEAL_MWALK
    PrintS("\n//** Mwalk: update weight vectors.\n");
#endif
    for(i=nV-1; i>=0; i--)
    {
      (*tmp_weight)[i] = (*curr_weight)[i];
      (*curr_weight)[i] = (*next_weight)[i];
    }
    delete next_weight;
  }
  rChangeCurrRing(XXRing);
  ideal result = idrMoveR(G,baseRing,currRing);
  idDelete(&G);
/*#ifdef CHECK_IDEAL_MWALK
  pDelete(&p);
#endif*/
  delete tmp_weight;
  delete ivNull;
  delete exivlp;
#ifndef BUCHBERGER_ALG
  delete last_omega;
#endif
#ifdef TIME_TEST
  Print("\n//** Mwalk: Groebner Walk took %d steps.\n", nstep);
  TimeString(tinput, tostd, tif, tstd, tlift, tred, tnw, nstep);
  Print("\n//** Mwalk: Ergebnis.\n");
  //Print("\n// pSetm_Error = (%d)", ErrorCheck());
  //Print("\n// Overflow_Error? (%d)\n", Overflow_Error);
#endif
  return(result);
}

// 07.11.2012
// THE RANDOM WALK ALGORITHM  ideal Go, intvec* orig_M, intvec* target_M, ring baseRing
ideal Mrwalk(ideal Go, intvec* orig_M, intvec* target_M, int weight_rad, int pert_deg, ring baseRing)
{
  BITSET save1 = si_opt_1; // save current options
  //si_opt_1 &= (~Sy_bit(OPT_REDSB)); // no reduced Groebner basis
  //si_opt_1 &= (~Sy_bit(OPT_REDTAIL)); // not tail reductions
  //si_opt_1|=(Sy_bit(OPT_REDTAIL)|Sy_bit(OPT_REDSB));
  Set_Error(FALSE);
  Overflow_Error = FALSE;
#ifdef TIME_TEST
  clock_t tinput, tostd, tif=0, tstd=0, tlift=0, tred=0, tnw=0;
  xtif=0; xtstd=0; xtlift=0; xtred=0; xtnw=0;
  tinput = clock();
  clock_t tim;
#endif
  nstep=0;
  int i,nwalk,endwalks = 0;
  int nV = baseRing->N;

  ideal Gomega, M, F, Gomega1, Gomega2, M1; //, F1;
  ring newRing;
  ring XXRing = baseRing;
  intvec* ivNull = new intvec(nV);
  intvec* curr_weight = new intvec(nV);
  intvec* target_weight = new intvec(nV);
  intvec* exivlp = Mivlp(nV);
  intvec* tmp_weight = new intvec(nV);
  for(i=0; i<nV; i++)
  {
    (*tmp_weight)[i] = (*target_M)[i];
  }
  for(i=0; i<nV; i++)
  {
    (*curr_weight)[i] = (*orig_M)[i];
    (*target_weight)[i] = (*target_M)[i];
  }
#ifndef BUCHBERGER_ALG
  intvec* hilb_func;
   // to avoid (1,0,...,0) as the target vector
  intvec* last_omega = new intvec(nV);
  for(i=nV-1; i>0; i--)
  {
    (*last_omega)[i] = 1;
  }
  (*last_omega)[0] = 10000;
#endif
  rComplete(currRing);
#ifdef CHECK_IDEAL_MWALK
    idString(Go,"Go");
#endif
#ifdef TIME_TEST
  to = clock();
#endif
     if(orig_M->length() == nV)
      {
        newRing = VMrDefault(curr_weight); // define a new ring with ordering "(a(curr_weight),lp)
      }
      else
      {
        newRing = VMatrDefault(orig_M);
      }
  rChangeCurrRing(newRing);
  ideal G = MstdCC(idrMoveR(Go,baseRing,currRing));
  baseRing = currRing;
#ifdef TIME_TEST
  tostd = clock()-to;
#endif

