/** @file factor.cpp
*
- * Polynomial factorization code (implementation).
+ * Polynomial factorization (implementation).
+ *
+ * The interface function factor() at the end of this file is defined in the
+ * GiNaC namespace. All other utility functions and classes are defined in an
+ * additional anonymous namespace.
+ *
+ * Factorization starts by doing a square free factorization and making the
+ * coefficients integer. Then, depending on the number of free variables it
+ * proceeds either in dedicated univariate or multivariate factorization code.
+ *
+ * Univariate factorization does a modular factorization via Berlekamp's
+ * algorithm and distinct degree factorization. Hensel lifting is used at the
+ * end.
+ *
+ * Multivariate factorization uses the univariate factorization (applying a
+ * evaluation homomorphism first) and Hensel lifting raises the answer to the
+ * multivariate domain. The Hensel lifting code is completely distinct from the
+ * code used by the univariate factorization.
*
* 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.
+ * [Wan] 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.
+ * K.O.Geddes, S.R.Czapor, G.Labahn,
+ * Springer Verlag, 1992.
+ * [Mig] Some Useful Bounds,
+ * M.Mignotte,
+ * In "Computer Algebra, Symbolic and Algebraic Computation" (B.Buchberger et al., eds.),
+ * pp. 259-263, Springer-Verlag, New York, 1982.
*/
/*
#define DCOUT(str) cout << #str << endl
#define DCOUTVAR(var) cout << #var << ": " << var << endl
#define DCOUT2(str,var) cout << #str << ": " << var << endl
-#else
-#define DCOUT(str)
-#define DCOUTVAR(var)
-#define DCOUT2(str,var)
-#endif
-
-// anonymous namespace to hide all utility functions
-namespace {
-
-typedef vector<cl_MI> mvec;
-#ifdef DEBUGFACTOR
ostream& operator<<(ostream& o, const vector<int>& v)
{
vector<int>::const_iterator i = v.begin(), end = v.end();
}
return o;
}
-ostream& operator<<(ostream& o, const vector<cl_I>& v)
+static ostream& operator<<(ostream& o, const vector<cl_I>& v)
{
vector<cl_I>::const_iterator i = v.begin(), end = v.end();
while ( i != end ) {
}
return o;
}
-ostream& operator<<(ostream& o, const vector<cl_MI>& v)
+static ostream& operator<<(ostream& o, const vector<cl_MI>& v)
{
vector<cl_MI>::const_iterator i = v.begin(), end = v.end();
while ( i != end ) {
}
return o;
}
+ostream& operator<<(ostream& o, const vector<numeric>& v)
+{
+ for ( size_t i=0; i<v.size(); ++i ) {
+ o << v[i] << " ";
+ }
+ return o;
+}
ostream& operator<<(ostream& o, const vector< vector<cl_MI> >& v)
{
vector< vector<cl_MI> >::const_iterator i = v.begin(), end = v.end();
}
return o;
}
-#endif
+#else
+#define DCOUT(str)
+#define DCOUTVAR(var)
+#define DCOUT2(str,var)
+#endif // def DEBUGFACTOR
+
+// anonymous namespace to hide all utility functions
+namespace {
////////////////////////////////////////////////////////////////////////////////
// modular univariate polynomial code
////////////////////////////////////////////////////////////////////////////////
// modular matrix
+typedef vector<cl_MI> mvec;
+
class modular_matrix
{
friend ostream& operator<<(ostream& o, const modular_matrix& m);
// END modular matrix
////////////////////////////////////////////////////////////////////////////////
+/** Calculates the Q matrix for a polynomial. Used by Berlekamp's algorithm.
+ *
+ * @param[in] a_ modular polynomial
+ * @param[out] Q Q matrix
+ */
static void q_matrix(const umodpoly& a_, modular_matrix& Q)
{
umodpoly a = a_;
}
}
+/** Determine the nullspace of a matrix M-1.
