| Copyright | © Anders Kaseorg 2013 |
|---|---|
| License | BSD-style |
| Maintainer | Anders Kaseorg <andersk@mit.edu> |
| Stability | experimental |
| Portability | Non-portable (GHC extensions) |
| Safe Haskell | None |
| Language | Haskell98 |
Data.Real.Constructible
Description
The constructible reals, Construct, are the subset of the real
numbers that can be represented exactly using field operations
(addition, subtraction, multiplication, division) and positive square
roots. They support exact computations, equality comparisons, and
ordering.
>>>[((1 + sqrt 5)/2)^n - ((1 - sqrt 5)/2)^n :: Construct | n <- [1..10]][sqrt 5,sqrt 5,2*sqrt 5,3*sqrt 5,5*sqrt 5,8*sqrt 5,13*sqrt 5,21*sqrt 5,34*sqrt 5,55*sqrt 5]
>>>let f (a, b, t, p) = ((a + b)/2, sqrt (a*b), t - p*((a - b)/2)^2, 2*p)>>>let (a, b, t, p) = f . f . f . f $ (1, 1/sqrt 2, 1/4, 1 :: Construct)>>>floor $ ((a + b)^2/(4*t))*10**4031415926535897932384626433832795028841971
>>>let qf (p, q) = ((p + sqrt (p^2 - 4*q))/2, (p - sqrt (p^2 - 4*q))/2 :: Construct)>>>let [(v, w), (x, _), (y, _), (z, _)] = map qf [(-1, -4), (v, -1), (w, -1), (x, y)]>>>z/2-1/16 + 1/16*sqrt 17 + 1/8*sqrt (17/2 - 1/2*sqrt 17) + 1/4*sqrt (17/4 + 3/4*sqrt 17 - (3/4 + 1/4*sqrt 17)*sqrt (17/2 - 1/2*sqrt 17))
Constructible complex numbers may be built from constructible reals
using Complex from the complex-generic library.
>>>(z/2 :+ sqrt (1 - (z/2)^2))^171 :+ 0
Synopsis
- data Construct
- deconstruct :: Construct -> Either Rational (Construct, Construct, Construct)
- fromConstruct :: Floating a => Construct -> a
- data ConstructException
Documentation
The type of constructible real numbers.
Instances
deconstruct :: Construct -> Either Rational (Construct, Construct, Construct) Source #
Deconstruct a rational constructible number as a Rational, or an
irrational constructible number as a triple (a, b, r) of simpler
constructible numbers representing a + b*sqrt r (with b /= 0 and
r > 0). Recursively calling deconstruct on all triples will yield
a finite tree that terminates in Rational leaves.
Note that two constructible numbers that compare as equal may deconstruct in different ways.
fromConstruct :: Floating a => Construct -> a Source #
Evaluate a floating-point approximation for a constructible number.
To improve numerical stability, addition of numbers with different signs is avoided using quadratic conjugation.
>>>fromConstruct $ sum (map sqrt [7, 14, 39, 70, 72, 76, 85]) - sum (map sqrt [13, 16, 46, 55, 67, 73, 79])1.8837969820815017e-19
data ConstructException Source #
The type of exceptions thrown by impossible Construct operations.
Constructors
| ConstructIrrational |
|
| ConstructSqrtNegative |
|
| Unconstructible String |
|