-- 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 -- Type signature determined by GHC. fromTaylorToCF :: Fractional a => [a] -> a -> [(a, a)] 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 -- -----------------------------------------------------------------------------