/**
* Univariate GenPolynomial algebraic partial fraction decomposition, Absolute factorization or
* Rothstein-Trager algorithm.
*
* @param A univariate GenPolynomial, deg(A) < deg(P).
* @param P univariate squarefree GenPolynomial, gcd(A,P) == 1.
* @return partial fraction container.
*/
public PartialFraction<C> baseAlgebraicPartialFraction(GenPolynomial<C> A, GenPolynomial<C> P) {
if (P == null || P.isZERO()) {
throw new RuntimeException(" P == null or P == 0");
}
if (A == null || A.isZERO()) {
throw new RuntimeException(" A == null or A == 0");
// PartialFraction(A,P,al,pl,empty,empty)
}
// System.out.println("\nP_base_algeb_part = " + P);
GenPolynomialRing<C> pfac = P.ring; // K[x]
if (pfac.nvar > 1) {
// System.out.println("facs_base_irred: univ");
throw new RuntimeException("only for univariate polynomials");
}
if (!pfac.coFac.isField()) {
// System.out.println("facs_base_irred: field");
throw new RuntimeException("only for field coefficients");
}
List<C> cfactors = new ArrayList<C>();
List<GenPolynomial<C>> cdenom = new ArrayList<GenPolynomial<C>>();
List<AlgebraicNumber<C>> afactors = new ArrayList<AlgebraicNumber<C>>();
List<GenPolynomial<AlgebraicNumber<C>>> adenom =
new ArrayList<GenPolynomial<AlgebraicNumber<C>>>();
// P linear
if (P.degree(0) <= 1) {
cfactors.add(A.leadingBaseCoefficient());
cdenom.add(P);
return new PartialFraction<C>(A, P, cfactors, cdenom, afactors, adenom);
}
List<GenPolynomial<C>> Pfac = baseFactorsSquarefree(P);
// System.out.println("\nPfac = " + Pfac);
List<GenPolynomial<C>> Afac = engine.basePartialFraction(A, Pfac);
GenPolynomial<C> A0 = Afac.remove(0);
if (!A0.isZERO()) {
throw new RuntimeException(" A0 != 0: deg(A)>= deg(P)");
}
// algebraic and linear factors
int i = 0;
for (GenPolynomial<C> pi : Pfac) {
GenPolynomial<C> ai = Afac.get(i++);
if (pi.degree(0) <= 1) {
cfactors.add(ai.leadingBaseCoefficient());
cdenom.add(pi);
continue;
}
PartialFraction<C> pf = baseAlgebraicPartialFractionIrreducibleAbsolute(ai, pi);
// PartialFraction<C> pf = baseAlgebraicPartialFractionIrreducible(ai,pi);
cfactors.addAll(pf.cfactors);
cdenom.addAll(pf.cdenom);
afactors.addAll(pf.afactors);
adenom.addAll(pf.adenom);
}
return new PartialFraction<C>(A, P, cfactors, cdenom, afactors, adenom);
}
// @Override
public FactorsList<C> baseFactorsAbsoluteSquarefree(GenPolynomial<C> P) {
if (P == null) {
throw new RuntimeException(this.getClass().getName() + " P == null");
}
List<GenPolynomial<C>> factors = new ArrayList<GenPolynomial<C>>();
if (P.isZERO()) {
return new FactorsList<C>(P, factors);
}
// System.out.println("\nP_base_sqf = " + P);
GenPolynomialRing<C> pfac = P.ring; // K[x]
if (pfac.nvar > 1) {
// System.out.println("facs_base_sqf: univ");
throw new RuntimeException("only for univariate polynomials");
}
if (!pfac.coFac.isField()) {
// System.out.println("facs_base_sqf: field");
throw new RuntimeException("only for field coefficients");
}
if (P.degree(0) <= 1) {
factors.add(P);
return new FactorsList<C>(P, factors);
}
// factor over K (=C)
List<GenPolynomial<C>> facs = baseFactorsSquarefree(P);
// System.out.println("facs_base_irred = " + facs);
if (debug && !isFactorization(P, facs)) {
throw new RuntimeException("isFactorization = false");
}
if (logger.isInfoEnabled()) {
logger.info("all K factors = " + facs); // Q[X]
// System.out.println("\nall K factors = " + facs); // Q[X]
}
// factor over K(alpha)
List<Factors<C>> afactors = new ArrayList<Factors<C>>();
for (GenPolynomial<C> p : facs) {
// System.out.println("facs_base_sqf_p = " + p);
if (p.degree(0) <= 1) {
factors.add(p);
} else {
Factors<C> afacs = baseFactorsAbsoluteIrreducible(p);
// System.out.println("afacs_base_sqf = " + afacs);
if (logger.isInfoEnabled()) {
logger.info("K(alpha) factors = " + afacs); // K(alpha)[X]
}
afactors.add(afacs);
}
}
// System.out.println("K(alpha) factors = " + factors);
return new FactorsList<C>(P, factors, afactors);
}
/**
* GenPolynomial absolute factorization of a polynomial.
