-- Modified by Lennart Augustsson to fit into Haskell numerical hierarchy.
--
-- Module:
--
-- Fraction.hs
--
-- Language:
--
-- Haskell
--
-- Description: Rational with transcendental functionalities
--
--
-- This is a generalized Rational in disguise. Rational, as a type
-- synonim, could not be directly made an instance of any new class
-- at all.
-- But we would like it to be an instance of Transcendental, where
-- trigonometry, hyperbolics, logarithms, etc. are defined.
-- So here we are tiptoe-ing around, re-defining everything from
-- scratch, before designing the transcendental functions -- which
-- is the main motivation for this module.
--
-- Aside from its ability to compute transcendentals, Fraction
-- allows for denominators zero. Unlike Rational, Fraction does
-- not produce run-time errors for zero denominators, but use such
-- entities as indicators of invalid results -- plus or minus
-- infinities. Operations on fractions never fail in principle.
--
-- However, some function may compute slowly when both numerators
-- and denominators of their arguments are chosen to be huge.
-- For example, periodicity relations are utilized with large
-- arguments in trigonometric functions to reduce the arguments
-- to smaller values and thus improve on the convergence
-- of continued fractions. Yet, if pi number is chosen to
-- be extremely accurate then the reduced argument would
-- become a fraction with huge numerator and denominator
-- -- thus slowing down the entire computation of a trigonometric
-- function.
--
-- Usage:
--
-- When computation speed is not an issue and accuracy is important
-- this module replaces some of the functionalities typically handled
-- by the floating point numbers: trigonometry, hyperbolics, roots
-- and some special functions. All computations, including definitions
-- of the basic constants pi and e, can be carried with any desired
-- accuracy. One suggested usage is for mathematical servers, where
-- safety might be more important than speed. See also the module
-- Numerus, which supports mixed arithmetic between Integer,
-- Fraction and Cofra (Complex fraction), and returns complex
-- legal answers in some cases where Fraction would produce
-- infinities: log (-5), sqrt (-1), etc.
--
--
-- Required:
--
-- Haskell Prelude
--
-- Author:
--
-- Jan Skibinski, Numeric Quest Inc.
--
-- Date:
--
-- 1998.08.16, last modified 2000.05.31
--
-- See also bottom of the page for description of the format used
-- for continued fractions, references, etc.
-------------------------------------------------------------------
module Data.Number.FixedFunctions where
import Prelude hiding (pi, sqrt, tan, atan, exp, log)
import Data.Ratio
approx :: Rational -> Rational -> Rational
approx eps x = approxRational x eps
------------------------------------------------------------------
-- Category: Conversion
-- from continued fraction to fraction and vice versa,
-- from Taylor series to continued fraction.
-------------------------------------------------------------------
type CF = [(Rational, Rational)]
fromCF :: CF -> Rational
fromCF x =
--
-- Convert finite continued fraction to fraction
-- evaluating from right to left. This is used
-- mainly for testing in conjunction with "toCF".
--
foldr g 1 x
where
g :: (Rational, Rational) -> Rational -> Rational
g u v = (fst u) + (snd u) / v
toCF :: Rational -> CF
toCF x =
--
-- Convert fraction to finite continued fraction
--
toCF' x []
where
toCF' u lst =
case r of
0 -> reverse (((q%1),(0%1)):lst)
_ -> toCF' (b%r) (((q%1),(1%1)):lst)
where
a = numerator u
b = denominator u
(q,r) = quotRem a b
approxCF :: Rational -> CF -> Rational
approxCF eps [] = 0
approxCF eps x
--
-- Approximate infinite continued fraction x by fraction,
-- evaluating from left to right, and stopping when
-- accuracy eps is achieved, or when a partial numerator
-- is zero -- as it indicates the end of CF.
--
-- This recursive function relates continued fraction
-- to rational approximation.
--
= approxCF' eps x 0 1 1 q' p' 1
where
h = fst (x!!0)
(q', p') = x!!0
approxCF' eps x v2 v1 u2 u1 a' n
| abs (1 - f1/f) < eps = approx eps f
| a == 0 = approx eps f
| otherwise = approxCF' eps x v1 v u1 u a (n+1)
where
(b, a) = x!!n
u = b*u1 + a'*u2
v = b*v1 + a'*v2
f = u/v
f1 = u1/v1
fromTaylorToCF s x =
--
-- Convert infinite number of terms of Taylor expansion of
-- a function f(x) to an infinite continued fraction,
-- where s = [s0,s1,s2,s3....] is a list of Taylor
-- series coefficients, such that f(x)=s0 + s1*x + s2*x^2....
