/** @file factor.cpp
*
- * Polynomial factorization routines.
- * Only univariate at the moment and completely non-optimized!
+ * Polynomial factorization code (Implementation).
+ *
+ * Algorithms used can be found in
+ * [W1] An Improved Multivariate Polynomial Factoring Algorithm,
+ * P.S.Wang, Mathematics of Computation, Vol. 32, No. 144 (1978) 1215--1231.
+ * [GCL] Algorithms for Computer Algebra,
+ * K.O.Geddes, S.R.Czapor, G.Labahn, Springer Verlag, 1992.
*/
/*
#define DCOUT2(str,var)
#endif
+// forward declaration
+ex factor(const ex& poly, unsigned options);
+
+// anonymous namespace to hide all utility functions
namespace {
-typedef vector<cl_MI> Vec;
-typedef vector<Vec> VecVec;
+typedef vector<cl_MI> mvec;
#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const Vec& v)
+ostream& operator<<(ostream& o, const mvec& v)
{
- Vec::const_iterator i = v.begin(), end = v.end();
+ mvec::const_iterator i = v.begin(), end = v.end();
while ( i != end ) {
o << *i++ << " ";
}
#endif // def DEBUGFACTOR
#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const VecVec& v)
+ostream& operator<<(ostream& o, const vector<mvec>& v)
{
- VecVec::const_iterator i = v.begin(), end = v.end();
+ vector<mvec>::const_iterator i = v.begin(), end = v.end();
while ( i != end ) {
o << *i++ << endl;
}
}
#endif // def DEBUGFACTOR
-struct Term
-{
- cl_MI c; // coefficient
- unsigned int exp; // exponent >=0
-};
+////////////////////////////////////////////////////////////////////////////////
+// modular univariate polynomial code
+
+typedef cl_UP_MI umod;
+typedef vector<umod> umodvec;
+
+#define COPY(to,from) from.ring()->create(degree(from)); \
+ for ( int II=0; II<=degree(from); ++II ) to.set_coeff(II, coeff(from, II)); \
+ to.finalize()
#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const Term& t)
+ostream& operator<<(ostream& o, const umodvec& v)
{
- if ( t.exp ) {
- o << "(" << t.c << ")x^" << t.exp;
- }
- else {
- o << "(" << t.c << ")";
+ umodvec::const_iterator i = v.begin(), end = v.end();
+ while ( i != end ) {
+ o << *i++ << " , " << endl;
}
return o;
}
#endif // def DEBUGFACTOR
-struct UniPoly
+static umod umod_from_ex(const ex& e, const ex& x, const cl_univpoly_modint_ring& UPR)
{
- cl_modint_ring R;
- list<Term> terms; // highest exponent first
-
- UniPoly(const cl_modint_ring& ring) : R(ring) { }
- UniPoly(const cl_modint_ring& ring, const ex& poly, const ex& x) : R(ring)
- {
- // assert: poly is in Z[x]
- Term t;
- for ( int i=poly.degree(x); i>=poly.ldegree(x); --i ) {
- cl_I coeff = the<cl_I>(ex_to<numeric>(poly.coeff(x,i)).to_cl_N());
- if ( !zerop(coeff) ) {
- t.c = R->canonhom(coeff);
- if ( !zerop(t.c) ) {
- t.exp = i;
- terms.push_back(t);
- }
- }
- }
- }
- UniPoly(const cl_modint_ring& ring, const UniPoly& poly) : R(ring)
- {
- if ( R->modulus == poly.R->modulus ) {
- terms = poly.terms;
- }
- else {
- list<Term>::const_iterator i=poly.terms.begin(), end=poly.terms.end();
- for ( ; i!=end; ++i ) {
- terms.push_back(*i);
- terms.back().c = R->canonhom(poly.R->retract(i->c));
- if ( zerop(terms.back().c) ) {
- terms.pop_back();
- }
- }
- }
- }
- UniPoly(const cl_modint_ring& ring, const Vec& v) : R(ring)
- {
- Term t;
- for ( unsigned int i=0; i<v.size(); ++i ) {
- if ( !zerop(v[i]) ) {
- t.c = v[i];
- t.exp = i;
- terms.push_front(t);
- }
- }
- }
- unsigned int degree() const
- {
- if ( terms.size() ) {
- return terms.front().exp;
- }
- else {
- return 0;
- }
- }
- bool zero() const { return (terms.size() == 0); }
- const cl_MI operator[](unsigned int deg) const
- {
- list<Term>::const_iterator i = terms.begin(), end = terms.end();
- for ( ; i != end; ++i ) {
- if ( i->exp == deg ) {
- return i->c;
- }
- if ( i->exp < deg ) {
- break;
- }
- }
- return R->zero();
- }
- void set(unsigned int deg, const cl_MI& c)
- {
- list<Term>::iterator i = terms.begin(), end = terms.end();
- while ( i != end ) {
- if ( i->exp == deg ) {
- if ( !zerop(c) ) {
- i->c = c;
- }
- else {
- terms.erase(i);
- }
- return;
- }
- if ( i->exp < deg ) {
- break;
- }
- ++i;
- }
- if ( !zerop(c) ) {
- Term t;
- t.c = c;
- t.exp = deg;
- terms.insert(i, t);
- }
- }
- ex to_ex(const ex& x, bool symmetric = true) const
- {
- ex r;
- list<Term>::const_iterator i = terms.begin(), end = terms.end();
- if ( symmetric ) {
- numeric mod(R->modulus);
- numeric halfmod = (mod-1)/2;
- for ( ; i != end; ++i ) {
- numeric n(R->retract(i->c));
- if ( n > halfmod ) {
- r += pow(x, i->exp) * (n-mod);
- }
- else {
- r += pow(x, i->exp) * n;
- }
- }
- }
- else {
- for ( ; i != end; ++i ) {
- r += pow(x, i->exp) * numeric(R->retract(i->c));
- }
- }
- return r;
- }
- void unit_normal()
- {
- if ( terms.size() ) {
- if ( terms.front().c != R->one() ) {
- list<Term>::iterator i = terms.begin(), end = terms.end();
- cl_MI cont = i->c;
- i->c = R->one();
- while ( ++i != end ) {
- i->c = div(i->c, cont);
- if ( zerop(i->c) ) {
- terms.erase(i);
- }
- }
- }
- }
- }
- cl_MI unit() const
- {
- return terms.front().c;
- }
- void divide(const cl_MI& x)
- {
- list<Term>::iterator i = terms.begin(), end = terms.end();
- for ( ; i != end; ++i ) {
- i->c = div(i->c, x);
- if ( zerop(i->c) ) {
- terms.erase(i);
- }
- }
- }
