/**
* 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;
}
/**
* 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);
}
/** Test compare sequential with parallel GBase. */
public void testSequentialParallelGBase() {
List<GenPolynomial<BigRational>> Gs, Gp;
L = new ArrayList<GenPolynomial<BigRational>>();
a = fac.random(kl, ll, el, q);
b = fac.random(kl, ll, el, q);
c = fac.random(kl, ll, el, q);
d = fac.random(kl, ll, el, q);
e = d; // fac.random(kl, ll, el, q );
if (a.isZERO() || b.isZERO() || c.isZERO() || d.isZERO()) {
return;
}
L.add(a);
Gs = bbseq.GB(L);
Gp = bbpar.GB(L);
assertTrue("Gs.containsAll(Gp)", Gs.containsAll(Gp));
assertTrue("Gp.containsAll(Gs)", Gp.containsAll(Gs));
L = Gs;
L.add(b);
Gs = bbseq.GB(L);
Gp = bbpar.GB(L);
assertTrue("Gs.containsAll(Gp)", Gs.containsAll(Gp));
assertTrue("Gp.containsAll(Gs)", Gp.containsAll(Gs));
L = Gs;
L.add(c);
Gs = bbseq.GB(L);
Gp = bbpar.GB(L);
assertTrue("Gs.containsAll(Gp)", Gs.containsAll(Gp));
assertTrue("Gp.containsAll(Gs)", Gp.containsAll(Gs));
L = Gs;
L.add(d);
Gs = bbseq.GB(L);
Gp = bbpar.GB(L);
assertTrue("Gs.containsAll(Gp)", Gs.containsAll(Gp));
assertTrue("Gp.containsAll(Gs)", Gp.containsAll(Gs));
L = Gs;
L.add(e);
Gs = bbseq.GB(L);
Gp = bbpar.GB(L);
assertTrue("Gs.containsAll(Gp)", Gs.containsAll(Gp));
assertTrue("Gp.containsAll(Gs)", Gp.containsAll(Gs));
}
/** Test constructor and toString. */
public void testConstruction() {
c = fac.getONE();
assertTrue("length( c ) = 1", c.length() == 1);
assertTrue("isZERO( c )", !c.isZERO());
assertTrue("isONE( c )", c.isONE());
d = fac.getZERO();
assertTrue("length( d ) = 0", d.length() == 0);
assertTrue("isZERO( d )", d.isZERO());
assertTrue("isONE( d )", !d.isONE());
}
/**
* 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 object multiplication. */
public void testMultiplication() {
a = fac.random(ll);
assertTrue("not isZERO( a )", !a.isZERO());
b = fac.random(ll);
assertTrue("not isZERO( b )", !b.isZERO());
c = b.multiply(a);
d = a.multiply(b);
assertTrue("not isZERO( c )", !c.isZERO());
assertTrue("not isZERO( d )", !d.isZERO());
// System.out.println("a = " + a);
// System.out.println("b = " + b);
e = d.subtract(c);
assertTrue("isZERO( a*b-b*a ) " + e, e.isZERO());
assertTrue("a*b = b*a", c.equals(d));
assertEquals("a*b = b*a", c, d);
c = fac.random(ll);
// System.out.println("c = " + c);
d = a.multiply(b.multiply(c));
e = (a.multiply(b)).multiply(c);
// System.out.println("d = " + d);
// System.out.println("e = " + e);
// System.out.println("d-e = " + d.subtract(c) );
assertEquals("a(bc) = (ab)c", d, e);
assertTrue("a(bc) = (ab)c", d.equals(e));
BigComplex x = a.leadingBaseCoefficient().inverse();
c = a.monic();
d = a.multiply(x);
assertEquals("a.monic() = a(1/ldcf(a))", c, d);
BigComplex y = b.leadingBaseCoefficient().inverse();
c = b.monic();
d = b.multiply(y);
assertEquals("b.monic() = b(1/ldcf(b))", c, d);
e = new GenPolynomial<BigComplex>(fac, y);
d = b.multiply(e);
assertEquals("b.monic() = b(1/ldcf(b))", c, d);
d = e.multiply(b);
assertEquals("b.monic() = (1/ldcf(b) (0))*b", c, d);
}
/**
* 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 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);
}
/**
* 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();
}
/**
* GenPolynomial squarefree factorization.