*
* @param P GenPolynomial.
* @return factors map container: [p_1 -> e_1, ..., p_k -> e_k] with P = prod_{i=1,...,k}
* p_i**e_i. <b>Note:</b> K(alpha) not yet minimal.
*/
public FactorsMap<C> factorsAbsolute(GenPolynomial<C> P) {
if (P == null) {
throw new RuntimeException(this.getClass().getName() + " P == null");
}
SortedMap<GenPolynomial<C>, Long> factors = new TreeMap<GenPolynomial<C>, Long>();
if (P.isZERO()) {
return new FactorsMap<C>(P, factors);
}
// System.out.println("\nP_mult = " + P);
GenPolynomialRing<C> pfac = P.ring; // K[x]
if (pfac.nvar <= 1) {
return baseFactorsAbsolute(P);
}
if (!pfac.coFac.isField()) {
throw new RuntimeException("only for field coefficients");
}
if (P.degree() <= 1) {
factors.put(P, 1L);
return new FactorsMap<C>(P, factors);
}
// factor over K (=C)
SortedMap<GenPolynomial<C>, Long> facs = factors(P);
if (debug && !isFactorization(P, facs)) {
throw new RuntimeException("isFactorization = false");
}
if (logger.isInfoEnabled()) {
logger.info("all K factors = " + facs); // Q[X]
// System.out.println("\nall K factors = " + facs); // Q[X]
}
SortedMap<Factors<C>, Long> afactors = new TreeMap<Factors<C>, Long>();
// factor over K(alpha)
for (GenPolynomial<C> p : facs.keySet()) {
Long e = facs.get(p);
if (p.degree() <= 1) {
factors.put(p, e);
} else {
Factors<C> afacs = factorsAbsoluteIrreducible(p);
if (afacs.afac == null) { // absolute irreducible
factors.put(p, e);
} else {
afactors.put(afacs, e);
}
}
}
// System.out.println("K(alpha) factors multi = " + factors);
return new FactorsMap<C>(P, factors, afactors);
}
/**
* GenPolynomial absolute factorization of a irreducible polynomial.
*
* @param P irreducible! GenPolynomial.
* @return factors container: [p_1,...,p_k] with P = prod_{i=1, ..., k} p_i in K(alpha)[x] for
* suitable alpha and p_i irreducible over L[x], where K \subset K(alpha) \subset L is an
* algebraically closed field over K. <b>Note:</b> K(alpha) not yet minimal.
