edu.jas.application
Class PolyUtilApp<C extends RingElem<C>>

java.lang.Object
  edu.jas.application.PolyUtilApp<C>

Type Parameters:

C - coefficient type

public class PolyUtilApp<C extends RingElem<C>>
extends java.lang.Object

Polynomial utilities for applications, for example conversion ExpVector to Product or zero dimensional ideal root computation.

Author:

Heinz Kredel

Constructor Summary

PolyUtilApp()

Method Summary

static <C extends RingElem<C> & Rational,D extends GcdRingElem<D> & Rational> IdealWithComplexAlgebraicRoots<C,D>	complexAlgebraicRoots(IdealWithUniv<D> I) Construct complex roots for zero dimensional ideal(G).
static <C extends RingElem<C> & Rational,D extends GcdRingElem<D> & Rational> java.util.List<IdealWithComplexAlgebraicRoots<C,D>>	complexAlgebraicRoots(java.util.List<IdealWithUniv<D>> I) Construct complex roots for zero dimensional ideal(G).
static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<edu.jas.application.IdealWithComplexRoots<D>>	complexRoots(Ideal<D> G, C eps) Construct superset of complex roots for zero dimensional ideal(G).
static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<java.util.List<Complex<BigDecimal>>>	complexRoots(Ideal<D> I, java.util.List<GenPolynomial<D>> univs, C eps) Construct superset of complex roots for zero dimensional ideal(G).
static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<edu.jas.application.IdealWithComplexRoots<D>>	complexRoots(java.util.List<IdealWithUniv<D>> Il, C eps) Construct superset of complex roots for zero dimensional ideal(G).
static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<java.util.List<Complex<BigDecimal>>>	complexRootTuples(Ideal<D> I, C eps) Construct superset of complex roots for zero dimensional ideal(G).
static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<java.util.List<Complex<BigDecimal>>>	complexRootTuples(java.util.List<IdealWithUniv<D>> Il, C eps) Construct superset of complex roots for zero dimensional ideal(G).
static <C extends GcdRingElem<C> & Rational> GenPolynomial<Complex<RealAlgebraicNumber<C>>>	convertToComplexRealCoefficients(GenPolynomialRing<Complex<RealAlgebraicNumber<C>>> pfac, GenPolynomial<Complex<C>> A) Convert to Complex<RealAlgebraicNumber> coefficients.
static <C extends GcdRingElem<C>> AlgebraicNumber<C>	convertToPrimitiveElem(AlgebraicNumberRing<C> cfac, AlgebraicNumber<C> A, AlgebraicNumber<C> a) Convert to primitive element ring.
static <C extends GcdRingElem<C>> AlgebraicNumber<C>	convertToPrimitiveElem(AlgebraicNumberRing<C> cfac, AlgebraicNumber<C> A, AlgebraicNumber<C> B, AlgebraicNumber<AlgebraicNumber<C>> a) Convert to primitive element ring.
static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>>	convertToPrimitiveElem(AlgebraicNumberRing<C> cfac, AlgebraicNumber<C> A, AlgebraicNumber<C> B, GenPolynomial<AlgebraicNumber<AlgebraicNumber<C>>> a) Convert to primitive element ring.
static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>>	convertToPrimitiveElem(AlgebraicNumberRing<C> cfac, AlgebraicNumber<C> A, GenPolynomial<AlgebraicNumber<C>> a) Convert coefficients to primitive element ring.
static <C extends GcdRingElem<C>> GenPolynomial<GenPolynomial<C>>	fromProduct(GenPolynomialRing<GenPolynomial<C>> pfac, GenPolynomial<Product<Residue<C>>> P, int i) From product representation.
static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<GenPolynomial<C>>>	fromProduct(GenPolynomialRing<GenPolynomial<C>> pfac, java.util.List<GenPolynomial<Product<Residue<C>>>> L, int i) From product representation.
static boolean	isComplexRoots(java.util.List<GenPolynomial<Complex<BigDecimal>>> L, java.util.List<java.util.List<Complex<BigDecimal>>> roots, BigDecimal eps) Test for complex roots of zero dimensional ideal(L).
static boolean	isRealRoots(java.util.List<GenPolynomial<BigDecimal>> L, java.util.List<java.util.List<BigDecimal>> roots, BigDecimal eps) Test for real roots of zero dimensional ideal(L).
static <C extends GcdRingElem<C>> PrimitiveElement<C>	primitiveElement(AlgebraicNumberRing<AlgebraicNumber<C>> b) Construct primitive element for double field extension.
