/**
* 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;
}
/**
* 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;
}
/**
* 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;
}
/**
* Get a Product element from a BigInteger value.
*
* @param a BigInteger.
* @return a Product.
*/
public Product<C> fromInteger(java.math.BigInteger a) {
SortedMap<Integer, C> elem = new TreeMap<Integer, C>();
if (nCopies != 0) {
C c = ring.fromInteger(a);
for (int i = 0; i < nCopies; i++) {
elem.put(i, c);
}
} else {
int i = 0;
for (RingFactory<C> f : ringList) {
elem.put(i, f.fromInteger(a));
i++;
}
}
return new Product<C>(this, elem);
}
/**
* 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;
}
/**
* 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;
}
/**
* 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();
}
public Odds(RingFactory<C> fac) {
this.fac = fac;
two = fac.fromInteger(2);
// System.out.println("two = " + two);
}
/**
* 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);
}