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MathObj.LaurentPolynomial

Contents

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 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

# Show

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

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

# 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.