Factory.Data.Polynomial

AUTHOR

Dr. Alistair Ward

DESCRIPTION

• Describes a http://en.wikipedia.org/wiki/Univariate polynomial and operations on it.
• http://en.wikipedia.org/wiki/Polynomial.
• http://mathworld.wolfram.com/Polynomial.html.

Types

Type-synonyms

Data-types,

data Polynomial coefficient exponent

• The type of an arbitrary univariate polynomial; actually it's more general, since it permits negative powers (http://en.wikipedia.org/wiki/Laurent_polynomials). It can't describe multivariate polynomials, which would require a list of exponents. Rather than requiring the exponent to implement the type-class Integral, this is implemented at the function-level, as required.
• The structure permits gaps between exponents, in which coefficients are inferred to be zero, thus enabling efficient representation of sparse polynomials.
• CAVEAT: the MonomialList is required to; be ordered by descending exponent (ie. reverse http://en.wikipedia.org/wiki/Monomial_order); have had zero coefficients removed; and to have had like terms merged; so the raw data-constructor isn't exported.

Instances

(Eq coefficient, Eq exponent) => Eq (Polynomial coefficient exponent)

(Show coefficient, Show exponent) => Show (Polynomial coefficient exponent)

(Eq c, Num c, Num e, Ord e) => Ring (Polynomial c e)

Makes Polynomial a Ring, over the field composed from all possible coefficients; http://en.wikipedia.org/wiki/Polynomial_ring.

(Eq c, Fractional c, Num e, Ord e) => QuotientRing (Polynomial c e)

Defines the ability to divide polynomials.

Constants

zero :: Polynomial c e

Constructs a polynomial with zero terms.

one :: (Eq c, Num c, Num e) => Polynomial c e

Constructs a constant monomial, independent of the indeterminate.

Functions

evaluate

Arguments

:: (Num n, Integral e, Show e)

=> n

The indeterminate.

-> Polynomial n e

-> n

The Result.

• Evaluate the polynomial at a specific indeterminate.
• CAVEAT: requires positive exponents; but it wouldn't really be a polynomial otherwise.
• If the polynomial is very sparse, this may be inefficient, since it memoizes the complete sequence of powers up to the polynomial's degree.

getDegree :: Num e => Polynomial c e -> e

• Return the degree (AKA order) of the polynomial.
• http://en.wikipedia.org/wiki/Degree_of_a_polynomial.
• http://mathworld.wolfram.com/PolynomialDegree.html.

getLeadingTerm :: Polynomial c e -> Monomial c e

Return the highest-degree monomial.

lift :: (MonomialList c1 e1 -> MonomialList c2 e2) -> Polynomial c1 e1 -> Polynomial c2 e2

• Transforms the data behind the constructor.
• CAVEAT: similar to fmap, but Polynomial isn't an instance of Functor since we may want to operate on both type-parameters.
• CAVEAT: the caller is required to re-normalise the resulting polynomial depending on the nature of the transformation of the data.

mod'

Arguments

:: Integral c

=> Polynomial c e

-> c

Modulus.

-> Polynomial c e

Reduces all the coefficients using modular arithmetic.

normalise :: (Eq c, Num c, Ord e) => Polynomial c e -> Polynomial c e

Sorts into descending order of exponents, groups like exponents, and calls pruneCoefficients.

raiseModulo

Arguments

:: (Integral c, Integral power, Num e, Ord e, Show power)

=> Polynomial c e

The base.

-> power

The exponent to which the base should be raised.

-> c

The modulus.

-> Polynomial c e

The result.

• Raise a polynomial to the specified positive integral power, but using modulo-arithmetic.
• Whilst one could naively implement this as (x Data.Ring.=^ n) mod m, this will result in arithmetic operatons on unnecessarily big integers.

realCoefficientsToFrac :: (Real r, Fractional f) => Polynomial r e -> Polynomial f e

Convert the type of the coefficients.

terms :: Polynomial c e -> Int

Returns the number of non-zero terms in the polynomial.

Constructors

mkConstant :: (Eq c, Num c, Num e) => c -> Polynomial c e

Constructs an arbitrary zeroeth-degree polynomial, ie. independent of the indeterminate.

mkLinear

Arguments

:: (Eq c, Num c, Num e)

=> c

Gradient.

-> c

Constant.

-> Polynomial c e

Constructs an arbitrary first-degree polynomial.

mkPolynomial :: (Eq c, Num c, Ord e) => MonomialList c e -> Polynomial c e

Smart constructor. Constructs an arbitrary polynomial.

Operators

(*=) :: (Eq c, Num c, Num e) => Polynomial c e -> Monomial c e -> Polynomial c e infixl 7

• Scale-up the specified polynomial by a constant monomial factor.
• http://en.wikipedia.org/wiki/Scalar_multiplication.

Predicates

areCongruentModulo

Arguments

:: (Integral c, Num e, Ord e)

=> Polynomial c e

LHS.

-> Polynomial c e

RHS.

-> c

Modulus.

-> Bool

• True if the two specified polynomials are congruent in modulo-arithmetic.
• http://planetmath.org/encyclopedia/PolynomialCongruence.html.

inAscendingOrder :: Ord e => Polynomial c e -> Bool

True if the exponents of successive terms are in ascending order.

inDescendingOrder :: Ord e => Polynomial c e -> Bool

True if the exponents of successive terms are in descending order.

isMonic :: (Eq c, Num c) => Polynomial c e -> Bool

• True if the leading coefficient is one.
• http://en.wikipedia.org/wiki/Monic_polynomial#Classifications.

isMonomial :: Polynomial c e -> Bool

True if there's exactly one term.

isNormalised :: (Eq c, Num c, Ord e) => Polynomial c e -> Bool

True if no term has a coefficient of zero and the exponents of successive terms are in descending order.

isPolynomial :: Integral e => Polynomial c e -> Bool

True if all exponents are positive integers as required.

isZero :: Polynomial c e -> Bool

True if there are zero terms.