\chapter{Continued Fractions}
\begin{verbatim}
$Log: ContinuedFractions.lhs,v $
Revision 1.1 2004/08/05 11:12:52 malcolm
Add a regression testsuite for the nhc98 compiler. It isn't very good,
but it is better than nothing. I've been using it for about four years
on nightly builds, so it's about time it entered the repository! It
includes a slightly altered version of the nofib suite.
Instructions are in the README.
Revision 1.2 1999/11/02 16:10:42 simonpj
Haskell 98 changes
Revision 1.1 1996/01/08 20:05:19 partain
Initial revision
\end{verbatim}
> module ContinuedFractions
> (module Maybe, module QRationals,
> ContinuedFraction(..), Homography(..), Interval(..), rat2cf,
> algebraicAlgorithm, algebraicOutput,
> quadraticAlgorithm, quadraticOutput,
> cfRat2CFSqrt, decimals, accuracy, cf2Rat, integerFraction)
> where
> import Maybe
> import List( sort )
> import QRationals
\section{Representation of Continued Fractions}
The {\em Continued Fraction} representation of a number $r$ is a list of numbers
\[r = [x_0,~x_1,~x_2,~\ldots~x_i,~]\]
with the following interpretation:
\[r = x_0 + \frac{1}{x_1 + \frac{1}{x_2 +
\frac{1}{\ddots +\frac{1}{x_i + \frac{1}{ddots}}}}}\]
In this section we will describe all of the representational issues
associated with continued fractions.
We represent continued fractions as a list of @Integer@. The
@Homography@ is described seperately under the Algebraic and Quadratic
Algorithms. An @Interval@ is simply an interval with rational
end-points.
> type ContinuedFraction = [Integer]
> type Homography = ([Integer],[Integer])
> type Interval = (QRational,QRational)
To make it easy to negate a continued fraction, we are assuming that
the continued fractions are symmetric in the following sense:
\[-[x_0,\,x1,\,\ldots] = [ -x_0,\,-x_1,\,\ldots ] \]
\subsection{Creating a (Finite) Continued Fraction from a QRational}
The function @rat2cf@ turns a @QRational@ into the equivalent {\bf
Z}-fraction. This is acceptable provided that the invariant that we
choose for the relationship between the leading term of a continued
fraction and its interval is such that:
\[[\lfloor r\rceil - \frac{1}{2},\,\lfloor r\rceil +\frac{1}{2}) \subseteq
@cfBound@\,r\, \]
> rat2cf :: QRational -> ContinuedFraction
> rat2cf q
> = if qUndefined q then error "rat2cf: undefined" else
> if qInfinite q then [] else
> rat2cf' q
> where rat2cf' q = x : if q' == 0 then [] else rat2cf' (1 / q')
> where x = qRound q
> q' = q - (x%%1)
\subsection{Enclosing Bounds for Continued Fractions}
The function @cfBound@ gives @QRational@ bounds that enclose the
Euclidean Continued Fraction with leading term @x@. Because @x@ is
known to be @QRational@ -- and hence the denominator of the solution
to the quadratic equation is non-zero -- both of the roots are finite.
We observe the following conventions:
\begin{itemize}
\item The interval $(i,\,s)$ where $s<i$ is the interval
$(-\infty,\,s)\cup(i,\,\infty)$.
\end{itemize}
> cfBound :: QRational -> Interval
> cfBound x
> = if qInfinite x then (1%%0, 1%%0) else
> if abs x < 2 then (-2%%1, 2%%1) else
> if x < 0 then (1%%2-x, x+1%%2) else
> (x-1%%2, -x-1%%2)
> cfBound x
> = if qInfinite x then (1%%0, 1%%0) else
> if n > 0 then (c-pm,c+pm) else
> intersect (c-pm,c+pm)
> where x2 = x*x
> n = 3 - x2 -- never 0 because x is QRational!