static <C extends GcdRingElem<C>> PrimitiveElement<C>	primitiveElement(AlgebraicNumberRing<C> a, AlgebraicNumberRing<C> b) Construct primitive element for double field extension.
static <C extends GcdRingElem<C>> java.util.Map<Ideal<C>,PolynomialList<GenPolynomial<C>>>	productSlice(PolynomialList<Product<Residue<C>>> L) Product slice.
static <C extends GcdRingElem<C>> PolynomialList<GenPolynomial<C>>	productSlice(PolynomialList<Product<Residue<C>>> L, int i) Product slice at i.
static <C extends GcdRingElem<C>> java.lang.String	productSliceToString(java.util.Map<Ideal<C>,PolynomialList<GenPolynomial<C>>> L) Product slice to String.
static <C extends GcdRingElem<C>> java.lang.String	productToString(PolynomialList<Product<Residue<C>>> L) Product slice to String.
static <C extends RingElem<C> & Rational,D extends GcdRingElem<D> & Rational> java.util.List<IdealWithRealAlgebraicRoots<C,D>>	realAlgebraicRoots(Ideal<D> I) Construct exact set of real roots for zero dimensional ideal(G).
static <C extends RingElem<C> & Rational,D extends GcdRingElem<D> & Rational> IdealWithRealAlgebraicRoots<C,D>	realAlgebraicRoots(IdealWithUniv<D> I) Construct real roots for zero dimensional ideal(G).
static <C extends RingElem<C> & Rational,D extends GcdRingElem<D> & Rational> java.util.List<IdealWithRealAlgebraicRoots<C,D>>	realAlgebraicRoots(java.util.List<IdealWithUniv<D>> I) Construct real roots for zero dimensional ideal(G).
static <C extends GcdRingElem<C> & Rational> GenPolynomial<RealAlgebraicNumber<C>>	realAlgFromRealCoefficients(GenPolynomialRing<RealAlgebraicNumber<C>> afac, GenPolynomial<RealAlgebraicNumber<C>> A) Convert to RealAlgebraicNumber coefficients.
static <C extends GcdRingElem<C> & Rational> GenPolynomial<RealAlgebraicNumber<C>>	realFromRealAlgCoefficients(GenPolynomialRing<RealAlgebraicNumber<C>> rfac, GenPolynomial<RealAlgebraicNumber<C>> A) Convert to RealAlgebraicNumber coefficients.
static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<IdealWithRealRoots<D>>	realRoots(Ideal<D> G, C eps) Construct superset of real roots for zero dimensional ideal(G).
static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<java.util.List<BigDecimal>>	realRoots(Ideal<D> I, java.util.List<GenPolynomial<D>> univs, C eps) Construct superset of real roots for zero dimensional ideal(G).
static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<IdealWithRealRoots<D>>	realRoots(java.util.List<IdealWithUniv<D>> Il, C eps) Construct superset of real roots for zero dimensional ideal(G).
static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<java.util.List<BigDecimal>>	realRootTuples(Ideal<D> I, C eps) Construct superset of real roots for zero dimensional ideal(G).
static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<java.util.List<BigDecimal>>	realRootTuples(java.util.List<IdealWithUniv<D>> Il, C eps) Construct superset of real roots for zero dimensional ideal(G).
static <C extends GcdRingElem<C>> GenPolynomial<Product<Residue<C>>>	toProductRes(GenPolynomialRing<Product<Residue<C>>> pfac, GenPolynomial<GenPolynomial<C>> A) Product representation.
static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<Product<Residue<C>>>>	toProductRes(GenPolynomialRing<Product<Residue<C>>> pfac, java.util.List<GenPolynomial<GenPolynomial<C>>> L) Product representation.
static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<Product<Residue<C>>>>	toProductRes(java.util.List<ColoredSystem<C>> CS) Product residue representation.
static <C extends GcdRingElem<C>> Product<Residue<C>>	toProductRes(ProductRing<Residue<C>> pfac, GenPolynomial<C> c) Product representation.
static <C extends GcdRingElem<C>> GenPolynomial<Residue<C>>	toResidue(GenPolynomialRing<Residue<C>> pfac, GenPolynomial<GenPolynomial<C>> A) Residue coefficient representation.
static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<Residue<C>>>	toResidue(GenPolynomialRing<Residue<C>> pfac, java.util.List<GenPolynomial<GenPolynomial<C>>> L) Residue coefficient representation.

