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Language	Haskell98

Factory.Data.Polynomial

Contents

Types
- Type-synonyms
- Data-types,
Constants
Functions

Description

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.

Synopsis

Types

Type-synonyms

Data-types,

data Polynomial coefficient exponent Source

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 Source

Constructs a polynomial with zero terms.

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

Constructs a constant monomial, independent of the indeterminate.

Functions

evaluate Source

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 Source

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 Source

Return the highest-degree monomial.

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

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' Source

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 Source

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

raiseModulo Source

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 Source

Convert the type of the coefficients.

terms :: Polynomial c e -> Int Source

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

Constructors

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

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

mkLinear Source

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 Source

Smart constructor. Constructs an arbitrary polynomial.

Operators

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

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

Predicates

areCongruentModulo Source

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 Source

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

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

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

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

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

isMonomial :: Polynomial c e -> Bool Source

True if there's exactly one term.

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

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 Source

True if all exponents are positive integers as required.

isZero :: Polynomial c e -> Bool Source

True if there are zero terms.