/********************************************************************/
/** **/
/** BIBLIOTHEQUE MATHEMATIQUE **/
/** (deuxieme partie) **/
/** **/
/********************************************************************/
/* $Id: bibli2.c,v 1.1.1.1 1999/09/16 13:47:24 karim Exp $ */
#include "pari.h"
/********************************************************************/
/** **/
/** DEVELOPPEMENTS LIMITES **/
/** **/
/********************************************************************/
GEN
tayl(GEN x, long v, long precdl)
{
long tetpil,i,vx = gvar9(x), av=avma;
GEN p1,y;
if (v <= vx)
{
long p1[] = { evaltyp(t_SER)|m_evallg(2), 0 };
p1[1] = evalvalp(precdl) | evalvarn(v);
return gadd(p1,x);
}
p1=cgetg(v+2,t_VEC);
for (i=0; i<v; i++) p1[i+1]=lpolx[i];
p1[vx+1]=lpolx[v]; p1[v+1]=lpolx[vx];
y = tayl(changevar(x,p1), vx,precdl); tetpil=avma;
return gerepile(av,tetpil, changevar(y,p1));
}
GEN
grando0(GEN x, long n, long do_clone)
{
long m, v, tx=typ(x);
GEN y;
if (gcmp0(x)) err(talker,"zero argument in O()");
if (tx == t_INT)
{
if (!gcmp1(x)) /* bug 3 + O(1). We suppose x is a truc() */
{
y=cgetg(5,t_PADIC);
y[1] = evalvalp(n) | evalprecp(0);
y[2] = do_clone? lclone(x): licopy(x);
y[3] = un; y[4] = zero; return y;
}
v=0; m=0; /* 1 = x^0 */
}
else
{
if (tx != t_POL && ! is_rfrac_t(tx))
err(talker,"incorrect argument in O()");
v=gvar(x); m=n*gval(x,v);
}
return zeroser(v,m);
}
/*******************************************************************/
/** **/
/** SPECIAL POLYNOMIALS **/
/** **/
/*******************************************************************/
GEN addshiftw(GEN x, GEN y, long d);
/* Tchebichev polynomial */
/* T0=1; T1=X; T(n)=2*X*T(n-1)-T(n-2) */
GEN
tchebi(long n, long v)
{
long av,tetpil,lim,m;
GEN p0,p1,p2,q,z;
if (v<0) v = 0;
if (n==0) return polun[v];
if (n==1) return polx[v];
p0=gneg(polun[v]); p1=polx[v]; z = zeropol(v);
av=avma; lim=stack_lim(av,2);
for (m=2; m<n; m++)
{
q = addshiftw(gmul2n(p1,1), z, 1); tetpil=avma;
setvarn(q,v);
p2 = gadd(q, p0); p0=gneg(p1); p1=p2;
if (low_stack(lim, stack_lim(av,2)))
{
GEN *gptr[2]; gptr[0]=&p0; gptr[1]=&p1;
if(DEBUGMEM>1) err(warnmem,"tchebi");
gerepilemanysp(av,tetpil,gptr,2);
}
}
q = addshiftw(gmul2n(p1,1), z, 1); tetpil=avma;
setvarn(q,v);
return gerepile(av,tetpil,gadd(q,p0));
}
/* Legendre polynomial */
/* L0=1; L1=X; (n+1)*L(n+1)=(2*n+1)*X*L(n)-n*L(n-1) */
GEN
legendre(long n, long v)
{
long av,tetpil,m,lim;
GEN p0,p1,p2;
if (v<0) v = 0;
if (n==0) return polun[v];
if (n==1) return polx[v];
p0=polun[v]; av=avma; lim=stack_lim(av,2);
p1=gmul2n(polx[v],1);
for (m=1; m<n; m++)
{
p2 = addshiftw(gmulsg(4*m+2,p1), gmulsg(-4*m,p0), 1);
setvarn(p2,v);
p0 = p1; tetpil=avma; p1 = gdivgs(p2,m+1);
if (low_stack(lim, stack_lim(av,2)))
{
GEN *gptr[2];
if(DEBUGMEM>1) err(warnmem,"legendre");
p0=gcopy(p0); gptr[0]=&p0; gptr[1]=&p1;
gerepilemanysp(av,tetpil,gptr,2);
}
}
tetpil=avma; return gerepile(av,tetpil,gmul2n(p1,-n));
}
/* cyclotomic polynomial */
GEN
cyclo(long n, long v)
{
long av=avma,tetpil,d,q,m;
GEN yn,yd;
if (n<=0) err(arither2);
if (v<0) v = 0;
yn = yd = polun[0];
for (d=1; d*d<=n; d++)
{
if (n%d) continue;
q=n/d;
m = mu(stoi(q));
if (m)
{ /* y *= (x^d - 1) */
if (m>0) yn = addshiftw(yn, gneg(yn), d);
else yd = addshiftw(yd, gneg(yd), d);
}
if (q==d) break;
m = mu(stoi(d));
if (m)
{ /* y *= (x^q - 1) */
if (m>0) yn = addshiftw(yn, gneg(yn), q);
else yd = addshiftw(yd, gneg(yd), q);
}
}
