NumHask.Data.Complex

Documentation

data Complex a

Complex numbers are an algebraic type.

For a complex number z, abs z is a number with the magnitude of z, but oriented in the positive real direction, whereas sign z has the phase of z, but unit magnitude.

The Foldable and Traversable instances traverse the real part first.

Constructors

!a :+ !a infix 6

forms a complex number from its real and imaginary rectangular components.

Instances

Functor Complex

Methods

fmap :: (a -> b) -> Complex a -> Complex b

(<$) :: a -> Complex b -> Complex a

Foldable Complex

Methods

fold :: Monoid m => Complex m -> m

foldMap :: Monoid m => (a -> m) -> Complex a -> m

foldr :: (a -> b -> b) -> b -> Complex a -> b

foldr' :: (a -> b -> b) -> b -> Complex a -> b

foldl :: (b -> a -> b) -> b -> Complex a -> b

foldl' :: (b -> a -> b) -> b -> Complex a -> b

foldr1 :: (a -> a -> a) -> Complex a -> a

foldl1 :: (a -> a -> a) -> Complex a -> a

toList :: Complex a -> [a]

null :: Complex a -> Bool

length :: Complex a -> Int

elem :: Eq a => a -> Complex a -> Bool

maximum :: Ord a => Complex a -> a

minimum :: Ord a => Complex a -> a

sum :: Num a => Complex a -> a

product :: Num a => Complex a -> a

Traversable Complex

Methods

traverse :: Applicative f => (a -> f b) -> Complex a -> f (Complex b)

sequenceA :: Applicative f => Complex (f a) -> f (Complex a)

mapM :: Monad m => (a -> m b) -> Complex a -> m (Complex b)

sequence :: Monad m => Complex (m a) -> m (Complex a)

Eq a => Eq (Complex a)

Methods

(==) :: Complex a -> Complex a -> Bool

(/=) :: Complex a -> Complex a -> Bool

Data a => Data (Complex a)

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Complex a -> c (Complex a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Complex a)

toConstr :: Complex a -> Constr

dataTypeOf :: Complex a -> DataType

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c (Complex a))

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Complex a))

gmapT :: (forall b. Data b => b -> b) -> Complex a -> Complex a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Complex a -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Complex a -> r

gmapQ :: (forall d. Data d => d -> u) -> Complex a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Complex a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Complex a -> m (Complex a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Complex a -> m (Complex a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Complex a -> m (Complex a)

Read a => Read (Complex a)

Methods

readsPrec :: Int -> ReadS (Complex a)

readList :: ReadS [Complex a]

readPrec :: ReadPrec (Complex a)

readListPrec :: ReadPrec [Complex a]

Show a => Show (Complex a)

Methods

showsPrec :: Int -> Complex a -> ShowS

show :: Complex a -> String

showList :: [Complex a] -> ShowS

Generic (Complex a)

Associated Types

type Rep (Complex a) :: * -> *

Methods

from :: Complex a -> Rep (Complex a) x

to :: Rep (Complex a) x -> Complex a

AdditiveGroup a => AdditiveGroup (Complex a)

Methods

(-) :: Complex a -> Complex a -> Complex a

Additive a => Additive (Complex a)

Methods

(+) :: Complex a -> Complex a -> Complex a

AdditiveInvertible a => AdditiveInvertible (Complex a)

Methods

negate :: Complex a -> Complex a

AdditiveCommutative a => AdditiveCommutative (Complex a)

AdditiveAssociative a => AdditiveAssociative (Complex a)

AdditiveUnital a => AdditiveUnital (Complex a)

Methods

zero :: Complex a

AdditiveMagma a => AdditiveMagma (Complex a)

Methods

plus :: Complex a -> Complex a -> Complex a

(MultiplicativeUnital a, MultiplicativeAssociative a, MultiplicativeInvertible a, AdditiveGroup a) => MultiplicativeGroup (Complex a)

Methods

(/) :: Complex a -> Complex a -> Complex a

(MultiplicativeUnital a, MultiplicativeAssociative a, AdditiveGroup a) => Multiplicative (Complex a)