*/
public Factors<C> factorsAbsoluteIrreducible(GenPolynomial<C> P) {
if (P == null) {
throw new RuntimeException(this.getClass().getName() + " P == null");
}
if (P.isZERO()) {
return new Factors<C>(P);
}
GenPolynomialRing<C> pfac = P.ring; // K[x]
if (pfac.nvar <= 1) {
return baseFactorsAbsoluteIrreducible(P);
}
if (!pfac.coFac.isField()) {
throw new RuntimeException("only for field coefficients");
}
List<GenPolynomial<C>> factors = new ArrayList<GenPolynomial<C>>();
if (P.degree() <= 1) {
return new Factors<C>(P);
}
// find field extension K(alpha)
GenPolynomial<C> up = P;
RingFactory<C> cf = pfac.coFac;
long cr = cf.characteristic().longValue(); // char might be larger
if (cr == 0L) {
cr = Long.MAX_VALUE;
}
long rp = 0L;
for (int i = 0; i < (pfac.nvar - 1); i++) {
rp = 0L;
GenPolynomialRing<C> nfac = pfac.contract(1);
String[] vn = new String[] {pfac.getVars()[pfac.nvar - 1]};
GenPolynomialRing<GenPolynomial<C>> rfac =
new GenPolynomialRing<GenPolynomial<C>>(nfac, 1, pfac.tord, vn);
GenPolynomial<GenPolynomial<C>> upr = PolyUtil.<C>recursive(rfac, up);
// System.out.println("upr = " + upr);
GenPolynomial<C> ep;
do {
if (rp >= cr) {
throw new RuntimeException("elements of prime field exhausted: " + cr);
}
C r = cf.fromInteger(rp); // cf.random(rp);
// System.out.println("r = " + r);
ep = PolyUtil.<C>evaluateMain(nfac, upr, r);
// System.out.println("ep = " + ep);
rp++;
} while (!isSquarefree(ep) /*todo: || ep.degree() <= 1*/); // max deg
up = ep;
pfac = nfac;
}
up = up.monic();
if (debug) {
logger.info("P(" + rp + ") = " + up);
// System.out.println("up = " + up);
}
if (debug && !isSquarefree(up)) {
throw new RuntimeException("not irreducible up = " + up);
}
if (up.degree(0) <= 1) {
return new Factors<C>(P);
}
// find irreducible factor of up
List<GenPolynomial<C>> UF = baseFactorsSquarefree(up);
// System.out.println("UF = " + UF);
FactorsList<C> aUF = baseFactorsAbsoluteSquarefree(up);
// System.out.println("aUF = " + aUF);
AlgebraicNumberRing<C> arfac = aUF.findExtensionField();
// System.out.println("arfac = " + arfac);
long e = up.degree(0);
// search factor polynomial with smallest degree
for (int i = 0; i < UF.size(); i++) {
GenPolynomial<C> upi = UF.get(i);
long d = upi.degree(0);
if (1 <= d && d <= e) {
up = upi;
e = up.degree(0);
}
}
if (up.degree(0) <= 1) {
return new Factors<C>(P);
}
if (debug) {
logger.info("field extension by " + up);
}
List<GenPolynomial<AlgebraicNumber<C>>> afactors =
new ArrayList<GenPolynomial<AlgebraicNumber<C>>>();
// setup field extension K(alpha)
// String[] vars = new String[] { "z_" + Math.abs(up.hashCode() % 1000) };
String[] vars = pfac.newVars("z_");
pfac = pfac.clone();
String[] ovars = pfac.setVars(vars); // side effects!
GenPolynomial<C> aup = pfac.copy(up); // hack to exchange the variables
// AlgebraicNumberRing<C> afac = new AlgebraicNumberRing<C>(aup,true); // since irreducible
AlgebraicNumberRing<C> afac = arfac;
int depth = afac.depth();
// System.out.println("afac = " + afac);
GenPolynomialRing<AlgebraicNumber<C>> pafac =
new GenPolynomialRing<AlgebraicNumber<C>>(afac, P.ring.nvar, P.ring.tord, P.ring.getVars());
// System.out.println("pafac = " + pafac);
// convert to K(alpha)
GenPolynomial<AlgebraicNumber<C>> Pa =
PolyUtil.<C>convertToRecAlgebraicCoefficients(depth, pafac, P);
// System.out.println("Pa = " + Pa);
// factor over K(alpha)
FactorAbstract<AlgebraicNumber<C>> engine = FactorFactory.<C>getImplementation(afac);
afactors = engine.factorsSquarefree(Pa);
if (debug) {
logger.info("K(alpha) factors multi = " + afactors);
// System.out.println("K(alpha) factors = " + afactors);
}
if (afactors.size() <= 1) {
return new Factors<C>(P);
}
// normalize first factor to monic
GenPolynomial<AlgebraicNumber<C>> p1 = afactors.get(0);
AlgebraicNumber<C> p1c = p1.leadingBaseCoefficient();
if (!p1c.isONE()) {
GenPolynomial<AlgebraicNumber<C>> p2 = afactors.get(1);
afactors.remove(p1);
afactors.remove(p2);
p1 = p1.divide(p1c);
p2 = p2.multiply(p1c);
afactors.add(p1);
afactors.add(p2);
}
// recursion for splitting field
// find minimal field extension K(beta) \subset K(alpha)
return new Factors<C>(P, afac, Pa, afactors);
}
/**
* Univariate GenPolynomial algebraic partial fraction decomposition, via absolute factorization
* to linear factors.
*
* @param A univariate GenPolynomial, deg(A) < deg(P).
