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

MathObj.PartialFraction

Description

Implementation of partial fractions. Useful e.g. for fractions of integers and fractions of polynomials.

For the considered ring the prime factorization must be unique.

Synopsis

# Documentation

data T a Source

`Cons z (indexMapFromList [(x0,[y00,y01]), (x1,[y10]), (x2,[y20,y21,y22])])` represents the partial fraction `z + y00x0 + y01x0^2 + y10x1 + y20x2 + y21x2^2 + y22x2^3` The denominators `x0, x1, x2, ...` must be irreducible, but we can't check this in general. It is also not enough to have relatively prime denominators, because when adding two partial fraction representations there might concur denominators that have non-trivial common divisors.

Constructors

 Cons a (Map (ToOrd a) [a])

Instances

 Eq a => Eq (T a) Show a => Show (T a) (C a, C a, C a) => C (T a) (C a, C a) => C (T a)

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

Unchecked construction.

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

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

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

`C` is not really necessary here and only due to invokation of `toFraction`.

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

`C` is not really necessary here and only due to invokation of `%`.

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

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

`fromFactoredFraction x y` computes the partial fraction representation of `y % product x`, where the elements of `x` must be irreducible. The function transforms the factors into their standard form with respect to unit factors.

There are more direct methods for special cases like polynomials over rational numbers where the denominators are linear factors.

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

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

The list of denominators must contain equal elements. Sorry for this hack.

fromValue :: a -> T a Source

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

A normalization step which separates the integer part from the leading fraction of each sub-list.

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

Cf. Number.Positional

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

A normalization step which reduces all elements in sub-lists modulo their denominators. Zeros might be the result, that must be remove with `removeZeros`.

removeZeros :: (C a, C a) => T a -> T a Source

Remove trailing zeros in sub-lists because if lists are converted to fractions by `multiToFraction` we must be sure that the denominator of the (cancelled) fraction is indeed the stored power of the irreducible denominator. Otherwise `mulFrac` leads to wrong results.

zipWith :: C a => (a -> a -> a) -> ([a] -> [a] -> [a]) -> T a -> T a -> T a Source

mulFrac :: C a => T a -> T a -> (a, a) Source

Transforms a product of two partial fractions into a sum of two fractions. The denominators must be at least relatively prime. Since `T` requires irreducible denominators, these are also relatively prime.

Example: `mulFrac (1%6) (1%4)` fails because of the common divisor `2`.

mulFrac' :: C a => T a -> T a -> (T a, T a) Source

mulFracStupid :: C a => T a -> T a -> ((T a, T a), T a) Source

Works always but simply puts the product into the last fraction.

mulFracOverlap :: C a => T a -> T a -> ((T a, T a), T a) Source

Also works if the operands share a non-trivial divisor. However the results are quite arbitrary.

scaleFrac :: (C a, C a) => T a -> T a -> T a Source

Expects an irreducible denominator as associate in standard form.

scaleInt :: (C a, C a) => a -> T a -> T a Source

mul :: (C a, C a) => T a -> T a -> T a Source

mulFast :: (C a, C a) => T a -> T a -> T a Source

# Helper functions for work with Maps with Indexable keys

indexMapMapWithKey :: (a -> b -> c) -> Map (ToOrd a) b -> Map (ToOrd a) c Source

indexMapToList :: Map (ToOrd a) b -> [(a, b)] Source

indexMapFromList :: C a => [(a, b)] -> Map (ToOrd a) b Source

mapApplySplit :: Ord a => a -> (c -> c -> c) -> (b -> c) -> (Map a b -> Map a c) -> Map a b -> Map a c Source

Apply a function on a specific element if it exists, and another function to the rest of the map.