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, CSCSparseMatrix, 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:
  • Method Summary

    Modifier and Type
    Method
    Description
    add(G that)
    + : G × G → G
    minus(G that)
    - : G × G → G
    For each a in G, there exists an element b in G such that a + b = b + a = 0.
    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 Details

    • 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:
    • 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