> c = 4*x / n
> pm = r*(x2+1)/n
> r = if x2 < 3 then rootThreePlus else rootThreeMinus
> ax = abs x - (1%%2)
> intersect (i,s)
> = if i >= s then (max i ax, min s (-ax)) else
> if i < ax then (ax,s) else
> if i > -ax then (i,-ax) else
> error "cfBound: multiple intervals"
Two guesses for $\sqrt 3$ are:
> rootThreeMinus = 1732050807 %% 1000000000 :: QRational
> rootThreePlus = 1732050808 %% 1000000000 :: QRational
These satisfy $@rootThreeMinus@ < \sqrt 3$ and $@rootThreePlus@ > \sqrt 3$.
\subsection{Generating Output Terms of Continued Fractions}
We need a way to generate the next term to be emmitted by the various
algorithms that process the continued fractions; this is accomplished
by the @generateOutput@ function.
If the answer is now known with sufficient accuracy, the function
returns @Just o@ where @o@ is the leading term of the output continued
fraction. If it is not possible to determine the leading term of the
output continued fraction, @generateOutput@ returns @Nothing@.
> generateOutput
> :: (Homography -> [QRational] -> Interval) -> -- Bound
> (Homography -> [QRational] -> QRational) -> -- Evaluator
> Homography -> -- Homography
> [QRational] -> -- Leading Input Terms
> Maybe Integer -- Possible Output
> generateOutput bound eval homography qs
> = if not (qInfinite q0 && qInfinite q1) -- && signum (o%%1) == signum o'
> && acceptable int (o%%1) then Just o else Nothing
> where int@(q0,q1) = bound homography qs
> o' = eval homography qs
> o = qRound (if abs o' < 2 then o' else
> if abs q0 < abs q1 then q0 else q1)
> acceptable :: Interval -> QRational -> Bool
> acceptable int o = cfBound o `encloses` int
\subsection{Interval Operations}
The function @encloses@ tests whether its first interval argument
encloses its second.
> encloses :: Interval -> Interval -> Bool
> (i0,s0) `encloses` (i1,s1)
> = if i0 > s0 then
> if i1 < s1 then i0 <= i1 || s1 <= s0
> else i0 <= i1 && s1 <= s0
> else i1 < s1 && i0 <= i1 && s1 <= s0
Using $A \cap B = (A^c \cup B^c)^c$ ...
> intersect :: Interval -> Interval -> Interval
> intersect (i,s) (i',s') = (rs,ri) where (ri,rs) = union [(s,i),(s',i')]
> union :: [Interval] -> Interval
> union = result . foldl join [] . sort . finitize
> where result [int] = int
> result [(i,_),(_,s)] = (i,s)
> result ints = (0,0)
Assuming that the intervals are finite and ordered....
Then $@i@ \leq @i'@$ and if they're equal $@s@ \leq @s'@$
> join :: [Interval] -> Interval -> [Interval]
> join [] y = [y]
> join xs@((i,s):iss) y@(i',s') = if s >= s' then xs else
> if s >= i' then (i,s'):iss else y:xs
> finitize
> = concat . map split
> where split x@(i,s) = if i < s then [x] else [(-1%%0,s),(i,1%%0)]
> finiteInterval (i,s) = i < s
> infiniteInterval (i,s) = i > s
\section{The Algebraic Algorithm}
The algebraic algorithm takes a homography and calculates the following:
\[@algebraicAlgorithm@\,\left(\begin{array}[c]{cc}
n_0 & n_1 \\ d_0 & d_1\end{array}\right)\,x \, = \,
\frac{n_0x+n_1}{d_0x+d1}\]
Given that $x$ isn't $\bot$, this is well-defined provided that the
determinant of the homography is non-zero. If the determinant {\em is}
zero then the result is independent of the value of $x$.
Because we notice that absorbing a (well-defined and finite) term from
the input continued fraction or emitting a (well-defined and finite)
term of the output continued fraction doesn't change the determinant
of the homography we need only check this once.
