numeric-prelude-0.4.3: An experimental alternative hierarchy of numeric type classes

Copyright (c) Henning Thielemann 2004-2005 numericprelude@henning-thielemann.de provisional requires multi-parameter type classes None Haskell98

MathObj.RootSet

Contents

Description

Computations on the set of roots of a polynomial. These are represented as the list of their elementar symmetric terms. The difference between a polynomial and the list of elementar symmetric terms is the reversed order and the alternated signs.

Cf. MathObj.PowerSum .

Synopsis

# Documentation

newtype T a Source #

Constructors

 Cons Fieldscoeffs :: [a]

Instances

 Show a => Show (T a) Source # MethodsshowsPrec :: Int -> T a -> ShowS #show :: T a -> String #showList :: [T a] -> ShowS # (C a, C a) => C (T a) Source # Methodszero :: T a Source #(+) :: T a -> T a -> T a Source #(-) :: T a -> T a -> T a Source #negate :: T a -> T a Source # (C a, C a) => C (T a) Source # Methods(*) :: T a -> T a -> T a Source #one :: T a Source #(^) :: T a -> Integer -> T a Source # (C a, C a) => C (T a) Source # Methods(/) :: T a -> T a -> T a Source #recip :: T a -> T a Source #(^-) :: T a -> Integer -> T a Source # (C a, C a) => C (T a) Source # Methodssqrt :: T a -> T a Source #root :: Integer -> T a -> T a Source #(^/) :: T a -> Rational -> T a Source #

# Conversions

lift0 :: [a] -> T a Source #

lift1 :: ([a] -> [a]) -> T a -> T a Source #

lift2 :: ([a] -> [a] -> [a]) -> T a -> T a -> T a Source #

const :: C a => a -> T a Source #

toPolynomial :: T a -> T a Source #

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

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

addRoot :: C a => a -> [a] -> [a] Source #

cf. mulLinearFactor

fromRoots :: C a => [a] -> [a] Source #

liftPowerSum1Gen :: ([a] -> [a]) -> ([a] -> [a]) -> ([a] -> [a]) -> [a] -> [a] Source #

liftPowerSum2Gen :: ([a] -> [a]) -> ([a] -> [a]) -> ([a] -> [a] -> [a]) -> [a] -> [a] -> [a] Source #

liftPowerSum1 :: (C a, C a) => ([a] -> [a]) -> [a] -> [a] Source #

liftPowerSum2 :: (C a, C a) => ([a] -> [a] -> [a]) -> [a] -> [a] -> [a] Source #

liftPowerSumInt1 :: (C a, Eq a, C a) => ([a] -> [a]) -> [a] -> [a] Source #

liftPowerSumInt2 :: (C a, Eq a, C a) => ([a] -> [a] -> [a]) -> [a] -> [a] -> [a] Source #

# Show

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

addInt :: (C a, Eq a, C a) => [a] -> [a] -> [a] Source #

# Ring

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

mulInt :: (C a, Eq a, C a) => [a] -> [a] -> [a] Source #

pow :: (C a, C a) => Integer -> [a] -> [a] Source #

powInt :: (C a, Eq a, C a) => Integer -> [a] -> [a] Source #

# Algebra

approxPolynomial :: C a => Int -> Integer -> a -> (a, T a) Source #

Given an approximation of a root, the degree of the polynomial and maximum value of coefficients, find candidates of polynomials that have approximately this root and show the actual value of the polynomial at the given root approximation.

This algorithm runs easily into a stack overflow, I do not know why. We may also employ a more sophisticated integer relation algorithm, like PSLQ and friends.