- void divide(const cl_I& x)
- {
- list<Term>::iterator i = terms.begin(), end = terms.end();
- for ( ; i != end; ++i ) {
- i->c = cl_MI(R, the<cl_I>(R->retract(i->c) / x));
- }
- }
- void reduce_exponents(unsigned int prime)
- {
- list<Term>::iterator i = terms.begin(), end = terms.end();
- while ( i != end ) {
- if ( i->exp > 0 ) {
- // assert: i->exp is multiple of prime
- i->exp /= prime;
- }
- ++i;
- }
- }
- void deriv(UniPoly& d) const
- {
- list<Term>::const_iterator i = terms.begin(), end = terms.end();
- while ( i != end ) {
- if ( i->exp ) {
- cl_MI newc = i->c * i->exp;
- if ( !zerop(newc) ) {
- Term t;
- t.c = newc;
- t.exp = i->exp-1;
- d.terms.push_back(t);
- }
- }
- ++i;
- }
- }
- bool operator<(const UniPoly& o) const
- {
- if ( terms.size() != o.terms.size() ) {
- return terms.size() < o.terms.size();
- }
- list<Term>::const_iterator i1 = terms.begin(), end = terms.end();
- list<Term>::const_iterator i2 = o.terms.begin();
- while ( i1 != end ) {
- if ( i1->exp != i2->exp ) {
- return i1->exp < i2->exp;
- }
- if ( i1->c != i2->c ) {
- return R->retract(i1->c) < R->retract(i2->c);
- }
- ++i1; ++i2;
- }
- return true;
- }
- bool operator==(const UniPoly& o) const
- {
- if ( terms.size() != o.terms.size() ) {
- return false;
- }
- list<Term>::const_iterator i1 = terms.begin(), end = terms.end();
- list<Term>::const_iterator i2 = o.terms.begin();
- while ( i1 != end ) {
- if ( i1->exp != i2->exp ) {
- return false;
- }
- if ( i1->c != i2->c ) {
- return false;
- }
- ++i1; ++i2;
- }
- return true;
- }
- bool operator!=(const UniPoly& o) const
- {
- bool res = !(*this == o);
- return res;
- }
-};
-
-static UniPoly operator*(const UniPoly& a, const UniPoly& b)
-{
- unsigned int n = a.degree()+b.degree();
- UniPoly c(a.R);
- Term t;
- for ( unsigned int i=0 ; i<=n; ++i ) {
- t.c = a.R->zero();
- for ( unsigned int j=0 ; j<=i; ++j ) {
- t.c = t.c + a[j] * b[i-j];
- }
- if ( !zerop(t.c) ) {
- t.exp = i;
- c.terms.push_front(t);
- }
- }
- return c;
+ // assert: e is in Z[x]
+ int deg = e.degree(x);
+ umod p = UPR->create(deg);
+ int ldeg = e.ldegree(x);
+ for ( ; deg>=ldeg; --deg ) {
+ cl_I coeff = the<cl_I>(ex_to<numeric>(e.coeff(x, deg)).to_cl_N());
+ p.set_coeff(deg, UPR->basering()->canonhom(coeff));
+ }
+ for ( ; deg>=0; --deg ) {
+ p.set_coeff(deg, UPR->basering()->zero());
+ }
+ p.finalize();
+ return p;
}
-static UniPoly operator-(const UniPoly& a, const UniPoly& b)
+static umod umod_from_ex(const ex& e, const ex& x, const cl_modint_ring& R)
{
- list<Term>::const_iterator ia = a.terms.begin(), aend = a.terms.end();
- list<Term>::const_iterator ib = b.terms.begin(), bend = b.terms.end();
- UniPoly c(a.R);
- while ( ia != aend && ib != bend ) {
- if ( ia->exp > ib->exp ) {
- c.terms.push_back(*ia);
- ++ia;
- }
- else if ( ia->exp < ib->exp ) {
- c.terms.push_back(*ib);
- c.terms.back().c = -c.terms.back().c;
- ++ib;
- }
- else {
- Term t;
- t.exp = ia->exp;
- t.c = ia->c - ib->c;
- if ( !zerop(t.c) ) {
- c.terms.push_back(t);
- }
- ++ia; ++ib;
- }
- }
- while ( ia != aend ) {
- c.terms.push_back(*ia);
- ++ia;
- }
- while ( ib != bend ) {
- c.terms.push_back(*ib);
- c.terms.back().c = -c.terms.back().c;
- ++ib;
- }
- return c;
+ return umod_from_ex(e, x, find_univpoly_ring(R));
}
-static UniPoly operator*(const UniPoly& a, const cl_MI& fac)
+static umod umod_from_ex(const ex& e, const ex& x, const cl_I& modulus)
{
- unsigned int n = a.degree();
- UniPoly c(a.R);
- Term t;
- for ( unsigned int i=0 ; i<=n; ++i ) {
- t.c = a[i] * fac;
- if ( !zerop(t.c) ) {
- t.exp = i;
- c.terms.push_front(t);
- }
- }
- return c;
+ return umod_from_ex(e, x, find_modint_ring(modulus));
}
-static UniPoly operator+(const UniPoly& a, const UniPoly& b)
+static umod umod_from_modvec(const mvec& mv)
{
- list<Term>::const_iterator ia = a.terms.begin(), aend = a.terms.end();
- list<Term>::const_iterator ib = b.terms.begin(), bend = b.terms.end();
- UniPoly c(a.R);
- while ( ia != aend && ib != bend ) {
- if ( ia->exp > ib->exp ) {
- c.terms.push_back(*ia);
- ++ia;
- }
- else if ( ia->exp < ib->exp ) {
- c.terms.push_back(*ib);
- ++ib;
- }
- else {
- Term t;
- t.exp = ia->exp;
- t.c = ia->c + ib->c;
- if ( !zerop(t.c) ) {
- c.terms.push_back(t);
- }
- ++ia; ++ib;
- }
- }
- while ( ia != aend ) {
- c.terms.push_back(*ia);
- ++ia;
+ size_t n = mv.size(); // assert: n>0
+ while ( n && zerop(mv[n-1]) ) --n;
+ cl_univpoly_modint_ring UPR = find_univpoly_ring(mv.front().ring());
+ if ( n == 0 ) {
+ umod p = UPR->create(-1);
+ p.finalize();
+ return p;
}
- while ( ib != bend ) {
- c.terms.push_back(*ib);
- ++ib;
+ umod p = UPR->create(n-1);
+ for ( size_t i=0; i<n; ++i ) {
+ p.set_coeff(i, mv[i]);
}
- return c;
+ p.finalize();
+ return p;
}
-// static UniPoly operator-(const UniPoly& a)
-// {
-// list<Term>::const_iterator ia = a.terms.begin(), aend = a.terms.end();
-// UniPoly c(a.R);
-// while ( ia != aend ) {
-// c.terms.push_back(*ia);
-// c.terms.back().c = -c.terms.back().c;
-// ++ia;
-// }
-// return c;
-// }
-
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const UniPoly& t)
+static umod divide(const umod& a, const cl_I& x)
{
- list<Term>::const_iterator i = t.terms.begin(), end = t.terms.end();