+ *
+ * @param[in,out] M matrix, will be modified
+ * @param[out] basis calculated nullspace of M-1
+ */
static void nullspace(modular_matrix& M, vector<mvec>& basis)
{
const size_t n = M.rowsize();
}
}
+/** Berlekamp's modular factorization.
+ *
+ * The implementation follows the algorithm in chapter 8 of [GCL].
+ *
+ * @param[in] a modular polynomial
+ * @param[out] upv vector containing modular factors. if upv was not empty the
+ * new elements are added at the end
+ */
static void berlekamp(const umodpoly& a, upvec& upv)
{
cl_modint_ring R = a[0].ring();
umodpoly one(1, R->one());
+ // find nullspace of Q matrix
modular_matrix Q(degree(a), degree(a), R->zero());
q_matrix(a, Q);
vector<mvec> nu;
const unsigned int k = nu.size();
if ( k == 1 ) {
+ // irreducible
return;
}
list<umodpoly>::iterator u = factors.begin();
+ // calculate all gcd's
while ( true ) {
for ( unsigned int s=0; s<q; ++s ) {
umodpoly nur = nu[r];
}
}
+// modular square free factorization is not used at the moment so we deactivate
+// the code
+#if 0
+
+/** Calculates a^(1/prime).
+ *
+ * @param[in] a polynomial
+ * @param[in] prime prime number -> exponent 1/prime
+ * @param[in] ap resulting polynomial
+ */
static void expt_1_over_p(const umodpoly& a, unsigned int prime, umodpoly& ap)
{
size_t newdeg = degree(a)/prime;
}
}
+/** Modular square free factorization.
+ *
+ * @param[in] a polynomial
+ * @param[out] factors modular factors
+ * @param[out] mult corresponding multiplicities (exponents)
+ */
static void modsqrfree(const umodpoly& a, upvec& factors, vector<int>& mult)
{
const unsigned int prime = cl_I_to_uint(a[0].ring()->modulus);
}
}
+#endif // deactivation of square free factorization
+
+/** Distinct degree factorization (DDF).
+ *
+ * The implementation follows the algorithm in chapter 8 of [GCL].
+ *
+ * @param[in] a_ modular polynomial
+ * @param[out] degrees vector containing the degrees of the factors of the
+ * corresponding polynomials in ddfactors.
+ * @param[out] ddfactors vector containing polynomials which factors have the
+ * degree given in degrees.
+ */
static void distinct_degree_factor(const umodpoly& a_, vector<int>& degrees, upvec& ddfactors)
{
umodpoly a = a_;
}
}
+/** Modular same degree factorization.
+ * Same degree factorization is a kind of misnomer. It performs distinct degree
+ * factorization, but instead of using the Cantor-Zassenhaus algorithm it
+ * (sub-optimally) uses Berlekamp's algorithm for the factors of the same
+ * degree.
+ *
+ * @param[in] a modular polynomial
+ * @param[out] upv vector containing modular factors. if upv was not empty the
+ * new elements are added at the end
+ */
static void same_degree_factor(const umodpoly& a, upvec& upv)
{
cl_modint_ring R = a[0].ring();
}
}
+// Yes, we can (choose).
#define USE_SAME_DEGREE_FACTOR
+/** Modular univariate factorization.
+ *
+ * In principle, we have two algorithms at our disposal: Berlekamp's algorithm
+ * and same degree factorization (SDF). SDF seems to be slightly faster in
+ * almost all cases so it is activated as default.
+ *
+ * @param[in] p modular polynomial
+ * @param[out] upv vector containing modular factors. if upv was not empty the
+ * new elements are added at the end
+ */
static void factor_modular(const umodpoly& p, upvec& upv)
{
#ifdef USE_SAME_DEGREE_FACTOR
#endif
}
-/** Calculates polynomials s and t such that a*s+b*t==1.
+/** Calculates modular polynomials s and t such that a*s+b*t==1.
* Assertion: a and b are relatively prime and not zero.