Methods inherited from class java.lang.Object

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

Constructor Detail

PolyUtilApp

public PolyUtilApp()

Method Detail

toProductRes

public static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<Product<Residue<C>>>> toProductRes(GenPolynomialRing<Product<Residue<C>>> pfac,
                                                                                                         java.util.List<GenPolynomial<GenPolynomial<C>>> L)

Product representation.

Type Parameters:

C - coefficient type.

Parameters:

pfac - polynomial ring factory.

L - list of polynomials to be represented.

Returns:

Product represenation of L in the polynomial ring pfac.

toProductRes

public static <C extends GcdRingElem<C>> GenPolynomial<Product<Residue<C>>> toProductRes(GenPolynomialRing<Product<Residue<C>>> pfac,
                                                                                         GenPolynomial<GenPolynomial<C>> A)

Product representation.

Type Parameters:

C - coefficient type.

Parameters:

pfac - polynomial ring factory.

A - polynomial to be represented.

Returns:

Product represenation of A in the polynomial ring pfac.

toProductRes

public static <C extends GcdRingElem<C>> Product<Residue<C>> toProductRes(ProductRing<Residue<C>> pfac,
                                                                          GenPolynomial<C> c)

Product representation.

Type Parameters:

C - coefficient type.

Parameters:

pfac - product ring factory.

c - coefficient to be represented.

Returns:

Product represenation of c in the ring pfac.

toProductRes

public static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<Product<Residue<C>>>> toProductRes(java.util.List<ColoredSystem<C>> CS)

Product residue representation.

Type Parameters:

C - coefficient type.

Parameters:

CS - list of ColoredSystems from comprehensive GB system.

Returns:

Product residue represenation of CS.

toResidue

public static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<Residue<C>>> toResidue(GenPolynomialRing<Residue<C>> pfac,
                                                                                             java.util.List<GenPolynomial<GenPolynomial<C>>> L)

Residue coefficient representation.

Parameters:

pfac - polynomial ring factory.

L - list of polynomials to be represented.

Returns:

Represenation of L in the polynomial ring pfac.

toResidue

public static <C extends GcdRingElem<C>> GenPolynomial<Residue<C>> toResidue(GenPolynomialRing<Residue<C>> pfac,
                                                                             GenPolynomial<GenPolynomial<C>> A)

Residue coefficient representation.

Parameters:

pfac - polynomial ring factory.

A - polynomial to be represented.

Returns:

Represenation of A in the polynomial ring pfac.

productSlice

public static <C extends GcdRingElem<C>> java.util.Map<Ideal<C>,PolynomialList<GenPolynomial<C>>> productSlice(PolynomialList<Product<Residue<C>>> L)

Product slice.

Type Parameters:

C - coefficient type.

Parameters:

L - list of polynomials with product coefficients.

Returns:

Slices represenation of L.

productSlice

public static <C extends GcdRingElem<C>> PolynomialList<GenPolynomial<C>> productSlice(PolynomialList<Product<Residue<C>>> L,
                                                                                       int i)

Product slice at i.

Type Parameters:

C - coefficient type.

Parameters:

L - list of polynomials with product coeffients.

i - index of slice.

Returns:

Slice of of L at i.

fromProduct

public static <C extends GcdRingElem<C>> java.util.List<GenPolynomial<GenPolynomial<C>>> fromProduct(GenPolynomialRing<GenPolynomial<C>> pfac,
                                                                                                     java.util.List<GenPolynomial<Product<Residue<C>>>> L,
                                                                                                     int i)

From product representation.

Type Parameters:

C - coefficient type.

Parameters:

pfac - polynomial ring factory.

L - list of polynomials to be converted from product representation.

i - index of product representation to be taken.

Returns:

Represenation of i-slice of L in the polynomial ring pfac.

fromProduct

public static <C extends GcdRingElem<C>> GenPolynomial<GenPolynomial<C>> fromProduct(GenPolynomialRing<GenPolynomial<C>> pfac,
                                                                                     GenPolynomial<Product<Residue<C>>> P,
                                                                                     int i)

From product representation.

Type Parameters:

C - coefficient type.

Parameters:

pfac - polynomial ring factory.

P - polynomial to be converted from product representation.

i - index of product representation to be taken.

