/**
* Invariant interval for algebraic number magnitude.
*
* @param iv root isolating interval for f, with f(left) * f(right) < 0.
* @param f univariate polynomial, non-zero.
* @param g univariate polynomial, gcd(f,g) == 1.
* @param eps length limit for interval length.
* @return v with v a new interval contained in iv such that |g(a) - g(b)| < eps for a, b in v
* in iv.
*/
public Interval<C> invariantMagnitudeInterval(
Interval<C> iv, GenPolynomial<C> f, GenPolynomial<C> g, C eps) {
Interval<C> v = iv;
if (g == null || g.isZERO()) {
return v;
}
if (g.isConstant()) {
return v;
}
if (f == null || f.isZERO() || f.isConstant()) { // ?
return v;
}
GenPolynomial<C> gp = PolyUtil.<C>baseDeriviative(g);
// System.out.println("g = " + g);
// System.out.println("gp = " + gp);
C B = magnitudeBound(iv, gp);
// System.out.println("B = " + B);
RingFactory<C> cfac = f.ring.coFac;
C two = cfac.fromInteger(2);
while (B.multiply(v.length()).compareTo(eps) >= 0) {
C c = v.left.sum(v.right);
c = c.divide(two);
Interval<C> im = new Interval<C>(c, v.right);
if (signChange(im, f)) {
v = im;
} else {
v = new Interval<C>(v.left, c);
}
// System.out.println("v = " + v.toDecimal());
}
return v;
}
/**
* Real algebraic number sign.
*
* @param iv root isolating interval for f, with f(left) * f(right) < 0.
* @param f univariate polynomial, non-zero.
* @param g univariate polynomial, gcd(f,g) == 1.
* @return sign(g(v)), with v a new interval contained in iv such that g(v) != 0.
*/
public int realSign(Interval<C> iv, GenPolynomial<C> f, GenPolynomial<C> g) {
if (g == null || g.isZERO()) {
return 0;
}
if (f == null || f.isZERO() || f.isConstant()) {
return g.signum();
}
if (g.isConstant()) {
return g.signum();
}
Interval<C> v = invariantSignInterval(iv, f, g);
return realIntervalSign(v, f, g);
}
/**
* Test if x is an approximate real root.
*
* @param x approximate real root.
* @param f univariate polynomial, non-zero.
* @param fp univariate polynomial, non-zero, deriviative of f.
* @param eps requested interval length.
* @return true if x is a decimal approximation of a real v with f(v) = 0 with |d-v| < eps,
* else false.
*/
public boolean isApproximateRoot(
BigDecimal x, GenPolynomial<BigDecimal> f, GenPolynomial<BigDecimal> fp, BigDecimal eps) {
if (x == null) {
throw new IllegalArgumentException("null root not allowed");
}
if (f == null || f.isZERO() || f.isConstant() || eps == null) {
return true;
}
BigDecimal dc = BigDecimal.ONE; // only for clarity
// f(x)
BigDecimal fx = PolyUtil.<BigDecimal>evaluateMain(dc, f, x);
// System.out.println("fx = " + fx);
if (fx.isZERO()) {
return true;
}
// f'(x)
BigDecimal fpx = PolyUtil.<BigDecimal>evaluateMain(dc, fp, x); // f'(d)
// System.out.println("fpx = " + fpx);
if (fpx.isZERO()) {
return false;
}
BigDecimal d = fx.divide(fpx);
if (d.isZERO()) {
return true;
}
if (d.abs().compareTo(eps) <= 0) {
return true;
}
System.out.println("x = " + x);
System.out.println("d = " + d);
return false;
}
/**
* Real algebraic number magnitude.
*
* @param iv root isolating interval for f, with f(left) * f(right) < 0.
* @param f univariate polynomial, non-zero.
* @param g univariate polynomial, gcd(f,g) == 1.
* @param eps length limit for interval length.
* @return g(iv) .
*/
public C realMagnitude(Interval<C> iv, GenPolynomial<C> f, GenPolynomial<C> g, C eps) {
if (g.isZERO() || g.isConstant()) {
return g.leadingBaseCoefficient();
}
Interval<C> v = invariantMagnitudeInterval(iv, f, g, eps);
return realIntervalMagnitude(v, f, g, eps);
}
/**
* Real root bound. With f(M) * f(-M) != 0.
*
* @param f univariate polynomial.
* @return M such that -M < root(f) < M.