--
-- Require: No Taylor coefficient is zero
--
zero:one:[higher m | m <- [2..]]
where
zero = (s!!0, s!!1 * x)
one = (1, -s!!2/s!!1 * x)
higher m = (1 + s!!m/s!!(m-1) * x, -s!!(m+1)/s!!m * x)
------------------------------------------------------------------
-- Category: Auxiliaries
------------------------------------------------------------------
fac :: Integer -> Integer
fac = product . enumFromTo 1
integerRoot2 :: Integer -> Integer
integerRoot2 1 = 1
integerRoot2 x =
--
-- Biggest integer m, such that x - m^2 >= 0,
-- where x is a positive integer
--
integerRoot2' 0 x (x `div` 2) x
where
integerRoot2' lo hi r y
| c > y = integerRoot2' lo r ((r + lo) `div` 2) y
| c == y = r
| otherwise =
if (r+1)^2 > y then
r
else
integerRoot2' r hi ((r + hi) `div` 2) y
where c = r^2
-------------------------------------------------------------------
-- Everything below is the instantiation of class Transcendental
-- for type Rational. See also modules Cofra and Numerus.
--
-- Category: Constants
-------------------------------------------------------------------
pi :: Rational -> Rational
pi eps =
--
-- pi with accuracy eps
--
-- Based on Ramanujan formula, as described in Ref. 3
-- Accuracy: extremely good, 10^-19 for one term of continued
-- fraction
--
(sqrt eps d) / (approxCF eps (fromTaylorToCF s x))
where
x = 1%(640320^3)::Rational
s = [((-1)^k*(fac (6*k))%((fac k)^3*(fac (3*k))))*((a*k+b)%c) | k<-[0..]]
a = 545140134
b = 13591409
c = 426880
d = 10005
---------------------------------------------------------------------
-- Category: Trigonometry
---------------------------------------------------------------------
tan :: Rational -> Rational -> Rational
tan eps 0 = 0
tan eps x
--
-- Tangent x computed with accuracy of eps.
--
-- Trigonometric identities are used first to reduce
-- the value of x to a value from within the range of [-pi/2,pi/2]
--
| x >= half_pi' = tan eps (x - ((1+m)%1)*xpi)
| x <= -half_pi' = tan eps (x + ((1+m)%1)*xpi)
--- | absx > 1 = 2 * t/(1 - t^2)
| otherwise = approxCF eps (cf x)
where
absx = abs x
t = tan eps (x/2)
m = floor ((absx - half_pi)/ xpi)
xpi = pi eps
half_pi'= 158%100
half_pi = xpi * (1%2)
cf u = ((0%1,1%1):[((2*r + 1)/u, -1) | r <- [0..]])
sin :: Rational -> Rational -> Rational
sin eps 0 = 0
sin eps x = 2*t/(1 + t*t)
where
t = tan eps (x/2)
cos :: Rational -> Rational -> Rational
cos eps 0 = 1
cos eps x = (1 - p)/(1 + p)
where
t = tan eps (x/2)
p = t*t
atan :: Rational -> Rational -> Rational
atan eps x
--
-- Inverse tangent of x with approximation eps
--
| x == 0 = 0
| x > 1 = (pi eps)/2 - atan eps (1/x)
| x < -1 = -(pi eps)/2 - atan eps (1/x)
| otherwise = approxCF eps ((0,x):[((2*m - 1),(m*x)^2) | m<- [1..]])
asin :: Rational -> Rational -> Rational
asin eps x
--
-- Inverse sine of x with approximation eps
--
| x == 0 = 0
| abs x > 1 = error "Fraction.asin"
| x == 1 = (pi eps) * (1%2)
| x == -1 = (pi eps) * (-1%2)
| otherwise = atan eps (x / (sqrt eps (1 - x^2)))
acos :: Rational -> Rational -> Rational
acos eps x
--
-- Inverse cosine of x with approximation eps
--
| x == 0 = (pi eps)*(1%2)
| abs x > 1 = error "Fraction.sin"
| x == 1 = 0
| x == -1 = pi eps
| otherwise = atan eps ((sqrt eps (1 - x^2)) / x)
---------------------------------------------------------------------
-- Category: Roots
---------------------------------------------------------------------
sqrt :: Rational -> Rational -> Rational
sqrt eps x
--
-- Square root of x with approximation eps
--
-- The CF pattern is: [(m,x-m^2),(2m,x-m^2),(2m,x-m^2)....]
-- where m is the biggest integer such that x-m^2 >= 0
--
| x < 0 = error "Fraction.sqrt"
| x == 0 = 0
| x < 1 = 1/(sqrt eps (1/x))
| otherwise = approxCF eps ((m,x-m^2):[(2*m,x-m^2) | r<-[0..]])
where
m = (integerRoot2 (floor x))%1
---------------------------------------------------------------------
-- Category: Exponentials and hyperbolics
---------------------------------------------------------------------
exp :: Rational -> Rational -> Rational
exp eps x
--
-- Exponent of x with approximation eps
--
-- Based on Jacobi type continued fraction for exponential,
-- with fractional terms:
-- n == 0 ==> (1,x)
-- n == 1 ==> (1 -x/2, x^2/12)
-- n >= 2 ==> (1, x^2/(16*n^2 - 4))
-- For x outside [-1,1] apply identity exp(x) = (exp(x/2))^2
--
| x == 0 = 1
| x > 1 = (approxCF eps (f (x*(1%p))))^p
| x < (-1) = (approxCF eps (f (x*(1%q))))^q
| otherwise = approxCF eps (f x)
where
p = ceiling x
q = -(floor x)
f y = (1,y):(1-y/2,y^2/12):[(1,y^2/(16*n^2-4)) | n<-[2..]]