tetpil=avma; yn = gerepile(av,tetpil,gdeuc(yn,yd));
setvarn(yn,v); return yn;
}
/* compute prod (L*x ± a[i]) */
GEN
roots_to_pol_intern(GEN L, GEN a, long v, int plus)
{
long i,k,lx = lg(a), code;
GEN p1,p2;
if (lx == 1) return polun[v];
p1 = cgetg(lx, t_VEC);
code = evalsigne(1)|evalvarn(v)|evallgef(5);
for (k=1,i=1; i<lx-1; i+=2)
{
p2 = cgetg(5,t_POL); p1[k++] = (long)p2;
p2[2] = lmul((GEN)a[i],(GEN)a[i+1]);
p2[3] = ladd((GEN)a[i],(GEN)a[i+1]);
if (plus == 0) p2[3] = lneg((GEN)p2[3]);
p2[4] = (long)L; p2[1] = code;
}
if (i < lx)
{
p2 = cgetg(4,t_POL); p1[k++] = (long)p2;
p2[1] = code = evalsigne(1)|evalvarn(v)|evallgef(4);
p2[2] = plus? a[i]: lneg((GEN)a[i]);
p2[3] = (long)L;
}
setlg(p1, k); return divide_conquer_prod(p1, gmul);
}
GEN
roots_to_pol(GEN a, long v)
{
return roots_to_pol_intern(gun,a,v,0);
}
/* prod_{i=1..r1} (x - a[i]) prod_{i=1..r2} (x - a[i])(x - conj(a[i]))*/
GEN
roots_to_pol_r1r2(GEN a, long r1, long v)
{
long i,k,lx = lg(a), code;
GEN p1;
if (lx == 1) return polun[v];
p1 = cgetg(lx, t_VEC);
code = evalsigne(1)|evalvarn(v)|evallgef(5);
for (k=1,i=1; i<r1; i+=2)
{
GEN p2 = cgetg(5,t_POL); p1[k++] = (long)p2;
p2[2] = lmul((GEN)a[i],(GEN)a[i+1]);
p2[3] = lneg(gadd((GEN)a[i],(GEN)a[i+1]));
p2[4] = un; p2[1] = code;
}
if (i < r1+1)
p1[k++] = ladd(polx[v], gneg((GEN)a[i]));
for (i=r1+1; i<lx; i++)
{
GEN p2 = cgetg(5,t_POL); p1[k++] = (long)p2;
p2[2] = lnorm((GEN)a[i]);
p2[3] = lneg(gtrace((GEN)a[i]));
p2[4] = un; p2[1] = code;
}
setlg(p1, k); return divide_conquer_prod(p1, gmul);
}
/* finds an equation for the d-th degree subfield of Q(zeta_n).
* (Z/nZ)* must be cyclic.
*/
GEN
subcyclo(GEN nn, GEN dd, int v)
{
long av=avma,tetpil,i,j,k,prec,q,d,p,pp,al,n,ex0,ex,aad,aa;
GEN a,z,pol,fa,powz,alpha;
if (typ(dd)!=t_INT || signe(dd)<=0) err(typeer,"subcyclo");
if (is_bigint(dd)) err(talker,"degree too large in subcyclo");
if (typ(nn)!=t_INT || signe(nn)<=0) err(typeer,"subcyclo");
if (v<0) v = 0;
d=itos(dd); if (d==1) return polx[v];
if (is_bigint(nn)) err(impl,"subcyclo for huge cyclotomic fields");
n = nn[2]; if ((n & 3) == 2) n >>= 1;
if (n == 1) err(talker,"degree does not divide phi(n) in subcyclo");
fa = factor(stoi(n));
p = itos(gmael(fa,1,1));
al= itos(gmael(fa,2,1));
if (lg((GEN)fa[1]) > 2 || (p==2 && al>2))
err(impl,"subcyclo in non-cyclic case");
if (d < n)
{
k = 1 + svaluation(d,p,&i);
if (k<al) { al = k; nn = gpowgs(stoi(p),al); n = nn[2]; }
}
avma=av; q = (n/p)*(p-1); /* = phi(n) */
if (q == d) return cyclo(n,v);
if (q % d) err(talker,"degree does not divide phi(n) in subcyclo");
q /= d;
if (p==2)
{
pol = powgi(polx[v],gdeux); pol[2]=un; /* replace gzero */
return pol; /* = x^2 + 1 */
}
a=gener(stoi(n)); aa = mael(a,2,2);
a=gpowgs(a,d); aad = mael(a,2,2);
#if 1
prec = expi(binome(stoi(d*q-1),d)) + expi(stoi(n));
prec = 2 + (prec>>TWOPOTBITS_IN_LONG);
if (prec<DEFAULTPREC) prec = DEFAULTPREC;
if (DEBUGLEVEL) fprintferr("subcyclo prec = %ld\n",prec);
z = cgetg(3,t_COMPLEX); a=mppi(prec); setexpo(a,2); /* a = 2\pi */
gsincos(divrs(a,n),(GEN*)(z+2),(GEN*)(z+1),prec); /* z = e_n(1) */
powz = cgetg(n,t_VEC); powz[1] = (long)z;
k = (n+3)>>1;
for (i=2; i<k; i++) powz[i] = lmul(z,(GEN)powz[i-1]);
if ((q&1) == 0) /* totally real field, take real part */
{
for (i=1; i<k; i++) powz[i] = mael(powz,i,1);
for ( ; i<n; i++) powz[i] = powz[n-i];
}
else
for ( ; i<n; i++) powz[i] = lconj((GEN)powz[n-i]);
alpha = cgetg(d+1,t_VEC) + 1; pol=gun;
