orbital.math
Interface UnivariatePolynomial

All Superinterfaces:

Arithmetic, Euclidean, Function, Functor, MathFunctor, Normed, Polynomial

public interface UnivariatePolynomial
extends Euclidean, Polynomial, Function

(Univariate) polynomial p in R[X].

Let R be a commutative ring with 1. The polynomial ring over R in one variable X is

R[X] := {Σ_{i in N} a_iXⁱ = (a_i)_{i in N} ¦ a_i=0 p.t. i in N and for all i in N a_i in R} with the convolution as multiplication. It is an associative, graded R-algebra, and as commutative or unital as R. R[X] inherits the properties of being an integrity domain, factorial (a unique factorization domain), Noetherian from R. Additionally, if R is an integrity domain, then R[X]^× = R^×.

The polynomial ring over a field in one variable even is Euclidean.

Author:

André Platzer

See Also:

Polynomial, ValueFactory.polynomial(Arithmetic[]), ValueFactory.polynomial(Object), ValueFactory.asPolynomial(Vector), NumericalAlgorithms.polynomialInterpolation(Matrix)

Nested Class Summary

Nested classes/interfaces inherited from interface orbital.math.functional.Function

Function.Composite

Nested classes/interfaces inherited from interface orbital.logic.functor.Functor

Functor.Specification

Field Summary

Fields inherited from interface orbital.logic.functor.Function

callTypeDeclaration

Method Summary

UnivariatePolynomial	add(UnivariatePolynomial b)
java.lang.Object	apply(java.lang.Object a) Evaluate this polynomial at a.
Integer	degree() Get the degree of this polynomial.
Arithmetic	get(int i) Get the coefficient of Xi.
Arithmetic[]	getCoefficients() Returns an array containing all the coefficients of this polynomial.
java.util.ListIterator	iterator() Returns an iterator over all coefficients (up to degree).
UnivariatePolynomial	modulo(UnivariatePolynomial g)
UnivariatePolynomial	multiply(UnivariatePolynomial b)
UnivariatePolynomial	quotient(UnivariatePolynomial g)
UnivariatePolynomial	subtract(UnivariatePolynomial b)

Methods inherited from interface orbital.math.Euclidean

modulo, quotient

Methods inherited from interface orbital.math.Polynomial

add, degrees, degreeValue, get, indexSet, indices, multiply, rank, subtract

Methods inherited from interface orbital.math.functional.Function

derive, integrate

Methods inherited from interface orbital.logic.functor.Functor

equals, hashCode, toString

Methods inherited from interface orbital.math.Arithmetic

add, divide, equals, inverse, isOne, isZero, minus, multiply, one, power, scale, subtract, toString, zero

Methods inherited from interface orbital.math.Normed

norm

Method Detail

degree

Integer degree()

Get the degree of this polynomial.

This is the Euclidean degree function δ and also the graduation function for polynomials. 0 is an element of undefined or all or none degrees. So for 0 we should return null (or Integer.MIN_VALUE, but this is not recommended).

Specified by:

degree in interface Euclidean

Specified by:

degree in interface Polynomial

Returns:

deg(this) = max {i in N ¦ a_i!=0}

get

Arithmetic get(int i)

Get the coefficient of Xⁱ. Convenience method for Polynomial.get(Arithmetic).

Returns:

a_i if i=<deg(this), or 0 if i>deg(this).

iterator

java.util.ListIterator iterator()

Returns an iterator over all coefficients (up to degree).

Specified by:

iterator in interface Polynomial

Postconditions:

always (RES.succeedes(#next()))

apply

java.lang.Object apply(java.lang.Object a)

Evaluate this polynomial at a. Using the "Einsetzungshomomorphismus".

Specified by:

apply in interface Function

Specified by:

apply in interface Polynomial

Parameters:

a - the index embedding a:S->(E,), encoded as a Function<S,E>, that determines to which element a(s) to map the index s in S.
With S=Nⁿ a can also be encoded as a Vector<E> a in Eⁿ=E^Nn.

Returns:

f(a) = f(X)|_X=a = (f(X) mod (X-a))

add

UnivariatePolynomial add(UnivariatePolynomial b)

subtract

UnivariatePolynomial subtract(UnivariatePolynomial b)

multiply

UnivariatePolynomial multiply(UnivariatePolynomial b)

quotient

UnivariatePolynomial quotient(UnivariatePolynomial g)

modulo

UnivariatePolynomial modulo(UnivariatePolynomial g)

getCoefficients

Arithmetic[] getCoefficients()

Returns an array containing all the coefficients of this polynomial.

Returns:

a new array containing all our coefficients.

See Also:

Object.clone()

Postconditions:

RES[i]==get(i) and RES.length==degree()+1 and RES!=RES