continued-fractions-0.9.1.0: src/Math/ContinuedFraction.hs
{-# LANGUAGE ParallelListComp #-}
module Math.ContinuedFraction
( CF
, cf, gcf
, asCF, asGCF
, truncateCF
, equiv
, setNumerators
, setDenominators
, partitionCF
, evenCF
, oddCF
, convergents
, steed
, lentz, lentzWith
, modifiedLentz, modifiedLentzWith
, sumPartialProducts
) where
import Control.Arrow ((***))
import Data.List (tails, mapAccumL)
-- * The 'CF' type and basic operations
-- I think I would like to try refactoring this stuff at some point to use
-- an "Inductive" CF type, something like:
--
-- > data CF a
-- > = CFZero -- eval CFZero = 0
-- > | CFAdd a (CF a) -- eval (CFAdd b x) = b + eval x
-- > | CFCont a a (CF a) -- eval (CFCont a b x) = a / (b + eval x)
-- |A continued fraction. Constructed by 'cf' or 'gcf'.
data CF a
= CF a [a]
-- ^ Not exported. See 'cf', the public constructor.
| GCF a [(a,a)]
-- ^ Not exported. See 'gcf', the public constructor.
-- |Construct a continued fraction from its first term and the
-- partial denominators in its canonical form, which is the form
-- where all the partial numerators are 1.
--
-- @cf a [b,c,d]@ corresponds to @a + (b \/ (1 + (c \/ (1 + d))))@,
-- or to @GCF a [(1,b),(1,c),(1,d)]@.
cf :: a -> [a] -> CF a
cf = CF
-- |Construct a continued fraction from its first term, its partial
-- numerators and its partial denominators.
--
-- @gcf b0 [(a1,b1), (a2,b2), (a3,b3)]@ corresponds to
-- @b0 + (a1 \/ (b1 + (a2 \/ (b2 + (a3 \/ b3)))))@
gcf :: a -> [(a,a)] -> CF a
gcf = GCF
instance Show a => Show (CF a) where
showsPrec p (CF b0 ab) = showParen (p>10)
( showString "cf "
. showsPrec 11 b0
. showChar ' '
. showsPrec 11 ab
)
showsPrec p (GCF b0 ab) = showParen (p>10)
( showString "gcf "
. showsPrec 11 b0
. showChar ' '
. showsPrec 11 ab
)
instance Functor CF where
fmap f (CF b0 cf) = CF (f b0) (map f cf)
fmap f (GCF b0 gcf) = GCF (f b0) (map (f *** f) gcf)
-- |Extract the partial denominators of a 'CF', normalizing it if necessary so
-- that all the partial numerators are 1.
asCF :: Fractional a => CF a -> (a, [a])
asCF (CF b0 cf) = (b0, cf)
asCF (GCF b0 []) = (b0, [])
asCF (GCF b0 cf) = (b0, zipWith (*) bs cs)
where
(a:as, bs) = unzip cf
cs = recip a : [recip (a*c) | c <- cs | a <- as]
-- |Extract all the partial numerators and partial denominators of a 'CF'.
asGCF :: Num a => CF a -> (a,[(a,a)])
asGCF (CF b0 cf) = (b0, [(1, b) | b <- cf])
asGCF (GCF b0 gcf) = (b0, takeWhile ((/=0).fst) gcf)
-- |Truncate a 'CF' to the specified number of partial numerators and denominators.
truncateCF :: Int -> CF a -> CF a
truncateCF n (CF b0 ab) = CF b0 (take n ab)
truncateCF n (GCF b0 ab) = GCF b0 (take n ab)
-- |Apply an equivalence transformation, multiplying each partial denominator
-- with the corresponding element of the supplied list and transforming
-- subsequent partial numerators and denominators as necessary. If the list
-- is too short, the rest of the 'CF' will be unscaled.
equiv :: Num a => [a] -> CF a -> CF a
equiv cs orig
= gcf b0 (zip as' bs')
where
(b0, terms) = asGCF orig
(as,bs) = unzip terms
as' = zipWith (*) (zipWith (*) cs' (1:cs')) as
bs' = zipWith (*) cs' bs
cs' = cs ++ repeat 1
-- |Apply an equivalence transformation that sets the partial denominators
-- of a 'CF' to the specfied values. If the input list is too short, the
-- rest of the 'CF' will be unscaled.
setDenominators :: Fractional a => [a] -> CF a -> CF a
setDenominators denoms orig
= gcf b0 (zip as' bs')
where
(b0, terms) = asGCF orig
(as,bs) = unzip terms
as' = zipWith (*) as (zipWith (*) cs (1:cs))
bs' = zipWith ($) (map const denoms ++ repeat id) bs
cs = zipWith (/) bs' bs
-- |Apply an equivalence transformation that sets the partial numerators
-- of a 'CF' to the specfied values. If the input list is too short, the
-- rest of the 'CF' will be unscaled.