*
* @param[in] a polynomial
canonicalize(t);
}
+/** Replaces the leading coefficient in a polynomial by a given number.
+ *
+ * @param[in] poly polynomial to change
+ * @param[in] lc new leading coefficient
+ * @return changed polynomial
+ */
static upoly replace_lc(const upoly& poly, const cl_I& lc)
{
if ( poly.empty() ) return poly;
return r;
}
+/** Calculates the bound for the modulus.
+ * See [Mig].
+ */
static inline cl_I calc_bound(const ex& a, const ex& x, int maxdeg)
{
cl_I maxcoeff = 0;
return ( B > maxcoeff ) ? B : maxcoeff;
}
+/** Calculates the bound for the modulus.
+ * See [Mig].
+ */
static inline cl_I calc_bound(const upoly& a, int maxdeg)
{
cl_I maxcoeff = 0;
return ( B > maxcoeff ) ? B : maxcoeff;
}
+/** Hensel lifting as used by factor_univariate().
+ *
+ * The implementation follows the algorithm in chapter 6 of [GCL].
+ *
+ * @param[in] a_ primitive univariate polynomials
+ * @param[in] p prime number that does not divide lcoeff(a)
+ * @param[in] u1_ modular factor of a (mod p)
+ * @param[in] w1_ modular factor of a (mod p), relatively prime to u1_,
+ * fulfilling u1_*w1_ == a mod p
+ * @param[out] u lifted factor
+ * @param[out] w lifted factor, u*w = a
+ */
static void hensel_univar(const upoly& a_, unsigned int p, const umodpoly& u1_, const umodpoly& w1_, upoly& u, upoly& w)
{
upoly a = a_;
}
}
+/** Returns a new prime number.
+ *
+ * @param[in] p prime number
+ * @return next prime number after p
+ */
static unsigned int next_prime(unsigned int p)
{
static vector<unsigned int> primes;
throw logic_error("next_prime: should not reach this point!");
}
+/** Manages the splitting a vector of of modular factors into two partitions.
+ */
class factor_partition
{
public:
+ /** Takes the vector of modular factors and initializes the first partition */
factor_partition(const upvec& factors_) : factors(factors_)
{
n = factors.size();
size_t size() const { return n; }
size_t size_left() const { return n-len; }
size_t size_right() const { return len; }
-#ifdef DEBUGFACTOR
- void get() const { DCOUTVAR(k); }
-#endif
+ /** Initializes the next partition.
+ Returns true, if there is one, false otherwise. */
bool next()
{
if ( last == n-1 ) {
split();
return true;
}
+ /** Get first partition */
umodpoly& left() { return lr[0]; }
+ /** Get second partition */
umodpoly& right() { return lr[1]; }
private:
void split_cached()
vector<int> k;
};
+/** Contains a pair of univariate polynomial and its modular factors.
+ * Used by factor_univariate().
+ */
struct ModFactors
{
upoly poly;
upvec factors;
};
+/** Univariate polynomial factorization.
+ *
+ * Modular factorization is tried for several primes to minimize the number of
+ * modular factors. Then, Hensel lifting is performed.
+ *
+ * @param[in] poly expanded square free univariate polynomial
+ * @param[in] x symbol
+ * @param[in,out] prime prime number to start trying modular factorization with,
+ * output value is the prime number actually used
+ */
static ex factor_univariate(const ex& poly, const ex& x, unsigned int& prime)
{
ex unit, cont, prim_ex;
const size_t n = tocheck.top().factors.size();
factor_partition part(tocheck.top().factors);
while ( true ) {
+ // call Hensel lifting
hensel_univar(tocheck.top().poly, prime, part.left(), part.right(), f1, f2);
if ( !f1.empty() ) {
+ // successful, update the stack and the result
if ( part.size_left() == 1 ) {
if ( part.size_right() == 1 ) {
result *= upoly_to_ex(f1, x) * upoly_to_ex(f2, x);
}
}
else {
+ // not successful
if ( !part.next() ) {
+ // if no more combinations left, return polynomial as
+ // irreducible
result *= upoly_to_ex(tocheck.top().poly, x);
tocheck.pop();
break;
return unit * cont * result;
}
+/** Second interface to factor_univariate() to be used if the information about
+ * the prime is not needed.
