Data.Complex.Cyclotomic

The cyclotomic numbers are a subset of the complex numbers with the following properties:

1. The cyclotomic numbers are represented exactly, enabling exact computations and equality comparisons.
2. The cyclotomic numbers contain the Gaussian rationals (complex numbers of the form p + q i with p and q rational). As a consequence, the cyclotomic numbers are a dense subset of the complex numbers.
3. The cyclotomic numbers contain the square roots of all rational numbers.
4. The cyclotomic numbers form a field: they are closed under addition, subtraction, multiplication, and division.
5. The cyclotomic numbers contain the sine and cosine of all rational multiples of pi.
6. The cyclotomic numbers can be thought of as the rational field extended with nth roots of unity for arbitrarily large integers n.

Floating point numbers do not do well with equality comparison:

(sqrt 2 + sqrt 3)^2 == 5 + 2 * sqrt 6
 -> False

Data.Complex.Cyclotomic represents these numbers exactly, allowing equality comparison:

(sqrtRat 2 + sqrtRat 3)^2 == 5 + 2 * sqrtRat 6
 -> True

Cyclotomics can be exported as inexact complex numbers using the toComplex function:

e 6
 -> -e(3)^2
real $ e 6
 -> 1/2
imag $ e 6
 -> -1/2*e(12)^7 + 1/2*e(12)^11
imag (e 6) == sqrtRat 3 / 2
 -> True
toComplex $ e 6
 -> 0.5000000000000003 :+ 0.8660254037844384

The algorithms for cyclotomic numbers are adapted from code by Martin Schoenert and Thomas Breuer in the GAP project http://www.gap-system.org/ (in particular source files gap4r4/src/cyclotom.c and gap4r4/lib/cyclotom.gi).

Documentation

data Cyclotomic

A cyclotomic number.

Instances

Num Cyclotomic

abs and signum are partial functions. A cyclotomic number is not guaranteed to have a cyclotomic absolute value. When defined, signum c is the complex number with magnitude 1 that has the same argument as c; signum c = c / abs c.

Defined in Data.Complex.Cyclotomic

Methods

(+) :: Cyclotomic -> Cyclotomic -> Cyclotomic

(-) :: Cyclotomic -> Cyclotomic -> Cyclotomic

(*) :: Cyclotomic -> Cyclotomic -> Cyclotomic

negate :: Cyclotomic -> Cyclotomic

abs :: Cyclotomic -> Cyclotomic

signum :: Cyclotomic -> Cyclotomic

fromInteger :: Integer -> Cyclotomic

Fractional Cyclotomic

Defined in Data.Complex.Cyclotomic

Methods

(/) :: Cyclotomic -> Cyclotomic -> Cyclotomic

recip :: Cyclotomic -> Cyclotomic

fromRational :: Rational -> Cyclotomic

Show Cyclotomic

Defined in Data.Complex.Cyclotomic

Methods

showsPrec :: Int -> Cyclotomic -> ShowS

show :: Cyclotomic -> String

showList :: [Cyclotomic] -> ShowS

Eq Cyclotomic

Defined in Data.Complex.Cyclotomic

Methods

(==) :: Cyclotomic -> Cyclotomic -> Bool

(/=) :: Cyclotomic -> Cyclotomic -> Bool

i :: Cyclotomic

The square root of -1.

e :: Integer -> Cyclotomic

The primitive nth root of unity. For example, e(4) = i is the primitive 4th root of unity, and e(5) = exp(2*pi*i/5) is the primitive 5th root of unity. In general, e n = exp(2*pi*i/n).

sqrtInteger :: Integer -> Cyclotomic

The square root of an Integer.

sqrtRat :: Rational -> Cyclotomic

The square root of a Rational number.

sinDeg :: Rational -> Cyclotomic

Sine function with argument in degrees.

cosDeg :: Rational -> Cyclotomic

Cosine function with argument in degrees.

sinRev :: Rational -> Cyclotomic

Sine function with argument in revolutions.

cosRev :: Rational -> Cyclotomic

Cosine function with argument in revolutions.

gaussianRat :: Rational -> Rational -> Cyclotomic

Make a Gaussian rational; gaussianRat p q is the same as p + q * i.

polarRat

Arguments

:: Rational

magnitude

-> Rational

angle, in revolutions

-> Cyclotomic

cyclotomic number

A complex number in polar form, with rational magnitude r and rational angle s of the form r * exp(2*pi*i*s); polarRat r s is the same as r * e q ^ p, where s = p/q. This function is the same as polarRatRev.

polarRatDeg

Arguments

:: Rational

magnitude

-> Rational

angle, in degrees

-> Cyclotomic

cyclotomic number

A complex number in polar form, with rational magnitude and rational angle in degrees.

polarRatRev

Arguments

:: Rational

magnitude

-> Rational

angle, in revolutions

-> Cyclotomic

cyclotomic number

A complex number in polar form, with rational magnitude and rational angle in revolutions.

conj :: Cyclotomic -> Cyclotomic

Complex conjugate.

real :: Cyclotomic -> Cyclotomic

Real part of the cyclotomic number.

imag :: Cyclotomic -> Cyclotomic

Imaginary part of the cyclotomic number.

isReal :: Cyclotomic -> Bool

Is the cyclotomic a real number?

isRat :: Cyclotomic -> Bool

Is the cyclotomic a rational?

isGaussianRat :: Cyclotomic -> Bool

Is the cyclotomic a Gaussian rational?

toComplex :: RealFloat a => Cyclotomic -> Complex a

Export as an inexact complex number.

toReal :: RealFloat a => Cyclotomic -> Maybe a

Export as an inexact real number if possible.

toRat :: Cyclotomic -> Maybe Rational

Return an exact rational number if possible.

goldenRatio :: Cyclotomic

The golden ratio, (1 + √5)/2.

dft :: [Cyclotomic] -> [Cyclotomic]

Discrete Fourier transform, X_k = sum_{n=0}^{N-1} x_n cdot e^{-i 2 pi frac{k}{N} n}.

dftInv :: [Cyclotomic] -> [Cyclotomic]

Inverse discrete Fourier transform, x_n = frac{1}{N} sum_{k=0}^{N-1} X_k cdot e^{i 2 pi frac{k}{N} n}.

rootsQuadEq

Arguments

:: Rational

a

-> Rational

b

-> Rational

c

-> Maybe (Cyclotomic, Cyclotomic)

roots

Solutions to the quadratic equation a x^2 + b x + c = 0. Returns Nothing if a == 0.

heron

Arguments

:: Rational

a

-> Rational

b

-> Rational

c

-> Cyclotomic

area of triangle

Heron's formula for the area of a triangle with side lengths a, b, c.