numeric-prelude-0.4.2: An experimental alternative hierarchy of numeric type classes

Copyright(c) The University of Glasgow 2001
LicenseBSD-style (see the file libraries/base/LICENSE)
Maintainernumericprelude@henning-thielemann.de
Stabilityprovisional
Portabilityportable (?)
Safe HaskellNone
LanguageHaskell98

Number.Complex

Contents

Description

Complex numbers.

Synopsis

Cartesian form

data T a Source

Complex numbers are an algebraic type.

Instances

Functor T 
C T 
C a b => C a (T b)

The '(*>)' method can't replace scale because it requires the Algebra.Module constraint

C a b => C a (T b) 
(C a, Sqr a b) => C a (T b) 
Sqr a b => Sqr a (T b) 
(Ord a, C a v) => C a (T v) 
(C a, C a v) => C a (T v) 
(Show v, C v, C v, C a v) => C a (T v) 
Eq a => Eq (T a) 
(Floating a, Eq a) => Fractional (T a) 
(Floating a, Eq a) => Num (T a) 
Read a => Read (T a) 
Show a => Show (T a) 
Arbitrary a => Arbitrary (T a) 
Storable a => Storable (T a) 
C a => C (T a) 
C a => C (T a) 
C a => C (T a) 
C a => C (T a) 
C a => C (T a) 
(Ord a, C a) => C (T a) 
(Ord a, C a, C a) => C (T a) 
(C a, C a, C a) => C (T a) 
C a => C (T a) 
(C a, C a, Power a) => C (T a) 
(C a, C a, C a, Power a) => C (T a) 

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

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