*
* @param P 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> squarefreeFactors(GenPolynomial<C> P) {
if (P == null) {
throw new IllegalArgumentException(this.getClass().getName() + " P != null");
}
GenPolynomialRing<C> pfac = P.ring;
if (pfac.nvar <= 1) {
return baseSquarefreeFactors(P);
}
SortedMap<GenPolynomial<C>, Long> sfactors = new TreeMap<GenPolynomial<C>, Long>();
if (P.isZERO()) {
return sfactors;
}
GenPolynomialRing<C> cfac = pfac.contract(1);
GenPolynomialRing<GenPolynomial<C>> rfac = new GenPolynomialRing<GenPolynomial<C>>(cfac, 1);
GenPolynomial<GenPolynomial<C>> Pr = PolyUtil.<C>recursive(rfac, P);
SortedMap<GenPolynomial<GenPolynomial<C>>, Long> PP = recursiveUnivariateSquarefreeFactors(Pr);
for (Map.Entry<GenPolynomial<GenPolynomial<C>>, Long> m : PP.entrySet()) {
Long i = m.getValue();
GenPolynomial<GenPolynomial<C>> Dr = m.getKey();
GenPolynomial<C> D = PolyUtil.<C>distribute(pfac, Dr);
sfactors.put(D, i);
}
return sfactors;
}
/**
* 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;
}
/**
* 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 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();
}
/** Test random polynomial. */
public void testRandom() {
for (int i = 0; i < 7; i++) {
a = fac.random(ll);
// fac.random(rl+i, kl*(i+1), ll+2*i, el+i, q );
assertTrue("length( a" + i + " ) <> 0", a.length() >= 0);
assertTrue(" not isZERO( a" + i + " )", !a.isZERO());
assertTrue(" not isONE( a" + i + " )", !a.isONE());
}
}
/**
* 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;
}
/**
* 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;
}
/** Test parallel GBase. */
public void testParallelGBase() {
L = new ArrayList<GenPolynomial<BigRational>>();
a = fac.random(kl, ll, el, q);
b = fac.random(kl, ll, el, q);
c = fac.random(kl, ll, el, q);
d = fac.random(kl, ll, el, q);
e = d; // fac.random(kl, ll, el, q );
if (a.isZERO() || b.isZERO() || c.isZERO() || d.isZERO()) {
return;
}
assertTrue("not isZERO( a )", !a.isZERO());
L.add(a);
L = bbpar.GB(L);
assertTrue("isGB( { a } )", bbpar.isGB(L));
assertTrue("not isZERO( b )", !b.isZERO());
L.add(b);
// System.out.println("L = " + L.size() );
L = bbpar.GB(L);
assertTrue("isGB( { a, b } )", bbpar.isGB(L));
assertTrue("not isZERO( c )", !c.isZERO());
L.add(c);
L = bbpar.GB(L);
assertTrue("isGB( { a, b, c } )", bbpar.isGB(L));
assertTrue("not isZERO( d )", !d.isZERO());
L.add(d);
L = bbpar.GB(L);
assertTrue("isGB( { a, b, c, d } )", bbpar.isGB(L));
assertTrue("not isZERO( e )", !e.isZERO());
L.add(e);
L = bbpar.GB(L);
assertTrue("isGB( { a, b, c, d, e } )", bbpar.isGB(L));
}
// @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);
}
/**
* 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);
}
/**
* GenPolynomial polynomial greatest squarefree divisor.
*
* @param P GenPolynomial.
* @return squarefree(pp(P)).
*/
@Override
public GenPolynomial<C> baseSquarefreePart(GenPolynomial<C> P) {
if (P == null || P.isZERO()) {
return P;
}
GenPolynomialRing<C> pfac = P.ring;
if (pfac.nvar > 1) {
throw new IllegalArgumentException(
this.getClass().getName() + " only for univariate polynomials");
}
// just for the moment:
GenPolynomial<C> s = pfac.getONE();
SortedMap<GenPolynomial<C>, Long> factors = baseSquarefreeFactors(P);
logger.info("sqfPart,factors = " + factors);
for (GenPolynomial<C> sp : factors.keySet()) {
s = s.multiply(sp);
}
return s.monic();
}
/**
* 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;
}
/** Test modular evaluation gcd. */
public void testModEvalGcd() {
GreatestCommonDivisorAbstract<ModInteger> ufd_me = new GreatestCommonDivisorModEval();
for (int i = 0; i < 1; i++) {
a = dfac.random(kl * (i + 2), ll + 2 * i, el + 0 * i, q);
b = dfac.random(kl * (i + 2), ll + 2 * i, el + 0 * i, q);
c = dfac.random(kl * (i + 2), ll + 2 * i, el + 0 * i, q);
c = c.multiply(dfac.univariate(0));
// a = ufd.basePrimitivePart(a);
// b = ufd.basePrimitivePart(b);
if (a.isZERO() || b.isZERO() || c.isZERO()) {
// skip for this turn
continue;
}
assertTrue("length( c" + i + " ) <> 0", c.length() > 0);
// assertTrue(" not isZERO( c"+i+" )", !c.isZERO() );
// assertTrue(" not isONE( c"+i+" )", !c.isONE() );
a = a.multiply(c);
b = b.multiply(c);
// System.out.println("a = " + a);
// System.out.println("b = " + b);
d = ufd_me.gcd(a, b);
c = ufd.basePrimitivePart(c).abs();
e = PolyUtil.<ModInteger>basePseudoRemainder(d, c);
// System.out.println("c = " + c);
// System.out.println("d = " + d);
assertTrue("c | gcd(ac,bc) " + e, e.isZERO());
e = PolyUtil.<ModInteger>basePseudoRemainder(a, d);
// System.out.println("e = " + e);
assertTrue("gcd(a,b) | a" + e, e.isZERO());
e = PolyUtil.<ModInteger>basePseudoRemainder(b, d);
// System.out.println("e = " + e);
assertTrue("gcd(a,b) | b" + e, e.isZERO());
}
}
/**
* 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);
}
/**
* 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");
}