Package edu.jas.root

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

java.lang.Object

edu.jas.root.ComplexRootsAbstract<C>

edu.jas.root.ComplexRootsSturm<C>

Type Parameters:

C - coefficient type.

All Implemented Interfaces:

ComplexRoots<C>, java.io.Serializable

public class ComplexRootsSturm<C extends RingElem<C> & Rational>
extends ComplexRootsAbstract<C>

Complex roots implemented by Sturm sequences. Algorithms use exact method derived from Wilf's numeric Routh-Hurwitz method.

Author:

Heinz Kredel

See Also:

Serialized Form

Field Summary

Fields inherited from class edu.jas.root.ComplexRootsAbstract

engine

Constructor Summary

Constructors

Constructor	Description
ComplexRootsSturm(RingFactory<Complex<C>> cf)	Constructor.

Method Summary

Modifier and Type	Method	Description
long	complexRootCount(Rectangle<C> rect, GenPolynomial<Complex<C>> a)	Complex root count of complex polynomial on rectangle.
java.util.List<Rectangle<C>>	complexRoots(Rectangle<C> rect, GenPolynomial<Complex<C>> a)	List of complex roots of complex polynomial a on rectangle.
Rectangle<C>	excludeZero(Rectangle<C> rect, GenPolynomial<Complex<C>> f)	Exclude zero.
long	indexOfCauchy(C a, C b, GenPolynomial<C> f, GenPolynomial<C> g)	Cauchy index of rational function f/g on interval.
long[]	indexOfRouth(C a, C b, GenPolynomial<C> f, GenPolynomial<C> g)	Routh index of complex function f + i g on interval.
Rectangle<C>	invariantRectangle(Rectangle<C> rect, GenPolynomial<Complex<C>> f, GenPolynomial<Complex<C>> g)	Invariant rectangle for algebraic number.
java.util.List<GenPolynomial<C>>	sturmSequence(GenPolynomial<C> f, GenPolynomial<C> g)	Sturm sequence.
long	windingNumber(Rectangle<C> rect, GenPolynomial<Complex<C>> A)	Winding number of complex function A on rectangle.

Methods inherited from class edu.jas.root.ComplexRootsAbstract

approximateRoot, approximateRoots, complexMagnitude, complexRectangleMagnitude, complexRootRefinement, complexRoots, complexRoots, copyOfComplex, invariantMagnitudeRectangle, magnitudeBound, rootBound, toDecimal

Methods inherited from class java.lang.Object

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

Constructor Detail

ComplexRootsSturm

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

Constructor.

Parameters:

cf - coefficient factory.

Method Detail

indexOfCauchy

public long indexOfCauchy(C a,
                          C b,
                          GenPolynomial<C> f,
                          GenPolynomial<C> g)

Cauchy index of rational function f/g on interval.

Parameters:

a - interval bound for I = [a,b].

b - interval bound for I = [a,b].

f - univariate polynomial.

g - univariate polynomial.

Returns:

winding number of f/g in I.

indexOfRouth

public long[] indexOfRouth(C a,
                           C b,
                           GenPolynomial<C> f,
                           GenPolynomial<C> g)

Routh index of complex function f + i g on interval.

Parameters:

a - interval bound for I = [a,b].

b - interval bound for I = [a,b].

f - univariate polynomial.

g - univariate polynomial != 0.

Returns:

index number of f + i g.

sturmSequence

public java.util.List<GenPolynomial<C>> sturmSequence(GenPolynomial<C> f,
                                                      GenPolynomial<C> g)

Sturm sequence.

Parameters:

f - univariate polynomial.

g - univariate polynomial.

Returns:

a Sturm sequence for f and g.

complexRootCount

public 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>

Specified by:

complexRootCount in class ComplexRootsAbstract<C extends RingElem<C> & Rational>

Parameters:

rect - rectangle.

a - univariate complex polynomial.

Returns:

root count of a in rectangle.

Throws:

InvalidBoundaryException

windingNumber

public long windingNumber(Rectangle<C> rect,
                          GenPolynomial<Complex<C>> A)
                   throws InvalidBoundaryException

Winding number of complex function A on rectangle.

Parameters:

rect - rectangle.

A - univariate complex polynomial.

Returns:

winding number of A around rect.

Throws:

InvalidBoundaryException

complexRoots

public 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>

Specified by:

complexRoots in class ComplexRootsAbstract<C extends RingElem<C> & Rational>

Parameters:

rect - rectangle.

a - univariate squarefree complex polynomial.

Returns:

list of complex roots.

Throws:

InvalidBoundaryException

invariantRectangle

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

Invariant rectangle for algebraic number.

Specified by:

invariantRectangle in class ComplexRootsAbstract<C extends RingElem<C> & Rational>

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

Throws:

InvalidBoundaryException

excludeZero

public Rectangle<C> excludeZero(Rectangle<C> rect,
                                GenPolynomial<Complex<C>> f)
                         throws InvalidBoundaryException

Exclude zero. If an axis intersects with the rectangle, it is shrunk to exclude the axis. Not used.

Parameters:

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

Returns:

a new rectangle r such that re(r) < 0 or (re)r > 0 and im(r) < 0 or (im)r > 0.

Throws:

InvalidBoundaryException