Example #1
0
 /**
  * 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;
 }
Example #2
0
  /**
   * Invariant interval for algebraic number magnitude.
   *
   * @param iv root isolating interval for f, with f(left) * f(right) &lt; 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)| &lt; 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;
  }
Example #3
0
 /**
  * Refine interval.
  *
  * @param iv root isolating interval with f(left) * f(right) &lt; 0.
  * @param f univariate polynomial, non-zero.
  * @param eps requested interval length.
  * @return a new interval v such that |v| &lt; 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;
 }
Example #4
0
 /**
  * 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);
 }
Example #6
0
 /**
  * Real algebraic number magnitude.
  *
  * @param iv root isolating interval for f, with f(left) * f(right) &lt; 0, with iv such that
  *     |g(a) - g(b)| &lt; 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;
 }
Example #7
0
  /**
   * Magnitude bound.
   *
   * @param iv interval.
   * @param f univariate polynomial.
   * @return B such that |f(c)| &lt; 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;
  }
Example #8
0
 /**
  * Real algebraic number sign.
  *
  * @param iv root isolating interval for f, with f(left) * f(right) &lt; 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();
 }
Example #9
0
 public Odds(RingFactory<C> fac) {
   this.fac = fac;
   two = fac.fromInteger(2);
   // System.out.println("two = " + two);
 }
Example #10
0
  /**
   * 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);
  }