* @param P univariate squarefree GenPolynomial, gcd(A,P) == 1.
* @return partial fraction container.
*/
public PartialFraction<C> baseAlgebraicPartialFractionIrreducibleAbsolute(
GenPolynomial<C> A, GenPolynomial<C> P) {
if (P == null || P.isZERO()) {
throw new RuntimeException(" P == null or P == 0");
}
// System.out.println("\nP_base_algeb_part = " + P);
GenPolynomialRing<C> pfac = P.ring; // K[x]
if (pfac.nvar > 1) {
// System.out.println("facs_base_irred: univ");
throw new RuntimeException("only for univariate polynomials");
}
if (!pfac.coFac.isField()) {
// System.out.println("facs_base_irred: field");
throw new RuntimeException("only for field coefficients");
}
List<C> cfactors = new ArrayList<C>();
List<GenPolynomial<C>> cdenom = new ArrayList<GenPolynomial<C>>();
List<AlgebraicNumber<C>> afactors = new ArrayList<AlgebraicNumber<C>>();
List<GenPolynomial<AlgebraicNumber<C>>> adenom =
new ArrayList<GenPolynomial<AlgebraicNumber<C>>>();
// P linear
if (P.degree(0) <= 1) {
cfactors.add(A.leadingBaseCoefficient());
cdenom.add(P);
return new PartialFraction<C>(A, P, cfactors, cdenom, afactors, adenom);
}
// non linear case
Factors<C> afacs = factorsAbsoluteIrreducible(P);
// System.out.println("linear algebraic factors = " + afacs);
// System.out.println("afactors = " + afacs.afactors);
// System.out.println("arfactors = " + afacs.arfactors);
// System.out.println("arfactors pol = " + afacs.arfactors.get(0).poly);
// System.out.println("arfactors2 = " + afacs.arfactors.get(0).afactors);
List<GenPolynomial<AlgebraicNumber<C>>> fact = afacs.getFactors();
// System.out.println("factors = " + fact);
GenPolynomial<AlgebraicNumber<C>> Pa = afacs.apoly;
GenPolynomial<AlgebraicNumber<C>> Aa =
PolyUtil.<C>convertToRecAlgebraicCoefficients(1, Pa.ring, A);
GreatestCommonDivisorAbstract<AlgebraicNumber<C>> aengine = GCDFactory.getProxy(afacs.afac);
// System.out.println("denom = " + Pa);
// System.out.println("numer = " + Aa);
List<GenPolynomial<AlgebraicNumber<C>>> numers = aengine.basePartialFraction(Aa, fact);
// System.out.println("part frac = " + numers);
GenPolynomial<AlgebraicNumber<C>> A0 = numers.remove(0);
if (!A0.isZERO()) {
throw new RuntimeException(" A0 != 0: deg(A)>= deg(P)");
}
int i = 0;
for (GenPolynomial<AlgebraicNumber<C>> fa : fact) {
GenPolynomial<AlgebraicNumber<C>> an = numers.get(i++);
if (fa.degree(0) <= 1) {
afactors.add(an.leadingBaseCoefficient());
adenom.add(fa);
continue;
}
System.out.println("fa = " + fa);
Factors<AlgebraicNumber<C>> faf = afacs.getFactor(fa);
System.out.println("faf = " + faf);
List<GenPolynomial<AlgebraicNumber<AlgebraicNumber<C>>>> fafact = faf.getFactors();
GenPolynomial<AlgebraicNumber<AlgebraicNumber<C>>> Aaa =
PolyUtil.<AlgebraicNumber<C>>convertToRecAlgebraicCoefficients(1, faf.apoly.ring, an);
GreatestCommonDivisorAbstract<AlgebraicNumber<AlgebraicNumber<C>>> aaengine =
GCDFactory.getImplementation(faf.afac);
List<GenPolynomial<AlgebraicNumber<AlgebraicNumber<C>>>> anumers =
aaengine.basePartialFraction(Aaa, fafact);
System.out.println("algeb part frac = " + anumers);
GenPolynomial<AlgebraicNumber<AlgebraicNumber<C>>> A0a = anumers.remove(0);
if (!A0a.isZERO()) {
throw new RuntimeException(" A0 != 0: deg(A)>= deg(P)");
}
int k = 0;
for (GenPolynomial<AlgebraicNumber<AlgebraicNumber<C>>> faa : fafact) {
GenPolynomial<AlgebraicNumber<AlgebraicNumber<C>>> ana = anumers.get(k++);
System.out.println("faa = " + faa);
System.out.println("ana = " + ana);
if (faa.degree(0) > 1) {
throw new RuntimeException(" faa not linear");
}
GenPolynomial<AlgebraicNumber<C>> ana1 =
(GenPolynomial<AlgebraicNumber<C>>) (GenPolynomial) ana;
GenPolynomial<AlgebraicNumber<C>> faa1 =
(GenPolynomial<AlgebraicNumber<C>>) (GenPolynomial) faa;
afactors.add(ana1.leadingBaseCoefficient());
adenom.add(faa1);
}
}
return new PartialFraction<C>(A, P, cfactors, cdenom, afactors, adenom);
}
/**
* Univariate GenPolynomial algebraic partial fraction decomposition, Rothstein-Trager algorithm.
