| NumericPrelude-0.0: An experimental alternative hierarchy of numeric type classes | Contents | Index |
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Description |
Power series, either finite or unbounded. (zipWith does exactly the
right thing to make it work almost transparently.)
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Synopsis |
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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 b => [b] -> a -> b | | evaluate :: C a b => T b -> a -> b | | approx :: C a b => [b] -> a -> [b] | | approximate :: C a b => T b -> a -> [b] | | alternate :: C a => [a] -> [a] | | holes2 :: C a => [a] -> [a] | | holes2alternate :: C a => [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]) | | 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) | | 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, 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]) |
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Documentation |
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newtype T a |
Constructors | | Instances | |
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fromCoeffs :: [a] -> T a |
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lift0 :: [a] -> T a |
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lift1 :: ([a] -> [a]) -> T a -> T a |
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lift2 :: ([a] -> [a] -> [a]) -> T a -> T a -> T a |
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const :: a -> T a |
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appPrec :: Int |
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truncate :: Int -> T a -> T a |
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eval :: C a b => [b] -> a -> b |
Evaluate (truncated) power series.
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evaluate :: C a b => T b -> a -> b |
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approx :: C a b => [b] -> a -> [b] |
Evaluate approximations that is evaluated all truncations of the series.
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approximate :: C a b => T b -> a -> [b] |
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Simple series manipulation
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alternate :: C a => [a] -> [a] |
For the series of a real function f
compute the series for x -> f (-x)
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holes2 :: C a => [a] -> [a] |
For the series of a real function f
compute the series for x -> (f x + f (-x)) / 2
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holes2alternate :: C a => [a] -> [a] |
For the series of a real function f
compute the real series for x -> (f (i*x) + f (-i*x)) / 2
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Series arithmetic
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add :: C a => [a] -> [a] -> [a] |
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sub :: C a => [a] -> [a] -> [a] |
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negate :: C a => [a] -> [a] |
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scale :: C a => a -> [a] -> [a] |
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mul :: C a => [a] -> [a] -> [a] |
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stripLeadZero :: C a => [a] -> [a] -> ([a], [a]) |
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divide :: C a => [a] -> [a] -> [a] |
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
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divideStripZero :: (C a, C a) => [a] -> [a] -> [a] |
Divide two series also if the divisor has leading zeros.
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divMod :: (C a, C a) => [a] -> [a] -> ([a], [a]) |
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progression :: C a => [a] |
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recipProgression :: C a => [a] |
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differentiate :: C a => [a] -> [a] |
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integrate :: C a => a -> [a] -> [a] |
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sqrt :: C a => (a -> a) -> [a] -> [a] |
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.
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pow :: C a => (a -> a) -> a -> [a] -> [a] |
Input series must start with non-zero term.
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exp :: C a => (a -> a) -> [a] -> [a] |
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'
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sinCos :: C a => (a -> (a, a)) -> [a] -> ([a], [a]) |
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sinCosScalar :: C a => a -> (a, a) |
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sin :: C a => (a -> (a, a)) -> [a] -> [a] |
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cos :: C a => (a -> (a, a)) -> [a] -> [a] |
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tan :: C a => (a -> (a, a)) -> [a] -> [a] |
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log :: C a => (a -> a) -> [a] -> [a] |
Input series must start with non-zero term.
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derivedLog :: C a => [a] -> [a] |
Computes (log x)', that is x'/x
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atan :: C a => (a -> a) -> [a] -> [a] |
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asin :: C a => (a -> a) -> (a -> a) -> [a] -> [a] |
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acos :: C a => (a -> a) -> (a -> a) -> [a] -> [a] |
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compose :: (C a, C a) => T a -> T a -> T a |
It fulfills
evaluate x . evaluate y == evaluate (compose x y)
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comp :: C a => [a] -> [a] -> [a] |
Since the inner series must start with a zero,
the first term is omitted in y.
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composeTaylor :: C a => (a -> [a]) -> [a] -> [a] |
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.
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inv :: C a => [a] -> (a, [a]) |
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.
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Produced by Haddock version 0.7 |