setNumerators :: Fractional a => [a] -> CF a -> CF a
setNumerators numers orig
= gcf b0 (zip as' bs')
where
(b0, terms) = asGCF orig
(as,bs) = unzip terms
as' = zipWith ($) (map const numers ++ repeat id) as
bs' = zipWith (*) bs cs
cs = zipWith (/) as' (zipWith (*) as (1:cs))
-- |Computes the even and odd parts, respectively, of a 'CF'. These are new
-- 'CF's that have the even-indexed and odd-indexed convergents of the
-- original, respectively.
partitionCF :: Fractional a => CF a -> (CF a, CF a)
partitionCF orig = case terms of
[] -> (orig, orig)
[(a1,b1)] ->
let final = cf (b0 + a1/b1) []
in (final, final)
_ -> (evenPart, oddPart)
where
(b0, terms) = asGCF orig
(as, bs) = unzip terms
pairs (a:b:rest) = (a,b) : pairs rest
pairs [a] = [(a,0)]
pairs _ = []
alphas@(alpha1:alpha2:_) = zipWith (/) as (zipWith (*) bs (1:bs))
evenPart = gcf b0 (zip cs ds)
where
cs = alpha1 : [(-aOdd) * aEven | (aEven, aOdd) <- pairs (tail alphas)]
ds = 1 + alpha2 : [1 + aOdd + aEven | (aOdd, aEven) <- tail (pairs alphas)]
oddPart = gcf (b0+alpha1) (zip cs ds)
where
cs = [(-aOdd) * aEven | (aOdd, aEven) <- pairs alphas]
ds = [1 + aOdd + aEven | (aEven, aOdd) <- pairs (tail alphas)]
-- |Computes the even part of a 'CF' (that is, a new 'CF' whose convergents are
-- the even-indexed convergents of the original).
evenCF :: Fractional a => CF a -> CF a
evenCF = fst . partitionCF
-- |Computes the odd part of a 'CF' (that is, a new 'CF' whose convergents are
-- the odd-indexed convergents of the original).
oddCF :: Fractional a => CF a -> CF a
oddCF = snd . partitionCF
-- * Evaluating continued fractions
-- |Evaluate the convergents of a continued fraction using the fundamental
-- recurrence formula:
--
-- A0 = b0, B0 = 1
--
-- A1 = b1b0 + a1, B1 = b1
--
-- A{n+1} = b{n+1}An + a{n+1}A{n-1}
--
-- B{n+1} = b{n+1}Bn + a{n+1}B{n-1}
--
-- The convergents are then Xn = An/Bn
convergents :: Fractional a => CF a -> [a]
convergents orig = drop 1 (zipWith (/) nums denoms)
where
(b0, terms) = asGCF orig
nums = 1:b0:[b * x1 + a * x0 | x0:x1:_ <- tails nums | (a,b) <- terms]
denoms = 0:1 :[b * x1 + a * x0 | x0:x1:_ <- tails denoms | (a,b) <- terms]
-- |Evaluate the convergents of a continued fraction using Steed's method.
-- Only valid if the denominator in the following recurrence for D_i never
-- goes to zero. If this method blows up, try 'modifiedLentz'.
--
-- D1 = 1/b1
--
-- D{i} = 1 / (b{i} + a{i} * D{i-1})
--
-- dx1 = a1 / b1
--
-- dx{i} = (b{i} * D{i} - 1) * dx{i-1}
--
-- x0 = b0
--
-- x{i} = x{i-1} + dx{i}
--
-- The convergents are given by @scanl (+) b0 dxs@
steed :: Fractional a => CF a -> [a]
steed (CF b0 []) = [b0]
steed (GCF b0 []) = [b0]
steed (CF 0 ( a :rest)) = map (1 /) (steed (CF a rest))
steed (GCF 0 ((a,b):rest)) = map (a /) (steed (GCF b rest))
steed orig
= scanl (+) b0 dxs
where
(b0, (a1,b1):gcf) = asGCF orig
dxs = a1/b1 : [(b * d - 1) * dx | (a,b) <- gcf | d <- ds | dx <- dxs]
ds = 1/b1 : [recip (b + a * d) | (a,b) <- gcf | d <- ds]
-- |Evaluate the convergents of a continued fraction using Lentz's method.
-- Only valid if the denominators in the following recurrence never go to
-- zero. If this method blows up, try 'modifiedLentz'.
--
-- C1 = b1 + a1 / b0
--
-- D1 = 1/b1
--
-- C{n} = b{n} + a{n} / C{n-1}
--
-- D{n} = 1 / (b{n} + a{n} * D{n-1})
--
-- The convergents are given by @scanl (*) b0 (zipWith (*) cs ds)@
lentz :: Fractional a => CF a -> [a]
lentz = lentzWith id (*) recip
-- |Evaluate the convergents of a continued fraction using Lentz's method,
-- mapping the terms in the final product to a new group before performing
-- the final multiplications. A useful group, for example, would be logarithms
-- under addition. In @lentzWith f op inv@, the arguments are:
--
-- * @f@, a group homomorphism (eg, 'log') from {@a@,(*),'recip'} to the group
-- in which you want to perform the multiplications.
