/****************************************************
*          GCD and SCM of two polynomials           *
*                                                   *
* ------------------------------------------------- *
* Ref.: "Mathématiques en Turbo-Pascal By M. Ducamp *
* and A. Reverchon (vol 2), Eyrolles, Paris, 1988"  *
* [BIBLI 05].                                       *
* ------------------------------------------------- *
* SAMPLE RUN:                                       *
*                                                   *
* GCD AND SCM OF TWO POLYNOMIALS:                   *
*                                                   *
* P(x) = x5 +5x4 +7x3 +5x2 +x -1                    *
* Q(x) = x4 +4x3 -7x +2                             *
*                                                   *
*                                                   *
* Greatest common divisor:                          *
*                                                   *
* + X2 +3X -1                                       *
*                                                   *
*                                                   *
* Smallest common multiple:                         *
*                                                   *
* + X7 +6 X6 +10 X5 +2 X4 -8 X3 -10 X2 -3 X +2      *
*                                                   *
*                                                   *
* ------------------------------------------------- *
* Functions used (of module Polynoms):              *
*                                                   *
*  AddNumber(), EnterPolynom(), DivNumber(),        *
*  DisplayPolynom(), MultNumber() and SetNumber().  *
*                                                   *
*                      C++ version by J-P Moreau.   *
*                  (To be linked with polynoms.cpp).*
*                          (www.jpmoreau.fr)        *
*****************************************************
 Explanations:
 The Greatest common divisor (GCD) and the Smallest Common
 multiple (SCM) of two polynomials are defined but for a
 coefficient. That is to say, if a polynomial R(x) is the
 GCD of P(x) and Q(x), any polynomial µ*R(x) is also a GCD.
 To find the GCD, after permutating P(x) and Q(x), if the
 degree of P is lower than the degree of Q, we first perform
 the Euclidian division  P/Q: P=Q.H1+R1 (see program Divpol.pas).
 if R1<>0, we then divide Q by R1: Q=R1.H2+R2.
 if R2<>0, we continue until Rn=0. Rn-1 is then the GCD of
 P(x) and Q(x).
 To find the SCM of P(x) and Q(x), we use the relation:
    P(x).Q(x)=GCD(P,Q).SCM(P,Q).  
---------------------------------------------------------------*/
#include <stdio.h>
#include <malloc.h>
#include <math.h>

#include "polynoms.h"


ar_polynom  *P, *Q, *R;


// P(X) * Q(X) = R(X) - R can be either P or Q
bool MultPolynom(ar_polynom *P, ar_polynom *Q, ar_polynom *R) {
  int i,j, n;
  ar_number u;
  ar_polynom *nr;
  nr = (ar_polynom *) calloc(1,sizeof(ar_polynom));
  //verify that P and Q are not void
  if (P->degree==0 && P->coeff[0].value==0) return FALSE;
  if (Q->degree==0 && Q->coeff[0].value==0) return FALSE;
  nr->degree=P->degree+Q->degree;
  if (nr->degree>AR_MAXPOL) return FALSE;  // R degree is too big
  for (n=0; n<=nr->degree; n++)  {
    if (!SetNumber(&nr->coeff[n],"0")) return FALSE;
    for (i=0; i<=P->degree; i++) {
      j=n-i;
      if (j>=0 && j<=Q->degree) {
        if (!MultNumber(P->coeff[i],Q->coeff[j],&u)) return FALSE;
        if (!AddNumber(nr->coeff[n],u,&nr->coeff[n])) return FALSE;
      }
    }
  }
  //copy nr in R
  R->degree=nr->degree;
  for (i=0; i<=nr->degree; i++)  R->coeff[i]=nr->coeff[i];
  free(nr);
  return TRUE;
}

bool DivPolynom(ar_polynom *P, ar_polynom *Q, ar_polynom *H, ar_polynom *R) {
  int i,j;
  ar_number u;
  ar_polynom *nh, *nr;
  nh = (ar_polynom *) calloc(1,sizeof(ar_polynom));
  nr = (ar_polynom *) calloc(1,sizeof(ar_polynom));
  //The Q polynomial must be <> zero
  if (Q->degree==0 && Q->coeff[0].value==0) return FALSE;;
  nr->degree=P->degree;
  for (i=0; i<=P->degree; i++)  nr->coeff[i]=P->coeff[i];
  nh->degree = P->degree - Q->degree;
  if (nh->degree<0) {
    nh->degree=0; if (!SetNumber(&nh->coeff[0],"0")) return FALSE;
  }
  else {
    for (i=nh->degree; i>=0; i--)  {
      if (!DivNumber(nr->coeff[nr->degree],Q->coeff[Q->degree],
                       &nh->coeff[i])) return FALSE;
	  for (j=i; j<=nr->degree; j++) {
        if (!MultNumber(nh->coeff[i],Q->coeff[j-i], &u)) return FALSE;
        u.p=-u.p; u.value=-u.value;
        if (!AddNumber(nr->coeff[j],u, &nr->coeff[j])) return FALSE;
      }
      if (nr->degree > 0)  nr->degree--;
    }
    while (fabs(nr->coeff[nr->degree].value) < AR_SMALL && 
              nr->degree>0)  nr->degree--;
  }
  H->degree=nh->degree;
  for (i=0; i<=nh->degree; i++)  H->coeff[i]=nh->coeff[i];
  R->degree=nr->degree;
  for (i=0; i<=nr->degree; i++)  R->coeff[i]=nr->coeff[i];
  free(nh); free(nr);
  return TRUE;
} //DivPolynom()