  nwalk = 0;
  while(1)
  {
    nwalk ++;
    nstep ++;
#ifdef TIME_TEST
    to = clock();
#endif
#ifdef CHECK_IDEAL_MWALK
    idString(G,"G");
#endif
    Gomega = MwalkInitialForm(G, curr_weight); // compute an initial form ideal of <G> w.r.t. "curr_vector"
#ifdef TIME_TEST
    tif = tif + clock()-to; //time for computing initial form ideal
#endif
#ifdef CHECK_IDEAL_MWALK
    idString(Gomega,"Gomega");
#endif
#ifndef  BUCHBERGER_ALG
    if(isNolVector(curr_weight) == 0)
    {
      hilb_func = hFirstSeries(Gomega,NULL,NULL,curr_weight,currRing);
    }
    else
    {
      hilb_func = hFirstSeries(Gomega,NULL,NULL,last_omega,currRing);
    }
#endif
    if(nwalk == 1)
    {
      if(orig_M->length() == nV)
      {
        newRing = VMrDefault(curr_weight); // define a new ring with ordering "(a(curr_weight),lp)
      }
      else
      {
        newRing = VMatrDefault(orig_M);
      }
    }
    else
    {
     if(target_M->length() == nV)
     {
       newRing = VMrRefine(curr_weight,target_weight); //define a new ring with ordering "(a(curr_weight),Wp(target_weight))"
     }
     else
     {
       newRing = VMatrRefine(target_M,curr_weight);
     }
    }
    rChangeCurrRing(newRing);
    Gomega1 = idrMoveR(Gomega, baseRing,currRing);
    idDelete(&Gomega);
    // compute a reduced Groebner basis of <Gomega> w.r.t. "newRing"
#ifdef TIME_TEST
    to = clock();
#endif
#ifndef  BUCHBERGER_ALG
    M=kStd(Gomega1,NULL,isHomog,NULL,hilb_func,0,NULL,curr_weight);
    delete hilb_func;
#else
    M = kStd(Gomega1,NULL,testHomog,NULL,NULL,0,0,NULL);
#endif
#ifdef TIME_TEST
    tstd = tstd + clock() - to;
#endif
    idSkipZeroes(M);
#ifdef CHECK_IDEAL_MWALK
    PrintS("\n//** Mwalk: computed M.\n");
    idString(M, "M");
#endif
    //change the ring to baseRing
    rChangeCurrRing(baseRing);
    M1 =  idrMoveR(M, newRing,currRing);
    idDelete(&M);
    Gomega2 = idrMoveR(Gomega1, newRing,currRing);
    idDelete(&Gomega1);
#ifdef TIME_TEST
    to = clock();
#endif
    // compute a representation of the generators of submod (M) with respect to those of mod (Gomega), where Gomega is a reduced Groebner basis w.r.t. the current ring
    F = MLifttwoIdeal(Gomega2, M1, G);
#ifdef TIME_TEST
    tlift = tlift + clock() - to;
#endif
#ifdef CHECK_IDEAL_MWALK
    idString(F, "F");
#endif
    idDelete(&Gomega2);
    idDelete(&M1);
    rChangeCurrRing(newRing); // change the ring to newRing
    G = idrMoveR(F,baseRing,currRing);
    idDelete(&F);
    baseRing = currRing;
#ifdef TIME_TEST
    to = clock();
#endif
    //G = kStd(F1,NULL,testHomog,NULL,NULL,0,0,NULL);
#ifdef TIME_TEST
    tstd = tstd + clock() - to;
#endif
    idSkipZeroes(G);
#ifdef CHECK_IDEAL_MWALK
    idString(G, "G");
#endif
#ifdef TIME_TEST
    to = clock();
#endif
    intvec* next_weight = MWalkRandomNextWeight(G, curr_weight, target_weight, weight_rad, pert_deg);//next_weight = MwalkNextWeightCC(curr_weight,target_weight,G);
#ifdef TIME_TEST
    tnw = tnw + clock() - to;
#endif
#ifdef PRINT_VECTORS
    MivString(curr_weight, target_weight, next_weight);
#endif
    if(MivComp(next_weight, ivNull) == 1 || MivComp(target_weight,curr_weight) == 1)// || test_w_in_ConeCC(G, target_weight) == 1 || MivComp(next_weight,curr_weight) == 1)
    {
#ifdef CHECK_IDEAL_MWALK
      PrintS("\n//** Mwalk: entering last cone.\n");
#endif
      Gomega = MwalkInitialForm(G, curr_weight); // compute an initial form ideal of <G> w.r.t. "curr_vector"
      if(target_M->length() == nV)
      {
        newRing = VMrDefault(target_weight); // define a new ring with ordering "(a(curr_weight),lp)
      }
      else
      {
        newRing = VMatrDefault(target_M);
      }
      rChangeCurrRing(newRing);
      Gomega1 = idrMoveR(Gomega, baseRing,currRing);
      idDelete(&Gomega);
#ifdef CHECK_IDEAL_MWALK
      idString(Gomega1, "Gomega");
#endif
      M = kStd(Gomega1,NULL,testHomog,NULL,NULL,0,0,NULL);
#ifdef CHECK_IDEAL_MWALK
      idString(M,"M");
#endif
      rChangeCurrRing(baseRing);
      M1 =  idrMoveR(M, newRing,currRing);
      idDelete(&M);
      Gomega2 = idrMoveR(Gomega1, newRing,currRing);
      idDelete(&Gomega1);
      F = MLifttwoIdeal(Gomega2, M1, G);
#ifdef CHECK_IDEAL_MWALK
      idString(F,"F");
#endif
      idDelete(&Gomega2);
      idDelete(&M1);
      rChangeCurrRing(newRing); // change the ring to newRing
      G = idrMoveR(F,baseRing,currRing);
      idDelete(&F);
      baseRing = currRing;
      si_opt_1 = save1; //set original options, e. g. option(RedSB)
      idSkipZeroes(G);
#ifdef TIME_TEST
      to = clock();
#endif
 //     if(si_opt_1 == (Sy_bit(OPT_REDSB)))
  //    {
        //G = kInterRedCC(G,NULL); //reduce the Groebner basis <G> w.r.t. currRing, if option(redSB) is set
  //    }
#ifdef TIME_TEST
      tred = tred + clock() - to;
#endif
      idSkipZeroes(G);
      delete next_weight;
      break;
#ifdef CHECK_IDEAL_MWALK
      PrintS("\n//** Mwalk: last cone.\n");
#endif
    }
#ifdef CHECK_IDEAL_MWALK
    PrintS("\n//** Mwalk: update weight vectors.\n");
#endif
    for(i=nV-1; i>=0; i--)
    {
      (*tmp_weight)[i] = (*curr_weight)[i];
      (*curr_weight)[i] = (*next_weight)[i];
    }
    delete next_weight;
  }
  rChangeCurrRing(XXRing);
  ideal result = idrMoveR(G,baseRing,currRing);
  idDelete(&G);
/*#ifdef CHECK_IDEAL_MWALK
  pDelete(&p);
#endif*/
  delete tmp_weight;
  delete ivNull;
  delete exivlp;
#ifndef BUCHBERGER_ALG
  delete last_omega;
#endif
#ifdef TIME_TEST
  Print("\n//** Mwalk: Groebner Walk took %d steps.\n", nstep);
  TimeString(tinput, tostd, tif, tstd, tlift, tred, tnw, nstep);
  Print("\n//** Mwalk: Ergebnis.\n");
  //Print("\n// pSetm_Error = (%d)", ErrorCheck());
  //Print("\n// Overflow_Error? (%d)\n", Overflow_Error);
#endif
  return(result);
}

//unused
#if 0
ideal Mwalk_tst(ideal Go, intvec* curr_weight, intvec* target_weight)
{
  //clock_t tinput=clock();
  //idString(Go,"Ginp");
  int i, nV = currRing->N;
  int nwalk=0, endwalks=0;

  ideal Gomega, M, F, Gomega1, Gomega2, M1, F1, G;
  // ideal G1; ring endRing;
  ring newRing, oldRing;
  intvec* ivNull = new intvec(nV);
  ring XXRing = currRing;

  intvec* tmp_weight = new intvec(nV);
  for(i=nV-1; i>=0; i--)
  {
    (*tmp_weight)[i] = (*curr_weight)[i];
  }
  /* the monomial ordering of this current ring would be "dp" */
  G = MstdCC(Go);
#ifndef BUCHBERGER_ALG
  intvec* hilb_func;
#endif
  /* to avoid (1,0,...,0) as the target vector */
  intvec* last_omega = new intvec(nV);
  for(i=nV-1; i>0; i--)
    (*last_omega)[i] = 1;
  (*last_omega)[0] = 10000;

  while(1)
  {
    nwalk ++;
    //Print("\n// Entering the %d-th step:", nwalk);
    //Print("\n// ring r[%d] = %s;", nwalk, rString(currRing));
    idString(G,"G");
    /* compute an initial form ideal of <G> w.r.t. "curr_vector" */
    Gomega = MwalkInitialForm(G, curr_weight);
    //ivString(curr_weight, "omega");
    idString(Gomega,"Gw");