--
-- * @op@, the group operation (eg., (+)).
--
-- * @inv@, the group inverse (eg., 'negate').
--
-- The 'lentz' function, for example, is given by the identity homomorphism:
-- @lentz@ = @lentzWith id (*) recip@.
--
-- The original motivation for this function is to allow computation of
-- the natural log of very large numbers that would overflow with the naive
-- implementation in 'lentz'. In this case, the arguments would be 'log', (+),
-- and 'negate', respectively.
--
-- In cases where terms of the product can be negative (i.e., the sequence of
-- convergents contains negative values), the following definitions could
-- be used instead:
--
-- > signLog x = (signum x, log (abs x))
-- > addSignLog (xS,xL) (yS,yL) = (xS*yS, xL+yL)
-- > negateSignLog (s,l) = (negate s, l)
{-# INLINE lentzWith #-}
lentzWith :: Fractional a => (a -> b) -> (b -> b -> b) -> (b -> b) -> CF a -> [b]
lentzWith f op inv (CF 0 ( a :rest)) = map inv (lentzWith f op inv (CF a rest))
lentzWith f op inv (GCF 0 ((a,b):rest)) = map (op (f a) . inv) (lentzWith f op inv (GCF b rest))
lentzWith f op inv c = scanl opF (f b0) (zipWith (*) cs ds)
where
opF x y = op x (f y)
(b0, cs, ds) = lentzRecurrence c
lentzRecurrence :: Fractional a => CF a -> (a,[a],[a])
lentzRecurrence orig
| null terms = (b0,[],[])
| otherwise = (b0, cs, ds)
where
(b0, terms) = asGCF orig
cs = [ b + a/c | (a,b) <- terms | c <- b0 : cs]
ds = [1/(b + a*d) | (a,b) <- terms | d <- 0 : ds]
-- |Evaluate the convergents of a continued fraction using Lentz's method,
-- (see 'lentz') with the additional rule that if a denominator ever goes
-- to zero, it will be replaced by a (very small) number of your choosing,
-- typically 1e-30 or so (this modification was proposed by Thompson and
-- Barnett).
--
-- Additionally splits the resulting list of convergents into sublists,
-- starting a new list every time the \'modification\' is invoked.
modifiedLentz :: Fractional a => a -> CF a -> [[a]]
modifiedLentz = modifiedLentzWith id (*) recip
-- |'modifiedLentz' with a group homomorphism (see 'lentzWith', it bears the
-- same relationship to 'lentz' as this function does to 'modifiedLentz',
-- and solves the same problems). Alternatively, 'lentzWith' with the same
-- modification to the recurrence as 'modifiedLentz'.
{-# INLINE modifiedLentzWith #-}
modifiedLentzWith :: Fractional a => (a -> b) -> (b -> b -> b) -> (b -> b) -> a -> CF a -> [[b]]
modifiedLentzWith f op inv z (CF 0 ( a :rest)) = map (map inv ) (modifiedLentzWith f op inv z (CF a rest))
modifiedLentzWith f op inv z (GCF 0 ((a,b):rest)) = map (map (op (f a) . inv)) (modifiedLentzWith f op inv z (GCF b rest))
modifiedLentzWith f op inv z orig = separate (scanl opF (False, f b0) cds)
where
(b0, cs, ds) = modifiedLentzRecurrence z orig
cds = zipWith mult cs ds
mult (xa,xb) (ya,yb) = (xa || ya, xb * yb)
opF (xa,xb) (ya,yb) = (xa || ya, op xb (f yb))
-- |Takes a list of (Bool,a) and breaks it into sublists, starting
-- a new one every time it encounters (True,_).
separate [] = []
separate ((_,x):xs) = case break fst xs of
(xs, ys) -> (x:map snd xs) : separate ys
modifiedLentzRecurrence :: Fractional a => a -> CF a -> (a,[(Bool, a)],[(Bool, a)])
modifiedLentzRecurrence z orig
| null terms = (b0, [], [])
| otherwise = (b0, cs, ds)
where
(b0, terms) = asGCF orig
cs = [reset (b + a/c) id | (a,b) <- terms | c <- b0 : map snd cs]
ds = [reset (b + a*d) recip | (a,b) <- terms | d <- 0 : map snd ds]
-- The sublist breaking is computed secondarily - initially,
-- 'cs' and 'ds' are constructed with this helper function that
-- adds a marker to the list whenever a term of interest goes to 0,
-- while also resetting that term to a small nonzero amount.
-- Then later, 'separate' breaks the list every time it sees one
-- of these markers.
reset x f
| x == 0 = (True, f z)
| otherwise = (False, f x)
-- |Euler's formula for computing @sum (scanl1 (*) xs)@.
-- Successive convergents of the resulting 'CF' are successive partial sums
-- in the series.
sumPartialProducts :: Num a => [a] -> CF a
sumPartialProducts [] = cf 0 []
sumPartialProducts (x:xs) = gcf 0 ((x,1):[(negate x, 1 + x) | x <- xs])