for (ex0=1,k=0; k<d; k++, ex0=(ex0*aa)%n)
{
GEN p1 = gzero;
long av1 = avma; (void)new_chunk(2*prec + 3);
for (ex=ex0,i=0; i<q; i++)
{
for (pp=ex,j=0; j<al; j++)
{
p1 = gadd(p1,(GEN)powz[pp]);
pp = mulssmod(pp,p, n);
}
ex = mulssmod(ex,aad, n);
}
/* p1 = sum z^{p^k*h}, k = 0..al-1, h runs through the subgroup of order
* q = phi(n)/d of (Z/nZ)^* */
avma = av1; alpha[k] = lneg(p1);
}
pol = roots_to_pol_intern(gun,alpha-1,v, 1);
if (q&1) pol=greal(pol); /* already done otherwise */
tetpil=avma; return gerepile(av,tetpil,ground(pol));
#else
{
/* exact computation (much slower) */
GEN p1 = cgetg(n+2,t_POL)+2; for (i=0; i<n; i++) p1[i]=0;
for (ex=1,i=0; i<q; i++, ex=(ex*aad)%n)
for (pp=ex,j=0; j<al; j++, pp=(pp*p)%n) p1[pp]++;
for (i=0; i<n; i++) p1[i] = lstoi(p1[i]);
p1 = normalizepol_i(p1-2,n+2); setvarn(p1,v);
z = cyclo(n,v); a = caract2(z,gres(p1,z),v);
a = gdeuc(a, modulargcd(a,derivpol(a)));
return gerepileupto(av, a);
}
#endif
}
/********************************************************************/
/** **/
/** HILBERT & PASCAL MATRICES **/
/** **/
/********************************************************************/
GEN addshiftpol(GEN x, GEN y, long d);
GEN
mathilbert(long n) /* Hilbert matrix of order n */
{
long i,j;
GEN a,p;
if (n<0) n = 0;
p = cgetg(n+1,t_MAT);
for (j=1; j<=n; j++)
{
p[j]=lgetg(n+1,t_COL);
for (i=1; i<=n; i++)
{
a=cgetg(3,t_FRAC); a[1]=un; a[2]=lstoi(i+j-1);
coeff(p,i,j)=(long)a;
}
}
return p;
}
/* q-Pascal triangle = (choose(i,j)_q) (ordinary binomial if q = NULL) */
GEN
matqpascal(long n, GEN q)
{
long i,j,I, av = avma;
GEN m, *qpow;
if (n<0) n = -1;
n++; m = cgetg(n+1,t_MAT);
for (j=1; j<=n; j++) m[j] = lgetg(n+1,t_COL);
if (q)
{
I = (n+1)/2;
if (I > 1) { qpow = (GEN*)new_chunk(I+1); qpow[2]=q; }
for (j=3; j<=I; j++) qpow[j] = gmul(q, qpow[j-1]);
}
for (i=1; i<=n; i++)
{
I = (i+1)/2; coeff(m,i,1)=un;
if (q)
{
for (j=2; j<=I; j++)
coeff(m,i,j) = ladd(gmul(qpow[j],gcoeff(m,i-1,j)), gcoeff(m,i-1,j-1));
}
else
{
for (j=2; j<=I; j++)
coeff(m,i,j) = laddii(gcoeff(m,i-1,j), gcoeff(m,i-1,j-1));
}
for ( ; j<=i; j++) coeff(m,i,j) = coeff(m,i,i+1-j);
for ( ; j<=n; j++) coeff(m,i,j) = zero;
}
return gerepileupto(av, gcopy(m));
}
/********************************************************************/
/** **/
/** LAPLACE TRANSFORM (OF A SERIES) **/
/** **/
/********************************************************************/
GEN
laplace(GEN x)
{
long i,l,ec,av,tetpil;
GEN y,p1;
if (typ(x)!=t_SER) err(talker,"not a series in laplace");
if (gcmp0(x)) return gcopy(x);
av=avma; ec=valp(x);
if (ec<0) err(talker,"negative valuation in laplace");
l=lg(x); y=cgetg(l,t_SER);
p1=mpfact(ec); y[1]=x[1];
for (i=2; i<l; i++)
{
y[i]=lmul(p1,(GEN)x[i]);
ec++; p1=mulsi(ec,p1);
}
tetpil=avma; return gerepile(av,tetpil,gcopy(y));
}
/********************************************************************/
/** **/
/** CONVOLUTION PRODUCT (OF TWO SERIES) **/
/** **/
/********************************************************************/
GEN
convol(GEN x, GEN y)
{
long l,i,j,v, vx=varn(x), lx=lg(x), ly=lg(y), ex=valp(x), ey=valp(y);
GEN z;
if (typ(x) != t_SER || typ(y) != t_SER)
err(talker,"not a series in convol");
if (gcmp0(x) || gcmp0(y))
err(talker,"zero series in convol");
if (varn(y) != vx)
err(talker,"different variables in convol");
v=ex; if (ey>v) v=ey;
l=ex+lx; i=ey+ly; if (i<l) l=i;
l -= v; if (l<3) err(talker,"non significant result in convol");
for (i=v+2; i < v+l; i++)