#ifndef  BUCHBERGER_ALG
    if(isNolVector(curr_weight) == 0)
      hilb_func = hFirstSeries(Gomega,NULL,NULL,curr_weight,currRing);
    else
      hilb_func = hFirstSeries(Gomega,NULL,NULL,last_omega,currRing);
#endif // BUCHBERGER_ALG


    oldRing = currRing;

    /* define a new ring that its ordering is "(a(curr_weight),lp) */
    VMrDefault(curr_weight);
    newRing = currRing;

    Gomega1 = idrMoveR(Gomega, oldRing,currRing);

    /* compute a reduced Groebner basis of <Gomega> w.r.t. "newRing" */
#ifdef  BUCHBERGER_ALG
    M = MstdhomCC(Gomega1);
#else
    M=kStd(Gomega1,NULL,isHomog,NULL,hilb_func,0,NULL,curr_weight);
    delete hilb_func;
#endif // BUCHBERGER_ALG

    idString(M,"M");

      /* change the ring to oldRing */
    rChangeCurrRing(oldRing);
    M1 =  idrMoveR(M, newRing,currRing);
    Gomega2 =  idrMoveR(Gomega1, newRing,currRing);

      /* compute a representation of the generators of submod (M)
         with respect to those of mod (Gomega).
         Gomega is a reduced Groebner basis w.r.t. the current ring */
    F = MLifttwoIdeal(Gomega2, M1, G);
    idDelete(&M1);
    idDelete(&Gomega2);
    idDelete(&G);
    idString(F,"F");

    /* change the ring to newRing */
    rChangeCurrRing(newRing);
    F1 = idrMoveR(F, oldRing,currRing);

    /* reduce the Groebner basis <G> w.r.t. new ring */
    G = kInterRedCC(F1, NULL);
    //idSkipZeroes(G);//done by kInterRed
    idDelete(&F1);
    idString(G,"G");
    if(endwalks == 1)
      break;

    /* compute a next weight vector */
    intvec* next_weight = MkInterRedNextWeight(curr_weight,target_weight,G);
#ifdef PRINT_VECTORS
    MivString(curr_weight, target_weight, next_weight);
#endif

    if(MivComp(next_weight, ivNull) == 1)
    {
      delete next_weight;
      break;
    }
    if(MivComp(next_weight, target_weight) == 1)
      endwalks = 1;

    for(i=nV-1; i>=0; i--)
      (*tmp_weight)[i] = (*curr_weight)[i];

    /* 06.11.01 to free the memory: NOT Changed!!*/
    for(i=nV-1; i>=0; i--)
      (*curr_weight)[i] = (*next_weight)[i];
    delete next_weight;
  }
  rChangeCurrRing(XXRing);
  G = idrMoveR(G, newRing,currRing);

  delete tmp_weight;
  delete ivNull;
  PrintLn();
  return(G);
}
#endif

/**************************************************************/
/*     Implementation of the perturbation walk algorithm      */
/**************************************************************/
/* If the perturbed target weight vector or an intermediate weight vector
   doesn't stay in the correct Groebner cone, we have only
   a reduced Groebner basis for the given ideal with respect to
   a monomial order which differs to the given order.
   Then we have to compute the wanted  reduced Groebner basis for it.
   For this, we can use
   1) the improved Buchberger algorithm or
   2) the changed perturbation walk algorithm with a decreased degree.
*/
// use kStd, if nP = 0, else call LastGB
ideal Mpwalk(ideal Go, int op_deg, int tp_deg,intvec* curr_weight,
             intvec* target_weight, int nP)
{
  Set_Error(FALSE  );
  Overflow_Error = FALSE;
  //Print("// pSetm_Error = (%d)", ErrorCheck());

  clock_t  tinput, tostd, tif=0, tstd=0, tlift=0, tred=0, tnw=0;
  xtextra=0;
  xtif=0; xtstd=0; xtlift=0; xtred=0; xtnw=0;
  tinput = clock();

  clock_t tim;

  nstep = 0;
  int i, ntwC=1, ntestw=1,  nV = currRing->N;
  int endwalks=0;

  ideal Gomega, M, F, G, Gomega1, Gomega2, M1,F1,Eresult,ssG;
  ring newRing, oldRing, TargetRing;
  intvec* iv_M_dp;
  intvec* iv_M_lp;
  intvec* exivlp = Mivlp(nV);
  intvec* orig_target = target_weight;
  intvec* pert_target_vector = target_weight;
  intvec* ivNull = new intvec(nV);
  intvec* iv_dp = MivUnit(nV);// define (1,1,...,1)
#ifndef BUCHBERGER_ALG
  intvec* hilb_func;
#endif
  intvec* next_weight;

  // to avoid (1,0,...,0) as the target vector
  intvec* last_omega = new intvec(nV);
  for(i=nV-1; i>0; i--)
    (*last_omega)[i] = 1;
  (*last_omega)[0] = 10000;

  ring XXRing = currRing;


  to = clock();
  /* perturbs the original vector */
  if(MivComp(curr_weight, iv_dp) == 1) //rOrdStr(currRing) := "dp"
  {
    G = MstdCC(Go);
    tostd = clock()-to;
    if(op_deg != 1){
      iv_M_dp = MivMatrixOrderdp(nV);
      //ivString(iv_M_dp, "iv_M_dp");
      curr_weight = MPertVectors(G, iv_M_dp, op_deg);
    }
  }
  else
  {
    //define ring order := (a(curr_weight),lp);
    if (rParameter(currRing) != NULL)
      DefRingPar(curr_weight);
    else
      rChangeCurrRing(VMrDefault(curr_weight)); //Aenderung 1

    G = idrMoveR(Go, XXRing,currRing);
    G = MstdCC(G);
    tostd = clock()-to;
    if(op_deg != 1){
      iv_M_dp = MivMatrixOrder(curr_weight);
      curr_weight = MPertVectors(G, iv_M_dp, op_deg);
    }
  }
  delete iv_dp;
  if(op_deg != 1) delete iv_M_dp;

  ring HelpRing = currRing;

  /* perturbs the target weight vector */
  if(tp_deg > 1 && tp_deg <= nV)
  {
    if (rParameter(currRing) != NULL)
      DefRingPar(target_weight);
    else
      rChangeCurrRing(VMrDefault(target_weight)); // Aenderung 2