Returns:

Represenation of i-slice of P in the polynomial ring pfac.

productSliceToString

public static <C extends GcdRingElem<C>> java.lang.String productSliceToString(java.util.Map<Ideal<C>,PolynomialList<GenPolynomial<C>>> L)

Product slice to String.

Type Parameters:

C - coefficient type.

Parameters:

L - list of polynomials with to be represented.

Returns:

Product represenation of L in the polynomial ring pfac.

productToString

public static <C extends GcdRingElem<C>> java.lang.String productToString(PolynomialList<Product<Residue<C>>> L)

Product slice to String.

Type Parameters:

C - coefficient type.

Parameters:

L - list of polynomials with product coefficients.

Returns:

string represenation of slices of L.

complexRootTuples

public static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<java.util.List<Complex<BigDecimal>>> complexRootTuples(Ideal<D> I,
                                                                                                                                               C eps)

Construct superset of complex roots for zero dimensional ideal(G).

Parameters:

I - zero dimensional ideal.

eps - desired precision.

Returns:

list of coordinates of complex roots for ideal(G)

complexRoots

public static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<java.util.List<Complex<BigDecimal>>> complexRoots(Ideal<D> I,
                                                                                                                                          java.util.List<GenPolynomial<D>> univs,
                                                                                                                                          C eps)

Construct superset of complex roots for zero dimensional ideal(G).

Parameters:

I - zero dimensional ideal.

univs - list of univariate polynomials.

eps - desired precision.

Returns:

list of coordinates of complex roots for ideal(G)

complexRootTuples

public static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<java.util.List<Complex<BigDecimal>>> complexRootTuples(java.util.List<IdealWithUniv<D>> Il,
                                                                                                                                               C eps)

Construct superset of complex roots for zero dimensional ideal(G).

Parameters:

Il - list of zero dimensional ideals with univariate polynomials.

eps - desired precision.

Returns:

list of coordinates of complex roots for ideal(cap_i(G_i))

complexRoots

public static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<edu.jas.application.IdealWithComplexRoots<D>> complexRoots(java.util.List<IdealWithUniv<D>> Il,
                                                                                                                                                   C eps)

Construct superset of complex roots for zero dimensional ideal(G).

Parameters:

Il - list of zero dimensional ideals with univariate polynomials.

eps - desired precision.

Returns:

list of ideals with coordinates of complex roots for ideal(cap_i(G_i))

complexRoots

public static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<edu.jas.application.IdealWithComplexRoots<D>> complexRoots(Ideal<D> G,
                                                                                                                                                   C eps)

Construct superset of complex roots for zero dimensional ideal(G).

Parameters:

G - list of polynomials of a of zero dimensional ideal.

eps - desired precision.

Returns:

list of ideals with coordinates of complex roots for ideal(G)

realRootTuples

public static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<java.util.List<BigDecimal>> realRootTuples(Ideal<D> I,
                                                                                                                                   C eps)

Construct superset of real roots for zero dimensional ideal(G).

Parameters:

I - zero dimensional ideal.

eps - desired precision.

Returns:

list of coordinates of real roots for ideal(G)

realRoots

public static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<java.util.List<BigDecimal>> realRoots(Ideal<D> I,
                                                                                                                              java.util.List<GenPolynomial<D>> univs,
                                                                                                                              C eps)

Construct superset of real roots for zero dimensional ideal(G).

Parameters:

I - zero dimensional ideal.

univs - list of univariate polynomials.

eps - desired precision.

Returns:

list of coordinates of real roots for ideal(G)

realRootTuples

public static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<java.util.List<BigDecimal>> realRootTuples(java.util.List<IdealWithUniv<D>> Il,
                                                                                                                                   C eps)

Construct superset of real roots for zero dimensional ideal(G).

Parameters:

Il - list of zero dimensional ideals with univariate polynomials.

eps - desired precision.

Returns:

list of coordinates of real roots for ideal(cap_i(G_i))

realRoots

public static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<IdealWithRealRoots<D>> realRoots(java.util.List<IdealWithUniv<D>> Il,
                                                                                                                         C eps)

Construct superset of real roots for zero dimensional ideal(G).

Parameters:

Il - list of zero dimensional ideals with univariate polynomials.

eps - desired precision.