if (!gcmp0((GEN)x[i-ex]) && !gcmp0((GEN)y[i-ey])) { i++; break; }
if (i == l+v) return zeroser(vx, v+l-2);
z = cgetg(l-i+3+v,t_SER);
z[1] = evalsigne(1) | evalvalp(i-3) | evalvarn(vx);
for (j=i-1; j<l+v; j++) z[j-i+3]=lmul((GEN)x[j-ex],(GEN)y[j-ey]);
return z;
}
/******************************************************************/
/** **/
/** PRECISION CHANGES **/
/** **/
/******************************************************************/
GEN
gprec(GEN x, long l)
{
long tx=typ(x),lx=lg(x),i,pr;
GEN y;
if (l<=0) err(talker,"precision<=0 in gprec");
switch(tx)
{
case t_REAL:
pr = (long) (l*pariK1+3); y=cgetr(pr); affrr(x,y); break;
case t_PADIC:
y=cgetg(lx,tx); copyifstack(x[2], y[2]);
if (!signe(x[4]))
{
y[1]=evalvalp(l+precp(x)) | evalprecp(0);
y[3]=un; y[4]=zero; return y;
}
y[1]=x[1]; setprecp(y,l);
y[3]=lpuigs((GEN)x[2],l);
y[4]=lmodii((GEN)x[4],(GEN)y[3]);
break;
case t_SER:
if (gcmp0(x)) return zeroser(varn(x), l);
y=cgetg(l+2,t_SER); y[1]=x[1]; l++; i=l;
if (l>=lx)
for ( ; i>=lx; i--) y[i]=zero;
for ( ; i>=2; i--) y[i]=lcopy((GEN)x[i]);
break;
case t_POL:
lx=lgef(x); y=cgetg(lx,tx); y[1]=x[1];
for (i=2; i<lx; i++) y[i]=lprec((GEN)x[i],l);
break;
case t_COMPLEX: case t_POLMOD: case t_RFRAC: case t_RFRACN:
case t_VEC: case t_COL: case t_MAT:
y=cgetg(lx,tx);
for (i=1; i<lx; i++) y[i]=lprec((GEN)x[i],l);
break;
default: y=gcopy(x);
}
return y;
}
/* internal: precision given in word length (including codewords) */
GEN
gprec_w(GEN x, long pr)
{
long tx=typ(x),lx=lg(x),i;
GEN y;
switch(tx)
{
case t_REAL:
y=cgetr(pr); affrr(x,y); break;
case t_POL:
lx=lgef(x); y=cgetg(lx,tx); y[1]=x[1];
for (i=2; i<lx; i++) y[i]=(long)gprec_w((GEN)x[i],pr);
break;
case t_COMPLEX: case t_POLMOD: case t_RFRAC: case t_RFRACN:
case t_VEC: case t_COL: case t_MAT:
y=cgetg(lx,tx);
for (i=1; i<lx; i++) y[i]=(long)gprec_w((GEN)x[i],pr);
break;
default: y=gprec(x,pr);
}
return y;
}
/*******************************************************************/
/** **/
/** RECIPROCAL POLYNOMIAL **/
/** **/
/*******************************************************************/
GEN
polrecip(GEN x)
{
long lx=lgef(x),i,j;
GEN y;
if (typ(x) != t_POL) err(typeer,"polrecip");
y=cgetg(lx,t_POL); y[1]=x[1];
for (i=2,j=lx-1; i<lx; i++,j--) y[i]=lcopy((GEN)x[j]);
return normalizepol_i(y,lx);
}
/* as above. Internal (don't copy or normalize) */
GEN
polrecip_i(GEN x)
{
long lx=lgef(x),i,j;
GEN y;
y=cgetg(lx,t_POL); y[1]=x[1];
for (i=2,j=lx-1; i<lx; i++,j--) y[i]=x[j];
return y;
}
/*******************************************************************/
/** **/
/** BINOMIAL COEFFICIENTS **/
/** **/
/*******************************************************************/
GEN
binome(GEN n, long k)
{
long av,i;
GEN y;
if (k <= 1)
{
if (k < 0) return gzero;
if (k == 0) return gun;
return gcopy(n);
}
av = avma; y = n;
if (typ(n) == t_INT)
{
if (signe(n) > 0)
{
GEN z = subis(n,k);
if (cmpis(z,k) < 0) k = itos(z);
avma = av;
if (k <= 1)
{
if (k < 0) return gzero;
if (k == 0) return gun;
return gcopy(n);
}
}
for (i=2; i<=k; i++)
y = gdivgs(gmul(y,addis(n,i-1-k)), i);
}
else
{
for (i=2; i<=k; i++)
y = gdivgs(gmul(y,gaddgs(n,i-1-k)), i);
}
return gerepileupto(av, y);
}
/********************************************************************/
/** **/
/** POLYNOMIAL INTERPOLATION **/
/** **/
/********************************************************************/
GEN
polint_i(GEN xa, GEN ya, GEN x, long n, GEN *ptdy)
