|
|
|
|
|
Description |
Power series, either finite or unbounded. (zipWith does exactly the
right thing to make it work almost transparently.)
|
|
Synopsis |
|
newtype T a = Cons {} | | 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 |
|
|
Constructors | | Instances | |
|
|
|
|
|
|
lift1 :: ([a] -> [a]) -> T a -> T a | Source |
|
|
lift2 :: ([a] -> [a] -> [a]) -> T a -> T a -> T a | Source |
|
|
|
|
|
|
|
|
|
Evaluate (truncated) power series.
|
|
|
|
evalCoeffVector :: C a v => [v] -> a -> v | Source |
|
Evaluate (truncated) power series.
|
|
evaluateCoeffVector :: C a v => T v -> a -> v | Source |
|
|
evalArgVector :: (C a v, C v) => [a] -> v -> v | Source |
|
|
evaluateArgVector :: (C a v, C v) => T a -> v -> v | Source |
|
|
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)
|
|
|
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
|
|
sub :: C a => [a] -> [a] -> [a] | Source |
|
|
add :: C a => [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 |
|
|
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.
|
|
divMod :: (C a, C a) => [a] -> [a] -> ([a], [a]) | Source |
|
|
|
|
recipProgression :: 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 non-zero term.
|
|
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 |
|
|
cos :: C a => (a -> (a, a)) -> [a] -> [a] | Source |
|
|
sin :: 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 |
|
Input series must start with non-zero term.
|
|
derivedLog :: C a => [a] -> [a] | Source |
|
Computes (log x)', that is x'/x
|
|
atan :: C a => (a -> a) -> [a] -> [a] | Source |
|
|
acos :: C a => (a -> a) -> (a -> a) -> [a] -> [a] | Source |
|
|
asin :: C a => (a -> a) -> (a -> a) -> [a] -> [a] | Source |
|
|
|
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.
|
|
inv :: C a => [a] -> (a, [a]) | Source |
|
This function returns the series of the function in the form:
(point of the expansion, power series)
This is exceptionally slow and needs cubic run-time.
|
|
Produced by Haddock version 2.4.2 |