*
* @param A univariate GenPolynomial, deg(A) < deg(P).
* @param P univariate squarefree GenPolynomial, gcd(A,P) == 1.
* @return partial fraction container.
*/
@Deprecated
public PartialFraction<C> baseAlgebraicPartialFractionIrreducible(
GenPolynomial<C> A, GenPolynomial<C> P) {
if (P == null || P.isZERO()) {
throw new RuntimeException(" P == null or P == 0");
}
// System.out.println("\nP_base_algeb_part = " + P);
GenPolynomialRing<C> pfac = P.ring; // K[x]
if (pfac.nvar > 1) {
// System.out.println("facs_base_irred: univ");
throw new RuntimeException("only for univariate polynomials");
}
if (!pfac.coFac.isField()) {
// System.out.println("facs_base_irred: field");
throw new RuntimeException("only for field coefficients");
}
List<C> cfactors = new ArrayList<C>();
List<GenPolynomial<C>> cdenom = new ArrayList<GenPolynomial<C>>();
List<AlgebraicNumber<C>> afactors = new ArrayList<AlgebraicNumber<C>>();
List<GenPolynomial<AlgebraicNumber<C>>> adenom =
new ArrayList<GenPolynomial<AlgebraicNumber<C>>>();
// P linear
if (P.degree(0) <= 1) {
cfactors.add(A.leadingBaseCoefficient());
cdenom.add(P);
return new PartialFraction<C>(A, P, cfactors, cdenom, afactors, adenom);
}
// deriviative
GenPolynomial<C> Pp = PolyUtil.<C>baseDeriviative(P);
// no: Pp = Pp.monic();
// System.out.println("\nP = " + P);
// System.out.println("Pp = " + Pp);
// Q[t]
String[] vars = new String[] {"t"};
GenPolynomialRing<C> cfac = new GenPolynomialRing<C>(pfac.coFac, 1, pfac.tord, vars);
GenPolynomial<C> t = cfac.univariate(0);
// System.out.println("t = " + t);
// Q[x][t]
GenPolynomialRing<GenPolynomial<C>> rfac =
new GenPolynomialRing<GenPolynomial<C>>(pfac, cfac); // sic
// System.out.println("rfac = " + rfac.toScript());
// transform polynomials to bi-variate polynomial
GenPolynomial<GenPolynomial<C>> Ac = PolyUfdUtil.<C>introduceLowerVariable(rfac, A);
// System.out.println("Ac = " + Ac);
GenPolynomial<GenPolynomial<C>> Pc = PolyUfdUtil.<C>introduceLowerVariable(rfac, P);
// System.out.println("Pc = " + Pc);
GenPolynomial<GenPolynomial<C>> Pcp = PolyUfdUtil.<C>introduceLowerVariable(rfac, Pp);
// System.out.println("Pcp = " + Pcp);
// Q[t][x]
GenPolynomialRing<GenPolynomial<C>> rfac1 = Pc.ring;
// System.out.println("rfac1 = " + rfac1.toScript());
// A - t P'
GenPolynomial<GenPolynomial<C>> tc = rfac1.getONE().multiply(t);
// System.out.println("tc = " + tc);
GenPolynomial<GenPolynomial<C>> At = Ac.subtract(tc.multiply(Pcp));
// System.out.println("At = " + At);
GreatestCommonDivisorSubres<C> engine = new GreatestCommonDivisorSubres<C>();
// = GCDFactory.<C>getImplementation( cfac.coFac );
GreatestCommonDivisorAbstract<AlgebraicNumber<C>> aengine = null;
GenPolynomial<GenPolynomial<C>> Rc = engine.recursiveResultant(Pc, At);
// System.out.println("Rc = " + Rc);
GenPolynomial<C> res = Rc.leadingBaseCoefficient();
// no: res = res.monic();
// System.out.println("\nres = " + res);