{
long av = avma,tetpil,i,m, ns=0, tx=typ(x);
GEN den,ho,hp,w,y,c,d,dy;
if (is_scalar_t(tx) && tx != t_INTMOD && tx != t_PADIC && tx != t_POLMOD)
{
GEN dif = NULL, dift;
for (i=0; i<n; i++)
{
dift = gabs(gsub(x,(GEN)xa[i]), MEDDEFAULTPREC);
if (!dif || gcmp(dift,dif)<0) { ns=i; dif=dift; }
}
}
c=new_chunk(n);
d=new_chunk(n); for (i=0; i<n; i++) c[i] = d[i] = ya[i];
y=(GEN)d[ns--];
for (m=1; m<n; m++)
{
for (i=0; i<n-m; i++)
{
ho = gsub((GEN)xa[i],x);
hp = gsub((GEN)xa[i+m],x); den = gsub(ho,hp);
if (gcmp0(den)) err(talker,"two abcissas are equal in polint");
w=gsub((GEN)c[i+1],(GEN)d[i]); den = gdiv(w,den);
c[i]=lmul(ho,den);
d[i]=lmul(hp,den);
}
dy = (2*(ns+1) < n-m)? (GEN)c[ns+1]: (GEN)d[ns--];
tetpil=avma; y=gadd(y,dy);
}
if (!ptdy) y = gerepile(av,tetpil,y);
else
{
GEN *gptr[2];
*ptdy=gcopy(dy); gptr[0]=&y; gptr[1]=ptdy;
gerepilemanysp(av,tetpil,gptr,2);
}
return y;
}
GEN
polint(GEN xa, GEN ya, GEN x, GEN *ptdy)
{
long tx=typ(xa), ty=typ(ya), lx=lg(xa);
if (! is_vec_t(tx) || ! is_vec_t(ty))
err(talker,"not vectors in polinterpolate");
if (lx != lg(ya))
err(talker,"different lengths in polinterpolate");
if (lx <= 2)
{
if (lx == 1) err(talker,"no data in polinterpolate");
ya=gcopy((GEN)ya[1]); if (ptdy) *ptdy = ya;
return ya;
}
if (!x) x = polx[0];
return polint_i(xa+1,ya+1,x,lx-1,ptdy);
}
/***********************************************************************/
/* */
/* SET OPERATIONS */
/* */
/***********************************************************************/
static GEN
gtostr(GEN x)
{
char *s=GENtostr(x);
x = strtoGENstr(s,0); free(s); return x;
}
GEN
gtoset(GEN x)
{
long i,c,av,tetpil,tx,lx;
GEN y;
if (!x) return cgetg(1, t_VEC);
tx = typ(x); lx = lg(x);
if (!is_vec_t(tx))
{
if (tx != t_LIST)
{ y=cgetg(2,t_VEC); y[1]=(long)gtostr(x); return y; }
lx = lgef(x)-1; x++;
}
if (lx==1) return cgetg(1,t_VEC);
av=avma; y=cgetg(lx,t_VEC);
for (i=1; i<lx; i++) y[i]=(long)gtostr((GEN)x[i]);
y = sort(y);
c=1;
for (i=2; i<lx; i++)
if (!gegal((GEN)y[i], (GEN)y[c])) y[++c] = y[i];
tetpil=avma; setlg(y,c+1);
return gerepile(av,tetpil,gcopy(y));
}
long
setisset(GEN x)
{
long lx,i;
if (typ(x)!=t_VEC) return 0;
lx=lg(x)-1; if (!lx) return 1;
for (i=1; i<lx; i++)
if (typ(x[i]) != t_STR || gcmp((GEN)x[i+1],(GEN)x[i])<=0) return 0;
return typ(x[i]) == t_STR;
}
/* looks if y belongs to the set x and returns the index if yes, 0 if no */
long
setsearch(GEN x, GEN y, long flag)
{
long av = avma,lx,j,li,ri,fl, tx = typ(x);
if (tx==t_VEC) lx = lg(x);
else
{
if (tx!=t_LIST) err(talker,"not a set in setsearch");
lx=lgef(x)-1; x++;
}
if (lx==1) return flag? 1: 0;
li=1; ri=lx-1;
if (typ(y) != t_STR) y = gtostr(y);
while (ri>=li)
{
j = (ri+li)>>1; fl = gcmp((GEN)x[j],y);
if (!fl) { avma=av; return flag? 0: j; }
if (fl<0) li=j+1; else ri=j-1;
}
avma=av; if (!flag) return 0;
return (fl<0)? j+1: j;
}
GEN
setunion(GEN x, GEN y)
{
long av=avma,tetpil;
GEN z;
if (typ(x) != t_VEC || typ(y) != t_VEC) err(talker,"not a set in setunion");
z=concatsp(x,y); tetpil=avma; return gerepile(av,tetpil,gtoset(z));
}
GEN
setintersect(GEN x, GEN y)
{
long av=avma,tetpil,i,lx,c;
GEN z;
if (!setisset(x) || !setisset(y)) err(talker,"not a set in setintersect");
lx=lg(x); z=cgetg(lx,t_VEC); c=1;
for (i=1; i<lx; i++)
if (setsearch(y, (GEN)x[i], 0)) z[c++] = x[i];
tetpil=avma; setlg(z,c);
return gerepile(av,tetpil,gcopy(z));
}
GEN
setminus(GEN x, GEN y)
{
long av=avma,tetpil,i,lx,c;
GEN z;
