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>, Serializable

Direct Known Subclasses:

ComplexRootsSturm

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

Complex roots abstract class.

See Also:

• Serialized Form

Field Summary

Fields

Modifier and Type	Field	Description
private static final boolean	debug
final Squarefree<Complex<C>>	engine	Engine for square free decomposition.
private static final org.apache.logging.log4j.Logger	logger

Constructor Summary

Constructors

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

Method Summary

Modifier and Type	Method	Description
Complex<BigDecimal>	approximateRoot(Rectangle<C> rt, GenPolynomial<Complex<C>> f, BigRational eps)	Approximate complex root.
List<Complex<BigDecimal>>	approximateRoots(GenPolynomial<Complex<C>> a, BigRational 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, BigRational 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.
List<Rectangle<C>>	complexRoots(GenPolynomial<Complex<C>> a)	List of complex roots of complex polynomial.
List<Rectangle<C>>	complexRoots(GenPolynomial<Complex<C>> a, BigRational len)	List of complex roots of complex polynomial.
abstract 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, BigRational 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.
String	toDecimal(Complex<C> a)	Get decimal approximation.

Methods inherited from class Object

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

Field Details

logger

private static final org.apache.logging.log4j.Logger logger

debug

private static final boolean debug

engine

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

Engine for square free decomposition.

Constructor Details

ComplexRootsAbstract

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

Constructor.

Parameters:

cf - coefficient factory.

Method Details

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 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 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 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 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,
 BigRational 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 List<Complex<BigDecimal>> approximateRoots(GenPolynomial<Complex<C>> a,
 BigRational 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,
 BigRational 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,
 BigRational 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