Stability | provisional |
---|---|
Maintainer | numericprelude@henning-thielemann.de |
Exact Real Arithmetic - Computable reals. Inspired by ''The most unreliable technique for computing pi.'' See also http://www.haskell.org/haskellwiki/Exact_real_arithmetic .
- type T = (Exponent, Mantissa)
- type FixedPoint = (Integer, Mantissa)
- type Mantissa = [Digit]
- type Digit = Int
- type Exponent = Int
- type Basis = Int
- moveToZero :: Basis -> Digit -> (Digit, Digit)
- checkPosDigit :: Basis -> Digit -> Digit
- checkDigit :: Basis -> Digit -> Digit
- nonNegative :: Basis -> T -> T
- nonNegativeMant :: Basis -> Mantissa -> Mantissa
- compress :: Basis -> T -> T
- compressFirst :: Basis -> T -> T
- compressMant :: Basis -> Mantissa -> Mantissa
- compressSecondMant :: Basis -> Mantissa -> Mantissa
- prependDigit :: Basis -> T -> T
- trim :: T -> T
- trimUntil :: Exponent -> T -> T
- trimOnce :: T -> T
- decreaseExp :: Basis -> T -> T
- pumpFirst :: Basis -> Mantissa -> Mantissa
- decreaseExpFP :: Basis -> (Exponent, FixedPoint) -> (Exponent, FixedPoint)
- pumpFirstFP :: Basis -> FixedPoint -> FixedPoint
- negativeExp :: Basis -> T -> T
- fromBaseCardinal :: Basis -> Integer -> T
- fromBaseInteger :: Basis -> Integer -> T
- mantissaToNum :: C a => Basis -> Mantissa -> a
- mantissaFromCard :: C a => Basis -> a -> Mantissa
- mantissaFromInt :: C a => Basis -> a -> Mantissa
- mantissaFromFixedInt :: Basis -> Exponent -> Integer -> Mantissa
- fromBaseRational :: Basis -> Rational -> T
- toFixedPoint :: Basis -> T -> FixedPoint
- fromFixedPoint :: Basis -> FixedPoint -> T
- toDouble :: Basis -> T -> Double
- fromDouble :: Basis -> Double -> T
- fromDoubleApprox :: Basis -> Double -> T
- fromDoubleRough :: Basis -> Double -> T
- mantLengthDouble :: Basis -> Exponent
- liftDoubleApprox :: Basis -> (Double -> Double) -> T -> T
- liftDoubleRough :: Basis -> (Double -> Double) -> T -> T
- showDec :: Exponent -> T -> String
- showHex :: Exponent -> T -> String
- showBin :: Exponent -> T -> String
- showBasis :: Basis -> Exponent -> T -> String
- powerBasis :: Basis -> Exponent -> T -> T
- rootBasis :: Basis -> Exponent -> T -> T
- fromBasis :: Basis -> Basis -> T -> T
- fromBasisMant :: Basis -> Basis -> Mantissa -> Mantissa
- cmp :: Basis -> T -> T -> Ordering
- lessApprox :: Basis -> Exponent -> T -> T -> Bool
- trunc :: Exponent -> T -> T
- equalApprox :: Basis -> Exponent -> T -> T -> Bool
- ifLazy :: Basis -> Bool -> T -> T -> T
- mean2 :: Basis -> (Digit, Digit) -> (Digit, Digit) -> Digit
- withTwoMantissas :: Mantissa -> Mantissa -> a -> ((Digit, Mantissa) -> (Digit, Mantissa) -> a) -> a
- align :: Basis -> Exponent -> T -> T
- alignMant :: Basis -> Exponent -> T -> Mantissa
- absolute :: T -> T
- absMant :: Mantissa -> Mantissa
- fromLaurent :: T Int -> T
- toLaurent :: T -> T Int
- liftLaurent2 :: (T Int -> T Int -> T Int) -> T -> T -> T