SortedMap<GenPolynomial<C>, Long> resfac = baseFactors(res);
// System.out.println("resfac = " + resfac + "\n");
for (GenPolynomial<C> r : resfac.keySet()) {
// System.out.println("\nr(t) = " + r);
if (r.isConstant()) {
continue;
}
// if ( r.degree(0) <= 1L ) {
// System.out.println("warning linear factor in resultant ignored");
// continue;
// //throw new RuntimeException("input not irreducible");
// }
// vars = new String[] { "z_" + Math.abs(r.hashCode() % 1000) };
vars = pfac.newVars("z_");
pfac = pfac.clone();
vars = pfac.setVars(vars);
r = pfac.copy(r); // hack to exchange the variables
// System.out.println("r(z_) = " + r);
AlgebraicNumberRing<C> afac = new AlgebraicNumberRing<C>(r, true); // since irreducible
logger.debug("afac = " + afac.toScript());
AlgebraicNumber<C> a = afac.getGenerator();
// no: a = a.negate();
// System.out.println("a = " + a);
// K(alpha)[x]
GenPolynomialRing<AlgebraicNumber<C>> pafac =
new GenPolynomialRing<AlgebraicNumber<C>>(afac, Pc.ring);
// System.out.println("pafac = " + pafac.toScript());
// convert to K(alpha)[x]
GenPolynomial<AlgebraicNumber<C>> Pa = PolyUtil.<C>convertToAlgebraicCoefficients(pafac, P);
// System.out.println("Pa = " + Pa);
GenPolynomial<AlgebraicNumber<C>> Pap = PolyUtil.<C>convertToAlgebraicCoefficients(pafac, Pp);
// System.out.println("Pap = " + Pap);
GenPolynomial<AlgebraicNumber<C>> Aa = PolyUtil.<C>convertToAlgebraicCoefficients(pafac, A);
// System.out.println("Aa = " + Aa);
// A - a P'
GenPolynomial<AlgebraicNumber<C>> Ap = Aa.subtract(Pap.multiply(a));
// System.out.println("Ap = " + Ap);
if (aengine == null) {
aengine = GCDFactory.<AlgebraicNumber<C>>getImplementation(afac);
// System.out.println("aengine = " + aengine);
}
GenPolynomial<AlgebraicNumber<C>> Ga = aengine.baseGcd(Pa, Ap);
// System.out.println("Ga = " + Ga);
if (Ga.isConstant()) {
// System.out.println("warning constant gcd ignored");
continue;
}
afactors.add(a);
adenom.add(Ga);
// quadratic case
if (P.degree(0) == 2 && Ga.degree(0) == 1) {
GenPolynomial<AlgebraicNumber<C>>[] qra =
PolyUtil.<AlgebraicNumber<C>>basePseudoQuotientRemainder(Pa, Ga);
GenPolynomial<AlgebraicNumber<C>> Qa = qra[0];
if (!qra[1].isZERO()) {
throw new RuntimeException("remainder not zero");
}
// System.out.println("Qa = " + Qa);
afactors.add(a.negate());
adenom.add(Qa);
}
if (false && P.degree(0) == 3 && Ga.degree(0) == 1) {
GenPolynomial<AlgebraicNumber<C>>[] qra =
PolyUtil.<AlgebraicNumber<C>>basePseudoQuotientRemainder(Pa, Ga);
GenPolynomial<AlgebraicNumber<C>> Qa = qra[0];
if (!qra[1].isZERO()) {
throw new RuntimeException("remainder not zero");
}
System.out.println("Qa3 = " + Qa);
// afactors.add( a.negate() );
// adenom.add( Qa );
}
}
return new PartialFraction<C>(A, P, cfactors, cdenom, afactors, adenom);
}
/**
* GenPolynomial base absolute factorization of a irreducible polynomial.
*
* @param P irreducible! univariate GenPolynomial.
