com.numericalmethod.suanshu.mathstructure
Interface AbelianGroup<G>

Type Parameters:

G - an Abelian group

All Known Subinterfaces:

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

All Known Implementing Classes:

Basis, BidiagonalMatrix, BorderedHessian, Complex, ComplexMatrix, CovarianceMatrix, CsrSparseMatrix, DenseMatrix, DenseVector, DiagonalMatrix, DokSparseMatrix, DPolynomial, GenericMatrix, GivensMatrix, GoldfeldQuandtTrotter, Gradient, Hessian, HilbertMatrix, ImmutableKroneckerProduct, ImmutableMatrix, ImmutableVector, Inverse, Jacobian, JordanExchange, KroneckerProduct, LilSparseMatrix, LowerTriangularMatrix, MatrixMathImpl, MatrixStorageImpl, MatthewsDavies, PermutationMatrix, Polynomial, Pow, PseudoInverse, Real, RealMatrix, SparseVector, SubMatrixRef, SymmetricMatrix, TriangularMatrix, TridiagonalMatrix, UpperTriangularMatrix

public interface AbelianGroup<G>

This interface represents an Abelian group.

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

G	add(G that) + : G × G → G
G	minus(G that) - : G × G → G - 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 That is, it is the object such as this.add(this.opposite()) == this.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.

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

- 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

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