cyclotomic-0.4.1: A subfield of the complex numbers for exact calculation.

Safe HaskellNone



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 (in particular source files gap4r4/src/cyclotom.c and gap4r4/lib/



data Cyclotomic Source

A cyclotomic number.


Eq Cyclotomic 
Fractional Cyclotomic 
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.

Show Cyclotomic 

i :: CyclotomicSource

The square root of -1.

e :: Integer -> CyclotomicSource

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 -> CyclotomicSource

The square root of an Integer.

sqrtRat :: Rational -> CyclotomicSource

The square root of a Rational number.

sinDeg :: Rational -> CyclotomicSource

Sine function with argument in degrees.

cosDeg :: Rational -> CyclotomicSource

Cosine function with argument in degrees.

gaussianRat :: Rational -> Rational -> CyclotomicSource

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

polarRat :: Rational -> Rational -> CyclotomicSource

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.

conj :: Cyclotomic -> CyclotomicSource

Complex conjugate.

real :: Cyclotomic -> CyclotomicSource

Real part of the cyclotomic number.

imag :: Cyclotomic -> CyclotomicSource

Imaginary part of the cyclotomic number.

isReal :: Cyclotomic -> BoolSource

Is the cyclotomic a real number?

isRat :: Cyclotomic -> BoolSource

Is the cyclotomic a rational?

isGaussianRat :: Cyclotomic -> BoolSource

Is the cyclotomic a Gaussian rational?

toComplex :: Cyclotomic -> Complex DoubleSource

Export as an inexact complex number.

toReal :: Cyclotomic -> Maybe DoubleSource

Export as an inexact real number if possible.

toRat :: Cyclotomic -> Maybe RationalSource

Return an exact rational number if possible.

goldenRatio :: CyclotomicSource

The golden ratio, (1 + √5)/2.

dft :: [Cyclotomic] -> [Cyclotomic]Source

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

dftInv :: [Cyclotomic] -> [Cyclotomic]Source

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}.