numeric-prelude-0.1.3.4: 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) (C a, Eq a, Show a, C a) => Fractional (T a) (C a, Eq a, Show a, C 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) (C a, C a) => C (T a) (C a, C a) => C (T a)

fromCoeffs :: [a] -> T aSource

coeffs :: T a -> [a]Source

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

const :: a -> T aSource

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

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

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 -> vSource

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

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

`compose` is the functional composition of polynomials.

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

equal :: (Eq a, C a) => [a] -> [a] -> BoolSource

add :: C a => [a] -> [a] -> [a]Source

sub :: C a => [a] -> [a] -> [a]Source

negate :: C a => [a] -> [a]Source

horner :: C a => a -> [a] -> aSource

Horner's scheme for evaluating a polynomial in a ring.

hornerCoeffVector :: C a v => a -> [v] -> vSource

Horner's scheme for evaluating a polynomial in a module.

hornerArgVector :: (C a v, C v) => v -> [a] -> vSource

shift :: C a => [a] -> [a]Source

Multiply by the variable, used internally.

unShift :: [a] -> [a]Source

mul :: C a => [a] -> [a] -> [a]Source

`mul` is fast if the second argument is a short polynomial, `MathObj.PowerSeries.**` relies on that fact.

scale :: C a => a -> [a] -> [a]Source

divMod :: (C a, C a) => [a] -> [a] -> ([a], [a])Source

divModRev :: (C a, C a) => [a] -> [a] -> ([a], [a])Source

tensorProduct :: C a => [a] -> [a] -> [[a]]Source

tensorProductAlt :: C a => [a] -> [a] -> [[a]]Source

mulShear :: C a => [a] -> [a] -> [a]Source

mulShearTranspose :: C a => [a] -> [a] -> [a]Source

progression :: C a => [a]Source

differentiate :: C a => [a] -> [a]Source

integrate :: C a => a -> [a] -> [a]Source

integrateInt :: (C a, C a) => a -> [a] -> [a]Source

Integrates if it is possible to represent the integrated polynomial in the given ring. Otherwise undefined coefficients occur.

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

alternate :: C a => [a] -> [a]Source

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

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

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

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