    TargetRing = currRing;
    ssG = idrMoveR(G,HelpRing,currRing);
    if(MivSame(target_weight, exivlp) == 1)
    {
      iv_M_lp = MivMatrixOrderlp(nV);
      //ivString(iv_M_lp, "iv_M_lp");
      //target_weight = MPertVectorslp(ssG, iv_M_lp, tp_deg);
      target_weight = MPertVectors(ssG, iv_M_lp, tp_deg);
    }
    else
    {
      iv_M_lp = MivMatrixOrder(target_weight);
      //target_weight = MPertVectorslp(ssG, iv_M_lp, tp_deg);
      target_weight = MPertVectors(ssG, iv_M_lp, tp_deg);
    }
    delete iv_M_lp;
    pert_target_vector = target_weight;
    rChangeCurrRing(HelpRing);
    G = idrMoveR(ssG, TargetRing,currRing);
  }
  /*
    Print("\n// Perturbationwalkalg. vom Gradpaar (%d,%d):",op_deg,tp_deg);
    ivString(curr_weight, "new sigma");
    ivString(target_weight, "new tau");
  */
  while(1)
  {
    nstep ++;
    to = clock();
    /* compute an initial form ideal of <G> w.r.t. the weight vector
       "curr_weight" */
    Gomega = MwalkInitialForm(G, curr_weight);


#ifdef ENDWALKS
    if(endwalks == 1){
      Print("\n// ring r%d = %s;\n", nstep, rString(currRing));
      idElements(G, "G");
      // idElements(Gomega, "Gw");
      headidString(G, "G");
      //headidString(Gomega, "Gw");
    }
#endif

    tif = tif + clock()-to;

#ifndef  BUCHBERGER_ALG
    if(isNolVector(curr_weight) == 0)
      hilb_func = hFirstSeries(Gomega,NULL,NULL,curr_weight,currRing);
    else
      hilb_func = hFirstSeries(Gomega,NULL,NULL,last_omega,currRing);
#endif // BUCHBERGER_ALG

    oldRing = currRing;

    // define a new ring with ordering "(a(curr_weight),lp)
    if (rParameter(currRing) != NULL)
      DefRingPar(curr_weight);
    else
      rChangeCurrRing(VMrDefault(curr_weight)); //Aenderung 3

    newRing = currRing;
    Gomega1 = idrMoveR(Gomega, oldRing,currRing);

#ifdef ENDWALKS
    if(endwalks==1)
    {
      Print("\n// ring r%d = %s;\n", nstep, rString(currRing));
      idElements(Gomega1, "Gw");
      headidString(Gomega1, "headGw");
      PrintS("\n// compute a rGB of Gw:\n");

#ifndef  BUCHBERGER_ALG
      ivString(hilb_func, "w");
#endif
    }
#endif

    tim = clock();
    to = clock();
    /* compute a reduced Groebner basis of <Gomega> w.r.t. "newRing" */
#ifdef  BUCHBERGER_ALG
    M = MstdhomCC(Gomega1);
#else
    M=kStd(Gomega1,NULL,isHomog,NULL,hilb_func,0,NULL,curr_weight);
    delete hilb_func;
#endif // BUCHBERGER_ALG

    if(endwalks == 1){
      xtstd = xtstd+clock()-to;
#ifdef ENDWALKS
      Print("\n// time for the last std(Gw)  = %.2f sec\n",
            ((double) clock())/1000000 -((double)tim) /1000000);
#endif
    }
    else
      tstd=tstd+clock()-to;

    /* change the ring to oldRing */
    rChangeCurrRing(oldRing);
    M1 =  idrMoveR(M, newRing,currRing);
    Gomega2 =  idrMoveR(Gomega1, newRing,currRing);

    //if(endwalks==1)  PrintS("\n// Lifting is working:..");

    to=clock();
    /* compute a representation of the generators of submod (M)
       with respect to those of mod (Gomega).
       Gomega is a reduced Groebner basis w.r.t. the current ring */
    F = MLifttwoIdeal(Gomega2, M1, G);
    if(endwalks != 1)
      tlift = tlift+clock()-to;
    else
      xtlift=clock()-to;

    idDelete(&M1);
    idDelete(&Gomega2);
    idDelete(&G);

    /* change the ring to newRing */
    rChangeCurrRing(newRing);
    F1 = idrMoveR(F, oldRing,currRing);

    //if(endwalks==1)PrintS("\n// InterRed is working now:");

    to=clock();
    /* reduce the Groebner basis <G> w.r.t. new ring */
    G = kInterRedCC(F1, NULL);
    if(endwalks != 1)
      tred = tred+clock()-to;
    else
      xtred=clock()-to;

    idDelete(&F1);

    if(endwalks == 1)
      break;

    to=clock();
    /* compute a next weight vector */
    next_weight = MkInterRedNextWeight(curr_weight,target_weight, G);
    tnw=tnw+clock()-to;
#ifdef PRINT_VECTORS
    MivString(curr_weight, target_weight, next_weight);
#endif

    if(Overflow_Error == TRUE)
    {
      ntwC = 0;
      //ntestomega = 1;
      //Print("\n// ring r%d = %s;\n", nstep, rString(currRing));
      //idElements(G, "G");
      delete next_weight;
      goto FINISH_160302;
    }
    if(MivComp(next_weight, ivNull) == 1){
      newRing = currRing;
      delete next_weight;
      //Print("\n// ring r%d = %s;\n", nstep, rString(currRing));
      break;
    }
    if(MivComp(next_weight, target_weight) == 1)
      endwalks = 1;

    for(i=nV-1; i>=0; i--)
      (*curr_weight)[i] = (*next_weight)[i];

    delete next_weight;
  }//while

  if(tp_deg != 1)
  {
  FINISH_160302:
    if(MivSame(orig_target, exivlp) == 1)
      if (rParameter(currRing) != NULL)
        DefRingParlp();
      else
        VMrDefaultlp();
    else
      if (rParameter(currRing) != NULL)
        DefRingPar(orig_target);
      else
        rChangeCurrRing(VMrDefault(orig_target)); //Aenderung

    TargetRing=currRing;
    F1 = idrMoveR(G, newRing,currRing);
#ifdef CHECK_IDEAL
      headidString(G, "G");
#endif


    // check whether the pertubed target vector stays in the correct cone
    if(ntwC != 0){
      ntestw = test_w_in_ConeCC(F1, pert_target_vector);
    }

    if( ntestw != 1 || ntwC == 0)
    {
      /*
      if(ntestw != 1){
        ivString(pert_target_vector, "tau");
        PrintS("\n// ** perturbed target vector doesn't stay in cone!!");
        Print("\n// ring r%d = %s;\n", nstep, rString(currRing));
        idElements(F1, "G");
      }
      */
      // LastGB is "better" than the kStd subroutine
      to=clock();
      ideal eF1;
      if(nP == 0 || tp_deg == 1 || MivSame(orig_target, exivlp) != 1){
        // PrintS("\n// ** calls \"std\" to compute a GB");
        eF1 = MstdCC(F1);
        idDelete(&F1);
      }
      else {
        // PrintS("\n// ** calls \"LastGB\" to compute a GB");
        rChangeCurrRing(newRing);
        ideal F2 = idrMoveR(F1, TargetRing,currRing);
        eF1 = LastGB(F2, curr_weight, tp_deg-1);
        F2=NULL;
      }
      xtextra=clock()-to;
      ring exTargetRing = currRing;

      rChangeCurrRing(XXRing);
      Eresult = idrMoveR(eF1, exTargetRing,currRing);
    }
    else{
      rChangeCurrRing(XXRing);
      Eresult = idrMoveR(F1, TargetRing,currRing);
    }
  }
  else {
    rChangeCurrRing(XXRing);
    Eresult = idrMoveR(G, newRing,currRing);
  }
  delete ivNull;
  if(tp_deg != 1)
    delete target_weight;

  if(op_deg != 1 )
    delete curr_weight;

  delete exivlp;
  delete last_omega;

