Copyright	(c) Henning Thielemann 2004-2005
Maintainer	numericprelude@henning-thielemann.de
Stability	provisional
Portability	requires multi-parameter type classes
Safe Haskell	None
Language	Haskell98

MathObj.RootSet

Contents

Conversions
Show
Additive
Ring
Field.C
Algebra

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

newtype T a = Cons {
- coeffs :: [a]
}
lift0 :: [a] -> T a
lift1 :: ([a] -> [a]) -> T a -> T a
lift2 :: ([a] -> [a] -> [a]) -> T a -> T a -> T a
const :: C a => a -> T a
toPolynomial :: T a -> T a
fromPolynomial :: T a -> T a
toPowerSums :: (C a, C a) => [a] -> [a]
fromPowerSums :: (C a, C a) => [a] -> [a]
addRoot :: C a => a -> [a] -> [a]
fromRoots :: C a => [a] -> [a]
liftPowerSum1Gen :: ([a] -> [a]) -> ([a] -> [a]) -> ([a] -> [a]) -> [a] -> [a]
liftPowerSum2Gen :: ([a] -> [a]) -> ([a] -> [a]) -> ([a] -> [a] -> [a]) -> [a] -> [a] -> [a]
liftPowerSum1 :: (C a, C a) => ([a] -> [a]) -> [a] -> [a]
liftPowerSum2 :: (C a, C a) => ([a] -> [a] -> [a]) -> [a] -> [a] -> [a]
liftPowerSumInt1 :: (C a, Eq a, C a) => ([a] -> [a]) -> [a] -> [a]
liftPowerSumInt2 :: (C a, Eq a, C a) => ([a] -> [a] -> [a]) -> [a] -> [a] -> [a]
appPrec :: Int
add :: (C a, C a) => [a] -> [a] -> [a]
addInt :: (C a, Eq a, C a) => [a] -> [a] -> [a]
mul :: (C a, C a) => [a] -> [a] -> [a]
mulInt :: (C a, Eq a, C a) => [a] -> [a] -> [a]
pow :: (C a, C a) => Integer -> [a] -> [a]
powInt :: (C a, Eq a, C a) => Integer -> [a] -> [a]
approxPolynomial :: C a => Int -> Integer -> a -> (a, T a)

Documentation

newtype T a Source #

Constructors

Cons
Fields coeffs :: [a]

Instances

Show a => Show (T a) Source #
Instance details Defined in MathObj.RootSet Methods showsPrec :: Int -> T a -> ShowS # show :: T a -> String # showList :: [T a] -> ShowS #
(C a, C a) => C (T a) Source #
Instance details Defined in MathObj.RootSet Methods zero :: 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 #
Instance details Defined in MathObj.RootSet Methods (*) :: T a -> T a -> T a Source # one :: T a Source # fromInteger :: Integer -> T a Source # (^) :: T a -> Integer -> T a Source #
(C a, C a) => C (T a) Source #
Instance details Defined in MathObj.RootSet Methods (/) :: T a -> T a -> T a Source # recip :: T a -> T a Source # fromRational' :: Rational -> T a Source # (^-) :: T a -> Integer -> T a Source #
(C a, C a) => C (T a) Source #
Instance details Defined in MathObj.RootSet Methods sqrt :: 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 #

fromPolynomial :: 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

appPrec :: Int Source #

Additive

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 #

Field.C

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.