if (!setisset(x) || !setisset(y)) err(talker,"not a set in setminus");
lx=lg(x); z=cgetg(lx,t_VEC); c=1;
for (i=1; i<lx; i++)
if (setsearch(y, (GEN)x[i], 1)) z[c++] = x[i];
tetpil=avma; setlg(z,c);
return gerepile(av,tetpil,gcopy(z));
}
/***********************************************************************/
/* */
/* OPERATIONS ON DIRICHLET SERIES */
/* */
/***********************************************************************/
/* Addition, subtraction and scalar multiplication of Dirichlet series
are done on the corresponding vectors */
static long
dirval(GEN x)
{
long i=1,lx=lg(x);
while (i<lx && gcmp0((GEN)x[i])) i++;
return i;
}
GEN
dirmul(GEN x, GEN y)
{
long lx,ly,lz,dx,dy,av,tetpil,i,j;
GEN z,p1;
if (typ(x)!=t_VEC || typ(y)!=t_VEC) err(talker,"not a dirseries in dirmul");
av=avma; dx=dirval(x); dy=dirval(y); lx=lg(x); ly=lg(y);
if (ly-dy<lx-dx) { z=y; y=x; x=z; lz=ly; ly=lx; lx=lz; lz=dy; dy=dx; dx=lz; }
lz=min(lx*dy,ly*dx);
z=cgetg(lz,t_VEC); for (i=1; i<lz; i++) z[i]=zero;
for (j=dx; j<lx; j++)
{
p1=(GEN)x[j];
if (!gcmp0(p1))
{
if (gcmp1(p1))
for (i=j*dy; i<lz; i+=j) z[i]=ladd((GEN)z[i],(GEN)y[i/j]);
else
{
if (gcmp_1(p1))
for (i=j*dy; i<lz; i+=j) z[i]=lsub((GEN)z[i],(GEN)y[i/j]);
else
for (i=j*dy; i<lz; i+=j) z[i]=ladd((GEN)z[i],gmul(p1,(GEN)y[i/j]));
}
}
}
tetpil=avma; return gerepile(av,tetpil,gcopy(z));
}
GEN
dirdiv(GEN x, GEN y)
{
long lx,ly,lz,dx,dy,av,tetpil,i,j;
GEN z,p1;
if (typ(x)!=t_VEC || typ(y)!=t_VEC) err(talker,"not a dirseries in dirmul");
av=avma; dx=dirval(x); dy=dirval(y); lx=lg(x); ly=lg(y);
if (dy!=1) err(talker,"not an invertible dirseries in dirdiv");
lz=min(lx,ly*dx); p1=(GEN)y[1];
if (!gcmp1(p1)) { y=gdiv(y,p1); x=gdiv(x,p1); }
else x=gcopy(x);
z=cgetg(lz,t_VEC); for (i=1; i<dx; i++) z[i]=zero;
for (j=dx; j<lz; j++)
{
p1=(GEN)x[j]; z[j]=(long)p1;
if (!gcmp0(p1))
{
if (gcmp1(p1))
for (i=j+j; i<lz; i+=j) x[i]=lsub((GEN)x[i],(GEN)y[i/j]);
else
{
if (gcmp_1(p1))
for (i=j+j; i<lz; i+=j) x[i]=ladd((GEN)x[i],(GEN)y[i/j]);
else
for (i=j+j; i<lz; i+=j) x[i]=lsub((GEN)x[i],gmul(p1,(GEN)y[i/j]));
}
}
}
tetpil=avma; return gerepile(av,tetpil,gcopy(z));
}
/*************************************************************************/
/** **/
/** RANDOM **/
/** **/
/*************************************************************************/
static long pari_randseed = 1;
/* BSD rand gives this: seed = 1103515245*seed + 12345 */
long
mymyrand()
{
#if BITS_IN_RANDOM == 64
pari_randseed = (1000000000000654397*pari_randseed + 12347) & ~HIGHBIT;
#else
pari_randseed = (1000276549*pari_randseed + 12347) & 0x7fffffff;
#endif
return pari_randseed;
}
GEN muluu(ulong x, ulong y);
static ulong
gp_rand()
{
#define GLUE2(hi, lo) (((hi) << BITS_IN_HALFULONG) | (lo))
#if !defined(LONG_IS_64BIT) || BITS_IN_RANDOM == 64
return GLUE2((mymyrand()>>12)&LOWMASK,
(mymyrand()>>12)&LOWMASK);
#else
#define GLUE4(hi1,hi2, lo1,lo2) GLUE2(((hi1)<<16)|(hi2), ((lo1)<<16)|(lo2))
# define LOWMASK2 0xffffUL
return GLUE4((mymyrand()>>12)&LOWMASK2,
(mymyrand()>>12)&LOWMASK2,
(mymyrand()>>12)&LOWMASK2,
(mymyrand()>>12)&LOWMASK2);
#endif
}
GEN
genrand(GEN N)
{
long lx,i;
GEN x;
if (!N) return stoi(mymyrand());
if (typ(N)!=t_INT || signe(N)<=0) err(talker,"invalid bound in random");
lx = lgefint(N); x = new_chunk(lx);
for (i=2; i<lx; i++)
{
long av = avma, n = N[i];
ulong r = gp_rand();
if (i < lx-1) n++; else if (!n) r = 0;
if (n) { GEN p1 = muluu(n,r); r = (lgefint(p1)<=3)? 0: p1[2]; }