+ */
static inline ex factor_univariate(const ex& poly, const ex& x)
{
unsigned int prime;
return factor_univariate(poly, x, prime);
}
+/** Represents an evaluation point (<symbol>==<integer>).
+ */
struct EvalPoint
{
ex x;
int evalpoint;
};
+#ifdef DEBUGFACTOR
+ostream& operator<<(ostream& o, const vector<EvalPoint>& v)
+{
+ for ( size_t i=0; i<v.size(); ++i ) {
+ o << "(" << v[i].x << "==" << v[i].evalpoint << ") ";
+ }
+ return o;
+}
+#endif // def DEBUGFACTOR
+
// forward declaration
static 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);
+/** Utility function for multivariate Hensel lifting.
+ *
+ * Solves the equation
+ * s_1*b_1 + ... + s_r*b_r == 1 mod p^k
+ * with deg(s_i) < deg(a_i)
+ * and with given b_1 = a_1 * ... * a_{i-1} * a_{i+1} * ... * a_r
+ *
+ * The implementation follows the algorithm in chapter 6 of [GCL].
+ *
+ * @param[in] a vector of modular univariate polynomials
+ * @param[in] x symbol
+ * @param[in] p prime number
+ * @param[in] k p^k is modulus
+ * @return vector of polynomials (s_i)
+ */
static upvec multiterm_eea_lift(const upvec& a, const ex& x, unsigned int p, unsigned int k)
{
const size_t r = a.size();
return s;
}
-/**
- * Assert: a not empty.
+/** Changes the modulus of a modular polynomial. Used by eea_lift().
+ *
+ * @param[in] R new modular ring
+ * @param[in,out] a polynomial to change (in situ)
*/
static void change_modulus(const cl_modint_ring& R, umodpoly& a)
{
canonicalize(a);
}
+/** Utility function for multivariate Hensel lifting.
+ *
+ * Solves s*a + t*b == 1 mod p^k given a,b.
+ *
+ * The implementation follows the algorithm in chapter 6 of [GCL].
+ *
+ * @param[in] a polynomial
+ * @param[in] b polynomial
+ * @param[in] x symbol
+ * @param[in] p prime number
+ * @param[in] k p^k is modulus
+ * @param[out] s_ output polynomial
+ * @param[out] t_ output polynomial
+ */
static void eea_lift(const umodpoly& a, const umodpoly& b, const ex& x, unsigned int p, unsigned int k, umodpoly& s_, umodpoly& t_)
{
cl_modint_ring R = find_modint_ring(p);
s_ = s; t_ = t;
}
+/** Utility function for multivariate Hensel lifting.
+ *
+ * Solves the equation
+ * s_1*b_1 + ... + s_r*b_r == x^m mod p^k
+ * with given b_1 = a_1 * ... * a_{i-1} * a_{i+1} * ... * a_r
+ *
+ * The implementation follows the algorithm in chapter 6 of [GCL].
+ *
+ * @param a vector with univariate polynomials mod p^k
+ * @param x symbol
+ * @param m exponent of x^m in the equation to solve
+ * @param p prime number
+ * @param k p^k is modulus
+ * @return vector of polynomials (s_i)
+ */
static upvec univar_diophant(const upvec& a, const ex& x, unsigned int m, unsigned int p, unsigned int k)
{
cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),k));
return result;
}
+/** Map used by function make_modular().
+ * Finds every coefficient in a polynomial and replaces it by is value in the
+ * given modular ring R (symmetric representation).
+ */
struct make_modular_map : public map_function {
cl_modint_ring R;
make_modular_map(const cl_modint_ring& R_) : R(R_) { }
}
};
+/** Helps mimicking modular multivariate polynomial arithmetic.