- liftLaurentMany :: ([T Int] -> T Int) -> [T] -> T
- add :: Basis -> T -> T -> T
- sub :: Basis -> T -> T -> T
- neg :: Basis -> T -> T
- addSome :: Basis -> [T] -> T
- addMany :: Basis -> [T] -> T
- type Series = [(Exponent, T)]
- series :: Basis -> Series -> T
- seriesPlain :: Basis -> Series -> T
- splitAtPadZero :: Int -> Mantissa -> (Mantissa, Mantissa)
- splitAtMatchPadZero :: [()] -> Mantissa -> (Mantissa, Mantissa)
- truncSeriesSummands :: Series -> Series
- scale :: Basis -> Digit -> T -> T
- scaleSimple :: Basis -> T -> T
- scaleMant :: Basis -> Digit -> Mantissa -> Mantissa
- mulSeries :: Basis -> T -> T -> Series
- mul :: Basis -> T -> T -> T
- divide :: Basis -> T -> T -> T
- divMant :: Basis -> Mantissa -> Mantissa -> Mantissa
- divMantSlow :: Basis -> Mantissa -> Mantissa -> Mantissa
- reciprocal :: Basis -> T -> T
- divIntMant :: Basis -> Int -> Mantissa -> Mantissa
- divIntMantInf :: Basis -> Int -> Mantissa -> Mantissa
- divInt :: Basis -> Digit -> T -> T
- sqrt :: Basis -> T -> T
- sqrtDriver :: Basis -> (Basis -> FixedPoint -> Mantissa) -> T -> T
- sqrtMant :: Basis -> Mantissa -> Mantissa
- sqrtFP :: Basis -> FixedPoint -> Mantissa
- sqrtNewton :: Basis -> T -> T
- sqrtFPNewton :: Basis -> FixedPoint -> Mantissa
- lazyInits :: [a] -> [[a]]
- expSeries :: Basis -> T -> Series
- expSmall :: Basis -> T -> T
- expSeriesLazy :: Basis -> T -> Series
- expSmallLazy :: Basis -> T -> T
- exp :: Basis -> T -> T
- intPower :: Basis -> Integer -> T -> T -> T -> T
- cardPower :: Basis -> Integer -> T -> T -> T
- powerSeries :: Basis -> Rational -> T -> Series
- powerSmall :: Basis -> Rational -> T -> T
- power :: Basis -> Rational -> T -> T
- root :: Basis -> Integer -> T -> T
- cosSinhSmall :: Basis -> T -> (T, T)
- cosSinSmall :: Basis -> T -> (T, T)
- cosSinFourth :: Basis -> T -> (T, T)
- cosSin :: Basis -> T -> (T, T)
- tan :: Basis -> T -> T
- cot :: Basis -> T -> T
- lnSeries :: Basis -> T -> Series
- lnSmall :: Basis -> T -> T
- lnNewton :: Basis -> T -> T
- lnNewton' :: Basis -> T -> T
- ln :: Basis -> T -> T
- angle :: Basis -> (T, T) -> T
- arctanSeries :: Basis -> T -> Series
- arctanSmall :: Basis -> T -> T
- arctanStem :: Basis -> Int -> T
- arctan :: Basis -> T -> T
- arctanClassic :: Basis -> T -> T
- zero :: T
- one :: T
- minusOne :: T
- eConst :: Basis -> T
- recipEConst :: Basis -> T
- piConst :: Basis -> T
- sliceVertPair :: [a] -> [(a, a)]
types
type FixedPoint = (Integer, Mantissa)Source
basic helpers
checkPosDigit :: Basis -> Digit -> DigitSource
checkDigit :: Basis -> Digit -> DigitSource
nonNegative :: Basis -> T -> TSource
Converts all digits to non-negative digits, that is the usual positional representation. However the conversion will fail when the remaining digits are all zero. (This cannot be improved!)
nonNegativeMant :: Basis -> Mantissa -> MantissaSource
Requires, that no digit is (basis-1)
or (1-basis)
.