*/
public C realRootBound(GenPolynomial<C> f) {
if (f == null) {
return null;
}
RingFactory<C> cfac = f.ring.coFac;
C M = cfac.getONE();
if (f.isZERO() || f.isConstant()) {
return M;
}
C a = f.leadingBaseCoefficient().abs();
for (C c : f.getMap().values()) {
C d = c.abs().divide(a);
if (M.compareTo(d) < 0) {
M = d;
}
}
// works also without this case, only for optimization
// to use rational number interval end points
// can fail if real root is in interval [r,r+1]
// for too low precision or too big r, since r is approximation
if ((Object) M instanceof RealAlgebraicNumber) {
RealAlgebraicNumber Mr = (RealAlgebraicNumber) M;
BigRational r = Mr.magnitude();
M = cfac.fromInteger(r.numerator()).divide(cfac.fromInteger(r.denominator()));
}
M = M.sum(f.ring.coFac.getONE());
// System.out.println("M = " + M);
return M;
}
/**
* GenPolynomial greatest squarefree divisor.
*
* @param P GenPolynomial.
* @return squarefree(pp(P)).
*/
@Override
public GenPolynomial<C> squarefreePart(GenPolynomial<C> P) {
if (P == null) {
throw new IllegalArgumentException(this.getClass().getName() + " P != null");
}
if (P.isZERO()) {
return P;
}
GenPolynomialRing<C> pfac = P.ring;
if (pfac.nvar <= 1) {
return baseSquarefreePart(P);
}
// just for the moment:
GenPolynomial<C> s = pfac.getONE();
SortedMap<GenPolynomial<C>, Long> factors = squarefreeFactors(P);
if (logger.isInfoEnabled()) {
logger.info("sqfPart,factors = " + factors);
}
for (GenPolynomial<C> sp : factors.keySet()) {
if (sp.isConstant()) {
continue;
}
s = s.multiply(sp);
}
return s.monic();
}
/**
* Refine interval.
*
* @param iv root isolating interval with f(left) * f(right) < 0.
* @param f univariate polynomial, non-zero.
* @param eps requested interval length.
* @return a new interval v such that |v| < eps.
*/
public Interval<C> refineInterval(Interval<C> iv, GenPolynomial<C> f, C eps) {
if (f == null || f.isZERO() || f.isConstant() || eps == null) {
return iv;
}
if (iv.length().compareTo(eps) < 0) {
return iv;
}
RingFactory<C> cfac = f.ring.coFac;
C two = cfac.fromInteger(2);
Interval<C> v = iv;
while (v.length().compareTo(eps) >= 0) {
C c = v.left.sum(v.right);
c = c.divide(two);
// System.out.println("c = " + c);
// c = RootUtil.<C>bisectionPoint(v,f);
if (PolyUtil.<C>evaluateMain(cfac, f, c).isZERO()) {
v = new Interval<C>(c, c);
break;
}
Interval<C> iv1 = new Interval<C>(v.left, c);
if (signChange(iv1, f)) {
v = iv1;
} else {
v = new Interval<C>(c, v.right);
}
}
return v;
}
/**
* Bi-section point.
*
* @param iv interval with f(left) * f(right) != 0.
* @param f univariate polynomial, non-zero.
* @return a point c in the interval iv such that f(c) != 0.
*/
public C bisectionPoint(Interval<C> iv, GenPolynomial<C> f) {
if (f == null) {
return null;
}
RingFactory<C> cfac = f.ring.coFac;
C two = cfac.fromInteger(2);
C c = iv.left.sum(iv.right);
c = c.divide(two);
if (f.isZERO() || f.isConstant()) {
return c;
}
C m = PolyUtil.<C>evaluateMain(cfac, f, c);
while (m.isZERO()) {
C d = iv.left.sum(c);
d = d.divide(two);
if (d.equals(c)) {
d = iv.right.sum(c);
d = d.divide(two);
if (d.equals(c)) {
throw new RuntimeException("should not happen " + iv);
}
}
c = d;
m = PolyUtil.<C>evaluateMain(cfac, f, c);
// System.out.println("c = " + c);
}
// System.out.println("c = " + c);
return c;
}
/**
* Real algebraic number sign.
*
* @param iv root isolating interval for f, with f(left) * f(right) < 0, with iv such that
* g(iv) != 0.
* @param f univariate polynomial, non-zero.
* @param g univariate polynomial, gcd(f,g) == 1.
* @return sign(g(iv)) .
*/
public int realIntervalSign(Interval<C> iv, GenPolynomial<C> f, GenPolynomial<C> g) {
if (g == null || g.isZERO()) {
return 0;
}
if (f == null || f.isZERO() || f.isConstant()) {
return g.signum();
}
if (g.isConstant()) {
return g.signum();
}
RingFactory<C> cfac = f.ring.coFac;
C c = iv.left.sum(iv.right);
c = c.divide(cfac.fromInteger(2));
C ev = PolyUtil.<C>evaluateMain(cfac, g, c);
// System.out.println("ev = " + ev);
return ev.signum();
}
/**
* Real algebraic number magnitude.