+ *
+ * @param e expression of which to make the coefficients equal to their value
+ * in the modular ring R (symmetric representation)
+ * @param R modular ring
+ * @return resulting expression
+ */
static ex make_modular(const ex& e, const cl_modint_ring& R)
{
make_modular_map map(R);
return map(e.expand());
}
+/** Utility function for multivariate Hensel lifting.
+ *
+ * Returns the polynomials s_i that fulfill
+ * s_1*b_1 + ... + s_r*b_r == c mod <I^(d+1),p^k>
+ * with given b_1 = a_1 * ... * a_{i-1} * a_{i+1} * ... * a_r
+ *
+ * The implementation follows the algorithm in chapter 6 of [GCL].
+ *
+ * @param a_ vector of multivariate factors mod p^k
+ * @param x symbol (equiv. x_1 in [GCL])
+ * @param c polynomial mod p^k
+ * @param I vector of evaluation points
+ * @param d maximum total degree of result
+ * @param p prime number
+ * @param k p^k is modulus
+ * @return vector of polynomials (s_i)
+ */
static 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)
{
return sigma;
}
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const vector<EvalPoint>& v)
-{
- for ( size_t i=0; i<v.size(); ++i ) {
- o << "(" << v[i].x << "==" << v[i].evalpoint << ") ";
- }
- return o;
-}
-#endif // def DEBUGFACTOR
-
-static ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I, unsigned int p, const cl_I& l, const upvec& u, const vector<ex>& lcU)
+/** Multivariate Hensel lifting.
+ * The implementation follows the algorithm in chapter 6 of [GCL].
+ * Since we don't have a data type for modular multivariate polynomials, the
+ * respective operations are done in a GiNaC::ex and the function
+ * make_modular() is then called to make the coefficient modular p^l.
+ *
+ * @param a multivariate polynomial primitive in x
+ * @param x symbol (equiv. x_1 in [GCL])
+ * @param I vector of evaluation points (x_2==a_2,x_3==a_3,...)
+ * @param p prime number (should not divide lcoeff(a mod I))
+ * @param l p^l is the modulus of the lifted univariate field
+ * @param u vector of modular (mod p^l) factors of a mod I
+ * @param lcU correct leading coefficient of the univariate factors of a mod I
+ * @return list GiNaC::lst with lifted factors (multivariate factors of a),
+ * empty if Hensel lifting did not succeed
+ */
+static ex hensel_multivar(const ex& a, const ex& x, const vector<EvalPoint>& I,
+ unsigned int p, const cl_I& l, const upvec& u, const vector<ex>& lcU)
{
const size_t nu = I.size() + 1;
const cl_modint_ring R = find_modint_ring(expt_pos(cl_I(p),l));
}
}
+/** Takes a factorized expression and puts the factors in a lst. The exponents
+ * of the factors are discarded, e.g. 7*x^2*(y+1)^4 --> {7,x,y+1}. The first
+ * element of the list is always the numeric coefficient.
+ */
static ex put_factors_into_lst(const ex& e)
{
lst result;
-
if ( is_a<numeric>(e) ) {
result.append(e);
return result;
throw runtime_error("put_factors_into_lst: bad term.");
}
-#ifdef DEBUGFACTOR
-ostream& operator<<(ostream& o, const vector<numeric>& v)
-{
- for ( size_t i=0; i<v.size(); ++i ) {
- o << v[i] << " ";
- }
- return o;
-}
-#endif // def DEBUGFACTOR
-
-/** Checks whether in a set of numbers each has a unique prime factor.
+/** Checks a set of numbers for whether each number has a unique prime factor.
*
* @param[in] f list of numbers to check
- * @return true: if number set is bad, false: otherwise
+ * @return true: if number set is bad, false: if set is okay (has unique
+ * prime factors)
*/
static bool checkdivisors(const lst& f)
{
* 1. lcoeff(evaluated_polynomial) does not vanish
* 2. factors of lcoeff(evaluated_polynomial) have each a unique prime factor
* 3. evaluated_polynomial is square free
- * See [W1] for more details.