The leading digit might be negative and might be -basis
or basis
.
compressFirst :: Basis -> T -> TSource
Compress first digit. May prepend a digit.
compressMant :: Basis -> Mantissa -> MantissaSource
Does not prepend a digit.
compressSecondMant :: Basis -> Mantissa -> MantissaSource
Compress second digit. Sometimes this is enough to keep the digits in the admissible range. Does not prepend a digit.
prependDigit :: Basis -> T -> TSource
decreaseExp :: Basis -> T -> TSource
Accept a high leading digit for the sake of a reduced exponent.
This eliminates one leading digit.
Like pumpFirst
but with exponent management.
pumpFirst :: Basis -> Mantissa -> MantissaSource
Merge leading and second digit.
This is somehow an inverse of compressMant
.
decreaseExpFP :: Basis -> (Exponent, FixedPoint) -> (Exponent, FixedPoint)Source
pumpFirstFP :: Basis -> FixedPoint -> FixedPointSource
negativeExp :: Basis -> T -> TSource
Make sure that a number with absolute value less than 1 has a (small) negative exponent. Also works with zero because it chooses an heuristic exponent for stopping.
conversions
integer
fromBaseCardinal :: Basis -> Integer -> TSource
fromBaseInteger :: Basis -> Integer -> TSource
mantissaToNum :: C a => Basis -> Mantissa -> aSource
mantissaFromCard :: C a => Basis -> a -> MantissaSource
mantissaFromInt :: C a => Basis -> a -> MantissaSource
rational
fromBaseRational :: Basis -> Rational -> TSource
fixed point
toFixedPoint :: Basis -> T -> FixedPointSource
Split into integer and fractional part.
fromFixedPoint :: Basis -> FixedPoint -> TSource
floating point
fromDouble :: Basis -> Double -> TSource
cf. Numeric.floatToDigits
fromDoubleApprox :: Basis -> Double -> TSource
Only return as much digits as are contained in Double. This will speedup further computations.
fromDoubleRough :: Basis -> Double -> TSource
text
basis
powerBasis :: Basis -> Exponent -> T -> TSource
Convert from a b
basis representation to a b^e
basis.
Works well with every exponent.
rootBasis :: Basis -> Exponent -> T -> TSource
Convert from a b^e
basis representation to a b
basis.
Works well with every exponent.
fromBasis :: Basis -> Basis -> T -> TSource
Convert between arbitrary bases. This conversion is expensive (quadratic time).
comparison
cmp :: Basis -> T -> T -> OrderingSource
The basis must be at least ***. Note: Equality cannot be asserted in finite time on infinite precise numbers. If you want to assert, that a number is below a certain threshold, you should not call this routine directly, because it will fail on equality. Better round the numbers before comparison.
ifLazy :: Basis -> Bool -> T -> T -> TSource
If all values are completely defined, then it holds
if b then x else y == ifLazy b x y
However if b
is undefined,
then it is at least known that the result is between x
and y
.
mean2 :: Basis -> (Digit, Digit) -> (Digit, Digit) -> DigitSource
mean2 b (x0,x1) (y0,y1)
computes round ((x0.x1 + y0.y1)/2)
,
where x0.x1
and y0.y1
are positional rational numbers
with respect to basis b
withTwoMantissas :: Mantissa -> Mantissa -> a -> ((Digit, Mantissa) -> (Digit, Mantissa) -> a) -> aSource
alignMant :: Basis -> Exponent -> T -> MantissaSource
Get the mantissa in such a form that it fits an expected exponent.
x
and (e, alignMant b e x)
represent the same number.
arithmetic
fromLaurent :: T Int -> TSource
addSome :: Basis -> [T] -> TSource
Add at most basis
summands.
