numeric-prelude-0.4: An experimental alternative hierarchy of numeric type classes

Portabilityrequires multi-parameter type classes
Stabilityprovisional
Maintainernumericprelude@henning-thielemann.de
Safe HaskellNone

MathObj.LaurentPolynomial

Contents

Description

Polynomials with negative and positive exponents.

Synopsis

Documentation

data T a Source

Polynomial including negative exponents

Constructors

Cons 

Fields

expon :: Int
 
coeffs :: [a]
 

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

Basic Operations

const :: a -> T aSource

(!) :: C a => T a -> Int -> aSource

fromCoeffs :: [a] -> T aSource

fromShiftCoeffs :: Int -> [a] -> T aSource

bounds :: T a -> (Int, Int)Source

shift :: Int -> T a -> T aSource

translate :: Int -> T a -> T aSource

Deprecated: In order to avoid confusion with Polynomial.translate, use shift instead

Show

Additive

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

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

addShiftedMany :: C a => [Int] -> [[a]] -> [a]Source

Add lists of numbers respecting a relative shift between the starts of the lists. The shifts must be non-negative. The list of relative shifts is one element shorter than the list of summands. Infinitely many summands are permitted, provided that runs of zero shifts are all finite.

We could add the lists either with foldl or with foldr, foldl would be straightforward, but more time consuming (quadratic time) whereas foldr is not so obvious but needs only linear time.

(stars denote the coefficients, frames denote what is contained in the interim results) foldl sums this way:

 | | | *******************************
 | | +--------------------------------
 | |          ************************
 | +----------------------------------
 |                        ************
 +------------------------------------

I.e. foldl would use much time find the time differences by successive subtraction 1.

foldr mixes this way:

     +--------------------------------
     | *******************************
     |      +-------------------------
     |      | ************************
     |      |           +-------------
     |      |           | ************

addShifted :: C a => Int -> [a] -> [a] -> [a]Source

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

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

Module

Ring

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

Field.C

div :: (C a, C a) => T a -> T a -> T aSource

Comparisons

equivalent :: (Eq a, C a) => T a -> T a -> BoolSource

Two polynomials may be stored differently. This function checks whether two values of type LaurentPolynomial actually represent the same polynomial.

identical :: Eq a => T a -> T a -> BoolSource

isAbsolute :: C a => T a -> BoolSource

Check whether a Laurent polynomial has only the absolute term, that is, it represents the constant polynomial.

Transformations of arguments

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

p(z) -> p(-z)

reverse :: T a -> T aSource

p(z) -> p(1/z)

adjoint :: C a => T (T a) -> T (T a)Source

p(exp(i·x)) -> conjugate(p(exp(i·x)))

If you interpret (p*) as a linear operator on the space of Laurent polynomials, then (adjoint p *) is the adjoint operator.