dev.nm.algebra.structure

Interface AbelianGroup<G>

Type Parameters:

G - an Abelian group

All Known Subinterfaces:

BanachSpace<B,F>, Field<F>, GenericMatrix<T,F>, HilbertSpace<H,F>, Matrix, MatrixRing, Ring<R>, SparseMatrix, Vector, VectorSpace<V,F>

All Known Implementing Classes:

Basis, BidiagonalMatrix, BorderedHessian, CauchyPolynomial, ColumnBindMatrix, CombinedVectorByRef, Complex, ComplexMatrix, CongruentMatrix, CorrelationMatrix, CSRSparseMatrix, DenseMatrix, DenseVector, DiagonalMatrix, DiagonalSum, DOKSparseMatrix, DPolynomial, FastKroneckerProduct, GenericFieldMatrix, GivensMatrix, GoldfeldQuandtTrotter, Gradient, Hessian, HilbertMatrix, ImmutableMatrix, ImmutableVector, Inverse, Jacobian, KroneckerProduct, LILSparseMatrix, LowerTriangularMatrix, MAT, MatrixRootByDiagonalization, MatthewsDavies, OuterProduct, PermutationMatrix, Polynomial, PositiveDefiniteMatrixByPositiveDiagonal, PositiveSemiDefiniteMatrixNonNegativeDiagonal, Pow, PseudoInverse, QuadraticMonomial, Real, RealMatrix, ReturnsMatrix, SampleCovariance, ScaledPolynomial, SimilarMatrix, SparseVector, SubMatrixRef, SubVectorRef, SVEC, SymmetricKronecker, SymmetricMatrix, TridiagonalMatrix, UpperTriangularMatrix, VariancebtX

public interface AbelianGroup<G>

An Abelian group is a group with a binary additive operation (+), satisfying the group axioms:

1. closure
2. associativity
3. existence of additive identity
4. existence of additive opposite
5. commutativity of addition

See Also:

Wikipedia: Abelian group

Method Summary

Modifier and Type	Method and Description
G	add(G that) + : G × G → G
G	minus(G that) - : G × G → G The operation "-" is not in the definition of of an additive group but can be deduced.
G	opposite() For each a in G, there exists an element b in G such that a + b = b + a = 0.
G	ZERO() The additive element 0 in the group, such that for all elements a in the group, the equation 0 + a = a + 0 = a holds.

Method Detail

add

G add(G that)

+ : G × G → G

Parameters:

that - the object to be added

Returns:

this + that

minus

G minus(G that)

- : G × G → G

The operation "-" is not in the definition of of an additive group but can be deduced. This function is provided for convenience purpose. It is equivalent to

this.add(that.opposite())

.

Parameters:

that - the object to be subtracted (subtrahend)

Returns:

this - that

opposite

G opposite()

For each a in G, there exists an element b in G such that a + b = b + a = 0. That is, it is the object such as

this.add(this.opposite()) == this.ZERO

Returns:

-this, the additive opposite

See Also:

Wikipedia: Additive inverse

ZERO

G ZERO()

The additive element 0 in the group, such that for all elements a in the group, the equation 0 + a = a + 0 = a holds.

Returns:

0, the additive identity