Power series, either finite or unbounded. (zipWith does exactly the right thing to make it work almost transparently.)

- newtype T a = Cons {
- coeffs :: [a]

- fromCoeffs :: [a] -> T a
- lift0 :: [a] -> T a
- lift1 :: ([a] -> [a]) -> T a -> T a
- lift2 :: ([a] -> [a] -> [a]) -> T a -> T a -> T a
- const :: a -> T a
- appPrec :: Int
- truncate :: Int -> T a -> T a
- eval :: C a => [a] -> a -> a
- evaluate :: C a => T a -> a -> a
- evalCoeffVector :: C a v => [v] -> a -> v
- evaluateCoeffVector :: C a v => T v -> a -> v
- evalArgVector :: (C a v, C v) => [a] -> v -> v
- evaluateArgVector :: (C a v, C v) => T a -> v -> v
- approx :: C a => [a] -> a -> [a]
- approximate :: C a => T a -> a -> [a]
- approxCoeffVector :: C a v => [v] -> a -> [v]
- approximateCoeffVector :: C a v => T v -> a -> [v]
- approxArgVector :: (C a v, C v) => [a] -> v -> [v]
- approximateArgVector :: (C a v, C v) => T a -> v -> [v]
- alternate :: C a => [a] -> [a]
- holes2 :: C a => [a] -> [a]
- holes2alternate :: C a => [a] -> [a]
- sub :: C a => [a] -> [a] -> [a]
- add :: C a => [a] -> [a] -> [a]
- negate :: C a => [a] -> [a]
- scale :: C a => a -> [a] -> [a]
- mul :: C a => [a] -> [a] -> [a]
- stripLeadZero :: C a => [a] -> [a] -> ([a], [a])
- divide :: C a => [a] -> [a] -> [a]
- divideStripZero :: (C a, C a) => [a] -> [a] -> [a]
- divMod :: (C a, C a) => [a] -> [a] -> ([a], [a])
- progression :: C a => [a]
- recipProgression :: C a => [a]
- differentiate :: C a => [a] -> [a]
- integrate :: C a => a -> [a] -> [a]
- sqrt :: C a => (a -> a) -> [a] -> [a]
- pow :: C a => (a -> a) -> a -> [a] -> [a]
- exp :: C a => (a -> a) -> [a] -> [a]
- sinCos :: C a => (a -> (a, a)) -> [a] -> ([a], [a])
- sinCosScalar :: C a => a -> (a, a)
- cos :: C a => (a -> (a, a)) -> [a] -> [a]
- sin :: C a => (a -> (a, a)) -> [a] -> [a]
- tan :: C a => (a -> (a, a)) -> [a] -> [a]
- log :: C a => (a -> a) -> [a] -> [a]
- derivedLog :: C a => [a] -> [a]
- atan :: C a => (a -> a) -> [a] -> [a]
- acos :: C a => (a -> a) -> (a -> a) -> [a] -> [a]
- asin :: C a => (a -> a) -> (a -> a) -> [a] -> [a]
- compose :: (C a, C a) => T a -> T a -> T a
- comp :: C a => [a] -> [a] -> [a]
- composeTaylor :: C a => (a -> [a]) -> [a] -> [a]
- inv :: C a => [a] -> (a, [a])

# Documentation

fromCoeffs :: [a] -> T aSource

evalCoeffVector :: C a v => [v] -> a -> vSource

Evaluate (truncated) power series.

evaluateCoeffVector :: C a v => T v -> a -> vSource

evalArgVector :: (C a v, C v) => [a] -> v -> vSource

evaluateArgVector :: (C a v, C v) => T a -> v -> vSource

approx :: C a => [a] -> a -> [a]Source

Evaluate approximations that is evaluate all truncations of the series.

approximate :: C a => T a -> a -> [a]Source

approxCoeffVector :: C a v => [v] -> a -> [v]Source

Evaluate approximations that is evaluate all truncations of the series.

approximateCoeffVector :: C a v => T v -> a -> [v]Source

approxArgVector :: (C a v, C v) => [a] -> v -> [v]Source

Evaluate approximations that is evaluate all truncations of the series.

approximateArgVector :: (C a v, C v) => T a -> v -> [v]Source

# Simple series manipulation

alternate :: C a => [a] -> [a]Source

For the series of a real function `f`

compute the series for `x -> f (-x)`

holes2 :: C a => [a] -> [a]Source

For the series of a real function `f`

compute the series for `x -> (f x + f (-x)) / 2`

holes2alternate :: C a => [a] -> [a]Source

For the series of a real function `f`

compute the real series for `x -> (f (i*x) + f (-i*x)) / 2`

# Series arithmetic

stripLeadZero :: C a => [a] -> [a] -> ([a], [a])Source

divide :: C a => [a] -> [a] -> [a]Source

Divide two series where the absolute term of the divisor is non-zero. That is, power series with leading non-zero terms are the units in the ring of power series.

Knuth: Seminumerical algorithms

divideStripZero :: (C a, C a) => [a] -> [a] -> [a]Source

Divide two series also if the divisor has leading zeros.

progression :: C a => [a]Source

recipProgression :: C a => [a]Source

differentiate :: C a => [a] -> [a]Source

sqrt :: C a => (a -> a) -> [a] -> [a]Source

We need to compute the square root only of the first term. That is, if the first term is rational, then all terms of the series are rational.

exp :: C a => (a -> a) -> [a] -> [a]Source

The first term needs a transcendent computation but the others do not. That's why we accept a function which computes the first term.

(exp . x)' = (exp . x) * x' (sin . x)' = (cos . x) * x' (cos . x)' = - (sin . x) * x'

sinCosScalar :: C a => a -> (a, a)Source

derivedLog :: C a => [a] -> [a]Source

Computes `(log x)'`

, that is `x'/x`

compose :: (C a, C a) => T a -> T a -> T aSource

It fulfills
` evaluate x . evaluate y == evaluate (compose x y) `

comp :: C a => [a] -> [a] -> [a]Source

Since the inner series must start with a zero, the first term is omitted in y.

composeTaylor :: C a => (a -> [a]) -> [a] -> [a]Source

Compose two power series where the outer series can be developed for any expansion point. To be more precise: The outer series must be expanded with respect to the leading term of the inner series.