More summands will violate the allowed digit range.
addMany :: Basis -> [T] -> TSource
Add many numbers efficiently by computing sums of sub lists with only little carry propagation.
series :: Basis -> Series -> TSource
Add an infinite number of numbers. You must provide a list of estimate of the current remainders. The estimates must be given as exponents of the remainder. If such an exponent is too small, the summation will be aborted. If exponents are too big, computation will become inefficient.
seriesPlain :: Basis -> Series -> TSource
splitAtPadZero :: Int -> Mantissa -> (Mantissa, Mantissa)Source
Like splitAt
,
but it pads with zeros if the list is too short.
This way it preserves
length (fst (splitAtPadZero n xs)) == n
truncSeriesSummands :: Series -> SeriesSource
help showing series summands
scaleSimple :: Basis -> T -> TSource
divide :: Basis -> T -> T -> TSource
Undefined if the divisor is zero - of course. Because it is impossible to assert that a real is zero, the routine will not throw an error in general.
ToDo: Rigorously derive the minimal required magnitude of the leading divisor digit.
reciprocal :: Basis -> T -> TSource
divIntMant :: Basis -> Int -> Mantissa -> MantissaSource
Fast division for small integral divisors, which occur for instance in summands of power series.
algebraic functions
sqrtDriver :: Basis -> (Basis -> FixedPoint -> Mantissa) -> T -> TSource
sqrtFP :: Basis -> FixedPoint -> MantissaSource
Square root.
We need a leading digit of type Integer,
because we have to collect up to 4 digits.
This presentation can also be considered as FixedPoint
.
ToDo: Rigorously derive the minimal required magnitude of the leading digit of the root.
Mathematically the n
th digit of the square root
depends roughly only on the first n
digits of the input.
This is because sqrt (1+eps)
.
However this implementation requires equalApprox
1 + eps/22*n
input digits
for emitting n
digits.
This is due to the repeated use of compressMant
.
It would suffice to fully compress only every basis
th iteration (digit)
and compress only the second leading digit in each iteration.
Can the involved operations be made lazy enough to solve
y = (x+frac)^2
by
frac = (y-x^2-frac^2) / (2*x)
?
sqrtNewton :: Basis -> T -> TSource
sqrtFPNewton :: Basis -> FixedPoint -> MantissaSource
Newton iteration doubles the number of correct digits in every step. Thus we process the data in chunks of sizes of powers of two. This way we get fastest computation possible with Newton but also more dependencies on input than necessary. The question arises whether this implementation still fits the needs of computational reals. The input is requested as larger and larger chunks, and the input itself might be computed this way, e.g. a repeated square root. Requesting one digit too much, requires the double amount of work for the input computation, which in turn multiplies time consumption by a factor of four, and so on.
Optimal fast implementation of one routine does not preserve fast computation of composed computations.
The routine assumes, that the integer parts is at least b^2.
lazyInits :: [a] -> [[a]]Source
List.inits is defined by
inits = foldr (x ys -> [] : map (x:) ys) [[]]
This is too strict for our application.
Prelude> List.inits (0:1:2:undefined) [[],[0],[0,1]*** Exception: Prelude.undefined
The following routine is more lazy than inits
and even lazier than Data.List.HT.inits
from utility-ht
package,
but it is restricted to infinite lists.
This degree of laziness is needed for sqrtFP
.
Prelude> lazyInits (0:1:2:undefined) [[],[0],[0,1],[0,1,2],[0,1,2,*** Exception: Prelude.undefined
transcendent functions
exponential functions
expSeriesLazy :: Basis -> T -> SeriesSource
expSmallLazy :: Basis -> T -> TSource
powerSeries :: Basis -> Rational -> T -> SeriesSource
Residue estimates will only hold for exponents with absolute value below one.
The computation is based on Int
,
thus the denominator should not be too big.
(Say, at most 1000 for 1000000 digits.)