> algebraicAlgorithm :: Homography -> -- Homography
> ContinuedFraction -> -- Input continued fraction
> ContinuedFraction -- Output continued fraction
> algebraicAlgorithm homography@([n0,n1],[d0,d1]) xs
> = if n0*d1 /= n1*d0 then algebraicAlgorithmNoCheck homography xs
> else if d1 /= 0 then rat2cf (n1%%d1)
> else if d0 /= 0 then rat2cf (n0%%d0)
> else error "algebraicAlgorithm: undefined homography"
> algebraicAlgorithmNoCheck :: Homography -> -- Homography
> ContinuedFraction -> -- Input CF
> ContinuedFraction -- Output CF
> algebraicAlgorithmNoCheck homography@([n0,n1],[d0,d1]) []
> = rat2cf (n0%%d0)
> algebraicAlgorithmNoCheck homography@([n0,n1],[d0,d1]) xs'@(x:xs)
> = case algebraicOutput homography (x%%1) of
> Just o -> o : algebraicAlgorithmNoCheck ([d0,d1],[n0-d0*o,n1-d1*o]) xs'
> Nothing -> algebraicAlgorithmNoCheck ([n0*x+n1,n0],[d0*x+d1,d0]) xs
> algebraicOutput :: Homography -> -- Homography
> QRational -> -- Leading Term of Input CF
> Maybe Integer -- Leading Term of Output CF
> algebraicOutput homography q
> = if q == 0 then Nothing else a
> where a = generateOutput algebraicBound algebraicEval homography [q]
\subsection{Bounds on the Output Continued Fraction}
The function @algebraicBound@, given a homography and the first
(@QRational@) term of a continued fraction determines the interval
within which the transformed continued fraction {\em must} lie.
> algebraicBound :: Homography -> -- Homography
> [QRational] -> -- Leading term of input CF
> Interval -- Interval
> algebraicBound homography@([n0,n1],[d0,d1]) [q]
> = if n0*d1 == n1*d0
> then (indp, indp) -- bound independent of x
> else if xi == dcross || xs == dcross -- infinity at endpoint
> then if xi > xs && (ncross <= xs || xi <= ncross) ||
> xi < xs && ncross <= xs && xi <= ncross
> then zint -- zero in interval
> else nzint -- zero not in interval
> else if xi > xs && (d0 == 0 || xi < dcross || dcross < xs) ||
> xi < xs && d0 /= 0 && xi < dcross && dcross < xs
> then (ys,yi) -- infinity in interval
> else (yi,ys) -- infinity not in interval
> where dcross = (-d1) %% d0
> ncross = (-n1) %% n0
> (xi,xs) = cfBound q
> yi' = algebraicEval homography [xi]
> ys' = algebraicEval homography [xs]
> (yi,ys) = (min yi' ys', max yi' ys')
> indp = if d0 /= 0 then n0%%d0 else
> if d1 /= 0 then n1%%d1 else
> error "algebraicBound: denominator 0"
> zint = if qInfinite yi then if ys < 0%%1 then (ys, 1%%0)
> else (-1%%0, ys)
> else if yi < 0%%1 then (yi, 1%%0)
> else (-1%%0, yi)
> nzint = if qInfinite yi then if ys < 0%%1 then (-1%%0, ys)
> else (ys, 1%%0)
> else if yi < 0%%1 then (-1%%0, yi)
> else (yi, 1%%0)
\subsection{Evaluating the homography}
The function @algebraicEval@ evaluates the homography applied to a
QRational number.
> algebraicEval :: Homography -> -- Homography
> [QRational] -> -- Input QRational
> QRational -- Output QRational
> algebraicEval ([n0,n1],[d0,d1]) [x] = (n0*xn + n1*xd) %% (d0*xn + d1*xd)
> where xn = qNumerator x
> xd = qDenominator x
\section{Quadratic Functions}
In a similar fashion we can define the Quadratic Algorithm. This
calculates the following sum:
\[@quadraticAlgorithm@~
\left(\begin{array}[c]{cccc}n_0 & n_1 & n_2 & n_3\\d_0 & d_1 & d_2 & d_3
\end{array}\right)~x~y
= \frac{n_0xy + n_1y + n_2x + n_3}{d_0xy +d_1y +d_2x + d_3}
\]
I could believe that the @quadraticAlgorithm@ might not work correctly
for finite continued fractions, as I haven't checked the conditions on
the homography for definedness. On the positive side, there shouldn't
be any finite continued fractions anyway.
