Ejemplo n.º 1
0
  /**
   * 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;
  }
Ejemplo n.º 2
0
  /**
   * 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());
  }
Ejemplo n.º 5
0
 /**
  * Real algebraic number sign.
  *
  * @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.
  * @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);
  }
Ejemplo n.º 7
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;
 }
Ejemplo n.º 8
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;
 }
Ejemplo n.º 9
0
 /**
  * Real 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 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);
 }
Ejemplo n.º 10
0
 /**
  * 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();
 }
Ejemplo n.º 11
0
  /**
   * GenPolynomial squarefree factorization.
   *
   * @param P GenPolynomial.
   * @return [p_1 -&gt; e_1, ..., p_k -&gt; 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;
  }
Ejemplo n.º 12
0
 /**
  * Real root bound. With f(M) * f(-M) != 0.
  *
  * @param f univariate polynomial.
  * @return M such that -M &lt; root(f) &lt; 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;
 }
Ejemplo n.º 13
0
 /**
  * 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| &lt; 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;
 }
Ejemplo n.º 14
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();
 }
 /** 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());
   }
 }
Ejemplo n.º 16
0
 /**
  * Refine intervals.
  *
  * @param V list of isolating intervals with f(left) * f(right) &lt; 0.
  * @param f univariate polynomial, non-zero.
  * @param eps requested intervals length.
  * @return a list of new intervals v such that |v| &lt; 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;
 }
Ejemplo n.º 17
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;
 }
  /** 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));
  }
Ejemplo n.º 19
0
 // @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);
 }
Ejemplo n.º 20
0
 /**
  * GenPolynomial absolute factorization of a polynomial.
  *
  * @param P GenPolynomial.
  * @return factors map container: [p_1 -&gt; e_1, ..., p_k -&gt; 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);
 }
Ejemplo n.º 21
0
 /**
  * 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| &lt; 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);
 }
Ejemplo n.º 22
0
 /**
  * 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();
 }
Ejemplo n.º 23
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;
  }
Ejemplo n.º 24
0
 /**
  * Polynomial is char-th root.
  *
  * @param P polynomial.
  * @param F = [p_1 -&gt; e_1, ..., p_k -&gt; 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;
 }
Ejemplo n.º 25
0
  /** 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());
    }
  }
Ejemplo n.º 26
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);
  }
Ejemplo n.º 27
0
  /**
   * 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);
  }
Ejemplo n.º 28
0
  /**
   * 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);
  }
Ejemplo n.º 29
0
 /**
  * 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);
 }
Ejemplo n.º 30
0
  /**
   * Approximate real root.
   *
   * @param iv real root isolating interval with f(left) * f(right) &lt; 0.
   * @param f univariate polynomial, non-zero.
   * @param eps requested interval length.
   * @return a decimal approximation d such that |d-v| &lt; 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");
  }