continued-fractions-0.9.1.0: Continued fractions.

Math.ContinuedFraction

Synopsis

Documentation

data CF a Source

A continued fraction. Constructed by `cf` or `gcf`.

Instances

 Functor CF Show a => Show (CF a)

cf :: a -> [a] -> CF aSource

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

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.

setDenominators :: Fractional a => [a] -> CF a -> CF aSource

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.

partitionCF :: Fractional a => CF a -> (CF a, CF a)Source

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.

evenCF :: Fractional a => CF a -> CF aSource

Computes the even part of a `CF` (that is, a new `CF` whose convergents are the even-indexed convergents of the original).

oddCF :: Fractional a => CF a -> CF aSource

Computes the odd part of a `CF` (that is, a new `CF` whose convergents are the odd-indexed convergents of the original).

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{n-1}

B{n+1} = b{n+1}Bn + a{n+1}B{n-1}

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{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`

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{n-1}

D{n} = 1 / (b{n} + a{n} * D{n-1})

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 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.

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.