> quadraticAlgorithm :: Homography -> -- Homography
> ContinuedFraction -> -- First Input CF
> ContinuedFraction -> -- Second Input CF
> ContinuedFraction -- Output CF
> quadraticAlgorithm homography@([n0,n1,n2,n3],[d0,d1,d2,d3]) [] []
> = rat2cf (n0 %% d0)
> quadraticAlgorithm homography@([n0,n1,n2,n3],[d0,d1,d2,d3]) [] ys'@(y:ys)
> = if aaDet homography' == 0
> then quadraticAlgorithm (qaInRight homography y) [] ys
> else algebraicAlgorithmNoCheck homography' ys'
> where homography' = ([n0,n2],[d0,d2])
> quadraticAlgorithm homography@([n0,n1,n2,n3],[d0,d1,d2,d3]) xs'@(x:xs) []
> = if aaDet homography' == 0
> then quadraticAlgorithm (qaInLeft homography x) xs []
> else algebraicAlgorithmNoCheck homography' xs'
> where homography' = ([n0,n1],[d0,d1])
> quadraticAlgorithm homography xs'@(x:xs) ys'@(y:ys)
> = case quadraticOutput homography [x%%1,y%%1] of
> Just o -> o : quadraticAlgorithm (qaOut homography o) xs' ys'
> Nothing -> quadraticAlgorithm (qaIn homography x y) xs ys
> quadraticOutput :: Homography -> -- Homography
> [QRational] -> -- Leading Term of Input CF
> Maybe Integer -- Leading Term of Output CF
> quadraticOutput = generateOutput quadraticBound quadraticEval
> qaIn :: Homography -> Integer -> Integer -> Homography
> qaIn ([n0,n1,n2,n3],[d0,d1,d2,d3]) x y
> = ([(n0*x+n1)*y+n2*x+n3, n0*y+n2, n0*x+n1, n0],
> [(d0*x+d1)*y+d2*x+d3, d0*y+d2, d0*x+d1, d0])
> qaOut :: Homography -> Integer -> Homography
> qaOut ([n0,n1,n2,n3],[d0,d1,d2,d3]) o
> = ([d0,d1,d2,d3], [n0-d0*o, n1-d1*o, n2-d2*o, n3-d3*o])
> qaInLeft :: Homography -> Integer -> Homography
> qaInLeft ([n0,n1,n2,n3],[d0,d1,d2,d3]) x
> = ([n0*x+n1, n0, n2*x+n3, n2], [d0*x+d1, d0, d2*x+d3, d2])
> qaInRight :: Homography -> Integer -> Homography
> qaInRight ([n0,n1,n2,n3],[d0,d1,d2,d3]) y
> = ([n0*y+n2, n1*y+n3, n0, n1], [d0*y+d2, d1*y+d3, d0, d1])
> aaDet :: Homography -> Integer
> aaDet ([n0,n1],[d0,d1]) = n0*d1 - n1*d0
[Worry that it would be quicker to do Integer arithmetic in qaEval.]
> quadraticEval :: Homography -> -- homography
> [QRational] -> -- QRational
> QRational -- result
> quadraticEval (ns,ds) [x, y]
> = ((n0*x+n1)*y + (n2*x+n3)) / ((d0*x+d1)*y + (d2*x+d3))
> where [n0,n1,n2,n3] = map (%% 1) ns
> [d0,d1,d2,d3] = map (%% 1) ds
> quadraticBound :: Homography -> -- homography
> [QRational] -> -- terms of CFs
> Interval -- interval
> quadraticBound ([n0,n1,n2,n3],[d0,d1,d2,d3]) [x, y]
> = union ints
> where (xi,xs) = cfBound x
> (yi,ys) = cfBound y
> ints = [algebraicBound (aax x) [y] | x <- [xi,xs]] ++
> [algebraicBound (aay y) [x] | y <- [yi,ys]]
> aax x = ([n0*xn+n1*xd,n2*xn+n3*xd],[d0*xn+d1*xd,d2*xn+d3*xd])
> where xn = qNumerator x
> xd = qDenominator x
> aay y = ([n0*yn+n2*yd,n1*yn+n3*yd],[d0*yn+d2*yd,d1*yn+d3*yd])
> where yn = qNumerator y
> yd = qDenominator y
\section{Square Roots}
\subsection{Rational Square Roots}
Notice that there is no way we can test for rationalness of these
continued fractions.