Returns:

list of ideals with coordinates of real roots for ideal(cap_i(G_i))

realRoots

public static <C extends RingElem<C> & Rational,D extends GcdRingElem<D>> java.util.List<IdealWithRealRoots<D>> realRoots(Ideal<D> G,
                                                                                                                         C eps)

Construct superset of real roots for zero dimensional ideal(G).

Parameters:

G - list of polynomials of a of zero dimensional ideal.

eps - desired precision.

Returns:

list of ideals with coordinates of real roots for ideal(G)

isRealRoots

public static boolean isRealRoots(java.util.List<GenPolynomial<BigDecimal>> L,
                                  java.util.List<java.util.List<BigDecimal>> roots,
                                  BigDecimal eps)

Test for real roots of zero dimensional ideal(L).

Parameters:

L - list of polynomials.

roots - list of real roots for ideal(G).

eps - desired precision.

Returns:

true if root is a list of coordinates of real roots for ideal(L)

isComplexRoots

public static boolean isComplexRoots(java.util.List<GenPolynomial<Complex<BigDecimal>>> L,
                                     java.util.List<java.util.List<Complex<BigDecimal>>> roots,
                                     BigDecimal eps)

Test for complex roots of zero dimensional ideal(L).

Parameters:

L - list of polynomials.

roots - list of real roots for ideal(G).

eps - desired precision.

Returns:

true if root is a list of coordinates of complex roots for ideal(L)

realAlgebraicRoots

public static <C extends RingElem<C> & Rational,D extends GcdRingElem<D> & Rational> IdealWithRealAlgebraicRoots<C,D> realAlgebraicRoots(IdealWithUniv<D> I)

Construct real roots for zero dimensional ideal(G).

Parameters:

I - zero dimensional ideal with univariate irreducible polynomials and bi-variate polynomials.

Returns:

real algebraic roots for ideal(G)

realAlgebraicRoots

public static <C extends RingElem<C> & Rational,D extends GcdRingElem<D> & Rational> java.util.List<IdealWithRealAlgebraicRoots<C,D>> realAlgebraicRoots(java.util.List<IdealWithUniv<D>> I)

Construct real roots for zero dimensional ideal(G).

Parameters:

I - list of zero dimensional ideal with univariate irreducible polynomials and bi-variate polynomials.

Returns:

list of real algebraic roots for all ideal(I_i)

complexAlgebraicRoots

public static <C extends RingElem<C> & Rational,D extends GcdRingElem<D> & Rational> IdealWithComplexAlgebraicRoots<C,D> complexAlgebraicRoots(IdealWithUniv<D> I)

Construct complex roots for zero dimensional ideal(G).

Parameters:

I - zero dimensional ideal with univariate irreducible polynomials and bi-variate polynomials.

Returns:

complex algebraic roots for ideal(G)

complexAlgebraicRoots

public static <C extends RingElem<C> & Rational,D extends GcdRingElem<D> & Rational> java.util.List<IdealWithComplexAlgebraicRoots<C,D>> complexAlgebraicRoots(java.util.List<IdealWithUniv<D>> I)

Construct complex roots for zero dimensional ideal(G).

Parameters:

I - list of zero dimensional ideal with univariate irreducible polynomials and bi-variate polynomials.

Returns:

list of complex algebraic roots for ideal(G)

realAlgebraicRoots

public static <C extends RingElem<C> & Rational,D extends GcdRingElem<D> & Rational> java.util.List<IdealWithRealAlgebraicRoots<C,D>> realAlgebraicRoots(Ideal<D> I)

Construct exact set of real roots for zero dimensional ideal(G).

Parameters:

I - zero dimensional ideal.

Returns:

list of coordinates of real roots for ideal(G)

primitiveElement

public static <C extends GcdRingElem<C>> PrimitiveElement<C> primitiveElement(AlgebraicNumberRing<C> a,
                                                                              AlgebraicNumberRing<C> b)

Construct primitive element for double field extension.

Parameters:

a - algebraic number ring with squarefree monic minimal polynomial

b - algebraic number ring with squarefree monic minimal polynomial

Returns:

primitive element container with algebraic number ring c, with Q(c) = Q(a,b)

convertToPrimitiveElem

public static <C extends GcdRingElem<C>> AlgebraicNumber<C> convertToPrimitiveElem(AlgebraicNumberRing<C> cfac,
                                                                                   AlgebraicNumber<C> A,
                                                                                   AlgebraicNumber<C> a)

Convert to primitive element ring.