* @return factors container: [p_1,...,p_k] with P = prod_{i=1, ..., k} p_i in K(alpha)[x] for
* suitable alpha and p_i irreducible over L[x], where K \subset K(alpha) \subset L is an
* algebraically closed field over K. <b>Note:</b> K(alpha) not yet minimal.
*/
public Factors<C> baseFactorsAbsoluteIrreducible(GenPolynomial<C> P) {
if (P == null) {
throw new RuntimeException(this.getClass().getName() + " P == null");
}
if (P.isZERO()) {
return new Factors<C>(P);
}
// System.out.println("\nP_base_irred = " + P);
GenPolynomialRing<C> pfac = P.ring; // K[x]
if (pfac.nvar > 1) {
// System.out.println("facs_base_irred: univ");
throw new RuntimeException("only for univariate polynomials");
}
if (!pfac.coFac.isField()) {
// System.out.println("facs_base_irred: field");
throw new RuntimeException("only for field coefficients");
}
if (P.degree(0) <= 1) {
return new Factors<C>(P);
}
// setup field extension K(alpha) where alpha = z_xx
// String[] vars = new String[] { "z_" + Math.abs(P.hashCode() % 1000) };
String[] vars = pfac.newVars("z_");
pfac = pfac.clone();
vars = pfac.setVars(vars);
GenPolynomial<C> aP = pfac.copy(P); // hack to exchange the variables
AlgebraicNumberRing<C> afac = new AlgebraicNumberRing<C>(aP, true); // since irreducible
if (logger.isInfoEnabled()) {
logger.info("K(alpha) = " + afac);
logger.info("K(alpha) = " + afac.toScript());
// System.out.println("K(alpha) = " + afac);
}
GenPolynomialRing<AlgebraicNumber<C>> pafac =
new GenPolynomialRing<AlgebraicNumber<C>>(afac, aP.ring.nvar, aP.ring.tord, /*old*/ vars);
// convert to K(alpha)
GenPolynomial<AlgebraicNumber<C>> Pa = PolyUtil.<C>convertToAlgebraicCoefficients(pafac, P);
if (logger.isInfoEnabled()) {
logger.info("P over K(alpha) = " + Pa);
// logger.info("P over K(alpha) = " + Pa.toScript());
// System.out.println("P in K(alpha) = " + Pa);
}
// factor over K(alpha)
FactorAbstract<AlgebraicNumber<C>> engine = FactorFactory.<C>getImplementation(afac);
// System.out.println("K(alpha) engine = " + engine);
List<GenPolynomial<AlgebraicNumber<C>>> factors = engine.baseFactorsSquarefree(Pa);
// System.out.println("factors = " + factors);
if (logger.isInfoEnabled()) {
logger.info("factors over K(alpha) = " + factors);
// System.out.println("factors over K(alpha) = " + factors);
}
List<GenPolynomial<AlgebraicNumber<C>>> faca =
new ArrayList<GenPolynomial<AlgebraicNumber<C>>>(factors.size());
;
List<Factors<AlgebraicNumber<C>>> facar = new ArrayList<Factors<AlgebraicNumber<C>>>();
for (GenPolynomial<AlgebraicNumber<C>> fi : factors) {
if (fi.degree(0) <= 1) {
faca.add(fi);
} else {
// System.out.println("fi.deg > 1 = " + fi);
FactorAbsolute<AlgebraicNumber<C>> aengine =
(FactorAbsolute<AlgebraicNumber<C>>) FactorFactory.<C>getImplementation(afac);
Factors<AlgebraicNumber<C>> fif = aengine.baseFactorsAbsoluteIrreducible(fi);
// System.out.println("fif = " + fif);
facar.add(fif);
}
}
if (facar.size() == 0) {
facar = null;
}
// find minimal field extension K(beta) \subset K(alpha)
return new Factors<C>(P, afac, Pa, faca, facar);
}
/**
* GenPolynomial polynomial squarefree factorization.
*
* @param A GenPolynomial.
* @return [p_1 -> e_1, ..., p_k -> e_k] with P = prod_{i=1,...,k} p_i^{e_i} and p_i
* squarefree.