> cfRat2CFSqrt :: QRational -> ContinuedFraction
> cfRat2CFSqrt q
> = map snd (next ([2*d*z,d,n-d*z*z,0], z))
> where n = qNumerator q
> d = qDenominator q
> s = intSquareRoot (n*d)
> z = qRound (s%%d)
> t = s - z*d
> next :: ([Integer],Integer) -> [([Integer],Integer)]
> next (s,z) = if d0 == 0
> then [(s,z)]
> else (s,z): next ([n0',n1',d0',d1'],z')
> where [n0,n1,d0,d1] = s
> n0' = d0*z'+d1
> n1' = d0
> d0' = n0*z'+n1-z'*n0'
> d1' = n0-z'*d0
> z' = qRound ((n0+t) %% d0)
\subsection{Integer Square Roots}
The function @intSquareRoot@ returns a non-negative integer $n$, such
that $n = \lfloor\sqrt{@x@}\, \rceil$.
> intSquareRoot :: Integer -> Integer
> intSquareRoot x = until satisfy (improve x) x
> where satisfy y = y2+y >= x && y2-y < x
> where y2 = y*y
> improve x y = (y*y+x) `ddiv` (2*y)
> ddiv x y = if (r <= y `div` 2) then q else q+1
> where (q,r) = divMod x y
> root2 = cfRat2CFSqrt (2%%1)
> root3 = cfRat2CFSqrt (3%%1)
> test0 :: ContinuedFraction
> test0 = f 1 where f n = 0: 10^n: f (n+1)
For debugging.
\section{Conversion of Continued Fractions to QRationals}
I've chosen to print out 10 decimal places; you can add more (or less)
by changing @decimals@.
> decimals = 10 :: Int
> accuracy = 1 %% (10^(toInteger decimals)) :: QRational
To convert a continued fraction to a rational within the given
accuracy, we use a variant of the Algebraic Algorithm.
> cf2Rat :: ContinuedFraction -> QRational
> cf2Rat = outputAlgorithm ([1,0],[0,1])
> outputAlgorithm :: Homography -> -- Homography
> ContinuedFraction -> -- Input CF
> QRational -- Final Approximation
> outputAlgorithm homography@([n0,n1],[d0,d1]) [] = n0 %% d0
> outputAlgorithm homography@([n0,n1],[d0,d1]) xs'@(x:xs)
> = if i <= s && s-i < accuracy
> then algebraicEval homography [x%%1]
> else outputAlgorithm ([n0*x+n1,n0],[d0*x+d1,d0]) xs
> where (i,s) = algebraicBound homography [x%%1]
> integerFraction :: ContinuedFraction -> -- Input CF
> (Integer,ContinuedFraction) -- Answer
> integerFraction x = (o, algebraicAlgorithm h' f)
> where (h,f) = intFrac (([1,0],[0,1]),x)
> o = qRound (algebraicEval h [head f%%1])
> h' = ([n0-d0*o,n1-d1*o],[d0,d1])
> where ([n0,n1],[d0,d1]) = h
> intFrac :: (Homography,ContinuedFraction) -> (Homography,ContinuedFraction)
> intFrac st@(_,[_]) = st
> intFrac st@(homography@([n0,n1],[d0,d1]),(x:xs))
> = if i <= s && s-i < 1 %% 10
> then st
> else intFrac (([n0*x+n1,n0],[d0*x+d1,d0]), xs)
> where (i,s) = algebraicBound homography [x%%1]
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