- if ( i == end ) {
- o << "0";
- return o;
- }
- for ( ; i != end; ) {
- o << *i++;
- if ( i != end ) {
- o << " + ";
- }
- }
- return o;
+ DCOUT(divide);
+ DCOUTVAR(a);
+ cl_univpoly_modint_ring UPR = a.ring();
+ cl_modint_ring R = UPR->basering();
+ int deg = degree(a);
+ umod newa = UPR->create(deg);
+ for ( int i=0; i<=deg; ++i ) {
+ cl_I c = R->retract(coeff(a, i));
+ newa.set_coeff(i, cl_MI(R, the<cl_I>(c / x)));
+ }
+ newa.finalize();
+ DCOUT(END divide);
+ return newa;
}
-#endif // def DEBUGFACTOR
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const list<UniPoly>& t)
+static ex umod_to_ex(const umod& a, const ex& x)
{
- list<UniPoly>::const_iterator i = t.begin(), end = t.end();
- o << "{" << endl;
- for ( ; i != end; ) {
- o << *i++ << endl;
- }
- o << "}" << endl;
- return o;
+ ex e;
+ cl_modint_ring R = a.ring()->basering();
+ cl_I mod = R->modulus;
+ cl_I halfmod = (mod-1) >> 1;
+ for ( int i=degree(a); i>=0; --i ) {
+ cl_I n = R->retract(coeff(a, i));
+ if ( n > halfmod ) {
+ e += numeric(n-mod) * pow(x, i);
+ } else {
+ e += numeric(n) * pow(x, i);
+ }
+ }
+ return e;
}
-#endif // def DEBUGFACTOR
-
-typedef vector<UniPoly> UniPolyVec;
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const UniPolyVec& v)
+static void unit_normal(umod& a)
{
- UniPolyVec::const_iterator i = v.begin(), end = v.end();
- while ( i != end ) {
- o << *i++ << " , " << endl;
+ int deg = degree(a);
+ if ( deg >= 0 ) {
+ cl_MI lc = coeff(a, deg);
+ cl_MI one = a.ring()->basering()->one();
+ if ( lc != one ) {
+ umod newa = a.ring()->create(deg);
+ newa.set_coeff(deg, one);
+ for ( --deg; deg>=0; --deg ) {
+ cl_MI nc = div(coeff(a, deg), lc);
+ newa.set_coeff(deg, nc);
+ }
+ newa.finalize();
+ a = newa;
+ }
}
- return o;
}
-#endif // def DEBUGFACTOR
-struct UniFactor
+static umod rem(const umod& a, const umod& b)
{
- UniPoly p;
- unsigned int exp;
-
- UniFactor(const cl_modint_ring& ring) : p(ring) { }
- UniFactor(const UniPoly& p_, unsigned int exp_) : p(p_), exp(exp_) { }
- bool operator<(const UniFactor& o) const
- {
- return p < o.p;
+ int k, n;
+ n = degree(b);
+ k = degree(a) - n;
+ if ( k < 0 ) {
+ umod c = COPY(c, a);
+ return c;
}
-};
-
-struct UniFactorVec
-{
- vector<UniFactor> factors;
- void unique()
- {
- sort(factors.begin(), factors.end());
- if ( factors.size() > 1 ) {
- vector<UniFactor>::iterator i = factors.begin();
- vector<UniFactor>::const_iterator cmp = factors.begin()+1;
- vector<UniFactor>::iterator end = factors.end();
- while ( cmp != end ) {
- if ( i->p != cmp->p ) {
- ++i;
- ++cmp;
- }
- else {
- i->exp += cmp->exp;
- ++cmp;
- }
- }
- if ( i != end-1 ) {
- factors.erase(i+1, end);
+ umod c = COPY(c, a);
+ do {
+ cl_MI qk = div(coeff(c, n+k), coeff(b, n));
+ if ( !zerop(qk) ) {
+ unsigned int j;
+ for ( int i=0; i<n; ++i ) {
+ j = n + k - 1 - i;
+ c.set_coeff(j, coeff(c, j) - qk * coeff(b, j-k));
}
}
- }
-};
+ } while ( k-- );
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const UniFactorVec& ufv)
-{
- for ( size_t i=0; i<ufv.factors.size(); ++i ) {
- if ( i != ufv.factors.size()-1 ) {
- o << "*";
- }
- else {
- o << " ";
- }
- o << "[ " << ufv.factors[i].p << " ]^" << ufv.factors[i].exp << endl;
+ cl_MI zero = a.ring()->basering()->zero();
+ for ( int i=degree(a); i>=n; --i ) {
+ c.set_coeff(i, zero);
}
- return o;
+
+ c.finalize();
+ return c;
}
-#endif // def DEBUGFACTOR
-static void rem(const UniPoly& a_, const UniPoly& b, UniPoly& c)
+static umod div(const umod& a, const umod& b)
{
- if ( a_.degree() < b.degree() ) {
- c = a_;
- return;
+ int k, n;
+ n = degree(b);
+ k = degree(a) - n;
+ if ( k < 0 ) {
+ umod q = a.ring()->create(-1);
+ q.finalize();
+ return q;
}
- unsigned int k, n;
- n = b.degree();
- k = a_.degree() - n;
-
- if ( n == 0 ) {
- c.terms.clear();
- return;
- }
-
- c = a_;
- Term termbuf;
-
- while ( true ) {
- cl_MI qk = div(c[n+k], b[n]);
+ umod c = COPY(c, a);
+ umod q = a.ring()->create(k);
+ do {
+ cl_MI qk = div(coeff(c, n+k), coeff(b, n));
if ( !zerop(qk) ) {
+ q.set_coeff(k, qk);
unsigned int j;
- for ( unsigned int i=0; i<n; ++i ) {
+ for ( int i=0; i<n; ++i ) {
j = n + k - 1 - i;
- c.set(j, c[j] - qk*b[j-k]);
+ c.set_coeff(j, coeff(c, j) - qk * coeff(b, j-k));
}
}
- if ( k == 0 ) break;
- --k;
- }
- list<Term>::iterator i = c.terms.begin(), end = c.terms.end();
- while ( i != end ) {
- if ( i->exp <= n-1 ) {
- break;
- }
- ++i;
- }
- c.terms.erase(c.terms.begin(), i);
+ } while ( k-- );
+
+ q.finalize();
+ return q;
}
-static void div(const UniPoly& a_, const UniPoly& b, UniPoly& q)
+static umod remdiv(const umod& a, const umod& b, umod& q)
{
- if ( a_.degree() < b.degree() ) {
- q.terms.clear();
- return;
- }
-
- unsigned int k, n;
- n = b.degree();
- k = a_.degree() - n;
-
- UniPoly c = a_;
- Term termbuf;
-
- while ( true ) {
- cl_MI qk = div(c[n+k], b[n]);
+ int k, n;
+ n = degree(b);
+ k = degree(a) - n;
+ if ( k < 0 ) {
+ q = a.ring()->create(-1);
+ q.finalize();
+ umod c = COPY(c, a);
+ return c;
+ }
+
+ umod c = COPY(c, a);
+ q = a.ring()->create(k);
+ do {
+ cl_MI qk = div(coeff(c, n+k), coeff(b, n));
if ( !zerop(qk) ) {
- Term t;
- t.c = qk;
- t.exp = k;
- q.terms.push_back(t);
+ q.set_coeff(k, qk);