#ifdef TIME_TEST
  TimeStringFractal(tinput, tostd, tif+xtif, tstd+xtstd,0, tlift+xtlift, tred+xtred,
             tnw+xtnw);

  Print("\n// pSetm_Error = (%d)", ErrorCheck());
  Print("\n// It took %d steps and Overflow_Error? (%d)\n", nstep,  Overflow_Error);
#endif
  return(Eresult);
}

intvec* XivNull;

// define a matrix (1 ... 1)
intvec* MMatrixone(int nV)
{
  int i,j;
  intvec* ivM = new intvec(nV*nV);

  for(i=0; i<nV; i++)
    for(j=0; j<nV; j++)
    (*ivM)[i*nV + j] = 1;

  return(ivM);
}

int nnflow;
int Xcall;
int Xngleich;

/***********************************************************************
 * Perturb the start weight vector at the top level, i.e. nlev = 1     *
 ***********************************************************************/
static ideal rec_fractal_call(ideal G, int nlev, intvec* omtmp)
{
  Overflow_Error =  FALSE;
  //Print("\n\n// Entering the %d-th recursion:", nlev);

  int i, nV = currRing->N;
  ring new_ring, testring;
  //ring extoRing;
  ideal Gomega, Gomega1, Gomega2, F, F1, Gresult, Gresult1, G1, Gt;
  int nwalks = 0;
  intvec* Mwlp;
#ifndef BUCHBERGER_ALG
  intvec* hilb_func;
#endif
//  intvec* extXtau;
  intvec* next_vect;
  intvec* omega2 = new intvec(nV);
  intvec* altomega = new intvec(nV);

  //BOOLEAN isnewtarget = FALSE;

  // to avoid (1,0,...,0) as the target vector (Hans)
  intvec* last_omega = new intvec(nV);
  for(i=nV-1; i>0; i--)
    (*last_omega)[i] = 1;
  (*last_omega)[0] = 10000;

  intvec* omega = new intvec(nV);
  for(i=0; i<nV; i++) {
    if(Xsigma->length() == nV)
      (*omega)[i] =  (*Xsigma)[i];
    else
      (*omega)[i] = (*Xsigma)[(nV*(nlev-1))+i];

    (*omega2)[i] = (*Xtau)[(nlev-1)*nV+i];
  }

   if(nlev == 1)  Xcall = 1;
   else Xcall = 0;

  ring oRing = currRing;

  while(1)
  {
#ifdef FIRST_STEP_FRACTAL
    // perturb the current weight vector only on the top level or
    // after perturbation of the both vectors, nlev = 2 as the top level
    if((nlev == 1 && Xcall == 0) || (nlev == 2 && Xngleich == 1))
      if(islengthpoly2(G) == 1)
      {
        Mwlp = MivWeightOrderlp(omega);
        Xsigma = Mfpertvector(G, Mwlp);
        delete Mwlp;
        Overflow_Error = FALSE;
      }
#endif
    nwalks ++;
    NEXT_VECTOR_FRACTAL:
    to=clock();
    /* determine the next border */
    next_vect = MkInterRedNextWeight(omega,omega2,G);
    xtnw=xtnw+clock()-to;
#ifdef PRINT_VECTORS
    MivString(omega, omega2, next_vect);
#endif
    oRing = currRing;

    /* We only perturb the current target vector at the recursion  level 1 */
    if(Xngleich == 0 && nlev == 1) //(ngleich == 0) important, e.g. ex2, ex3
      if (MivComp(next_vect, omega2) != 1)
      {
        /* to dispense with taking initial (and lifting/interreducing
           after the call of recursion */
        //Print("\n\n// ** Perturb the both vectors with degree %d with",nlev);
        //idElements(G, "G");

        Xngleich = 1;
        nlev +=1;

        if (rParameter(currRing) != NULL)
          DefRingPar(omtmp);
        else
          rChangeCurrRing(VMrDefault1(omtmp)); //Aenderung3

        testring = currRing;
        Gt = idrMoveR(G, oRing,currRing);

        /* perturb the original target vector w.r.t. the current GB */
        delete Xtau;
        Xtau = NewVectorlp(Gt);

        rChangeCurrRing(oRing);
        G = idrMoveR(Gt, testring,currRing);

        /* perturb the current vector w.r.t. the current GB */
        Mwlp = MivWeightOrderlp(omega);
        Xsigma = Mfpertvector(G, Mwlp);
        delete Mwlp;

        for(i=nV-1; i>=0; i--) {
          (*omega2)[i] = (*Xtau)[nV+i];
          (*omega)[i] = (*Xsigma)[nV+i];
        }

        delete next_vect;
        to=clock();

        /* to avoid the value of Overflow_Error that occur in Mfpertvector*/
        Overflow_Error = FALSE;

        next_vect = MkInterRedNextWeight(omega,omega2,G);
        xtnw=xtnw+clock()-to;

#ifdef PRINT_VECTORS
        MivString(omega, omega2, next_vect);
#endif
      }


    /* check whether the the computed vector is in the correct cone */
    /* If no, the reduced GB of an omega-homogeneous ideal will be
       computed by Buchberger algorithm and stop this recursion step*/
    //if(test_w_in_ConeCC(G, next_vect) != 1) //e.g. Example s7, cyc6
    if(Overflow_Error == TRUE)
    {
      delete next_vect;
      if (rParameter(currRing) != NULL)
      {
        DefRingPar(omtmp);
      }
      else
      {
        rChangeCurrRing(VMrDefault1(omtmp)); // Aenderung4
      }
#ifdef TEST_OVERFLOW
      Gt = idrMoveR(G, oRing,currRing);
      Gt = NULL; return(Gt);
#endif