cosh :: Rational -> Rational -> Rational
cosh eps x =
--
-- Hyperbolic cosine with approximation eps
--
(a + b)*(1%2)
where
a = exp eps x
b = 1/a
sinh :: Rational -> Rational -> Rational
sinh eps x =
--
-- Hyperbolic sine with approximation eps
--
(a - b)*(1%2)
where
a = exp eps x
b = 1/a
tanh :: Rational -> Rational -> Rational
tanh eps x =
--
-- Hyperbolic tangent with approximation eps
--
(a - b)/ (a + b)
where
a = exp eps x
b = 1/a
atanh :: Rational -> Rational -> Rational
atanh eps x
--
-- Inverse hyperbolic tangent with approximation eps
--
-- | x >= 1 = 1%0
-- | x <= -1 = -1%0
| otherwise = (1%2) * (log eps ((1 + x) / (1 - x)))
asinh :: Rational -> Rational -> Rational
asinh eps x
--
-- Inverse hyperbolic sine
--
-- | x == 1%0 = 1%0
-- | x == -1%0 = -1%0
| otherwise = log eps (x + (sqrt eps (x^2 + 1)))
acosh :: Rational -> Rational -> Rational
acosh eps x
--
-- Inverse hyperbolic cosine
--
-- | x == 1%0 = 1%0
-- | x < 1 = 1%0
| otherwise = log eps (x + (sqrt eps (x^2 - 1)))
---------------------------------------------------------------------
-- Category: Logarithms
---------------------------------------------------------------------
log :: Rational -> Rational -> Rational
log eps x
--
-- Natural logarithm of strictly positive x
--
-- Based on Stieltjes type continued fraction for log (1+y)
-- (0,y):(1,y/2):[(1,my/(4m+2)),(1,(m+1)y/(4m+2)),....
-- (m >= 1, two elements per m)
-- Efficient only for x close to one. For larger x we recursively
-- apply the identity log(x) = log(x/2) + log(2)
--
| x <= 0 = error "Fraction.log"
| x < 1 = -log eps (1/x)
| x == 1 = 0
| otherwise =
case (scaled (x,0)) of
(1,s) -> (s%1) * approxCF eps (series 1)
(y,0) -> approxCF eps (series (y-1))
(y,s) -> approxCF eps (series (y-1)) + (s%1)*approxCF eps (series 1)
where
series :: Rational -> CF
series u = (0,u):(1,u/2):[(1,u*((m+n)%(4*m + 2)))|m<-[1..],n<-[0,1]]
scaled :: (Rational,Integer) -> (Rational, Integer)
scaled (x, n)
| x == 2 = (1,n+1)
| x < 2 = (x, n)
| otherwise = scaled (x*(1%2), n+1)
---------------------------------------------------------------------------
-- References:
--
-- 1. Classical Gosper notes on continued fraction arithmetic:
-- http:%www.inwap.com/pdp10/hbaker/hakmem/cf.html
-- 2. Pages on numerical constants represented as continued fractions:
-- http:%www.mathsoft.com/asolve/constant/cntfrc/cntfrc.html
-- 3. "Efficient on-line computation of real functions using exact floating
-- point", by Peter John Potts, Imperial College
-- http:%theory.doc.ic.ac.uk/~pjp/ieee.html
--------------------------------------------------------------------------
--------------------------------------------------------------------------
-- The following representation of continued fractions is used:
--
-- Continued fraction: CF representation:
-- ================== ====================
-- b0 + a0
-- ------- ==> [(b0, a0), (b1, a1), (b2, a2).....]
-- b1 + a1
-- -------
-- b2 + ...
--
-- where "a's" and "b's" are Rationals.
--
-- Many continued fractions could be represented by much simpler form
-- [b1,b2,b3,b4..], where all coefficients "a" would have the same value 1
-- and would not need to be explicitely listed; and the coefficients "b"
-- could be chosen as integers.
-- However, there are some useful continued fractions that are
-- given with fraction coefficients: "a", "b" or both.
-- A fractional form can always be converted to an integer form, but
-- a conversion process is not always simple and such an effort is not
-- always worth of the achieved savings in the storage space or the
-- computational efficiency.
--
----------------------------------------------------------------------------
--
-- Copyright:
--
-- (C) 1998 Numeric Quest, All rights reserved
--
--
--
-- http://www.numeric-quest.com
--
-- License:
--
-- GNU General Public License, GPL
--
-----------------------------------------------------------------------------