unsigned int j;
- for ( unsigned int i=0; i<n; ++i ) {
+ for ( int i=0; i<n; ++i ) {
j = n + k - 1 - i;
- c.set(j, c[j] - qk*b[j-k]);
+ c.set_coeff(j, coeff(c, j) - qk * coeff(b, j-k));
}
}
- if ( k == 0 ) break;
- --k;
- }
-}
-
-static void gcd(const UniPoly& a, const UniPoly& b, UniPoly& c)
-{
- c = a;
- c.unit_normal();
- UniPoly d = b;
- d.unit_normal();
+ } while ( k-- );
- if ( c.degree() < d.degree() ) {
- gcd(b, a, c);
- return;
+ cl_MI zero = a.ring()->basering()->zero();
+ for ( int i=degree(a); i>=n; --i ) {
+ c.set_coeff(i, zero);
}
- while ( !d.zero() ) {
- UniPoly r(a.R);
- rem(c, d, r);
- c = d;
- d = r;
- }
- c.unit_normal();
+ q.finalize();
+ c.finalize();
+ return c;
}
-static bool is_one(const UniPoly& w)
+static umod gcd(const umod& a, const umod& b)
{
- if ( w.terms.size() == 1 && w[0] == w.R->one() ) {
- return true;
- }
- return false;
+ if ( degree(a) < degree(b) ) return gcd(b, a);
+
+ umod c = COPY(c, a);
+ unit_normal(c);
+ umod d = COPY(d, b);
+ unit_normal(d);
+ while ( !zerop(d) ) {
+ umod r = rem(c, d);
+ c = COPY(c, d);
+ d = COPY(d, r);
+ }
+ unit_normal(c);
+ return c;
}
-static void sqrfree_main(const UniPoly& a, UniFactorVec& fvec)
+static bool squarefree(const umod& a)
{
- unsigned int i = 1;
- UniPoly b(a.R);
- a.deriv(b);
- if ( !b.zero() ) {
- UniPoly c(a.R), w(a.R);
- gcd(a, b, c);
- div(a, c, w);
- while ( !is_one(w) ) {
- UniPoly y(a.R), z(a.R);
- gcd(w, c, y);
- div(w, y, z);
- if ( !is_one(z) ) {
- UniFactor uf(z, i);
- fvec.factors.push_back(uf);
- }
- ++i;
- w = y;
- UniPoly cbuf(a.R);
- div(c, y, cbuf);
- c = cbuf;
- }
- if ( !is_one(c) ) {
- unsigned int prime = cl_I_to_uint(c.R->modulus);
- c.reduce_exponents(prime);
- unsigned int pos = fvec.factors.size();
- sqrfree_main(c, fvec);
- for ( unsigned int p=pos; p<fvec.factors.size(); ++p ) {
- fvec.factors[p].exp *= prime;
- }
- return;
- }
- }
- else {
- unsigned int prime = cl_I_to_uint(a.R->modulus);
- UniPoly amod = a;
- amod.reduce_exponents(prime);
- unsigned int pos = fvec.factors.size();
- sqrfree_main(amod, fvec);
- for ( unsigned int p=pos; p<fvec.factors.size(); ++p ) {
- fvec.factors[p].exp *= prime;
- }
- return;
+ umod b = deriv(a);
+ if ( zerop(b) ) {
+ return false;
}
+ umod one = a.ring()->one();
+ umod c = gcd(a, b);
+ return c == one;
}
-static void squarefree(const UniPoly& a, UniFactorVec& fvec)
-{
- sqrfree_main(a, fvec);
- fvec.unique();
-}
+// END modular univariate polynomial code
+////////////////////////////////////////////////////////////////////////////////
-class Matrix
+////////////////////////////////////////////////////////////////////////////////
+// modular matrix
+
+class modular_matrix
{
- friend ostream& operator<<(ostream& o, const Matrix& m);
+ friend ostream& operator<<(ostream& o, const modular_matrix& m);
public:
- Matrix(size_t r_, size_t c_, const cl_MI& init) : r(r_), c(c_)
+ modular_matrix(size_t r_, size_t c_, const cl_MI& init) : r(r_), c(c_)
{
m.resize(c*r, init);
}
cl_MI operator()(size_t row, size_t col) const { return m[row*c + col]; }
void mul_col(size_t col, const cl_MI x)
{
- Vec::iterator i = m.begin() + col;
+ mvec::iterator i = m.begin() + col;
for ( size_t rc=0; rc<r; ++rc ) {
*i = *i * x;
i += c;
}
void sub_col(size_t col1, size_t col2, const cl_MI fac)
{
- Vec::iterator i1 = m.begin() + col1;
- Vec::iterator i2 = m.begin() + col2;
+ mvec::iterator i1 = m.begin() + col1;
+ mvec::iterator i2 = m.begin() + col2;
for ( size_t rc=0; rc<r; ++rc ) {
*i1 = *i1 - *i2 * fac;
i1 += c;
void switch_col(size_t col1, size_t col2)
{
cl_MI buf;
- Vec::iterator i1 = m.begin() + col1;
- Vec::iterator i2 = m.begin() + col2;
+ mvec::iterator i1 = m.begin() + col1;
+ mvec::iterator i2 = m.begin() + col2;
for ( size_t rc=0; rc<r; ++rc ) {
buf = *i1; *i1 = *i2; *i2 = buf;
i1 += c;
}
bool is_col_zero(size_t col) const
{
- Vec::const_iterator i = m.begin() + col;
+ mvec::const_iterator i = m.begin() + col;
for ( size_t rr=0; rr<r; ++rr ) {
if ( !zerop(*i) ) {
return false;
}
bool is_row_zero(size_t row) const
{
- Vec::const_iterator i = m.begin() + row*c;
+ mvec::const_iterator i = m.begin() + row*c;
for ( size_t cc=0; cc<c; ++cc ) {
if ( !zerop(*i) ) {
return false;
}
void set_row(size_t row, const vector<cl_MI>& newrow)
{
- Vec::iterator i1 = m.begin() + row*c;
- Vec::const_iterator i2 = newrow.begin(), end = newrow.end();
+ mvec::iterator i1 = m.begin() + row*c;
+ mvec::const_iterator i2 = newrow.begin(), end = newrow.end();
for ( ; i2 != end; ++i1, ++i2 ) {
*i1 = *i2;
}
}
- Vec::const_iterator row_begin(size_t row) const { return m.begin()+row*c; }
- Vec::const_iterator row_end(size_t row) const { return m.begin()+row*c+r; }
+ mvec::const_iterator row_begin(size_t row) const { return m.begin()+row*c; }
+ mvec::const_iterator row_end(size_t row) const { return m.begin()+row*c+r; }
private:
size_t r, c;
- Vec m;
+ mvec m;
};
#ifdef DEBUGFACTOR
-Matrix operator*(const Matrix& m1, const Matrix& m2)
+modular_matrix operator*(const modular_matrix& m1, const modular_matrix& m2)
{
const unsigned int r = m1.rowsize();
const unsigned int c = m2.colsize();
- Matrix o(r,c,m1(0,0));
+ modular_matrix o(r,c,m1(0,0));
for ( size_t i=0; i<r; ++i ) {
