orbital.math
Interface Normed

All Known Subinterfaces:

Arithmetic, BinaryFunction, BinaryFunction.Composite, Complex, Euclidean, Fraction, Function, Function.Composite, Integer, MathFunctor, MathFunctor.Composite, Matrix, Polynomial, Quotient, Rational, Real, Scalar, Symbol, Tensor, UnivariatePolynomial, Vector

public interface Normed

This interface imposes a norm on the objects of each class that implements it. A norm is a measure for lengths.

Let A be an S-module.

||.||:A→R is a norm if ∀a,b∈A, α∈S:
(pdef)	||a||≥0 and ||a||=0 ⇔ a=0	"positive definite"
(Δ)	||a+b|| ≤ ||a|| + ||b||	"triangular inequality"
(hom)	||λa|| = |λ|·||a||	"absolute homogenous"
⇒	Properties
(Δ)	|||a|| - ||b||| ≤ ||a - b||	"inverse triangular inequality"

A norm ||.|| induces a metric d:A×A→R; (a,b)↦d(a,b) := ||a-b||.

In turn, a norm itself can be induced by a scalar product as ||a|| := √⟨a,a⟩. It is induced by a scalar product ⇔ ||a+b||² + ||a-b||² = 2||a||² + 2||b||². This is the parallelogram identity.

Author:

André Platzer

See Also:

Metric, Comparable

Method Summary

Real	norm() Returns a norm ||.|| of this arithmetic object.

Method Detail

norm

Real norm()

Returns a norm ||.|| of this arithmetic object.

Returns:

the norm of this object, or perhaps Double.NaN if it is symbolic and really does not have a numeric norm or a useful symbolic norm.

Preconditions:

true

Postconditions:

RES &ge 0 ∧ (RES=0 ⇔ this=0) ∧ a.add(b).norm(x,y) ≤ a.norm() + b.norm() ∧ a.multiply(λ).norm() == Math.abs(λ) * a.norm() ∧ RES≠null