+ * See [Wan] for more details.
*
* @param[in] u multivariate polynomial to be factored
* @param[in] vn leading coefficient of u in x (x==first symbol in syms)
const ex& x = *syms.begin();
while ( true ) {
++modulus;
- /* generate a set of integers ... */
+ // generate a set of integers ...
u0 = u;
ex vna = vn;
ex vnatry;
do {
a[i] = mod(numeric(rand()), 2*modulus) - modulus;
vnatry = vna.subs(*s == a[i]);
- /* ... for which the leading coefficient doesn't vanish ... */
+ // ... for which the leading coefficient doesn't vanish ...
} while ( vnatry == 0 );
vna = vnatry;
u0 = u0.subs(*s == a[i]);
++s;
}
- /* ... for which u0 is square free ... */
+ // ... for which u0 is square free ...
ex g = gcd(u0, u0.diff(ex_to<symbol>(x)));
if ( !is_a<numeric>(g) ) {
continue;
}
if ( !is_a<numeric>(vn) ) {
- /* ... and for which the evaluated factors have each an unique prime factor */
+ // ... and for which the evaluated factors have each an unique prime factor
lst fnum = f;
fnum.let_op(0) = fnum.op(0) * u0.content(x);
for ( size_t i=1; i<fnum.nops(); ++i ) {
continue;
}
}
- /* ok, we have a valid set now */
+ // ok, we have a valid set now
return;
}
}
// forward declaration
static ex factor_sqrfree(const ex& poly);
-/**
- * ASSERT: poly is expanded
+/** Multivariate factorization.
+ *
+ * The implementation is based on the algorithm described in [Wan].
+ * An evaluation homomorphism (a set of integers) is determined that fulfills
+ * certain criteria. The evaluated polynomial is univariate and is factorized
+ * by factor_univariate(). The main work then is to find the correct leading
+ * coefficients of the univariate factors. They have to correspond to the
+ * factors of the (multivariate) leading coefficient of the input polynomial
+ * (as defined for a specific variable x). After that the Hensel lifting can be
+ * performed.
+ *
+ * @param[in] poly expanded, square free polynomial
+ * @param[in] syms contains the symbols in the polynomial
+ * @return factorized polynomial
*/
static ex factor_multivariate(const ex& poly, const exset& syms)
{
exset::const_iterator s;
const ex& x = *syms.begin();
- /* make polynomial primitive */
- ex p = poly.collect(x);
- ex cont = p.lcoeff(x);
- for ( int i=p.degree(x)-1; i>=p.ldegree(x); --i ) {
- cont = gcd(cont, p.coeff(x,i));
- if ( cont == 1 ) break;
- }
- ex pp = expand(normal(p / cont));
+ // make polynomial primitive
+ ex unit, cont, pp;
+ poly.unitcontprim(x, unit, cont, pp);
if ( !is_a<numeric>(cont) ) {
return factor_sqrfree(cont) * factor_sqrfree(pp);
}
- /* factor leading coefficient */
+ // factor leading coefficient
ex vn = pp.collect(x).lcoeff(x);
ex vnlst;
if ( is_a<numeric>(vn) ) {
numeric modulus = (vnlst.nops() > 3) ? vnlst.nops() : 3;
vector<numeric> a(syms.size()-1, 0);
- /* try now to factorize until we are successful */
+ // try now to factorize until we are successful
while ( true ) {
unsigned int trialcount = 0;
ex u, delta;
ex ufac, ufaclst;
- /* try several evaluation points to reduce the number of modular factors */
+ // try several evaluation points to reduce the number of factors
while ( trialcount < maxtrials ) {
- /* generate a set of valid evaluation points */
+ // generate a set of valid evaluation points
generate_set(pp, vn, syms, ex_to<lst>(vnlst), modulus, u, a);
ufac = factor_univariate(u, x, prime);
delta = ufaclst.op(0);
if ( factor_count <= 1 ) {
- /* irreducible */
+ // irreducible
return poly;
}
if ( min_factor_count < 0 ) {
- /* first time here */
+ // first time here
min_factor_count = factor_count;
}
else if ( min_factor_count == factor_count ) {
- /* one less to try */
+ // one less to try
++trialcount;
}
else if ( min_factor_count > factor_count ) {
- /* new minimum, reset trial counter */
+ // new minimum, reset trial counter
min_factor_count = factor_count;
trialcount = 0;
}
// determine true leading coefficients for the Hensel lifting
vector<ex> C(factor_count);
if ( is_a<numeric>(vn) ) {
+ // easy case
for ( size_t i=1; i<ufaclst.nops(); ++i ) {
C[i-1] = ufaclst.op(i).lcoeff(x);
}
}
else {
+ // difficult case.
