Copyright	(c) The University of Glasgow 2001
License	BSD-style (see the file libraries/base/LICENSE)
Maintainer	numericprelude@henning-thielemann.de
Stability	provisional
Portability	portable (?)
Safe Haskell	None
Language	Haskell98

Number.Complex

Contents

Cartesian form
Polar form
Conjugate
Properties
Auxiliary classes

Description

Complex numbers.

Synopsis

data T a
imaginaryUnit :: C a => T a
fromReal :: C a => a -> T a
(+:) :: a -> a -> T a
(-:) :: C a => a -> a -> T a
scale :: C a => a -> T a -> T a
exp :: C a => T a -> T a
quarterLeft :: C a => T a -> T a
quarterRight :: C a => T a -> T a
fromPolar :: C a => a -> a -> T a
cis :: C a => a -> T a
signum :: (C a, C a) => T a -> T a
signumNorm :: (C a, C a a, C a) => T a -> T a
toPolar :: (C a, C a) => T a -> (a, a)
magnitude :: C a => T a -> a
magnitudeSqr :: C a => T a -> a
phase :: (C a, C a) => T a -> a
conjugate :: C a => T a -> T a
propPolar :: (C a, C a) => T a -> Bool
class C a => Power a where
- power :: Rational -> T a -> T a
defltPow :: (C a, C a) => Rational -> T a -> T a

Cartesian form

data T a Source #

Complex numbers are an algebraic type.

Instances

Functor T Source #
Instance details Defined in Number.Complex Methods fmap :: (a -> b) -> T a -> T b # (<$) :: a -> T b -> T a #
C T Source #
Instance details Defined in Number.Complex Methods zero :: C a => T a Source # (<+>) :: C a => T a -> T a -> T a Source # (*>) :: C a => a -> T a -> T a Source #
C a b => C a (T b) Source #	The '(*>)' method can't replace `scale` because it requires the Algebra.Module constraint
Instance details Defined in Number.Complex Methods (*>) :: a -> T b -> T b Source #
C a b => C a (T b) Source #
Instance details Defined in Number.Complex
(Show v, C v, C v, C a v) => C a (T v) Source #
Instance details Defined in Number.Complex Methods toScalar :: T v -> a Source # toMaybeScalar :: T v -> Maybe a Source # fromScalar :: a -> T v Source #
(C a, C a v) => C a (T v) Source #
Instance details Defined in Number.Complex Methods norm :: T v -> a Source #
(Ord a, C a v) => C a (T v) Source #
Instance details Defined in Number.Complex Methods norm :: T v -> a Source #
(C a, Sqr a b) => C a (T b) Source #
Instance details Defined in Number.Complex Methods norm :: T b -> a Source #
Sqr a b => Sqr a (T b) Source #
Instance details Defined in Number.Complex Methods normSqr :: T b -> a Source #
Eq a => Eq (T a) Source #
Instance details Defined in Number.Complex Methods (==) :: T a -> T a -> Bool # (/=) :: T a -> T a -> Bool #
(Floating a, Eq a) => Fractional (T a) Source #
Instance details Defined in Number.Complex Methods (/) :: T a -> T a -> T a # recip :: T a -> T a # fromRational :: Rational -> T a #
(Floating a, Eq a) => Num (T a) Source #
Instance details Defined in Number.Complex Methods (+) :: T a -> T a -> T a # (-) :: T a -> T a -> T a # (*) :: T a -> T a -> T a # negate :: T a -> T a # abs :: T a -> T a # signum :: T a -> T a # fromInteger :: Integer -> T a #
Read a => Read (T a) Source #
Instance details Defined in Number.Complex Methods readsPrec :: Int -> ReadS (T a) # readList :: ReadS [T a] # readPrec :: ReadPrec (T a) # readListPrec :: ReadPrec [T a] #
Show a => Show (T a) Source #
Instance details Defined in Number.Complex Methods showsPrec :: Int -> T a -> ShowS # show :: T a -> String # showList :: [T a] -> ShowS #
Arbitrary a => Arbitrary (T a) Source #
Instance details Defined in Number.Complex Methods arbitrary :: Gen (T a) # shrink :: T a -> [T a] #
Storable a => Storable (T a) Source #
Instance details Defined in Number.Complex Methods sizeOf :: T a -> Int # alignment :: T a -> Int # peekElemOff :: Ptr (T a) -> Int -> IO (T a) # pokeElemOff :: Ptr (T a) -> Int -> T a -> IO () # peekByteOff :: Ptr b -> Int -> IO (T a) # pokeByteOff :: Ptr b -> Int -> T a -> IO () # peek :: Ptr (T a) -> IO (T a) # poke :: Ptr (T a) -> T a -> IO () #
C a => C (T a) Source #
Instance details Defined in Number.Complex Methods compare :: T a -> T a -> Ordering Source #
C a => C (T a) Source #
Instance details Defined in Number.Complex 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 (T a) Source #
Instance details Defined in Number.Complex Methods isZero :: T a -> Bool Source #
C a => C (T a) Source #
Instance details Defined in Number.Complex 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 (T a) Source #
Instance details Defined in Number.Complex Methods div :: T a -> T a -> T a Source # mod :: T a -> T a -> T a Source # divMod :: T a -> T a -> (T a, T a) Source #
(Ord a, C a) => C (T a) Source #
Instance details Defined in Number.Complex Methods isUnit :: T a -> Bool Source # stdAssociate :: T a -> T a Source # stdUnit :: T a -> T a Source # stdUnitInv :: T a -> T a Source #
(Ord a, C a, C a) => C (T a) Source #
Instance details Defined in Number.Complex Methods extendedGCD :: T a -> T a -> (T a, (T a, T a)) Source # gcd :: T a -> T a -> T a Source # lcm :: T a -> T a -> T a Source #
(C a, C a, C a) => C (T a) Source #
Instance details Defined in Number.Complex Methods abs :: T a -> T a Source # signum :: T a -> T a Source #
C a => C (T a) Source #
Instance details Defined in Number.Complex 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, Power a) => C (T a) Source #
Instance details Defined in Number.Complex Methods sqrt :: T a -> T a Source # root :: Integer -> T a -> T a Source # (^/) :: T a -> Rational -> T a Source #
(C a, C a, C a, Power a) => C (T a) Source #
Instance details Defined in Number.Complex Methods pi :: T a Source # exp :: T a -> T a Source # log :: T a -> T a Source # logBase :: T a -> T a -> T a Source # (**) :: T a -> T a -> T a Source # sin :: T a -> T a Source # cos :: T a -> T a Source # tan :: T a -> T a Source # asin :: T a -> T a Source # acos :: T a -> T a Source # atan :: T a -> T a Source # sinh :: T a -> T a Source # cosh :: T a -> T a Source # tanh :: T a -> T a Source # asinh :: T a -> T a Source # acosh :: T a -> T a Source # atanh :: T a -> T a Source #