avma = av; x[i] = r;
if (r < (ulong)N[i]) break;
}
for (i++; i<lx; i++) x[i] = gp_rand();
i=2; while (i<lx && !x[i]) i++;
i -= 2; x += i; lx -= i;
x[1] = evalsigne(lx>2) | evallgefint(lx);
x[0] = evaltyp(t_INT) | evallg(lx);
avma = (long)x; return x;
}
long
setrand(long seed) { return (pari_randseed = seed); }
long
getrand() { return pari_randseed; }
long
getstack() { return top-avma; }
long
gettime() { return timer2(); }
/***********************************************************************/
/** **/
/** PERMUTATIONS **/
/** **/
/***********************************************************************/
GEN
permute(long n, GEN x)
{
long av=avma,i,a,r;
GEN v,w,y;
v=(GEN)gpmalloc((n+1)*sizeof(long)); v[1]=1;
for (r=2; r<=n; r++)
{
x=dvmdis(x,r,&w); a=itos(w);
for (i=r; i>=a+2; i--) v[i]=v[i-1];
v[i]=r;
}
avma=av; y=cgetg(n+1,t_VEC);
for (i=1; i<=n; i++) y[i]=lstoi(v[i]);
free(v); return y;
}
GEN
permuteInv(GEN x)
{
long av=avma,tetpil, len=lg(x)-1, ini=len, last, ind;
GEN ary,res;
if (typ(x)!=t_VEC && typ(x)!=t_COL) err(talker,"not a vector in permuteInv");
res=gzero; ary=cgetg(len+1,t_VEC);
for (ind=1; ind<=len; ind++) ary[ind]=*++x;
ary++;
for (last=len; last>0; last--)
{
len--; ind=len;
while (ind>0 && itos((GEN)ary[ind])!=last) ind--;
res=mulis(res,last); tetpil=avma; res=addis(res,ind);
while (ind++<len) ary[ind-1]=ary[ind];
}
if (!signe(res)) { tetpil=avma; res=mpfact(ini); }
return gerepile(av,tetpil,res);
}
/********************************************************************/
/** **/
/** MODREVERSE **/
/** **/
/********************************************************************/
GEN
polymodrecip(GEN x)
{
long v,i,j,n,av,tetpil,lx;
GEN p1,p2,p3,p,phi,y,col;
if (typ(x)!=t_POLMOD) err(talker,"not a polymod in polymodrecip");
p=(GEN)x[1]; phi=(GEN)x[2];
v=varn(p); n=lgef(p)-3; if (n<=0) return gcopy(x);
if (n==1)
{
y=cgetg(3,t_POLMOD);
if (typ(phi)==t_POL) phi = (GEN)phi[2];
p1=cgetg(4,t_POL); p1[1]=p[1]; p1[2]=lneg(phi); p1[3]=un;
y[1]=(long)p1;
if (gcmp0((GEN)p[2])) p1 = zeropol(v);
else
{
p1=cgetg(3,t_POL); av=avma;
p1[1] = evalsigne(1) | evalvarn(n) | evallgef(3);
p2=gdiv((GEN)p[2],(GEN)p[3]); tetpil=avma;
p1[2] = lpile(av,tetpil,gneg(p2));
}
y[2]=(long)p1; return y;
}
if (gcmp0(phi) || typ(phi) != t_POL)
err(talker,"reverse polymod does not exist");
av=avma; y=cgetg(n+1,t_MAT);
y[1]=(long)gscalcol_i(gun,n);
p2=phi;
for (j=2; j<=n; j++)
{
lx=lgef(p2); p1=cgetg(n+1,t_COL); y[j]=(long)p1;
for (i=1; i<=lx-2; i++) p1[i]=p2[i+1];
for ( ; i<=n; i++) p1[i]=zero;
if (j<n) p2 = gmod(gmul(p2,phi), p);
}
col=cgetg(n+1,t_COL); col[1]=zero; col[2]=un;
for (i=3; i<=n; i++) col[i]=zero;
p1=gauss(y,col); p2=gtopolyrev(p1,v); p3=caract(x,v);
tetpil=avma; return gerepile(av,tetpil,gmodulcp(p2,p3));
}
/********************************************************************/
/** **/
/** HEAPSORT **/
/** **/
/********************************************************************/
static GEN vcmp_k;
static int vcmp_lk;
static int (*vcmp_cmp)(GEN,GEN);
int
pari_compare_int(int *a,int *b)
{
return *a - *b;
}
int
pari_compare_long(long *a,long *b)
{
return *a - *b;
}
static int
veccmp(GEN x, GEN y)
{
int i,s;
for (i=1; i<vcmp_lk; i++)
{
s = vcmp_cmp((GEN) x[vcmp_k[i]], (GEN) y[vcmp_k[i]]);
if (s) return s;
}
return 0;
}
static int
longcmp(GEN x, GEN y)
{
return ((long)x > (long)y)? 1: ((x == y)? 0: -1);
}
/* Sort x = vector of elts, using cmp to compare them.