*
* @param iv root isolating interval for f, with f(left) * f(right) < 0, with iv such that
* |g(a) - g(b)| < eps for a, b in iv.
* @param f univariate polynomial, non-zero.
* @param g univariate polynomial, gcd(f,g) == 1.
* @param eps length limit for interval length.
* @return g(iv) .
*/
public C realIntervalMagnitude(Interval<C> iv, GenPolynomial<C> f, GenPolynomial<C> g, C eps) {
if (g.isZERO() || g.isConstant()) {
return g.leadingBaseCoefficient();
}
RingFactory<C> cfac = g.ring.coFac;
C c = iv.left.sum(iv.right);
c = c.divide(cfac.fromInteger(2));
C ev = PolyUtil.<C>evaluateMain(cfac, g, c);
// System.out.println("ev = " + ev);
return ev;
}
/**
* Refine intervals.
*
* @param V list of isolating intervals with f(left) * f(right) < 0.
* @param f univariate polynomial, non-zero.
* @param eps requested intervals length.
* @return a list of new intervals v such that |v| < eps.
*/
public List<Interval<C>> refineIntervals(List<Interval<C>> V, GenPolynomial<C> f, C eps) {
if (f == null || f.isZERO() || f.isConstant() || eps == null) {
return V;
}
List<Interval<C>> IV = new ArrayList<Interval<C>>();
for (Interval<C> v : V) {
Interval<C> iv = refineInterval(v, f, eps);
IV.add(iv);
}
return IV;
}
/**
* Test if x is an approximate real root.
*
* @param x approximate real root.
* @param f univariate polynomial, non-zero.
* @param eps requested interval length.
* @return true if x is a decimal approximation of a real v with f(v) = 0 with |d-v| < eps,
* else false.
*/
public boolean isApproximateRoot(BigDecimal x, GenPolynomial<C> f, C eps) {
if (x == null) {
throw new IllegalArgumentException("null root not allowed");
}
if (f == null || f.isZERO() || f.isConstant() || eps == null) {
return true;
}
BigDecimal e = new BigDecimal(eps.getRational());
e = e.multiply(new BigDecimal("1000")); // relax
BigDecimal dc = BigDecimal.ONE;
// polynomials with decimal coefficients
GenPolynomialRing<BigDecimal> dfac = new GenPolynomialRing<BigDecimal>(dc, f.ring);
GenPolynomial<BigDecimal> df = PolyUtil.<C>decimalFromRational(dfac, f);
GenPolynomial<C> fp = PolyUtil.<C>baseDeriviative(f);
GenPolynomial<BigDecimal> dfp = PolyUtil.<C>decimalFromRational(dfac, fp);
//
return isApproximateRoot(x, df, dfp, e);
}
/**
* Magnitude bound.
*
* @param iv interval.
* @param f univariate polynomial.
* @return B such that |f(c)| < B for c in iv.
*/
public C magnitudeBound(Interval<C> iv, GenPolynomial<C> f) {
if (f == null) {
return null;
}
if (f.isZERO()) {
return f.ring.coFac.getONE();
}
if (f.isConstant()) {
return f.leadingBaseCoefficient().abs();
}
GenPolynomial<C> fa =
f.map(
new UnaryFunctor<C, C>() {
public C eval(C a) {
return a.abs();
}
});
// System.out.println("fa = " + fa);
C M = iv.left.abs();
if (M.compareTo(iv.right.abs()) < 0) {
M = iv.right.abs();
}
// System.out.println("M = " + M);
RingFactory<C> cfac = f.ring.coFac;
C B = PolyUtil.<C>evaluateMain(cfac, fa, M);
// works also without this case, only for optimization
// to use rational number interval end points
// can fail if real root is in interval [r,r+1]
// for too low precision or too big r, since r is approximation
if ((Object) B instanceof RealAlgebraicNumber) {
RealAlgebraicNumber Br = (RealAlgebraicNumber) B;
BigRational r = Br.magnitude();
B = cfac.fromInteger(r.numerator()).divide(cfac.fromInteger(r.denominator()));
}
// System.out.println("B = " + B);
return B;
}
/**
* Polynomial is char-th root.
*
* @param P polynomial.
* @param F = [p_1 -> e_1, ..., p_k -> e_k].
* @return true if P = prod_{i=1,...,k} p_i**(e_i*p), else false.
