 data CF a
 cf :: a > [a] > CF a
 gcf :: a > [(a, a)] > CF a
 asCF :: Fractional a => CF a > (a, [a])
 asGCF :: Num a => CF a > (a, [(a, a)])
 truncateCF :: Int > CF a > CF a
 equiv :: Num a => [a] > CF a > CF a
 setNumerators :: Fractional a => [a] > CF a > CF a
 setDenominators :: Fractional a => [a] > CF a > CF a
 partitionCF :: Fractional a => CF a > (CF a, CF a)
 evenCF :: Fractional a => CF a > CF a
 oddCF :: Fractional a => CF a > CF a
 convergents :: Fractional a => CF a > [a]
 steed :: Fractional a => CF a > [a]
 lentz :: Fractional a => CF a > [a]
 lentzWith :: Fractional a => (a > b) > (b > b > b) > (b > b) > CF a > [b]
 modifiedLentz :: Fractional a => a > CF a > [[a]]
 modifiedLentzWith :: Fractional a => (a > b) > (b > b > b) > (b > b) > a > CF a > [[b]]
 sumPartialProducts :: Num a => [a] > CF a
Documentation
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)]
.
gcf :: a > [(a, a)] > CF aSource
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)))))
asCF :: Fractional a => CF a > (a, [a])Source
Extract the partial denominators of a CF
, normalizing it if necessary so
that all the partial numerators are 1.
asGCF :: Num a => CF a > (a, [(a, a)])Source
Extract all the partial numerators and partial denominators of a CF
.
truncateCF :: Int > CF a > CF aSource
Truncate a CF
to the specified number of partial numerators and denominators.
equiv :: Num a => [a] > CF a > CF aSource
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.
setNumerators :: Fractional a => [a] > CF a > CF aSource
setDenominators :: Fractional a => [a] > CF a > CF aSource
partitionCF :: Fractional a => CF a > (CF a, CF a)Source
evenCF :: Fractional a => CF a > CF aSource
oddCF :: Fractional a => CF a > CF aSource
convergents :: Fractional a => CF a > [a]Source
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{n1}
B{n+1} = b{n+1}Bn + a{n+1}B{n1}
The convergents are then Xn = An/Bn
steed :: Fractional a => CF a > [a]Source
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{i1})
dx1 = a1 / b1
dx{i} = (b{i} * D{i}  1) * dx{i1}
x0 = b0
x{i} = x{i1} + dx{i}
The convergents are given by scanl (+) b0 dxs
lentz :: Fractional a => CF a > [a]Source
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{n1}
D{n} = 1 / (b{n} + a{n} * D{n1})
The convergents are given by scanl (*) b0 (zipWith (*) cs ds)
lentzWith :: Fractional a => (a > b) > (b > b > b) > (b > b) > CF a > [b]Source
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)
modifiedLentz :: Fractional a => a > CF a > [[a]]Source
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 1e30 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.
modifiedLentzWith :: Fractional a => (a > b) > (b > b > b) > (b > b) > a > CF a > [[b]]Source
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
.
sumPartialProducts :: Num a => [a] > CF aSource
Euler's formula for computing sum (scanl1 (*) xs)
.
Successive convergents of the resulting CF
are successive partial sums
in the series.