//copy polynomial q in polynomial p
void polcpy(ar_polynom *p, ar_polynom *q) {
  int i;
  p->degree=q->degree;
  for (i=0; i<=q->degree; i++) p->coeff[i]=q->coeff[i];
}

// GCD of two polynomials
bool GCDPolynom(ar_polynom *P,ar_polynom *Q, ar_polynom *R) {
  ar_number z;
  long pg,pp;
  int i;
  ar_polynom *v, *np, *nq;
  //dynamic memory allocation of polynomials
  v = (ar_polynom *) calloc(1,sizeof(ar_polynom));
  np = (ar_polynom *) calloc(1,sizeof(ar_polynom));
  nq = (ar_polynom *) calloc(1,sizeof(ar_polynom));  
  if (P->degree==0 && P->coeff[P->degree].value==0) return FALSE;
  if (Q->degree==0 && Q->coeff[Q->degree].value==0) return FALSE;
  polcpy(np,P); polcpy(nq,Q); polcpy(R,nq);
  while(R->degree>0) {
	if (!DivPolynom(np,nq,v,R)) return FALSE;
	while(R->degree>0 && fabs(R->coeff[R->degree].value)<AR_SMALL)
	  R->degree--;
	if (R->degree==0) {
	  if (fabs(R->coeff[0].value)>AR_SMALL) nq->degree=0;
    }
	else {
      polcpy(np,nq); polcpy(nq,R);
    }
	for (i=0; i<nq->degree; i++)
	  if (!DivNumber(nq->coeff[i],nq->coeff[nq->degree],
		&nq->coeff[i])) return FALSE;
    SetNumber(&nq->coeff[nq->degree], "1");
  }
  polcpy(R,nq);
  if (nq->degree==0)
	SetNumber(&R->coeff[0], "1");
  else {
	for (pg=0, i=0; i<=R->degree; i++)
	  if (!R->coeff[i].is_real)
		pg = (pg) ? GCD(pg,R->coeff[i].p) : R->coeff[i].p;
	for (pp=0, i=0; i<=R->degree; i++)
	  if (!R->coeff[i].is_real)
		pp = (pp==0) ? R->coeff[i].q : 
		  pp * R->coeff[i].q / GCD(pp,R->coeff[i].q);
	if (pg) {
	  z.is_real = FALSE;
	  z.p=pp; z.q=pg;
      z.value=(double) pp / (double) pg;
	  for (i=0; i<=R->degree; i++)
		if (!MultNumber(R->coeff[i], z, &R->coeff[i])) return FALSE;
    }
  }			
  free(v); free(np); free(nq);
  return TRUE;
}

// SCM of two polynomials
bool SCMPolynom(ar_polynom *P,ar_polynom *Q,ar_polynom *R)  {
  ar_polynom *u, *v;
  u = (ar_polynom *) calloc(1,sizeof(ar_polynom));
  v = (ar_polynom *) calloc(1,sizeof(ar_polynom));
  if (!GCDPolynom(P,Q,u)) return FALSE;
  if (!MultPolynom(P,Q,v)) return FALSE;
  if (!DivPolynom(v,u,R,v)) return FALSE;
  free(u); free(v);
  return TRUE;
}


void main()  {
  P = (ar_polynom *) calloc(1,sizeof(ar_polynom));
  Q = (ar_polynom *) calloc(1,sizeof(ar_polynom));
  R = (ar_polynom *) calloc(1,sizeof(ar_polynom));   
  printf("\n GCD AND SCM OF TWO POLYNOMIALS:\n\n");
  if (!EnterPolynom(" P(X) = ", P)) return;
  if (!EnterPolynom(" Q(X) = ", Q)) return;
  printf("\n\n");
  if (GCDPolynom(P,Q,R))  {
    printf(" Greatest common divisor:\n");
    DisplayPolynom(R);
  }
  else
    printf(" Error in looking for GCD.");
  printf("\n\n");
  if (SCMPolynom(P,Q,R))  {
    printf(" Smallest common multiple:\n");
    DisplayPolynom(R);
  }
  else
    printf(" Error in looking for SCM.");
  printf("\n\n");
  free(P); free(Q); free(R);
}

// end of file gcdpol.cpp