orbital.math
Interface Polynomial

All Superinterfaces:

Arithmetic, Function, Functor, MathFunctor, Normed

All Known Subinterfaces:

UnivariatePolynomial

public interface Polynomial
extends Arithmetic, Function

Polynomial p∈R[S] := R^(S).

R^(S) := ⊕_s∈S R is the R-algebra of the magma S over R.

With S=Nⁿ these are the multivariate polynomials in n variables. Of course, the multivariate polynomials are polynomials over polynomial rings:

R[X₀,...,X_n-1] = R[X₀][X₁]...[X_n-1] The importance of polynomial rings comes from the fact that they are the free commutative and associative R-algebras Libasc_R(I) := R^(N(I)) = R[(X_i)_i∈I] and enjoy corresponding universal properties. Although, of course, this is usually restricted to a finite number of variables for the purpose of computation.

Author:

André Platzer

See Also:

ValueFactory.polynomial(Object), ValueFactory.asPolynomial(Tensor), "N. Bourbaki, Algebra III.2.6: Algebra of a magma, a monoid, a group.", "N. Bourbaki, Algebra III.2.7: Free algebras."

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

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

Polynomial	add(Polynomial b)
java.lang.Object	apply(java.lang.Object a) Evaluate this polynomial at a.
Integer	degree() Get the total degree of this polynomial.
int[]	degrees() Returns the multi-degree, i.e., the vector of partial degrees of this polynomial for the individual variables Xi.
int	degreeValue() Get the (int value of the) degree of this polynomial.
Arithmetic	get(Arithmetic i) Get the i-th coefficient.
java.lang.Object	indexSet() Describes the index magma S of our polynomial ring R[S].
java.util.Iterator	indices() Returns an iterator over the (relevant) indices.
java.util.ListIterator	iterator() Returns an iterator over all coefficients (up to degree).
java.util.Iterator	monomials() Returns an iterator over our (relevant) monomials, i.e., pairs of (exponent, coefficient).
Polynomial	multiply(Polynomial b) Multiplies two polynomials.
int	rank() Get the rank of this polynomial, i.e., the number of distinct variables.
Polynomial	subtract(Polynomial b)

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.logic.functor.Functor

equals, hashCode, toString

Method Detail

indexSet

java.lang.Object indexSet()

Describes the index magma S of our polynomial ring R[S].

The index set specifies what form indices of coefficients have. Since there usually is no computer representation of the full index set, this method will only return a description object that can be compared to other index sets via Object.equals(Object).

The precise structure of this object is not defined but, for example, for the polynomial ring R[Nⁿ]=R[X₀,...,X_n-1] in n variables, it may simply be the integer n.

Returns:

a (memento) description of the index set S which is even a magma.

Postconditions:

RES.supports(#equals(Object)) || RES.getClass().isArray()

indices

java.util.Iterator indices()

Returns an iterator over the (relevant) indices.

The order of this iterator is not generally defined, but should be deterministic. Particularly, the iterator may - but need not - be restricted to occurring indices with coefficients ≠0.

Returns:

an iterator over a finite set of indices in S at least containing all indices of coefficients ≠0.

Postconditions:

∀i∈S∖RES get(i)=0

monomials

java.util.Iterator monomials()

Returns an iterator over our (relevant) monomials, i.e., pairs of (exponent, coefficient).

The order of this iterator is not generally defined, but should be deterministic. Particularly, the iterator may - but need not - be restricted to occurring indices with coefficients ≠0.

Returns:

an iterator over a finite set of pairs of exponents in S and coefficients in R at least containing all pairs for coefficients ≠0.

Postconditions:

∀(i,c)∈S×R∖RES get(i)=0

get

Arithmetic get(Arithmetic i)

Get the i-th coefficient. The i-th coefficient α_i is the coefficient of ι(i) for i∈S. For example, if S=Nⁿ that is the coefficient of the monomial X₀^i₀⋅X₁^i₁⋅...⋅X_n-1^i_n-1, which is 0 if |i|>deg(this).

Returns:

α_i.

Preconditions:

i∈S

rank

int rank()

Get the rank of this polynomial, i.e., the number of distinct variables. The rank is the number of different variables occurring in the polynomial, irrespective of their respective degree. In the polynomial ring R[X₀,...,X_n-1], the rank is n.

Throws:

java.lang.UnsupportedOperationException - if R[S] is not a ring with a meaningful finite rank.

See Also:

Tensor.rank()

Postconditions:

RES≥0

degree

Integer degree()

Get the total degree of this polynomial.

For example, if S=Nⁿ then this method returns the total degree deg(this) := max {|i| := ∑_j=0,...,n-1 i_j ¦ i∈Nⁿ ∧ a_i≠0}. Further, deg(0) < 0

Throws:

java.lang.UnsupportedOperationException - if R[S] is not a graded ring with a very meaningful graduation. By providing this option, implementations are not forced to use trivial graduations if no meaningful graduation exists.

Note:

though graduations may have a more general set of degrees than Z, we restrict ourselves to that case, which is by far the most usual one.

degreeValue

int degreeValue()

Get the (int value of the) degree of this polynomial.

See Also:

degreeValue()

degrees

int[] degrees()

Returns the multi-degree, i.e., the vector of partial degrees of this polynomial for the individual variables X_i.

For example, if S=Nⁿ then this method returns the vector of partial degrees deg_p(this) := (max {j ¦ there is an i∈Nⁿ with i_k=j ∧ a_i≠0})_k=0,...,n-1. Further, deg_p(f)_k < 0 iff X_k does not occur in f.

Returns:

an array d containing the partial degrees d[i] of variable X_i.

Throws:

java.lang.UnsupportedOperationException - if R[S] is not a ring with a meaningful finite rank.

See Also:

Tensor.dimensions()

Postconditions:

RES.length == rank()

iterator

java.util.ListIterator iterator()

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

apply

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

Evaluate this polynomial at a. Using the "Einsetzungshomomorphismus" from the universal mapping property.

Specified by:

apply in interface Function

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∈S.
With S=Nⁿ a can also be encoded as a Vector<E> a∈Eⁿ≅E^Nn.

Returns:

f(a) = f((X_k)_k)|_(Xk)k=a.

add

Polynomial add(Polynomial b)

subtract

Polynomial subtract(Polynomial b)

multiply

Polynomial multiply(Polynomial b)

Multiplies two polynomials.

Returns:

(∑_s∈S α_s·ι(s))⋅(∑_s∈S β_s·ι(s)) = ∑_s∈S (∑_t⋅u=s α_t β_u)·ι(s) = ∑_s∈S α_s·(∑_t∈S β_t·ι(t))⋅ι(s) = ∑_s,t∈S (α_s β_t)·(ι(s)⋅ι(t))