Parameters:

cfac - primitive element ring.

A - algebraic number representing the generating element of a in the new ring.

a - algebraic number to convert.

Returns:

a converted to the primitive element ring

convertToPrimitiveElem

public static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>> convertToPrimitiveElem(AlgebraicNumberRing<C> cfac,
                                                                                                  AlgebraicNumber<C> A,
                                                                                                  GenPolynomial<AlgebraicNumber<C>> a)

Convert coefficients to primitive element ring.

Parameters:

cfac - primitive element ring.

A - algebraic number representing the generating element of a in the new ring.

a - polynomial with coefficients algebraic number to convert.

Returns:

a with coefficients converted to the primitive element ring

convertToPrimitiveElem

public static <C extends GcdRingElem<C>> AlgebraicNumber<C> convertToPrimitiveElem(AlgebraicNumberRing<C> cfac,
                                                                                   AlgebraicNumber<C> A,
                                                                                   AlgebraicNumber<C> B,
                                                                                   AlgebraicNumber<AlgebraicNumber<C>> a)

Convert to primitive element ring.

Parameters:

cfac - primitive element ring.

A - algebraic number representing the generating element of a in the new ring.

a - recursive algebraic number to convert.

Returns:

a converted to the primitive element ring

primitiveElement

public static <C extends GcdRingElem<C>> PrimitiveElement<C> primitiveElement(AlgebraicNumberRing<AlgebraicNumber<C>> b)

Construct primitive element for double field extension.

Parameters:

b - algebraic number ring with squarefree monic minimal polynomial over Q(a)

Returns:

primitive element container with algebraic number ring c, with Q(c) = Q(a)(b)

convertToPrimitiveElem

public static <C extends GcdRingElem<C>> GenPolynomial<AlgebraicNumber<C>> convertToPrimitiveElem(AlgebraicNumberRing<C> cfac,
                                                                                                  AlgebraicNumber<C> A,
                                                                                                  AlgebraicNumber<C> B,
                                                                                                  GenPolynomial<AlgebraicNumber<AlgebraicNumber<C>>> a)

Convert to primitive element ring.

Parameters:

cfac - primitive element ring.

A - algebraic number representing the generating element of a in the new ring.

a - polynomial with recursive algebraic number coefficients to convert.

Returns:

a converted to the primitive element ring

realAlgFromRealCoefficients

public static <C extends GcdRingElem<C> & Rational> GenPolynomial<RealAlgebraicNumber<C>> realAlgFromRealCoefficients(GenPolynomialRing<RealAlgebraicNumber<C>> afac,
                                                                                                                     GenPolynomial<RealAlgebraicNumber<C>> A)

Convert to RealAlgebraicNumber coefficients. Represent as polynomial with RealAlgebraicNumber coefficients from package

 edu.jas.root
 

.

Parameters:

afac - result polynomial factory.

A - polynomial with RealAlgebraicNumber<C> coefficients to be converted.

Returns:

polynomial with RealAlgebraicNumber<C> coefficients.

realFromRealAlgCoefficients

public static <C extends GcdRingElem<C> & Rational> GenPolynomial<RealAlgebraicNumber<C>> realFromRealAlgCoefficients(GenPolynomialRing<RealAlgebraicNumber<C>> rfac,
                                                                                                                     GenPolynomial<RealAlgebraicNumber<C>> A)

Convert to RealAlgebraicNumber coefficients. Represent as polynomial with RealAlgebraicNumber coefficients from package

 edu.jas.application
 

.

Parameters:

rfac - result polynomial factory.

A - polynomial with RealAlgebraicNumber<C> coefficients to be converted.

Returns:

polynomial with RealAlgebraicNumber<C> coefficients.

convertToComplexRealCoefficients

public static <C extends GcdRingElem<C> & Rational> GenPolynomial<Complex<RealAlgebraicNumber<C>>> convertToComplexRealCoefficients(GenPolynomialRing<Complex<RealAlgebraicNumber<C>>> pfac,
                                                                                                                                   GenPolynomial<Complex<C>> A)

Convert to Complex<RealAlgebraicNumber> coefficients. Represent as polynomial with Complex<RealAlgebraicNumber> coefficients, C is e.g. BigRational.

Parameters:

pfac - result polynomial factory.

A - polynomial with Complex coefficients to be converted.

Returns:

polynomial with Complex<RealAlgebraicNumber> coefficients.