for ( size_t j=0; j<c; ++j ) {
return o;
}
-ostream& operator<<(ostream& o, const Matrix& m)
+ostream& operator<<(ostream& o, const modular_matrix& m)
{
vector<cl_MI>::const_iterator i = m.m.begin(), end = m.m.end();
size_t wrap = 1;
}
#endif // def DEBUGFACTOR
-static void q_matrix(const UniPoly& a, Matrix& Q)
+// END modular matrix
+////////////////////////////////////////////////////////////////////////////////
+
+static void q_matrix(const umod& a, modular_matrix& Q)
{
- unsigned int n = a.degree();
- unsigned int q = cl_I_to_uint(a.R->modulus);
+ int n = degree(a);
+ unsigned int q = cl_I_to_uint(a.ring()->basering()->modulus);
// fast and buggy
// vector<cl_MI> r(n, a.R->zero());
// r[0] = a.R->one();
// }
// }
// slow and (hopefully) correct
- for ( size_t i=0; i<n; ++i ) {
- UniPoly qk(a.R);
- qk.set(i*q, a.R->one());
- UniPoly r(a.R);
- rem(qk, a, r);
- Vec rvec;
- for ( size_t j=0; j<n; ++j ) {
- rvec.push_back(r[j]);
+ cl_MI one = a.ring()->basering()->one();
+ for ( int i=0; i<n; ++i ) {
+ umod qk = a.ring()->create(i*q);
+ qk.set_coeff(i*q, one);
+ qk.finalize();
+ umod r = rem(qk, a);
+ mvec rvec;
+ for ( int j=0; j<n; ++j ) {
+ rvec.push_back(coeff(r, j));
}
Q.set_row(i, rvec);
}
}
-static void nullspace(Matrix& M, vector<Vec>& basis)
+static void nullspace(modular_matrix& M, vector<mvec>& basis)
{
const size_t n = M.rowsize();
const cl_MI one = M(0,0).ring()->one();
}
for ( size_t i=0; i<n; ++i ) {
if ( !M.is_row_zero(i) ) {
- Vec nu(M.row_begin(i), M.row_end(i));
+ mvec nu(M.row_begin(i), M.row_end(i));
basis.push_back(nu);
}
}
}
-static void berlekamp(const UniPoly& a, UniPolyVec& upv)
+static void berlekamp(const umod& a, umodvec& upv)
{
- Matrix Q(a.degree(), a.degree(), a.R->zero());
+ cl_modint_ring R = a.ring()->basering();
+ const umod one = a.ring()->one();
+
+ modular_matrix Q(degree(a), degree(a), R->zero());
q_matrix(a, Q);
- VecVec nu;
+ vector<mvec> nu;
nullspace(Q, nu);
const unsigned int k = nu.size();
if ( k == 1 ) {
return;
}
- list<UniPoly> factors;
+ list<umod> factors;
factors.push_back(a);
unsigned int size = 1;
unsigned int r = 1;
- unsigned int q = cl_I_to_uint(a.R->modulus);
+ unsigned int q = cl_I_to_uint(R->modulus);
- list<UniPoly>::iterator u = factors.begin();
+ list<umod>::iterator u = factors.begin();
while ( true ) {
for ( unsigned int s=0; s<q; ++s ) {
- UniPoly g(a.R);
- UniPoly nur(a.R, nu[r]);
- nur.set(0, nur[0] - cl_MI(a.R, s));
- gcd(nur, *u, g);
- if ( !is_one(g) && g != *u ) {
- UniPoly uo(a.R);
- div(*u, g, uo);
- if ( is_one(uo) ) {
+ umod nur = umod_from_modvec(nu[r]);
+ cl_MI buf = coeff(nur, 0) - cl_MI(R, s);
+ nur.set_coeff(0, buf);
+ nur.finalize();
+ umod g = gcd(nur, *u);
+ if ( g != one && g != *u ) {
+ umod uo = div(*u, g);
+ if ( uo == one ) {
throw logic_error("berlekamp: unexpected divisor.");
}
else {
- *u = uo;
+ *u = COPY((*u), uo);
}
factors.push_back(g);
size = 0;
- list<UniPoly>::const_iterator i = factors.begin(), end = factors.end();
+ list<umod>::const_iterator i = factors.begin(), end = factors.end();
while ( i != end ) {
- if ( i->degree() ) ++size;
+ if ( degree(*i) ) ++size;
++i;
}
if ( size == k ) {
- list<UniPoly>::const_iterator i = factors.begin(), end = factors.end();
+ list<umod>::const_iterator i = factors.begin(), end = factors.end();
while ( i != end ) {
upv.push_back(*i++);
}
return;
}
-// if ( u->degree() < nur.degree() ) {
-// break;
-// }
}
}
if ( ++r == k ) {
}
}
-static void factor_modular(const UniPoly& p, UniPolyVec& upv)
+static void factor_modular(const umod& p, umodvec& upv)
{
berlekamp(p, upv);
return;
}
-static void exteuclid(const UniPoly& a, const UniPoly& b, UniPoly& g, UniPoly& s, UniPoly& t)
+static void exteuclid(const umod& a, const umod& b, umod& g, umod& s, umod& t)
{
- if ( a.degree() < b.degree() ) {
+ if ( degree(a) < degree(b) ) {
exteuclid(b, a, g, t, s);
return;
}
- UniPoly c1(a.R), c2(a.R), d1(a.R), d2(a.R), q(a.R), r(a.R), r1(a.R), r2(a.R);
- UniPoly c = a; c.unit_normal();
- UniPoly d = b; d.unit_normal();
- c1.set(0, a.R->one());
- d2.set(0, a.R->one());
- while ( !d.zero() ) {
- q.terms.clear();
- div(c, d, q);
- r = c - q * d;
- r1 = c1 - q * d1;
- r2 = c2 - q * d2;
- c = d;
- c1 = d1;
- c2 = d2;
- d = r;
- d1 = r1;
- d2 = r2;
- }
- g = c; g.unit_normal();
- s = c1;
- s.divide(a.unit());
- s.divide(c.unit());
- t = c2;
- t.divide(b.unit());
- t.divide(c.unit());
+ umod c = COPY(c, a); unit_normal(c);
+ umod d = COPY(d, b); unit_normal(d);
+ umod c1 = a.ring()->one();
+ umod c2 = a.ring()->create(-1);
+ umod d1 = a.ring()->create(-1);
+ umod d2 = a.ring()->one();
+ while ( !zerop(d) ) {
+ umod q = div(c, d);
+ umod r = c - q * d;
+ umod r1 = c1 - q * d1;
+ umod r2 = c2 - q * d2;
+ c = COPY(c, d);
+ c1 = COPY(c1, d1);
+ c2 = COPY(c2, d2);
+ d = COPY(d, r);
+ d1 = COPY(d1, r1);
+ d2 = COPY(d2, r2);
+ }
+ g = COPY(g, c); unit_normal(g);
+ s = COPY(s, c1);
+ for ( int i=0; i<=degree(s); ++i ) {
+ s.set_coeff(i, coeff(s, i) * recip(coeff(a, degree(a)) * coeff(c, degree(c))));
+ }
+ s.finalize();
+ t = COPY(t, c2);
+ for ( int i=0; i<=degree(t); ++i ) {
+ t.set_coeff(i, coeff(t, i) * recip(coeff(b, degree(b)) * coeff(c, degree(c))));
+ }
+ t.finalize();
