Safe Haskell	None
Language	Haskell2010

Algorithms.Geometry.SoS.Symbolic

Synopsis

data EpsFold i
eps :: i -> EpsFold i
mkEpsFold :: Ord i => [i] -> EpsFold i
hasNoPertubation :: EpsFold i -> Bool
factors :: EpsFold i -> Bag i
suitableBase :: EpsFold i -> Int
data Term i r = Term r (EpsFold i)
term :: r -> i -> Term i r
constantFactor :: Lens' (Term i r) r
data Symbolic i r
constant :: Ord i => r -> Symbolic i r
symbolic :: Ord i => r -> i -> Symbolic i r
perturb :: (Num r, Ord i) => r -> i -> Symbolic i r
toTerms :: Symbolic i r -> [Term i r]
signOf :: (Num r, Eq r) => Symbolic i r -> Maybe Sign

Documentation

data EpsFold i Source #

Instances

Ord i => Eq (EpsFold i) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods (==) :: EpsFold i -> EpsFold i -> Bool # (/=) :: EpsFold i -> EpsFold i -> Bool #
Ord i => Ord (EpsFold i) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods compare :: EpsFold i -> EpsFold i -> Ordering # (<) :: EpsFold i -> EpsFold i -> Bool # (<=) :: EpsFold i -> EpsFold i -> Bool # (>) :: EpsFold i -> EpsFold i -> Bool # (>=) :: EpsFold i -> EpsFold i -> Bool # max :: EpsFold i -> EpsFold i -> EpsFold i # min :: EpsFold i -> EpsFold i -> EpsFold i #
Show i => Show (EpsFold i) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods showsPrec :: Int -> EpsFold i -> ShowS # show :: EpsFold i -> String # showList :: [EpsFold i] -> ShowS #
Ord i => Semigroup (EpsFold i) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods (<>) :: EpsFold i -> EpsFold i -> EpsFold i # sconcat :: NonEmpty (EpsFold i) -> EpsFold i # stimes :: Integral b => b -> EpsFold i -> EpsFold i #
Ord i => Monoid (EpsFold i) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods mempty :: EpsFold i # mappend :: EpsFold i -> EpsFold i -> EpsFold i # mconcat :: [EpsFold i] -> EpsFold i #
(Arbitrary i, Ord i) => Arbitrary (EpsFold i) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods arbitrary :: Gen (EpsFold i) # shrink :: EpsFold i -> [EpsFold i] #

eps :: i -> EpsFold i Source #

Creates the term $\varepsilon(i)$

mkEpsFold :: Ord i => [i] -> EpsFold i Source #

hasNoPertubation :: EpsFold i -> Bool Source #

Test if the epsfold has no pertubation at all (i.e. if it is $\Pi_{\emptyset}$

factors :: EpsFold i -> Bag i Source #

Gets the factors

suitableBase :: EpsFold i -> Int Source #

computes a base d that can be used as:

$ \varepsilon(i) = \varepsilon^{d^i} $

data Term i r Source #

A term 'Term c es' represents a term:

\[ c \Pi_{i \in es} \varepsilon(i) \]

for a constant c and an arbitrarily small value $\varepsilon$, parameterized by i.

Constructors

Term r (EpsFold i)

Instances

Functor (Term i) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods fmap :: (a -> b) -> Term i a -> Term i b # (<$) :: a -> Term i b -> Term i a #
(Ord i, Eq r) => Eq (Term i r) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods (==) :: Term i r -> Term i r -> Bool # (/=) :: Term i r -> Term i r -> Bool #
(Ord i, Ord r, Num r) => Ord (Term i r) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods compare :: Term i r -> Term i r -> Ordering # (<) :: Term i r -> Term i r -> Bool # (<=) :: Term i r -> Term i r -> Bool # (>) :: Term i r -> Term i r -> Bool # (>=) :: Term i r -> Term i r -> Bool # max :: Term i r -> Term i r -> Term i r # min :: Term i r -> Term i r -> Term i r #
(Show i, Show r) => Show (Term i r) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods showsPrec :: Int -> Term i r -> ShowS # show :: Term i r -> String # showList :: [Term i r] -> ShowS #
(Arbitrary r, Arbitrary (EpsFold i), Ord i) => Arbitrary (Term i r) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods arbitrary :: Gen (Term i r) # shrink :: Term i r -> [Term i r] #

term :: r -> i -> Term i r Source #

Creates a singleton term

constantFactor :: Lens' (Term i r) r Source #

Lens to access the constant c in the term.

data Symbolic i r Source #

Represents a Sum of terms, i.e. a value that has the form:

\[ \sum c \Pi_i \varepsilon(i) \]

The terms are represented in order of decreasing significance.

The main idea in this type is that, if symbolic values contains $\varepsilon(i)$ terms we can always order them. That is, two Symbolic terms will be equal only if:

they contain *only* a constant term (that is equal)
they contain the exact same $\varepsilon$-fold.

Instances

Functor (Symbolic i) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods fmap :: (a -> b) -> Symbolic i a -> Symbolic i b # (<$) :: a -> Symbolic i b -> Symbolic i a #
(Ord i, Eq r, Num r) => Eq (Symbolic i r) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods (==) :: Symbolic i r -> Symbolic i r -> Bool # (/=) :: Symbolic i r -> Symbolic i r -> Bool #
(Ord i, Num r, Eq r) => Num (Symbolic i r) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods (+) :: Symbolic i r -> Symbolic i r -> Symbolic i r # (-) :: Symbolic i r -> Symbolic i r -> Symbolic i r # (*) :: Symbolic i r -> Symbolic i r -> Symbolic i r # negate :: Symbolic i r -> Symbolic i r # abs :: Symbolic i r -> Symbolic i r # signum :: Symbolic i r -> Symbolic i r # fromInteger :: Integer -> Symbolic i r #
(Ord i, Ord r, Num r) => Ord (Symbolic i r) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods compare :: Symbolic i r -> Symbolic i r -> Ordering # (<) :: Symbolic i r -> Symbolic i r -> Bool # (<=) :: Symbolic i r -> Symbolic i r -> Bool # (>) :: Symbolic i r -> Symbolic i r -> Bool # (>=) :: Symbolic i r -> Symbolic i r -> Bool # max :: Symbolic i r -> Symbolic i r -> Symbolic i r # min :: Symbolic i r -> Symbolic i r -> Symbolic i r #
(Show i, Show r) => Show (Symbolic i r) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods showsPrec :: Int -> Symbolic i r -> ShowS # show :: Symbolic i r -> String # showList :: [Symbolic i r] -> ShowS #
(Arbitrary r, Ord i, Arbitrary (EpsFold i)) => Arbitrary (Symbolic i r) Source #
Instance details Defined in Algorithms.Geometry.SoS.Symbolic Methods arbitrary :: Gen (Symbolic i r) # shrink :: Symbolic i r -> [Symbolic i r] #

constant :: Ord i => r -> Symbolic i r Source #

Creates a constant symbolic value

symbolic :: Ord i => r -> i -> Symbolic i r Source #

Creates a symbolic vlaue with a single indexed term. If you just need a constant (i.e. non-indexed), use constant

perturb :: (Num r, Ord i) => r -> i -> Symbolic i r Source #

given the value c and the index i, creates the perturbed value $c + \varepsilon(i)$

toTerms :: Symbolic i r -> [Term i r] Source #

Produces a list of terms, in decreasing order of significance

signOf :: (Num r, Eq r) => Symbolic i r -> Maybe Sign Source #

Computing the Sign of an expression. (Nothing represents zero)