* flag & cmp_IND: indirect sort: return permutation that would sort x
* For private use:
* flag & cmp_C : as cmp_IND, but return permutation as vector of C-longs
*/
GEN
gen_sort(GEN x, int flag, int (*cmp)(GEN,GEN))
{
long i,j,indxt,ir,l,tx=typ(x),lx=lg(x);
GEN q,y,indx;
if (!is_matvec_t(tx) && tx != t_VECSMALL) err(typeer,"gen_sort");
if (flag & cmp_C) tx = t_VECSMALL;
else if (flag & cmp_IND) tx = t_VEC;
y = cgetg(lx, tx);
if (lx==1) return y;
if (lx==2)
{
if (flag & cmp_C)
y[1] = 1;
else if (flag & cmp_IND)
y[1] = un;
else
y[1] = lcopy((GEN)x[1]);
return y;
}
if (!cmp) cmp = &longcmp;
indx = (GEN) gpmalloc(lx*sizeof(long));
for (j=1; j<lx; j++) indx[j]=j;
ir=lx-1; l=(ir>>1)+1;
for(;;)
{
if (l>1)
{ l--; indxt = indx[l]; }
else
{
indxt = indx[ir]; indx[ir]=indx[1]; ir--;
if (ir == 1)
{
indx[1] = indxt;
if (flag & cmp_C)
for (i=1; i<lx; i++) y[i]=indx[i];
else if (flag & cmp_IND)
for (i=1; i<lx; i++) y[i]=lstoi(indx[i]);
else
for (i=1; i<lx; i++) y[i]=lcopy((GEN)x[indx[i]]);
free(indx); return y;
}
}
q = (GEN)x[indxt]; i=l;
for (j=i<<1; j<=ir; j<<=1)
{
if (j<ir && cmp((GEN)x[indx[j]],(GEN)x[indx[j+1]]) < 0) j++;
if (cmp(q,(GEN)x[indx[j]]) >= 0) break;
indx[i]=indx[j]; i=j;
}
indx[i]=indxt;
}
}
#define sort_fun(flag) ((flag & cmp_LEX)? &lexcmp: &gcmp)
GEN
gen_vecsort(GEN x, GEN k, long flag)
{
long i,j,l,t, lx = lg(x), tmp[2];
if (lx<=2) return gen_sort(x,flag,sort_fun(flag));
t = typ(k); vcmp_cmp = sort_fun(flag);
if (t==t_INT)
{
tmp[1] = (long)k; k = tmp;
vcmp_lk = 2;
}
else
{
if (! is_vec_t(t)) err(talker,"incorrect lextype in vecsort");
vcmp_lk = lg(k);
}
l = 0;
vcmp_k = (GEN)gpmalloc(vcmp_lk * sizeof(long));
for (i=1; i<vcmp_lk; i++)
{
j = itos((GEN)k[i]);
if (j<=0) err(talker,"negative index in vecsort");
vcmp_k[i]=j; if (j>l) l=j;
}
t = typ(x);
if (! is_matvec_t(t)) err(typeer,"vecsort");
for (j=1; j<lx; j++)
{
t = typ(x[j]);
if (! is_vec_t(t)) err(typeer,"vecsort");
if (lg((GEN)x[j]) <= l) err(talker,"index too large in vecsort");
}
x = gen_sort(x, flag, veccmp);
free(vcmp_k); return x;
}
GEN
vecsort0(GEN x, GEN k, long flag)
{
if (flag < 0 || flag >= cmp_C) err(flagerr,"vecsort");
return k? gen_vecsort(x,k,flag): gen_sort(x,flag, sort_fun(flag));
}
GEN
vecsort(GEN x, GEN k)
{
return gen_vecsort(x,k, 0);
}
GEN
sindexsort(GEN x)
{
return gen_sort(x, cmp_IND | cmp_C, gcmp);
}
GEN
sindexlexsort(GEN x)
{
return gen_sort(x, cmp_IND | cmp_C, lexcmp);
}
GEN
indexsort(GEN x)
{
return gen_sort(x, cmp_IND, gcmp);
}
GEN
indexlexsort(GEN x)
{
return gen_sort(x, cmp_IND, lexcmp);
}
GEN
sort(GEN x)
{
return gen_sort(x, 0, gcmp);
}
GEN
lexsort(GEN x)
{
return gen_sort(x, 0, lexcmp);
}
/* index of x in table T, 0 otherwise */
long
tablesearch(GEN T, GEN x, int (*cmp)(GEN,GEN))
{
long l=1,u=lg(T)-1,i,s;
while (u>=l)
{
i = (l+u)>>1; s = cmp(x,(GEN)T[i]);
if (!s) return i;
if (s<0) u=i-1; else l=i+1;
}
return 0;
}
/* assume lg(x) = lg(y), x,y in Z^n */
int
cmp_vecint(GEN x, GEN y)
{
long fl,i, lx = lg(x);
for (i=1; i<lx; i++)
if (( fl = cmpii((GEN)x[i], (GEN)y[i]) )) return fl;
return 0;
}
/* assume x and y in primedec format. */
int
cmp_prime_over_p(GEN x, GEN y)
{
int k = mael(x,4,2) - mael(y,4,2); /* diff. between residue degree */
return k? ((k > 0)? 1: -1)
: cmp_vecint((GEN)x[2], (GEN)y[2]);
}
int
cmp_prime_ideal(GEN x, GEN y)
{
int k = cmpii((GEN)x[1], (GEN)y[1]);
return k? k: cmp_prime_over_p(x,y);
}
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