}
static ex replace_lc(const ex& poly, const ex& x, const ex& lc)
return r;
}
-static ex hensel_univar(const ex& a_, const ex& x, unsigned int p, const UniPoly& u1_, const UniPoly& w1_, const ex& gamma_ = 0)
+static ex hensel_univar(const ex& a_, const ex& x, unsigned int p, const umod& u1_, const umod& w1_, const ex& gamma_ = 0)
{
ex a = a_;
- const cl_modint_ring& R = u1_.R;
+ const cl_univpoly_modint_ring& UPR = u1_.ring();
+ const cl_modint_ring& R = UPR->basering();
// calc bound B
ex maxcoeff;
maxcoeff += pow(abs(a.coeff(x, i)),2);
}
cl_I normmc = ceiling1(the<cl_R>(cln::sqrt(ex_to<numeric>(maxcoeff).to_cl_N())));
- cl_I maxdegree = (u1_.degree() > w1_.degree()) ? u1_.degree() : w1_.degree();
+ cl_I maxdegree = (degree(u1_) > degree(w1_)) ? degree(u1_) : degree(w1_);
cl_I B = normmc * expt_pos(cl_I(2), maxdegree);
// step 1
}
numeric gamma_ui = ex_to<numeric>(abs(gamma));
a = a * gamma;
- UniPoly nu1 = u1_;
- nu1.unit_normal();
- UniPoly nw1 = w1_;
- nw1.unit_normal();
+ umod nu1 = COPY(nu1, u1_);
+ unit_normal(nu1);
+ umod nw1 = COPY(nw1, w1_);
+ unit_normal(nw1);
ex phi;
- phi = expand(gamma * nu1.to_ex(x));
- UniPoly u1(R, phi, x);
- phi = expand(alpha * nw1.to_ex(x));
- UniPoly w1(R, phi, x);
+ phi = gamma * umod_to_ex(nu1, x);
+ umod u1 = umod_from_ex(phi, x, R);
+ phi = alpha * umod_to_ex(nw1, x);
+ umod w1 = umod_from_ex(phi, x, R);
// step 2
- UniPoly s(R), t(R), g(R);
+ umod g = UPR->create(-1);
+ umod s = UPR->create(-1);
+ umod t = UPR->create(-1);
exteuclid(u1, w1, g, s, t);
// step 3
- ex u = replace_lc(u1.to_ex(x), x, gamma);
- ex w = replace_lc(w1.to_ex(x), x, alpha);
+ ex u = replace_lc(umod_to_ex(u1, x), x, gamma);
+ ex w = replace_lc(umod_to_ex(w1, x), x, alpha);
ex e = expand(a - u * w);
numeric modulus = p;
const numeric maxmodulus = 2*numeric(B)*gamma_ui;
// step 4
while ( !e.is_zero() && modulus < maxmodulus ) {
ex c = e / modulus;
- phi = expand(s.to_ex(x)*c);
- UniPoly sigmatilde(R, phi, x);
- phi = expand(t.to_ex(x)*c);
- UniPoly tautilde(R, phi, x);
- UniPoly q(R), r(R);
- div(sigmatilde, w1, q);
- rem(sigmatilde, w1, r);
- UniPoly sigma = r;
- phi = expand(tautilde.to_ex(x) + q.to_ex(x) * u1.to_ex(x));
- UniPoly tau(R, phi, x);
- u = expand(u + tau.to_ex(x) * modulus);
- w = expand(w + sigma.to_ex(x) * modulus);
+ phi = expand(umod_to_ex(s, x) * c);
+ umod sigmatilde = umod_from_ex(phi, x, R);
+ phi = expand(umod_to_ex(t, x) * c);
+ umod tautilde = umod_from_ex(phi, x, R);
+ umod q = div(sigmatilde, w1);
+ umod r = rem(sigmatilde, w1);
+ umod sigma = COPY(sigma, r);
+ phi = expand(umod_to_ex(tautilde, x) + umod_to_ex(q, x) * umod_to_ex(u1, x));
+ umod tau = umod_from_ex(phi, x, R);
+ u = expand(u + umod_to_ex(tau, x) * modulus);
+ w = expand(w + umod_to_ex(sigma, x) * modulus);
e = expand(a - u * w);
modulus = modulus * p;
}
vector<int> k;
};
-static void split(const UniPolyVec& factors, const Partition& part, UniPoly& a, UniPoly& b)
+static void split(const umodvec& factors, const Partition& part, umod& a, umod& b)
{
- a.set(0, a.R->one());
- b.set(0, a.R->one());
+ a = factors.front().ring()->one();
+ b = factors.front().ring()->one();
for ( size_t i=0; i<part.size(); ++i ) {
if ( part[i] ) {
b = b * factors[i];
struct ModFactors
{
ex poly;
- UniPolyVec factors;
+ umodvec factors;
};
static ex factor_univariate(const ex& poly, const ex& x)
ex unit, cont, prim;
poly.unitcontprim(x, unit, cont, prim);
- // determine proper prime
- unsigned int p = 3;
- cl_modint_ring R = find_modint_ring(p);
- while ( true ) {
- if ( irem(ex_to<numeric>(prim.lcoeff(x)), p) != 0 ) {
- UniPoly modpoly(R, prim, x);
- UniFactorVec sqrfree_ufv;
- squarefree(modpoly, sqrfree_ufv);
- if ( sqrfree_ufv.factors.size() == 1 && sqrfree_ufv.factors.front().exp == 1 ) break;
- }
- p = next_prime(p);
- R = find_modint_ring(p);
- }
-
- // do modular factorization
- UniPoly modpoly(R, prim, x);
- UniPolyVec factors;
- factor_modular(modpoly, factors);
- if ( factors.size() <= 1 ) {
- // irreducible for sure
- return poly;
+ // determine proper prime and minimize number of modular factors
+ unsigned int p = 3, lastp;
+ cl_modint_ring R;
+ unsigned int trials = 0;
+ unsigned int minfactors = 0;
+ numeric lcoeff = ex_to<numeric>(prim.lcoeff(x));
+ umodvec factors;
+ while ( trials < 2 ) {
+ while ( true ) {
+ p = next_prime(p);
+ if ( irem(lcoeff, p) != 0 ) {
+ R = find_modint_ring(p);
+ umod modpoly = umod_from_ex(prim, x, R);
+ if ( squarefree(modpoly) ) break;
+ }
+ }
+
+ // do modular factorization
+ umod modpoly = umod_from_ex(prim, x, R);
+ umodvec trialfactors;
+ factor_modular(modpoly, trialfactors);
+ if ( trialfactors.size() <= 1 ) {
+ // irreducible for sure
+ return poly;
+ }
+
+ if ( minfactors == 0 || trialfactors.size() < minfactors ) {
+ factors = trialfactors;
+ minfactors = factors.size();
+ lastp = p;
+ trials = 1;
+ }
+ else {
+ ++trials;
+ }
}
+ p = lastp;
+ R = find_modint_ring(p);
+ cl_univpoly_modint_ring UPR = find_univpoly_ring(R);
// lift all factor combinations
stack<ModFactors> tocheck;
const size_t n = tocheck.top().factors.size();
Partition part(n);
while ( true ) {
- UniPoly a(R), b(R);
+ umod a = UPR->create(-1);
+ umod b = UPR->create(-1);