+ // we use the property of the ftilde having a unique prime factor.
+ // details can be found in [Wan].
+ // calculate ftilde
vector<numeric> ftilde(vnlst.nops()-1);
for ( size_t i=0; i<ftilde.size(); ++i ) {
ex ft = vnlst.op(i+1);
}
ftilde[i] = ex_to<numeric>(ft);
}
-
+ // calculate D and C
vector<bool> used_flag(ftilde.size(), false);
vector<ex> D(factor_count, 1);
if ( delta == 1 ) {
C[i] = D[i] * prefac;
}
}
-
+ // check if something went wrong
bool some_factor_unused = false;
for ( size_t i=0; i<used_flag.size(); ++i ) {
if ( !used_flag[i] ) {
continue;
}
}
-
+
+ // multiply the remaining content of the univariate polynomial into the
+ // first factor
if ( delta != 1 ) {
C[0] = C[0] * delta;
ufaclst.let_op(1) = ufaclst.op(1) * delta;
}
+ // set up evaluation points
EvalPoint ep;
vector<EvalPoint> epv;
s = syms.begin();
l = l + 1;
pl = pl * prime;
}
+
+ // set up modular factors (mod p^l)
cl_modint_ring R = find_modint_ring(expt_pos(cl_I(prime),l));
upvec modfactors(ufaclst.nops()-1);
for ( size_t i=1; i<ufaclst.nops(); ++i ) {
umodpoly_from_ex(modfactors[i-1], ufaclst.op(i), x, R);
}
+ // try Hensel lifting
ex res = hensel_multivar(pp, x, epv, prime, l, modfactors, C);
if ( res != lst() ) {
- ex result = cont;
+ ex result = cont * unit;
for ( size_t i=0; i<res.nops(); ++i ) {
result *= res.op(i).content(x) * res.op(i).unit(x);
result *= res.op(i).primpart(x);
}
}
+/** Finds all symbols in an expression. Used by factor_sqrfree() and factor().
+ */
struct find_symbols_map : public map_function {
exset syms;
ex operator()(const ex& e)
}
};
+/** Factorizes a polynomial that is square free. It calls either the univariate
+ * or the multivariate factorization functions.
+ */
static ex factor_sqrfree(const ex& poly)
{
// determine all symbols in poly
// univariate case
const ex& x = *(findsymbols.syms.begin());
if ( poly.ldegree(x) > 0 ) {
+ // pull out direct factors
int ld = poly.ldegree(x);
ex res = factor_univariate(expand(poly/pow(x, ld)), x);
return res * pow(x,ld);
return res;
}
+/** Map used by factor() when factor_options::all is given to access all
+ * subexpressions and to call factor() on them.
+ */
struct apply_factor_map : public map_function {
unsigned options;
apply_factor_map(unsigned options_) : options(options_) { }
} // anonymous namespace
+/** Interface function to the outside world. It checks the arguments, tries a
+ * square free factorization, and then calls factor_sqrfree to do the hard
+ * work.
+ */
ex factor(const ex& poly, unsigned options)
{
// check arguments