      //Print("\n\n// apply BB's alg. in ring r = %s;", rString(currRing));
      to=clock();
      Gt = idrMoveR(G, oRing,currRing);
      G1 = MstdCC(Gt);
      xtextra=xtextra+clock()-to;
      Gt = NULL;

      delete omega2;
      delete altomega;

      //Print("\n// Leaving the %d-th recursion with %d steps", nlev, nwalks);
      //Print("  ** Overflow_Error? (%d)", Overflow_Error);
      nnflow ++;

      Overflow_Error = FALSE;
      return (G1);
    }


    /* If the perturbed target vector stays in the correct cone,
       return the current GB,
       otherwise, return the computed  GB by the Buchberger-algorithm.
       Then we update the perturbed target vectors w.r.t. this GB. */

    /* the computed vector is equal to the origin vector, since
       t is not defined */
    if (MivComp(next_vect, XivNull) == 1)
    {
      if (rParameter(currRing) != NULL)
        DefRingPar(omtmp);
      else
        rChangeCurrRing(VMrDefault1(omtmp)); //Aenderung5

      testring = currRing;
      Gt = idrMoveR(G, oRing,currRing);

      if(test_w_in_ConeCC(Gt, omega2) == 1) {
        delete omega2;
        delete next_vect;
        delete altomega;
        //Print("\n// Leaving the %d-th recursion with %d steps ",nlev, nwalks);
        //Print(" ** Overflow_Error? (%d)", Overflow_Error);

        return (Gt);
      }
      else
      {
        //ivString(omega2, "tau'");
        //Print("\n//  tau' doesn't stay in the correct cone!!");

#ifndef  MSTDCC_FRACTAL
        //07.08.03
        //ivString(Xtau, "old Xtau");
        intvec* Xtautmp = Mfpertvector(Gt, MivMatrixOrder(omtmp));
#ifdef TEST_OVERFLOW
      if(Overflow_Error == TRUE)
      Gt = NULL; return(Gt);
#endif

        if(MivSame(Xtau, Xtautmp) == 1)
        {
          //PrintS("\n// Update vectors are equal to the old vectors!!");
          delete Xtautmp;
          goto FRACTAL_MSTDCC;
        }

        Xtau = Xtautmp;
        Xtautmp = NULL;
        //ivString(Xtau, "new  Xtau");

        for(i=nV-1; i>=0; i--)
          (*omega2)[i] = (*Xtau)[(nlev-1)*nV+i];

        //Print("\n//  ring tau = %s;", rString(currRing));
        rChangeCurrRing(oRing);
        G = idrMoveR(Gt, testring,currRing);

        goto NEXT_VECTOR_FRACTAL;
#endif

      FRACTAL_MSTDCC:
        //Print("\n//  apply BB-Alg in ring = %s;", rString(currRing));
        to=clock();
        G = MstdCC(Gt);
        xtextra=xtextra+clock()-to;

        oRing = currRing;

        // update the original target vector w.r.t. the current GB
        if(MivSame(Xivinput, Xivlp) == 1)
          if (rParameter(currRing) != NULL)
            DefRingParlp();
          else
            VMrDefaultlp();
        else
          if (rParameter(currRing) != NULL)
            DefRingPar(Xivinput);
          else
            rChangeCurrRing(VMrDefault1(Xivinput)); //Aenderung6

        testring = currRing;
        Gt = idrMoveR(G, oRing,currRing);

        delete Xtau;
        Xtau = NewVectorlp(Gt);

        rChangeCurrRing(oRing);
        G = idrMoveR(Gt, testring,currRing);

        delete omega2;
        delete next_vect;
        delete altomega;
        /*
          Print("\n// Leaving the %d-th recursion with %d steps,", nlev,nwalks);
          Print(" ** Overflow_Error? (%d)", Overflow_Error);
        */
        if(Overflow_Error == TRUE)
          nnflow ++;

        Overflow_Error = FALSE;
        return(G);
      }
    }

    for(i=nV-1; i>=0; i--) {
      (*altomega)[i] = (*omega)[i];
      (*omega)[i] = (*next_vect)[i];
    }
    delete next_vect;

    to=clock();
    /* Take the initial form of <G> w.r.t. omega */
    Gomega = MwalkInitialForm(G, omega);
    xtif=xtif+clock()-to;

#ifndef  BUCHBERGER_ALG
    if(isNolVector(omega) == 0)
      hilb_func = hFirstSeries(Gomega,NULL,NULL,omega,currRing);
    else
      hilb_func = hFirstSeries(Gomega,NULL,NULL,last_omega,currRing);
#endif // BUCHBERGER_ALG

    if (rParameter(currRing) != NULL)
      DefRingPar(omega);
    else
      rChangeCurrRing(VMrDefault1(omega)); //Aenderung7

    Gomega1 = idrMoveR(Gomega, oRing,currRing);

    /* Maximal recursion depth, to compute a red. GB */
    /* Fractal walk with the alternative recursion */
    /* alternative recursion */
    // if(nlev == nV || lengthpoly(Gomega1) == 0)
    if(nlev == Xnlev || lengthpoly(Gomega1) == 0)
      //if(nlev == nV) // blind recursion
    {
      /*
      if(Xnlev != nV)
      {
        Print("\n// ** Xnlev = %d", Xnlev);
        ivString(Xtau, "Xtau");
      }
      */
      to=clock();
#ifdef  BUCHBERGER_ALG
      Gresult = MstdhomCC(Gomega1);
#else
      Gresult =kStd(Gomega1,NULL,isHomog,NULL,hilb_func,0,NULL,omega);
      delete hilb_func;
#endif // BUCHBERGER_ALG
      xtstd=xtstd+clock()-to;
    }
    else {
      rChangeCurrRing(oRing);
      Gomega1 = idrMoveR(Gomega1, oRing,currRing);
      Gresult = rec_fractal_call(idCopy(Gomega1),nlev+1,omega);
    }

    //convert a Groebner basis from a ring to another ring,
    new_ring = currRing;

    rChangeCurrRing(oRing);
    Gresult1 = idrMoveR(Gresult, new_ring,currRing);
    Gomega2 = idrMoveR(Gomega1, new_ring,currRing);

    to=clock();
    /* Lifting process */
    F = MLifttwoIdeal(Gomega2, Gresult1, G);
    xtlift=xtlift+clock()-to;
    idDelete(&Gresult1);
    idDelete(&Gomega2);
    idDelete(&G);

    rChangeCurrRing(new_ring);
    F1 = idrMoveR(F, oRing,currRing);

    to=clock();
    /* Interreduce G */
    G = kInterRedCC(F1, NULL);
    xtred=xtred+clock()-to;
    idDelete(&F1);
  }
}