imaginaryUnit :: C a => T a Source #

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

(+:) :: a -> a -> T a infix 6 Source #

Construct a complex number from real and imaginary part.

(-:) :: C a => a -> a -> T a Source #

Construct a complex number with negated imaginary part.

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

Scale a complex number by a real number.

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

Exponential of a complex number with minimal type class constraints.

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

Turn the point one quarter to the right.

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

Turn the point one quarter to the right.

Polar form

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

Form a complex number from polar components of magnitude and phase.

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

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

signum :: (C a, C a) => T a -> T a Source #

Scale a complex number to magnitude 1.

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

signumNorm :: (C a, C a a, C a) => T a -> T a Source #

toPolar :: (C a, C a) => T a -> (a, a) Source #

The function toPolar 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 :: C a => T a -> a Source #

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

phase :: (C a, C a) => T a -> a Source #

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

Conjugate

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

The conjugate of a complex number.

Properties

propPolar :: (C a, C a) => T a -> Bool Source #

Auxiliary classes

class C a => Power a where Source #

We like to build the Complex Algebraic instance on top of the Algebraic instance of the scalar type. This poses no problem to sqrt. However, root requires computing the complex argument which is a transcendent operation. In order to keep the type class dependencies clean for more sophisticated algebraic number types, we introduce a type class which actually performs the radix operation.

Methods

power :: Rational -> T a -> T a Source #

Instances

Power Double Source #
Instance details Defined in Number.Complex Methods power :: Rational -> T Double -> T Double Source #
Power Float Source #
Instance details Defined in Number.Complex Methods power :: Rational -> T Float -> T Float Source #
Power T Source #
Instance details Defined in Number.Positional.Check Methods power :: Rational -> T0 T -> T0 T Source #

defltPow :: (C a, C a) => Rational -> T a -> T a Source #