Safe Haskell | None |
---|---|

Language | Haskell98 |

- evaluate :: C a => [a] -> a -> a
- evaluateCoeffVector :: C a v => [v] -> a -> v
- evaluateArgVector :: (C a v, C v) => [a] -> v -> v
- approximate :: C a => [a] -> a -> [a]
- approximateCoeffVector :: C a v => [v] -> a -> [v]
- approximateArgVector :: (C a v, C v) => [a] -> v -> [v]
- alternate :: C a => [a] -> [a]
- holes2 :: C a => [a] -> [a]
- holes2alternate :: C a => [a] -> [a]
- insertHoles :: C a => Int -> [a] -> [a]
- add :: C a => [a] -> [a] -> [a]
- sub :: 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])
- divMod :: (C a, C a) => [a] -> [a] -> ([a], [a])
- divide :: C a => [a] -> [a] -> [a]
- divideStripZero :: (C a, C 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)
- sin :: C a => (a -> (a, a)) -> [a] -> [a]
- cos :: 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]
- asin :: C a => (a -> a) -> (a -> a) -> [a] -> [a]
- acos :: C a => (a -> a) -> (a -> a) -> [a] -> [a]
- compose :: C a => [a] -> [a] -> [a]
- composeTaylor :: C a => (a -> [a]) -> [a] -> [a]
- inv :: (Eq a, C a) => [a] -> (a, [a])
- invDiff :: C a => [a] -> (a, [a])

# Documentation

evaluateCoeffVector :: C a v => [v] -> a -> v Source #

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

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

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

approximateArgVector :: (C a v, C v) => [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`

insertHoles :: C a => Int -> [a] -> [a] Source #

For power series of `f x`

, compute the power series of `f(x^n)`

.

# 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.

pow :: C a => (a -> a) -> a -> [a] -> [a] Source #

Input series must start with a non-zero term, even better with a positive one.

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 => [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.

inv :: (Eq a, C a) => [a] -> (a, [a]) Source #

This function returns the series of the inverse function in the form: (point of the expansion, power series).

That is, say we have the equation:

y = a + f(x)

where function f is given by a power series with f(0) = 0. We want to solve for x:

x = f^-1(y-a)

If you pass the power series of `a+f(x)`

to `inv`

,
you get `(a, f^-1)`

as answer, where `f^-1`

is a power series.

The linear term of `f`

(the coefficient of `x`

) must be non-zero.

This needs cubic run-time and thus is exceptionally slow. Computing inverse series for special power series might be faster.