numeric-prelude-0.4.1: An experimental alternative hierarchy of numeric type classes

MathObj.PowerSeries.Core

Synopsis

# Documentation

evaluate :: C a => [a] -> a -> aSource

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

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

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

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

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

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

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

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

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

divMod :: (C a, 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

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

integrate :: C a => 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'
```

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

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

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

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

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

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

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

Computes `(log x)'`, that is `x'/x`

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

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

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

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.

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