numeric-prelude-0.4.2: An experimental alternative hierarchy of numeric type classes

MathObj.Polynomial

Description

Polynomials and rational functions in a single indeterminate. Polynomials are represented by a list of coefficients. All non-zero coefficients are listed, but there may be extra '0's at the end.

Usage: Say you have the ring of `Integer` numbers and you want to add a transcendental element `x`, that is an element, which does not allow for simplifications. More precisely, for all positive integer exponents `n` the power `x^n` cannot be rewritten as a sum of powers with smaller exponents. The element `x` must be represented by the polynomial `[0,1]`.

In principle, you can have more than one transcendental element by using polynomials whose coefficients are polynomials as well. However, most algorithms on multi-variate polynomials prefer a different (sparse) representation, where the ordering of elements is not so fixed.

If you want division, you need Number.Ratios of polynomials with coefficients from a Algebra.Field.

You can also compute with an algebraic element, that is an element which satisfies an algebraic equation like `x^3-x-1==0`. Actually, powers of `x` with exponents above `3` can be simplified, since it holds `x^3==x+1`. You can perform these computations with Number.ResidueClass of polynomials, where the divisor is the polynomial equation that determines `x`. If the polynomial is irreducible (in our case `x^3-x-1` cannot be written as a non-trivial product) then the residue classes also allow unrestricted division (except by zero, of course). That is, using residue classes of polynomials you can work with roots of polynomial equations without representing them by radicals (powers with fractional exponents). It is well-known, that roots of polynomials of degree above 4 may not be representable by radicals.

Synopsis

# Documentation

data T a Source

Instances

 Functor T C T C a b => C a (T b) (C a, C a b) => C a (T b) (Eq a, C a) => Eq (T a) Fractional a => Fractional (T a) Num a => Num (T a) Show a => Show (T a) (Arbitrary a, C a) => Arbitrary (T a) (C a, C a) => C (T a) C a => C (T a) C a => C (T a) C a => C (T a) C a => C (T a) (C a, C a) => C (T a) The `C` instance is intensionally built from the `C` structure of the polynomial coefficients. If we would use `Integral.C a` superclass, then the Euclidean algorithm could not determine the greatest common divisor of e.g. `[1,1]` and ``. (C a, C a) => C (T a) (C a, C a) => C (T a)

fromCoeffs :: [a] -> T a Source

coeffs :: T a -> [a] Source

degree :: C a => T a -> Maybe Int Source

showsExpressionPrec :: (Show a, C a, C a) => Int -> String -> T a -> String -> String Source

const :: a -> T a Source

evaluate :: C a => T a -> a -> a Source

evaluateCoeffVector :: C a v => T v -> a -> v Source

Here the coefficients are vectors, for example the coefficients are real and the coefficents are real vectors.

evaluateArgVector :: (C a v, C v) => T a -> v -> v Source

Here the argument is a vector, for example the coefficients are complex numbers or square matrices and the coefficents are reals.

collinear :: (Eq a, C a) => T a -> T a -> Bool Source

integrate :: C a => a -> T a -> T a Source

compose :: C a => T a -> T a -> T a Source

`compose` is the functional composition of polynomials.

It fulfills ` eval x . eval y == eval (compose x y) `

fromRoots :: C a => [a] -> T a Source

reverse :: C a => T a -> T a Source

translate :: C a => a -> T a -> T a Source

dilate :: C a => a -> T a -> T a Source

shrink :: C a => a -> T a -> T a Source