edu.jas.root
Class ComplexRootsAbstract<C extends RingElem<C> & Rational>

java.lang.Object
  edu.jas.root.ComplexRootsAbstract<C>

Type Parameters:

C - coefficient type.

All Implemented Interfaces:

ComplexRoots<C>

Direct Known Subclasses:

ComplexRootsSturm

public abstract class ComplexRootsAbstract<C extends RingElem<C> & Rational>
extends java.lang.Object
implements ComplexRoots<C>

Complex roots abstract class.

Author:

Heinz Kredel

Field Summary

Squarefree<Complex<C>>

engine Engine for square free decomposition.

Constructor Summary

ComplexRootsAbstract(RingFactory<Complex<C>> cf)
Constructor.

Method Summary

Complex<BigDecimal>	approximateRoot(Rectangle<C> rt, GenPolynomial<Complex<C>> f, C eps) Approximate complex root.
java.util.List<Complex<BigDecimal>>	approximateRoots(GenPolynomial<Complex<C>> a, C eps) List of decimal approximations of complex roots of complex polynomial.
Complex<C>	complexMagnitude(Rectangle<C> rect, GenPolynomial<Complex<C>> f, GenPolynomial<Complex<C>> g, C eps) Complex algebraic number magnitude.
Complex<C>	complexRectangleMagnitude(Rectangle<C> rect, GenPolynomial<Complex<C>> f, GenPolynomial<Complex<C>> g) Complex algebraic number magnitude.
abstract long	complexRootCount(Rectangle<C> rect, GenPolynomial<Complex<C>> a) Complex root count of complex polynomial on rectangle.
Rectangle<C>	complexRootRefinement(Rectangle<C> rect, GenPolynomial<Complex<C>> a, BigRational len) Complex root refinement of complex polynomial a on rectangle.
java.util.List<Rectangle<C>>	complexRoots(GenPolynomial<Complex<C>> a) List of complex roots of complex polynomial.
java.util.List<Rectangle<C>>	complexRoots(GenPolynomial<Complex<C>> a, BigRational len) List of complex roots of complex polynomial.
abstract java.util.List<Rectangle<C>>	complexRoots(Rectangle<C> rect, GenPolynomial<Complex<C>> a) List of complex roots of complex polynomial a on rectangle.
Complex[]	copyOfComplex(Complex[] original, int newLength) Copy the specified array.
Rectangle<C>	invariantMagnitudeRectangle(Rectangle<C> rect, GenPolynomial<Complex<C>> f, GenPolynomial<Complex<C>> g, C eps) Invariant rectangle for algebraic number magnitude.
abstract Rectangle<C>	invariantRectangle(Rectangle<C> rect, GenPolynomial<Complex<C>> f, GenPolynomial<Complex<C>> g) Invariant rectangle for algebraic number.
C	magnitudeBound(Rectangle<C> rect, GenPolynomial<Complex<C>> f) Magnitude bound.
Complex<C>	rootBound(GenPolynomial<Complex<C>> f) Root bound.
java.lang.String	toDecimal(Complex<C> a) Get decimal approximation.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

engine

public final Squarefree<Complex<C extends RingElem<C> & Rational>> engine

Engine for square free decomposition.

Constructor Detail

ComplexRootsAbstract

public ComplexRootsAbstract(RingFactory<Complex<C>> cf)

Constructor.

Parameters:

cf - coefficient factory.

Method Detail

rootBound

public Complex<C> rootBound(GenPolynomial<Complex<C>> f)

Root bound. With f(-M + i M) * f(-M - i M) * f(M - i M) * f(M + i M) != 0.

Specified by:

rootBound in interface ComplexRoots<C extends RingElem<C> & Rational>

Parameters:

f - univariate polynomial.

Returns:

M such that root(f) is contained in the rectangle spanned by M.

magnitudeBound

public C magnitudeBound(Rectangle<C> rect,
                        GenPolynomial<Complex<C>> f)

Magnitude bound.

Parameters:

rect - rectangle.

f - univariate polynomial.

Returns:

B such that |f(c)| < B for c in rect.

complexRootCount

public abstract long complexRootCount(Rectangle<C> rect,
                                      GenPolynomial<Complex<C>> a)
                               throws InvalidBoundaryException

Complex root count of complex polynomial on rectangle.

Specified by:

complexRootCount in interface ComplexRoots<C extends RingElem<C> & Rational>

Parameters:

rect - rectangle.

a - univariate complex polynomial.

Returns:

root count of a in rectangle.

Throws:

InvalidBoundaryException

complexRoots

public abstract java.util.List<Rectangle<C>> complexRoots(Rectangle<C> rect,
                                                          GenPolynomial<Complex<C>> a)
                                                                                 throws InvalidBoundaryException

List of complex roots of complex polynomial a on rectangle.

Specified by:

complexRoots in interface ComplexRoots<C extends RingElem<C> & Rational>

Parameters:

rect - rectangle.

a - univariate squarefree complex polynomial.