*/
public boolean isCharRoot(GenPolynomial<C> P, SortedMap<GenPolynomial<C>, Long> F) {
if (P == null || F == null) {
throw new IllegalArgumentException("P and F may not be null");
}
if (P.isZERO() && F.size() == 0) {
return true;
}
GenPolynomial<C> t = P.ring.getONE();
long p = P.ring.characteristic().longValue();
for (Map.Entry<GenPolynomial<C>, Long> me : F.entrySet()) {
GenPolynomial<C> f = me.getKey();
Long E = me.getValue(); // F.get(f);
long e = E.longValue();
GenPolynomial<C> g = Power.<GenPolynomial<C>>positivePower(f, e);
if (!f.isConstant()) {
g = Power.<GenPolynomial<C>>positivePower(g, p);
}
t = t.multiply(g);
}
boolean f = P.equals(t) || P.equals(t.negate());
if (!f) {
System.out.println("\nfactorization(map): " + f);
System.out.println("P = " + P);
System.out.println("t = " + t);
P = P.monic();
t = t.monic();
f = P.equals(t) || P.equals(t.negate());
if (f) {
return f;
}
System.out.println("\nfactorization(map): " + f);
System.out.println("P = " + P);
System.out.println("t = " + t);
}
return f;
}
/**
* Approximate real root.
*
* @param iv real root isolating interval with f(left) * f(right) < 0.
* @param f univariate polynomial, non-zero.
* @param eps requested interval length.
* @return a decimal approximation d such that |d-v| < eps, for f(v) = 0, v real.
*/
public BigDecimal approximateRoot(Interval<C> iv, GenPolynomial<C> f, C eps)
throws NoConvergenceException {
if (iv == null) {
throw new IllegalArgumentException("null interval not allowed");
}
BigDecimal d = iv.toDecimal();
if (f == null || f.isZERO() || f.isConstant() || eps == null) {
return d;
}
if (iv.length().compareTo(eps) < 0) {
return d;
}
BigDecimal left = new BigDecimal(iv.left.getRational());
BigDecimal right = new BigDecimal(iv.right.getRational());
BigDecimal e = new BigDecimal(eps.getRational());
BigDecimal q = new BigDecimal("0.25");
// System.out.println("left = " + left);
// System.out.println("right = " + right);
e = e.multiply(d); // relative error
// System.out.println("e = " + e);
BigDecimal dc = BigDecimal.ONE;
// polynomials with decimal coefficients
GenPolynomialRing<BigDecimal> dfac = new GenPolynomialRing<BigDecimal>(dc, f.ring);
GenPolynomial<BigDecimal> df = PolyUtil.<C>decimalFromRational(dfac, f);
GenPolynomial<C> fp = PolyUtil.<C>baseDeriviative(f);
GenPolynomial<BigDecimal> dfp = PolyUtil.<C>decimalFromRational(dfac, fp);
// Newton Raphson iteration: x_{n+1} = x_n - f(x_n)/f'(x_n)
int i = 0;
final int MITER = 50;
int dir = 0;
while (i++ < MITER) {
BigDecimal fx = PolyUtil.<BigDecimal>evaluateMain(dc, df, d); // f(d)
if (fx.isZERO()) {
return d;
}
BigDecimal fpx = PolyUtil.<BigDecimal>evaluateMain(dc, dfp, d); // f'(d)
if (fpx.isZERO()) {
throw new NoConvergenceException("zero deriviative should not happen");
}
BigDecimal x = fx.divide(fpx);
BigDecimal dx = d.subtract(x);
// System.out.println("dx = " + dx);
if (d.subtract(dx).abs().compareTo(e) <= 0) {
return dx;
}
while (dx.compareTo(left) < 0 || dx.compareTo(right) > 0) { // dx < left: dx - left < 0
// dx > right: dx - right > 0
// System.out.println("trying to leave interval");
if (i++ > MITER) { // dx > right: dx - right > 0
throw new NoConvergenceException("no convergence after " + i + " steps");
}
if (i > MITER / 2 && dir == 0) {
BigDecimal sd = new BigDecimal(iv.randomPoint().getRational());
d = sd;
x = sd.getZERO();
logger.info("trying new random starting point " + d);
i = 0;
dir = 1;
}
if (i > MITER / 2 && dir == 1) {
BigDecimal sd = new BigDecimal(iv.randomPoint().getRational());
d = sd;
x = sd.getZERO();
logger.info("trying new random starting point " + d);
// i = 0;
dir = 2; // end
}
x = x.multiply(q); // x * 1/4
dx = d.subtract(x);
// System.out.println(" x = " + x);
// System.out.println("dx = " + dx);
}
d = dx;
}
throw new NoConvergenceException("no convergence after " + i + " steps");
}
/**
* 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 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;
}