split(tocheck.top().factors, part, a, b);
ex answer = hensel_univar(tocheck.top().poly, x, p, a, b);
break;
}
else {
- UniPolyVec newfactors1(part.size_first(), R), newfactors2(part.size_second(), R);
- UniPolyVec::iterator i1 = newfactors1.begin(), i2 = newfactors2.begin();
+ umodvec newfactors1(part.size_first(), UPR->create(-1)), newfactors2(part.size_second(), UPR->create(-1));
+ umodvec::iterator i1 = newfactors1.begin(), i2 = newfactors2.begin();
for ( size_t i=0; i<n; ++i ) {
if ( part[i] ) {
*i2++ = tocheck.top().factors[i];
return unit * cont * result;
}
-struct FindSymbolsMap : public map_function {
- exset syms;
- ex operator()(const ex& e)
- {
- if ( is_a<symbol>(e) ) {
- syms.insert(e);
- return e;
- }
- return e.map(*this);
- }
-};
-
struct EvalPoint
{
ex x;
int evalpoint;
};
+// MARK
+
// forward declaration
vector<ex> multivar_diophant(const vector<ex>& a_, const ex& x, const ex& c, const vector<EvalPoint>& I, unsigned int d, unsigned int p, unsigned int k);
-UniPolyVec multiterm_eea_lift(const UniPolyVec& a, const ex& x, unsigned int p, unsigned int k)
+umodvec multiterm_eea_lift(const umodvec& a, const ex& x, unsigned int p, unsigned int k)
{
DCOUT(multiterm_eea_lift);
DCOUTVAR(a);
const size_t r = a.size();
DCOUTVAR(r);
cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
- UniPoly fill(R);
- UniPolyVec q(r-1, fill);
+ cl_univpoly_modint_ring UPR = find_univpoly_ring(R);
+ umodvec q(r-1, UPR->create(-1));
q[r-2] = a[r-1];
for ( size_t j=r-2; j>=1; --j ) {
q[j-1] = a[j] * q[j];
}
DCOUTVAR(q);
- UniPoly beta(R);
- beta.set(0, R->one());
- UniPolyVec s;
+ umod beta = UPR->one();
+ umodvec s;
for ( size_t j=1; j<r; ++j ) {
DCOUTVAR(j);
DCOUTVAR(beta);
vector<ex> mdarg(2);
- mdarg[0] = q[j-1].to_ex(x);
- mdarg[1] = a[j-1].to_ex(x);
+ mdarg[0] = umod_to_ex(q[j-1], x);
+ mdarg[1] = umod_to_ex(a[j-1], x);
vector<EvalPoint> empty;
- vector<ex> exsigma = multivar_diophant(mdarg, x, beta.to_ex(x), empty, 0, p, k);
- UniPoly sigma1(R, exsigma[0], x);
- UniPoly sigma2(R, exsigma[1], x);
- beta = sigma1;
+ vector<ex> exsigma = multivar_diophant(mdarg, x, umod_to_ex(beta, x), empty, 0, p, k);
+ umod sigma1 = umod_from_ex(exsigma[0], x, R);
+ umod sigma2 = umod_from_ex(exsigma[1], x, R);
+ beta = COPY(beta, sigma1);
s.push_back(sigma2);
}
s.push_back(beta);
return s;
}
-void eea_lift(const UniPoly& a, const UniPoly& b, const ex& x, unsigned int p, unsigned int k, UniPoly& s_, UniPoly& t_)
+void change_modulus(umod& out, const umod& in)
+{
+ // ASSERT: out and in have same degree
+ if ( out.ring() == in.ring() ) {
+ out = COPY(out, in);
+ }
+ else {
+ for ( int i=0; i<=degree(in); ++i ) {
+ out.set_coeff(i, out.ring()->basering()->canonhom(in.ring()->basering()->retract(coeff(in, i))));
+ }
+ out.finalize();
+ }
+}
+
+void eea_lift(const umod& a, const umod& b, const ex& x, unsigned int p, unsigned int k, umod& s_, umod& t_)
{
DCOUT(eea_lift);
DCOUTVAR(a);
DCOUTVAR(k);
cl_modint_ring R = find_modint_ring(p);
- UniPoly amod(R, a);
- UniPoly bmod(R, b);
+ cl_univpoly_modint_ring UPR = find_univpoly_ring(R);
+ umod amod = UPR->create(degree(a));
+ DCOUTVAR(a);
+ change_modulus(amod, a);
+ umod bmod = UPR->create(degree(b));
+ change_modulus(bmod, b);
DCOUTVAR(amod);
DCOUTVAR(bmod);
- UniPoly smod(R), tmod(R), g(R);
+ umod g = UPR->create(-1);
+ umod smod = UPR->create(-1);
+ umod tmod = UPR->create(-1);
exteuclid(amod, bmod, g, smod, tmod);
DCOUTVAR(smod);
DCOUTVAR(tmod);
DCOUTVAR(g);
+ DCOUTVAR(a);
cl_modint_ring Rpk = find_modint_ring(expt_pos(cl_I(p),k));
- UniPoly s(Rpk, smod);
- UniPoly t(Rpk, tmod);
+ cl_univpoly_modint_ring UPRpk = find_univpoly_ring(Rpk);
+ umod s = UPRpk->create(degree(smod));
+ change_modulus(s, smod);
+ umod t = UPRpk->create(degree(tmod));
+ change_modulus(t, tmod);
DCOUTVAR(s);
DCOUTVAR(t);
cl_I modulus(p);
+ DCOUTVAR(a);
- UniPoly one(Rpk);
- one.set(0, Rpk->one());
+ umod one = UPRpk->one();
for ( size_t j=1; j<k; ++j ) {
- UniPoly e = one - a * s - b * t;
- e.divide(modulus);
- UniPoly c(R, e);
- UniPoly sigmabar(R);
- sigmabar = smod * c;
- UniPoly taubar(R);
- taubar = tmod * c;
- UniPoly q(R);
- div(sigmabar, bmod, q);
- UniPoly sigma(R);
- rem(sigmabar, bmod, sigma);
- UniPoly tau(R);
- tau = taubar + q * amod;
- UniPoly sadd(Rpk, sigma);
+ DCOUTVAR(a);
+ umod e = one - a * s - b * t;
+ DCOUTVAR(one);
+ DCOUTVAR(a*s);
+ DCOUTVAR(b*t);
+ DCOUTVAR(e);
+ e = divide(e, modulus);
+ umod c = UPR->create(degree(e));
+ change_modulus(c, e);
+ umod sigmabar = smod * c;
+ umod taubar = tmod * c;
+ umod q = div(sigmabar, bmod);
+ umod sigma = rem(sigmabar, bmod);
+ umod tau = taubar + q * amod;
+ umod sadd = UPRpk->create(degree(sigma));
+ change_modulus(sadd, sigma);
cl_MI modmodulus(Rpk, modulus);
s = s + sadd * modmodulus;
- UniPoly tadd(Rpk, tau);
+ umod tadd = UPRpk->create(degree(tau));
+ change_modulus(tadd, tau);
t = t + tadd * modmodulus;
modulus = modulus * p;
}
DCOUT(END eea_lift);
}
-UniPolyVec univar_diophant(const UniPolyVec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
+umodvec univar_diophant(const umodvec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
{
DCOUT(univar_diophant);
DCOUTVAR(a);
DCOUTVAR(k);
cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
+ cl_univpoly_modint_ring UPR = find_univpoly_ring(R);
const size_t r = a.size();
- UniPolyVec result;
+ umodvec result;
if ( r > 2 ) {
- UniPolyVec s = multiterm_eea_lift(a, x, p, k);
+ umodvec s = multiterm_eea_lift(a, x, p, k);
for ( size_t j=0; j<r; ++j ) {
- ex phi = expand(pow(x,m)*s[j].to_ex(x));
- UniPoly bmod(R, phi, x);
- UniPoly buf(R);
- rem(bmod, a[j], buf);
+ ex phi = expand(pow(x,m) * umod_to_ex(s[j], x));
+ umod bmod = umod_from_ex(phi, x, R);
+ umod buf = rem(bmod, a[j]);
result.push_back(buf);
}
}
else {
- UniPoly s(R), t(R);
+ umod s = UPR->create(-1);
+ umod t = UPR->create(-1);
eea_lift(a[1], a[0], x, p, k, s, t);
- ex phi = expand(pow(x,m)*s.to_ex(x));
- UniPoly bmod(R, phi, x);
- UniPoly buf(R);
- rem(bmod, a[0], buf);
- result.push_back(buf);
- UniPoly q(R);
- div(bmod, a[0], q);
- phi = expand(pow(x,m)*t.to_ex(x));
- UniPoly t1mod(R, phi, x);
- buf = t1mod + q * a[1];
+ ex phi = expand(pow(x,m) * umod_to_ex(s, x));
+ umod bmod = umod_from_ex(phi, x, R);
+ umod buf = rem(bmod, a[0]);
result.push_back(buf);
+ umod q = div(bmod, a[0]);
+ phi = expand(pow(x,m) * umod_to_ex(t, x));
+ umod t1mod = umod_from_ex(phi, x, R);
+ umod buf2 = t1mod + q * a[1];
+ result.push_back(buf2);
}
DCOUTVAR(result);
}
else {
DCOUT(uniterm left);
- UniPolyVec amod;
+ umodvec amod;
for ( size_t i=0; i<a.size(); ++i ) {
- UniPoly up(R, a[i], x);
+ umod up = umod_from_ex(a[i], x, R);
amod.push_back(up);
}
DCOUTVAR(m);
cl_I cm = the<cl_I>(ex_to<numeric>(z.lcoeff(x)).to_cl_N());
DCOUTVAR(cm);
- UniPolyVec delta_s = univar_diophant(amod, x, m, p, k);
+ umodvec delta_s = univar_diophant(amod, x, m, p, k);
cl_MI modcm;
cl_I poscm = cm;
while ( poscm < 0 ) {
DCOUTVAR(modcm);
for ( size_t j=0; j<delta_s.size(); ++j ) {
delta_s[j] = delta_s[j] * modcm;
- sigma[j] = sigma[j] + delta_s[j].to_ex(x);
+ sigma[j] = sigma[j] + umod_to_ex(delta_s[j], x);
}
DCOUTVAR(delta_s);
#ifdef DEBUGFACTOR
#endif // def DEBUGFACTOR
-ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I, unsigned int p, const cl_I& l, const UniPolyVec& u, const vector<ex>& lcU)
+ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I, unsigned int p, const cl_I& l, const umodvec& u, const vector<ex>& lcU)
{
DCOUT(hensel_multivar);
DCOUTVAR(a);
DCOUTVAR(n);
vector<ex> U(n);
for ( size_t i=0; i<n; ++i ) {
- U[i] = u[i].to_ex(x);
+ U[i] = umod_to_ex(u[i], x);
}
#ifdef DEBUGFACTOR
cout << "U ";
return false;
}
-#ifdef DEBUGFACTOR
-ex factor(const ex&);
-#endif
-
static ex factor_multivariate(const ex& poly, const exset& syms)
{
DCOUT(factor_multivariate);
DCOUTVAR(u.lcoeff(x));
while ( true ) {
if ( irem(ex_to<numeric>(u.lcoeff(x)), prime) != 0 ) {
- UniPoly modpoly(R, u, x);
- UniFactorVec sqrfree_ufv;
- squarefree(modpoly, sqrfree_ufv);
- DCOUTVAR(sqrfree_ufv);
- if ( sqrfree_ufv.factors.size() == 1 && sqrfree_ufv.factors.front().exp == 1 ) break;
+ umod modpoly = umod_from_ex(u, x, R);
+ if ( squarefree(modpoly) ) break;
}
prime = next_prime(prime);
DCOUTVAR(prime);
pl = pl * prime;
}
- UniPolyVec uvec;
+ umodvec uvec;
cl_modint_ring R = find_modint_ring(expt_pos(cl_I(prime),l));
for ( size_t i=0; i<(ufaclst.nops()-1)/2; ++i ) {
- UniPoly newu(R, ufaclst.op(i*2+1), x);
+ umod newu = umod_from_ex(ufaclst.op(i*2+1), x, R);
uvec.push_back(newu);
}
DCOUTVAR(uvec);
}
}
+struct find_symbols_map : public map_function {
+ exset syms;
+ ex operator()(const ex& e)
+ {
+ if ( is_a<symbol>(e) ) {
+ syms.insert(e);
+ return e;
+ }
+ return e.map(*this);
+ }
+};
+
static ex factor_sqrfree(const ex& poly)
{
// determine all symbols in poly
- FindSymbolsMap findsymbols;
+ find_symbols_map findsymbols;
findsymbols(poly);
if ( findsymbols.syms.size() == 0 ) {
return poly;
return res;
}
+struct apply_factor_map : public map_function {
+ unsigned options;
+ apply_factor_map(unsigned options_) : options(options_) { }
+ ex operator()(const ex& e)
+ {
+ if ( e.info(info_flags::polynomial) ) {
+#ifdef DEBUGFACTOR
+ return ::factor(e, options);
+#else
+ return factor(e, options);
+#endif
+ }
+ if ( is_a<add>(e) ) {
+ ex s1, s2;
+ for ( size_t i=0; i<e.nops(); ++i ) {
+ if ( e.op(i).info(info_flags::polynomial) ) {
+ s1 += e.op(i);
+ }
+ else {
+ s2 += e.op(i);
+ }
+ }
+ s1 = s1.eval();
+ s2 = s2.eval();
+#ifdef DEBUGFACTOR
+ return ::factor(s1, options) + s2.map(*this);
+#else
+ return factor(s1, options) + s2.map(*this);
+#endif
+ }
+ return e.map(*this);
+ }
+};
+
} // anonymous namespace
-ex factor(const ex& poly)
+#ifdef DEBUGFACTOR
+ex factor(const ex& poly, unsigned options = 0)
+#else
+ex factor(const ex& poly, unsigned options)
+#endif
{
+ // check arguments
+ if ( !poly.info(info_flags::polynomial) ) {
+ if ( options & factor_options::all ) {
+ options &= ~factor_options::all;
+ apply_factor_map factor_map(options);
+ return factor_map(poly);
+ }
+ return poly;
+ }
+
// determine all symbols in poly
- FindSymbolsMap findsymbols;
+ find_symbols_map findsymbols;
findsymbols(poly);
if ( findsymbols.syms.size() == 0 ) {
return poly;
return pow(f, sfpoly.op(1));
}
if ( is_a<mul>(sfpoly) ) {
+ // case: multiple factors
ex res = 1;
for ( size_t i=0; i<sfpoly.nops(); ++i ) {
const ex& t = sfpoly.op(i);