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

Copyright (c) Henning Thielemann 2004-2006 numericprelude@henning-thielemann.de provisional requires multi-parameter type classes None Haskell98

MathObj.LaurentPolynomial

Description

Polynomials with negative and positive exponents.

Synopsis

# Documentation

data T a Source

Polynomial including negative exponents

Constructors

 Cons Fieldsexpon :: 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 a Source

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

fromCoeffs :: [a] -> T a Source

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

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

shift :: Int -> T a -> T a Source

translate :: Int -> T a -> T a Source

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

# Show

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

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

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 a Source

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

# Ring

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

# Field.C

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

# Comparisons

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

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 -> Bool Source

isAbsolute :: C a => T a -> Bool Source

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 a Source

p(z) -> p(-z)

reverse :: T a -> T a Source

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.