Returns:

list of complex roots.

Throws:

InvalidBoundaryException

complexRoots

public java.util.List<Rectangle<C>> complexRoots(GenPolynomial<Complex<C>> a)

List of complex roots of complex polynomial.

Specified by:

complexRoots in interface ComplexRoots<C extends RingElem<C> & Rational>

Parameters:

a - univariate complex polynomial.

Returns:

list of complex roots.

complexRootRefinement

public Rectangle<C> complexRootRefinement(Rectangle<C> rect,
                                          GenPolynomial<Complex<C>> a,
                                          BigRational len)
                                                                 throws InvalidBoundaryException

Complex root refinement of complex polynomial a on rectangle.

Specified by:

complexRootRefinement in interface ComplexRoots<C extends RingElem<C> & Rational>

Parameters:

rect - rectangle containing exactly one complex root.

a - univariate squarefree complex polynomial.

len - rational length for refinement.

Returns:

refined complex root.

Throws:

InvalidBoundaryException

complexRoots

public java.util.List<Rectangle<C>> complexRoots(GenPolynomial<Complex<C>> a,
                                                 BigRational len)

List of complex roots of complex polynomial.

Parameters:

a - univariate complex polynomial.

len - rational length for refinement.

Returns:

list of complex roots to desired precision.

invariantRectangle

public abstract Rectangle<C> invariantRectangle(Rectangle<C> rect,
                                                GenPolynomial<Complex<C>> f,
                                                GenPolynomial<Complex<C>> g)
                                                                       throws InvalidBoundaryException

Invariant rectangle for algebraic number.

Parameters:

rect - root isolating rectangle for f which contains exactly one root.

f - univariate polynomial, non-zero.

g - univariate polynomial, gcd(f,g) == 1.

Returns:

v with v a new rectangle contained in iv such that g(w) != 0 for w in v.

Throws:

InvalidBoundaryException

toDecimal

public java.lang.String toDecimal(Complex<C> a)

Get decimal approximation.

Parameters:

a - complex number.

Returns:

decimal(a).

approximateRoot

public Complex<BigDecimal> approximateRoot(Rectangle<C> rt,
                                           GenPolynomial<Complex<C>> f,
                                           C eps)
                                    throws NoConvergenceException

Approximate complex root.

Parameters:

rt - root isolating rectangle.

f - univariate polynomial, non-zero.

eps - requested interval length.

Returns:

a decimal approximation d such that |d-v| < eps, for f(v) = 0, v in rt.

Throws:

NoConvergenceException

approximateRoots

public java.util.List<Complex<BigDecimal>> approximateRoots(GenPolynomial<Complex<C>> a,
                                                            C eps)

List of decimal approximations of complex roots of complex polynomial.

Parameters:

a - univariate complex polynomial.

eps - length for refinement.

Returns:

list of complex decimal root approximations to desired precision.

copyOfComplex

public Complex[] copyOfComplex(Complex[] original,
                               int newLength)

Copy the specified array.

Parameters:

original - array.

newLength - new array length.

Returns:

copy of this.

invariantMagnitudeRectangle

public Rectangle<C> invariantMagnitudeRectangle(Rectangle<C> rect,
                                                GenPolynomial<Complex<C>> f,
                                                GenPolynomial<Complex<C>> g,
                                                C eps)
                                                                       throws InvalidBoundaryException

Invariant rectangle for algebraic number magnitude.

Parameters:

rect - root isolating rectangle for f which contains exactly one root.

f - univariate polynomial, non-zero.

g - univariate polynomial, gcd(f,g) == 1.

eps - length limit for rectangle length.

Returns:

v with v a new rectangle contained in rect such that |g(a) - g(b)| < eps for a, b in v in rect.

Throws:

InvalidBoundaryException

complexRectangleMagnitude

public Complex<C> complexRectangleMagnitude(Rectangle<C> rect,
                                            GenPolynomial<Complex<C>> f,
                                            GenPolynomial<Complex<C>> g)

Complex algebraic number magnitude.

Parameters:

rect - root isolating rectangle for f which contains exactly one root, with rect such that |g(a) - g(b)| < eps for a, b in rect.

f - univariate polynomial, non-zero.

g - univariate polynomial, gcd(f,g) == 1.

Returns:

g(rect) .

complexMagnitude

public Complex<C> complexMagnitude(Rectangle<C> rect,
                                   GenPolynomial<Complex<C>> f,
                                   GenPolynomial<Complex<C>> g,
                                   C eps)
                                                          throws InvalidBoundaryException

Complex algebraic number magnitude.

Parameters:

rect - root isolating rectangle for f which contains exactly one root, with rect such that |g(a) - g(b)| < eps for a, b in rect.

f - univariate polynomial, non-zero.

g - univariate polynomial, gcd(f,g) == 1.

eps - length limit for rectangle length.

Returns:

g(rect) .

Throws:

InvalidBoundaryException