Methods

(*) :: Complex a -> Complex a -> Complex a

(AdditiveGroup a, MultiplicativeInvertible a) => MultiplicativeInvertible (Complex a)

Methods

recip :: Complex a -> Complex a

(MultiplicativeMagma a, AdditiveGroup a) => MultiplicativeCommutative (Complex a)

(AdditiveGroup a, MultiplicativeAssociative a) => MultiplicativeAssociative (Complex a)

(AdditiveUnital a, AdditiveGroup a, MultiplicativeUnital a) => MultiplicativeUnital (Complex a)

Methods

one :: Complex a

(MultiplicativeMagma a, AdditiveGroup a) => MultiplicativeMagma (Complex a)

Methods

times :: Complex a -> Complex a -> Complex a

(Distribution a, AdditiveGroup a) => Distribution (Complex a)

(Semiring a, AdditiveGroup a) => InvolutiveRing (Complex a)

Methods

adj :: Complex a -> Complex a

(MultiplicativeAssociative a, MultiplicativeUnital a, AdditiveGroup a, Semiring a) => CRing (Complex a)

(Semiring a, AdditiveGroup a) => Ring (Complex a)

(Semiring a, AdditiveGroup a) => Semiring (Complex a)

(Ord a, TrigField a, ExpField a) => ExpField (Complex a)

todo: bottom is here somewhere???

Methods

exp :: Complex a -> Complex a

log :: Complex a -> Complex a

logBase :: Complex a -> Complex a -> Complex a

(**) :: Complex a -> Complex a -> Complex a

sqrt :: Complex a -> Complex a

(MultiplicativeGroup a, AdditiveGroup a, Semiring a) => Field (Complex a)

Generic1 * Complex

Associated Types

type Rep1 Complex (f :: Complex -> *) :: k -> *

Methods

from1 :: f a -> Rep1 Complex f a

to1 :: Rep1 Complex f a -> f a

(Multiplicative a, ExpField a, Normed a a) => Metric (Complex a) a

Methods

distanceL1 :: Complex a -> Complex a -> a

distanceL2 :: Complex a -> Complex a -> a

distanceLp :: a -> Complex a -> Complex a -> a

(Multiplicative a, ExpField a, Normed a a) => Normed (Complex a) a

Methods

normL1 :: Complex a -> a

normL2 :: Complex a -> a

normLp :: a -> Complex a -> a

type Rep (Complex a)

type Rep (Complex a) = D1 * (MetaData "Complex" "NumHask.Data.Complex" "numhask-0.2.3.1-5X1V1rZSUVsE9M1b0C2H1i" False) (C1 * (MetaCons ":+" (InfixI NotAssociative 6) False) ((:*:) * (S1 * (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 * a)) (S1 * (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 * a))))

type Rep1 * Complex

type Rep1 * Complex = D1 * (MetaData "Complex" "NumHask.Data.Complex" "numhask-0.2.3.1-5X1V1rZSUVsE9M1b0C2H1i" False) (C1 * (MetaCons ":+" (InfixI NotAssociative 6) False) ((:*:) * (S1 * (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1) (S1 * (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1)))

realPart :: Complex a -> a

Extracts the real part of a complex number.

imagPart :: Complex a -> a

Extracts the imaginary part of a complex number.

Helpers from Data.Complex

mkPolar :: TrigField a => a -> a -> Complex a

cis :: TrigField a => a -> Complex a

cis t is a complex value with magnitude 1 and phase t (modulo 2*pi).

polar :: (RealFloat a, ExpField a) => Complex a -> (a, a)

The function polar takes a complex number and returns a (magnitude, phase) pair in canonical form: the magnitude is nonnegative, and the phase in the range (-pi, pi]; if the magnitude is zero, then so is the phase.

magnitude :: (ExpField a, RealFloat a) => Complex a -> a

The nonnegative magnitude of a complex number.

phase :: RealFloat a => Complex a -> a

The phase of a complex number, in the range (-pi, pi]. If the magnitude is zero, then so is the phase.