Package edu.jas.root

Class RealRootsAbstract<C extends RingElem<C> & Rational>

    • Method Detail

      • magnitudeBound

        public C magnitudeBound​(Interval<C> iv,
                        GenPolynomial<C> f)
        Magnitude bound.
        Parameters:
        iv - interval.
        f - univariate polynomial.
        Returns:
        B such that |f(c)| < B for c in iv.

      • realMinimalRootBound

        public C realMinimalRootBound​(GenPolynomial<C> f)
        Real minimal root bound.
        Parameters:
        f - univariate polynomial.
        Returns:
        M such that abs(xi) > M for f(xi) == 0.

      • realMinimalRootSeparation

        public C realMinimalRootSeparation​(GenPolynomial<C> f)
        Real minimal root separation.
        Parameters:
        f - univariate polynomial.
        Returns:
        M such that abs(xi-xj) > M for roots xi, xj of f.

      • bisectionPoint

        public C bisectionPoint​(Interval<C> iv,
                        GenPolynomial<C> f)
        Bi-section point.
        Parameters:
        iv - interval with f(left) * f(right) != 0.
        f - univariate polynomial, non-zero.
        Returns:
        a point c in the interval iv such that f(c) != 0.

      • realRoots

        public java.util.List<Interval<C>> realRoots​(GenPolynomial<C> f,
                                             C eps)
        Isolating intervals for the real roots.
        Specified by:
        realRoots in interface RealRoots<C extends RingElem<C> & Rational>
        Parameters:
        f - univariate polynomial.
        eps - requested intervals length.
        Returns:
        a list of isolating intervals v such that |v| < eps.

      • halfInterval

        public Interval<C> halfInterval​(Interval<C> iv,
                                GenPolynomial<C> f)
        Half interval.
        Parameters:
        iv - root isolating interval with f(left) * f(right) < 0.
        f - univariate polynomial, non-zero.
        Returns:
        a new interval v such that |v| < |iv|/2.

      • invariantSignInterval

        public abstract Interval<C> invariantSignInterval​(Interval<C> iv,
                                                  GenPolynomial<C> f,
                                                  GenPolynomial<C> g)
        Invariant interval for algebraic number sign.
        Parameters:
        iv - root isolating interval for f, with f(left) * f(right) < 0.
        f - univariate polynomial, non-zero.
        g - univariate polynomial, gcd(f,g) == 1.
        Returns:
        v with v a new interval contained in iv such that g(v) != 0.

      • realIntervalSign

        public int realIntervalSign​(Interval<C> iv,
                            GenPolynomial<C> f,
                            GenPolynomial<C> g)
        Real algebraic number sign.
        Parameters:
        iv - root isolating interval for f, with f(left) * f(right) < 0, with iv such that g(iv) != 0.
        f - univariate polynomial, non-zero.
        g - univariate polynomial, gcd(f,g) == 1.
        Returns:
        sign(g(iv)) .

      • realSign

        public int realSign​(Interval<C> iv,
                    GenPolynomial<C> f,
                    GenPolynomial<C> g)
        Real algebraic number sign.
        Specified by:
        realSign in interface RealRoots<C extends RingElem<C> & Rational>
        Parameters:
        iv - root isolating interval for f, with f(left) * f(right) < 0.
        f - univariate polynomial, non-zero.
        g - univariate polynomial, gcd(f,g) == 1.
        Returns:
        sign(g(v)), with v a new interval contained in iv such that g(v) != 0.

      • invariantMagnitudeInterval

        public Interval<C> invariantMagnitudeInterval​(Interval<C> iv,
                                              GenPolynomial<C> f,
                                              GenPolynomial<C> g,
                                              BigRational eps)
        Invariant interval for algebraic number magnitude.
        Parameters:
        iv - root isolating interval for f, with f(left) * f(right) < 0.
        f - univariate polynomial, non-zero.
        g - univariate polynomial, gcd(f,g) == 1.
        eps - length limit for interval length.
        Returns:
        v with v a new interval contained in iv such that |g(a) - g(b)| < eps for a, b in v in iv.

      • realIntervalMagnitude

        public C realIntervalMagnitude​(Interval<C> iv,
                               GenPolynomial<C> f,
                               GenPolynomial<C> g)
        Real algebraic number magnitude.
        Parameters:
        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.
        f - univariate polynomial, non-zero.
        g - univariate polynomial, gcd(f,g) == 1.
        Returns:
        g(iv) .

      • realIntervalMagnitudeInterval

        public Interval<C> realIntervalMagnitudeInterval​(Interval<C> iv,
                                                 GenPolynomial<C> f,
                                                 GenPolynomial<C> g)
        Real algebraic number magnitude.
        Parameters:
        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.
        f - univariate polynomial, non-zero.
        g - univariate polynomial, gcd(f,g) == 1.
        Returns:
        Interval( g(iv.left), g(iv.right) ) .

      • approximateRoots

        public java.util.List<BigDecimal> approximateRoots​(GenPolynomial<C> f,
                                                   BigRational eps)
        Approximate real roots.
        Parameters:
        f - univariate polynomial, non-zero.
        eps - requested interval length.
        Returns:
        a list of decimal approximations d such that |d-v| < eps for all real v with f(v) = 0.

      • isApproximateRoot

        public boolean isApproximateRoot​(BigDecimal x,
                                 GenPolynomial<C> f,
                                 C eps)
        Test if x is an approximate real root.
        Parameters:
        x - approximate real root.
        f - univariate polynomial, non-zero.
        eps - requested interval length.
        Returns:
        true if x is a decimal approximation of a real v with f(v) = 0 with |d-v| < eps, else false.

      • isApproximateRoot

        public boolean isApproximateRoot​(BigDecimal x,
                                 GenPolynomial<BigDecimal> f,
                                 GenPolynomial<BigDecimal> fp,
                                 BigDecimal eps)
        Test if x is an approximate real root.
        Parameters:
        x - approximate real root.
        f - univariate polynomial, non-zero.
        fp - univariate polynomial, non-zero, derivative of f.
        eps - requested interval length.
        Returns:
        true if x is a decimal approximation of a real v with f(v) = 0 with |d-v| < eps, else false.

      • isApproximateRoot

        public boolean isApproximateRoot​(java.util.List<BigDecimal> R,
                                 GenPolynomial<C> f,
                                 BigRational eps)
        Test if each x in R is an approximate real root.
        Parameters:
        R - ist of approximate real roots.
        f - univariate polynomial, non-zero.
        eps - requested interval length.
        Returns:
        true if each x in R is a decimal approximation of a real v with f(v) = 0 with |d-v| < eps, else false.

      • signSequence

        public java.util.List<java.lang.Integer> signSequence​(GenPolynomial<C> f,
                                                      Interval<C> v)
        Thom sign sequence.
        Parameters:
        f - univariate polynomial.
        v - interval for a real root, f(v.left) * f(v.right) < 0.
        Returns:
        (s1, s2, ..., sn) = (sign(f'(v)), .... sign(f(n)(v))) a Thom sign sequence for the real root in v of f.

      • realRootNumber

        public java.lang.Long realRootNumber​(GenPolynomial<C> f,
                                     Interval<C> v)
        Root number.
        Parameters:
        f - univariate polynomial.
        v - interval for a real root, f(v.left) * f(v.right) < 0.
        Returns:
        r the number of this root in the sequence a1 < a2 < ..., < am of all real roots of f