/************************************************************************
 * Perturb the start weight vector at the top level with random element *
 ************************************************************************/
static ideal rec_r_fractal_call(ideal G, int nlev, intvec* omtmp, int weight_rad)
{
  Overflow_Error =  FALSE;
  //Print("\n\n// Entering the %d-th recursion:", nlev);

  int i, nV = currRing->N;
  ring new_ring, testring;
  //ring extoRing;
  ideal Gomega, Gomega1, Gomega2, F, F1, Gresult, Gresult1, G1, Gt;
  int nwalks = 0;
  intvec* Mwlp;
#ifndef BUCHBERGER_ALG
  intvec* hilb_func;
#endif
//  intvec* extXtau;
  intvec* next_vect;
  intvec* omega2 = new intvec(nV);
  intvec* altomega = new intvec(nV);

  //BOOLEAN isnewtarget = FALSE;

  // to avoid (1,0,...,0) as the target vector (Hans)
  intvec* last_omega = new intvec(nV);
  for(i=nV-1; i>0; i--)
    (*last_omega)[i] = 1;
  (*last_omega)[0] = 10000;

  intvec* omega = new intvec(nV);
  for(i=0; i<nV; i++) {
    if(Xsigma->length() == nV)
      (*omega)[i] =  (*Xsigma)[i];
    else
      (*omega)[i] = (*Xsigma)[(nV*(nlev-1))+i];

    (*omega2)[i] = (*Xtau)[(nlev-1)*nV+i];
  }

   if(nlev == 1)  Xcall = 1;
   else Xcall = 0;

  ring oRing = currRing;

  while(1)
  {
#ifdef FIRST_STEP_FRACTAL
    // perturb the current weight vector only on the top level or
    // after perturbation of the both vectors, nlev = 2 as the top level
    if((nlev == 1 && Xcall == 0) || (nlev == 2 && Xngleich == 1))
      if(islengthpoly2(G) == 1)
      {
        Mwlp = MivWeightOrderlp(omega);
        Xsigma = Mfpertvector(G, Mwlp);
        delete Mwlp;
        Overflow_Error = FALSE;
      }
#endif
    nwalks ++;
    NEXT_VECTOR_FRACTAL:
    to=clock();
    /* determine the next border */
    next_vect = MWalkRandomNextWeight(G, omega,omega2, weight_rad, 1+nlev);
    //next_vect = MkInterRedNextWeight(omega,omega2,G);
    xtnw=xtnw+clock()-to;
#ifdef PRINT_VECTORS
    MivString(omega, omega2, next_vect);
#endif
    oRing = currRing;

    /* We only perturb the current target vector at the recursion  level 1 */
    if(Xngleich == 0 && nlev == 1) //(ngleich == 0) important, e.g. ex2, ex3
      if (MivComp(next_vect, omega2) == 1)
      {
        /* to dispense with taking initial (and lifting/interreducing
           after the call of recursion */
        //Print("\n\n// ** Perturb the both vectors with degree %d with",nlev);
        //idElements(G, "G");

        Xngleich = 1;
        nlev +=1;

        if (rParameter(currRing) != NULL)
          DefRingPar(omtmp);
        else
          rChangeCurrRing(VMrDefault1(omtmp)); //Aenderung3

        testring = currRing;
        Gt = idrMoveR(G, oRing,currRing);

        /* perturb the original target vector w.r.t. the current GB */
        delete Xtau;
        Xtau = NewVectorlp(Gt);

        rChangeCurrRing(oRing);
        G = idrMoveR(Gt, testring,currRing);

        /* perturb the current vector w.r.t. the current GB */
        Mwlp = MivWeightOrderlp(omega);
        Xsigma = Mfpertvector(G, Mwlp);
        delete Mwlp;

        for(i=nV-1; i>=0; i--) {
          (*omega2)[i] = (*Xtau)[nV+i];
          (*omega)[i] = (*Xsigma)[nV+i];
        }

        delete next_vect;
        to=clock();

        /* to avoid the value of Overflow_Error that occur in Mfpertvector*/
        Overflow_Error = FALSE;

        next_vect = MkInterRedNextWeight(omega,omega2,G);
        xtnw=xtnw+clock()-to;

#ifdef PRINT_VECTORS
        MivString(omega, omega2, next_vect);
#endif
      }


    /* check whether the the computed vector is in the correct cone */
    /* If no, the reduced GB of an omega-homogeneous ideal will be
       computed by Buchberger algorithm and stop this recursion step*/
    //if(test_w_in_ConeCC(G, next_vect) != 1) //e.g. Example s7, cyc6
    if(Overflow_Error == TRUE)
    {
      delete next_vect;
      if (rParameter(currRing) != NULL)
      {
        DefRingPar(omtmp);
      }
      else
      {
        rChangeCurrRing(VMrDefault1(omtmp)); // Aenderung4
      }
#ifdef TEST_OVERFLOW
      Gt = idrMoveR(G, oRing,currRing);
      Gt = NULL; return(Gt);
#endif

      //Print("\n\n// apply BB's alg. in ring r = %s;", rString(currRing));
      to=clock();
      Gt = idrMoveR(G, oRing,currRing);
      G1 = MstdCC(Gt);
      xtextra=xtextra+clock()-to;
      Gt = NULL;

      delete omega2;
      delete altomega;

      //Print("\n// Leaving the %d-th recursion with %d steps", nlev, nwalks);
      //Print("  ** Overflow_Error? (%d)", Overflow_Error);
      nnflow ++;

      Overflow_Error = FALSE;
      return (G1);
    }


    /* If the perturbed target vector stays in the correct cone,
       return the current GB,
       otherwise, return the computed  GB by the Buchberger-algorithm.
       Then we update the perturbed target vectors w.r.t. this GB. */

    /* the computed vector is equal to the origin vector, since
       t is not defined */
    if (MivComp(next_vect, XivNull) == 1)
    {
      if (rParameter(currRing) != NULL)
        DefRingPar(omtmp);
      else
        rChangeCurrRing(VMrDefault1(omtmp)); //Aenderung5

      testring = currRing;
      Gt = idrMoveR(G, oRing,currRing);

      if(test_w_in_ConeCC(Gt, omega2) == 1) {
        delete omega2;
        delete next_vect;
        delete altomega;
        //Print("\n// Leaving the %d-th recursion with %d steps ",nlev, nwalks);
        //Print(" ** Overflow_Error? (%d)", Overflow_Error);

        return (Gt);
      }
      else
      {
        //ivString(omega2, "tau'");
        //Print("\n//  tau' doesn't stay in the correct cone!!");