*/
@Override
public SortedMap<GenPolynomial<C>, Long> baseSquarefreeFactors(GenPolynomial<C> A) {
SortedMap<GenPolynomial<C>, Long> sfactors = new TreeMap<GenPolynomial<C>, Long>();
if (A == null || A.isZERO()) {
return sfactors;
}
GenPolynomialRing<C> pfac = A.ring;
if (A.isConstant()) {
C coeff = A.leadingBaseCoefficient();
// System.out.println("coeff = " + coeff + " @ " + coeff.factory());
SortedMap<C, Long> rfactors = squarefreeFactors(coeff);
// System.out.println("rfactors,const = " + rfactors);
if (rfactors != null && rfactors.size() > 0) {
for (Map.Entry<C, Long> me : rfactors.entrySet()) {
C c = me.getKey();
if (!c.isONE()) {
GenPolynomial<C> cr = pfac.getONE().multiply(c);
Long rk = me.getValue(); // rfactors.get(c);
sfactors.put(cr, rk);
}
}
} else {
sfactors.put(A, 1L);
}
return sfactors;
}
if (pfac.nvar > 1) {
throw new IllegalArgumentException(
this.getClass().getName() + " only for univariate polynomials");
}
C ldbcf = A.leadingBaseCoefficient();
if (!ldbcf.isONE()) {
A = A.divide(ldbcf);
SortedMap<C, Long> rfactors = squarefreeFactors(ldbcf);
// System.out.println("rfactors,ldbcf = " + rfactors);
if (rfactors != null && rfactors.size() > 0) {
for (Map.Entry<C, Long> me : rfactors.entrySet()) {
C c = me.getKey();
if (!c.isONE()) {
GenPolynomial<C> cr = pfac.getONE().multiply(c);
Long rk = me.getValue(); // rfactors.get(c);
sfactors.put(cr, rk);
}
}
} else {
GenPolynomial<C> f1 = pfac.getONE().multiply(ldbcf);
// System.out.println("gcda sqf f1 = " + f1);
sfactors.put(f1, 1L);
}
ldbcf = pfac.coFac.getONE();
}
GenPolynomial<C> T0 = A;
long e = 1L;
GenPolynomial<C> Tp;
GenPolynomial<C> T = null;
GenPolynomial<C> V = null;
long k = 0L;
long mp = 0L;
boolean init = true;
while (true) {
// System.out.println("T0 = " + T0);
if (init) {
if (T0.isConstant() || T0.isZERO()) {
break;
}
Tp = PolyUtil.<C>baseDeriviative(T0);
T = engine.baseGcd(T0, Tp);
T = T.monic();
V = PolyUtil.<C>basePseudoDivide(T0, T);
// System.out.println("iT0 = " + T0);
// System.out.println("iTp = " + Tp);
// System.out.println("iT = " + T);
// System.out.println("iV = " + V);
// System.out.println("const(iV) = " + V.isConstant());
k = 0L;
mp = 0L;
init = false;
}
if (V.isConstant()) {
mp = pfac.characteristic().longValue(); // assert != 0
// T0 = PolyUtil.<C> baseModRoot(T,mp);
T0 = baseRootCharacteristic(T);
logger.info("char root: T0 = " + T0 + ", T = " + T);
if (T0 == null) {
// break;
T0 = pfac.getZERO();
}
e = e * mp;
init = true;
continue;
}
k++;
if (mp != 0L && k % mp == 0L) {
T = PolyUtil.<C>basePseudoDivide(T, V);
System.out.println("k = " + k);
// System.out.println("T = " + T);
k++;
}
GenPolynomial<C> W = engine.baseGcd(T, V);
W = W.monic();
GenPolynomial<C> z = PolyUtil.<C>basePseudoDivide(V, W);
// System.out.println("W = " + W);
// System.out.println("z = " + z);
V = W;
T = PolyUtil.<C>basePseudoDivide(T, V);
// System.out.println("V = " + V);
// System.out.println("T = " + T);
if (z.degree(0) > 0) {
if (ldbcf.isONE() && !z.leadingBaseCoefficient().isONE()) {
z = z.monic();
logger.info("z,monic = " + z);
}
sfactors.put(z, (e * k));
}
}
// look, a stupid error:
// if ( !ldbcf.isONE() ) {
// GenPolynomial<C> f1 = sfactors.firstKey();
// long e1 = sfactors.remove(f1);
// System.out.println("gcda sqf c = " + c);
// f1 = f1.multiply(c);
// //System.out.println("gcda sqf f1e = " + f1);
// sfactors.put(f1,e1);
// }
logger.info("exit char root: T0 = " + T0 + ", T = " + T);
return sfactors;
}