It is not optimal to split the power into pure root and pure power (that means, with integer exponents). The root series can nicely handle all exponents, but for exponents above 1 the series summands rises at the beginning and thus make the residue estimate complicated. For powers with integer exponents the root series turns into the binomial formula, which is just a complicated way to compute a power which can also be determined by simple multiplication.
cosSinFourth :: Basis -> T -> (T, T)Source
Like cosSinSmall
but converges faster.
It calls cosSinSmall
with reduced arguments
using the trigonometric identities
cos (4*x) = 8 * cos x ^ 2 * (cos x ^ 2 - 1) + 1
sin (4*x) = 4 * sin x * cos x * (1 - 2 * sin x ^ 2)
Note that the faster convergence is hidden by the overhead.
The same could be achieved with a fourth power of a complex number.
logarithmic functions
lnNewton :: Basis -> T -> TSource
x' = x - (exp x - y) / exp x = x + (y * exp (-x) - 1)
First, the dependencies on low-significant places are currently
much more than mathematically necessary.
Check
*Number.Positional> expSmall 1000 (-1,100 : replicate 16 0 ++ [undefined])
(0,[1,105,171,-82,76*** Exception: Prelude.undefined
Every multiplication cut off two trailing digits.
*Number.Positional> nest 8 (mul 1000 (-1,repeat 1)) (-1,100 : replicate 16 0 ++ [undefined])
(-9,[101,*** Exception: Prelude.undefined
Possibly the dependencies of expSmall
could be resolved by not computing mul
immediately
but computing mul
series which are merged and subsequently added.
But this would lead to an explosion of series.
Second, even if the dependencies of all atomic operations
are reduced to a minimum,
the mathematical dependencies of the whole iteration function
are less than the sums of the parts.
Lets demonstrate this with the square root iteration.
It is
(1.4140 + 21.4140) 2 == 1.414213578500707
(1.4149 + 21.4149) 2 == 1.4142137288854335
That is, the digits 213
do not depend mathematically on x
of 1.414x
,
but their computation depends.
Maybe there is a glorious trick to reduce the computational dependencies
to the mathematical ones.
angle :: Basis -> (T, T) -> TSource
This is an inverse of cosSin
,
also known as atan2
with flipped arguments.
It's very slow because of the computation of sinus and cosinus.
However, because it uses the atan2
implementation as estimator,
the final application of arctan series should converge rapidly.
It could be certainly accelerated by not using cosSin and its fiddling with pi. Instead we could analyse quadrants before calling atan2, then calling cosSinSmall immediately.
arctanSeries :: Basis -> T -> SeriesSource
Arcus tangens of arguments with absolute value less than 1 / sqrt 3
.
arctanSmall :: Basis -> T -> TSource
arctanStem :: Basis -> Int -> TSource
Efficient computation of Arcus tangens of an argument of the form 1/n
.
arctan :: Basis -> T -> TSource
This implementation gets the first decimal place for free
by calling the arcus tangens implementation for Double
s.
arctanClassic :: Basis -> T -> TSource
A classic implementation without ''cheating'' with floating point implementations.
For x < 1 / sqrt 3
(1 / sqrt 3 == tan (pi/6)
)
use arctan
power series.
(sqrt 3
is approximately 19/11
.)
For x > sqrt 3
use
arctan x = pi/2 - arctan (1/x)
For other x
use
arctan x = pi/4 - 0.5*arctan ((1-x^2)/2*x)
(which follows from
arctan x + arctan y == arctan ((x+y) / (1-x*y))
which in turn follows from complex multiplication
(1:+x)*(1:+y) == ((1-x*y):+(x+y))
If x
is close to sqrt 3
or 1 / sqrt 3
the computation is quite inefficient.
constants
elementary
transcendental
recipEConst :: Basis -> TSource
auxilary functions
sliceVertPair :: [a] -> [(a, a)]Source