#ifndef  MSTDCC_FRACTAL
        //07.08.03
        //ivString(Xtau, "old Xtau");
        intvec* Xtautmp = Mfpertvector(Gt, MivMatrixOrder(omtmp));
#ifdef TEST_OVERFLOW
      if(Overflow_Error == TRUE)
      Gt = NULL; return(Gt);
#endif

        if(MivSame(Xtau, Xtautmp) == 1)
        {
          //PrintS("\n// Update vectors are equal to the old vectors!!");
          delete Xtautmp;
          goto FRACTAL_MSTDCC;
        }

        Xtau = Xtautmp;
        Xtautmp = NULL;
        //ivString(Xtau, "new  Xtau");

        for(i=nV-1; i>=0; i--)
          (*omega2)[i] = (*Xtau)[(nlev-1)*nV+i];

        //Print("\n//  ring tau = %s;", rString(currRing));
        rChangeCurrRing(oRing);
        G = idrMoveR(Gt, testring,currRing);

        goto NEXT_VECTOR_FRACTAL;
#endif

      FRACTAL_MSTDCC:
        //Print("\n//  apply BB-Alg in ring = %s;", rString(currRing));
        to=clock();
        G = MstdCC(Gt);
        xtextra=xtextra+clock()-to;

        oRing = currRing;

        // update the original target vector w.r.t. the current GB
        if(MivSame(Xivinput, Xivlp) == 1)
          if (rParameter(currRing) != NULL)
            DefRingParlp();
          else
            VMrDefaultlp();
        else
          if (rParameter(currRing) != NULL)
            DefRingPar(Xivinput);
          else
            rChangeCurrRing(VMrDefault1(Xivinput)); //Aenderung6

        testring = currRing;
        Gt = idrMoveR(G, oRing,currRing);

        delete Xtau;
        Xtau = NewVectorlp(Gt);

        rChangeCurrRing(oRing);
        G = idrMoveR(Gt, testring,currRing);

        delete omega2;
        delete next_vect;
        delete altomega;
        /*
          Print("\n// Leaving the %d-th recursion with %d steps,", nlev,nwalks);
          Print(" ** Overflow_Error? (%d)", Overflow_Error);
        */
        if(Overflow_Error == TRUE)
          nnflow ++;

        Overflow_Error = FALSE;
        return(G);
      }
    }

    for(i=nV-1; i>=0; i--) {
      (*altomega)[i] = (*omega)[i];
      (*omega)[i] = (*next_vect)[i];
    }
    delete next_vect;

    to=clock();
    /* Take the initial form of <G> w.r.t. omega */
    Gomega = MwalkInitialForm(G, omega);
    xtif=xtif+clock()-to;

#ifndef  BUCHBERGER_ALG
    if(isNolVector(omega) == 0)
      hilb_func = hFirstSeries(Gomega,NULL,NULL,omega,currRing);
    else
      hilb_func = hFirstSeries(Gomega,NULL,NULL,last_omega,currRing);
#endif // BUCHBERGER_ALG

    if (rParameter(currRing) != NULL)
      DefRingPar(omega);
    else
      rChangeCurrRing(VMrDefault1(omega)); //Aenderung7

    Gomega1 = idrMoveR(Gomega, oRing,currRing);

    /* Maximal recursion depth, to compute a red. GB */
    /* Fractal walk with the alternative recursion */
    /* alternative recursion */
    // if(nlev == nV || lengthpoly(Gomega1) == 0)
    if(nlev == Xnlev || lengthpoly(Gomega1) == 0)
      //if(nlev == nV) // blind recursion
    {
      /*
      if(Xnlev != nV)
      {
        Print("\n// ** Xnlev = %d", Xnlev);
        ivString(Xtau, "Xtau");
      }
      */
      to=clock();
#ifdef  BUCHBERGER_ALG
      Gresult = MstdhomCC(Gomega1);
#else
      Gresult =kStd(Gomega1,NULL,isHomog,NULL,hilb_func,0,NULL,omega);
      delete hilb_func;
#endif // BUCHBERGER_ALG
      xtstd=xtstd+clock()-to;
    }
    else {
      rChangeCurrRing(oRing);
      Gomega1 = idrMoveR(Gomega1, oRing,currRing);
      Gresult = rec_fractal_call(idCopy(Gomega1),nlev+1,omega);
    }

    //convert a Groebner basis from a ring to another ring,
    new_ring = currRing;

    rChangeCurrRing(oRing);
    Gresult1 = idrMoveR(Gresult, new_ring,currRing);
    Gomega2 = idrMoveR(Gomega1, new_ring,currRing);

    to=clock();
    /* Lifting process */
    F = MLifttwoIdeal(Gomega2, Gresult1, G);
    xtlift=xtlift+clock()-to;
    idDelete(&Gresult1);
    idDelete(&Gomega2);
    idDelete(&G);

    rChangeCurrRing(new_ring);
    F1 = idrMoveR(F, oRing,currRing);

    to=clock();
    /* Interreduce G */
    G = kInterRedCC(F1, NULL);
    xtred=xtred+clock()-to;
    idDelete(&F1);
  }
}




/*******************************************************************************
 * The implementation of the fractal walk algorithm                            *
 *                                                                             *
 * The main procedur Mfwalk calls the recursive Subroutine                     *
 * rec_fractal_call to compute the wanted Grï¿œbner basis.                       *
 * At the main procedur we compute the reduced Grï¿œbner basis w.r.t. a "fast"   *
 * order, e.g. "dp" and a sequence of weight vectors which are row vectors     *
 * of a matrix. This matrix defines the given monomial order, e.g. "lp"        *
 *******************************************************************************/
ideal Mfwalk(ideal G, intvec* ivstart, intvec* ivtarget)
{
  Set_Error(FALSE);
  Overflow_Error = FALSE;
  //Print("// pSetm_Error = (%d)", ErrorCheck());
  //Print("\n// ring ro = %s;", rString(currRing));

  nnflow = 0;
  Xngleich = 0;
  Xcall = 0;
  xtif=0; xtstd=0; xtlift=0; xtred=0; xtnw=0; xtextra=0;
  xftinput = clock();

  ring  oldRing = currRing;
  int i, nV = currRing->N;
  XivNull = new intvec(nV);
  Xivinput = ivtarget;
  ngleich = 0;
  to=clock();
  ideal I = MstdCC(G);
  G = NULL;
  xftostd=clock()-to;
  Xsigma = ivstart;

  Xnlev=nV;

#ifdef FIRST_STEP_FRACTAL
  ideal Gw = MwalkInitialForm(I, ivstart);
  for(i=IDELEMS(Gw)-1; i>=0; i--)
  {
    if((Gw->m[i]!=NULL) // len >=0
    && (