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

Cartesian form

data T a #

Complex numbers are an algebraic type.

Instances

Functor T #
Methods fmap :: (a -> b) -> T a -> T b # (<$) :: a -> T b -> T a #
C T #
Methods zero :: C a => T a # (<+>) :: C a => T a -> T a -> T a # (*>) :: C a => a -> T a -> T a #
C a b => C a (T b) #	The '(*>)' method can't replace `scale` because it requires the Algebra.Module constraint
Methods (*>) :: a -> T b -> T b #
C a b => C a (T b) #

(Show v, C v, C v, C a v) => C a (T v) #
Methods toScalar :: T v -> a # toMaybeScalar :: T v -> Maybe a # fromScalar :: a -> T v #
(C a, C a v) => C a (T v) #
Methods norm :: T v -> a #
(Ord a, C a v) => C a (T v) #
Methods norm :: T v -> a #
(C a, Sqr a b) => C a (T b) #
Methods norm :: T b -> a #
Sqr a b => Sqr a (T b) #
Methods normSqr :: T b -> a #
Eq a => Eq (T a) #
Methods (==) :: T a -> T a -> Bool # (/=) :: T a -> T a -> Bool #
(Floating a, Eq a) => Fractional (T a) #
Methods (/) :: T a -> T a -> T a # recip :: T a -> T a # fromRational :: Rational -> T a #
(Floating a, Eq a) => Num (T a) #
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) #
Methods readsPrec :: Int -> ReadS (T a) # readList :: ReadS [T a] # readPrec :: ReadPrec (T a) # readListPrec :: ReadPrec [T a] #
Show a => Show (T a) #
Methods showsPrec :: Int -> T a -> ShowS # show :: T a -> String # showList :: [T a] -> ShowS #
Arbitrary a => Arbitrary (T a) #
Methods arbitrary :: Gen (T a) # shrink :: T a -> [T a] #
Storable a => Storable (T a) #
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) #
Methods compare :: T a -> T a -> Ordering #
C a => C (T a) #
Methods zero :: T a # (+) :: T a -> T a -> T a # (-) :: T a -> T a -> T a # negate :: T a -> T a #
C a => C (T a) #
Methods isZero :: T a -> Bool #
C a => C (T a) #
Methods (*) :: T a -> T a -> T a # one :: T a # fromInteger :: Integer -> T a # (^) :: T a -> Integer -> T a #
C a => C (T a) #
Methods div :: T a -> T a -> T a # mod :: T a -> T a -> T a # divMod :: T a -> T a -> (T a, T a) #
(Ord a, C a) => C (T a) #
Methods isUnit :: T a -> Bool # stdAssociate :: T a -> T a # stdUnit :: T a -> T a # stdUnitInv :: T a -> T a #
(Ord a, C a, C a) => C (T a) #
Methods extendedGCD :: T a -> T a -> (T a, (T a, T a)) # gcd :: T a -> T a -> T a # lcm :: T a -> T a -> T a #
(C a, C a, C a) => C (T a) #
Methods abs :: T a -> T a # signum :: T a -> T a #
C a => C (T a) #
Methods (/) :: T a -> T a -> T a # recip :: T a -> T a # fromRational' :: Rational -> T a # (^-) :: T a -> Integer -> T a #
(C a, C a, Power a) => C (T a) #
Methods sqrt :: T a -> T a # root :: Integer -> T a -> T a # (^/) :: T a -> Rational -> T a #
(C a, C a, C a, Power a) => C (T a) #
Methods pi :: T a # exp :: T a -> T a # log :: T a -> T a # logBase :: T a -> T a -> T a # (**) :: T a -> T a -> T a # sin :: T a -> T a # cos :: T a -> T a # tan :: T a -> T a # asin :: T a -> T a # acos :: T a -> T a # atan :: T a -> T a # sinh :: T a -> T a # cosh :: T a -> T a # tanh :: T a -> T a # asinh :: T a -> T a # acosh :: T a -> T a # atanh :: T a -> T a #

imaginaryUnit :: C a => T a #

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

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

Construct a complex number from real and imaginary part.

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

Construct a complex number with negated imaginary part.

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

Scale a complex number by a real number.

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

Exponential of a complex number with minimal type class constraints.

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

Turn the point one quarter to the right.

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

Turn the point one quarter to the right.

Polar form

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

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

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

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

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

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 #

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

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 #

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

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

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 #

The conjugate of a complex number.

Properties

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

Auxiliary classes

class C a => Power a where #

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.

Minimal complete definition

power

Methods

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

Instances

Power Double #
Methods power :: Rational -> T Double -> T Double #
Power Float #
Methods power :: Rational -> T Float -> T Float #
Power T #
Methods power :: Rational -> T T -> T T #

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