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
- 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.

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/2`2*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 + 2
```

That is, the digits *1.4140) * 2 == 1.414213578500707
(1.4149 + 2*1.4149) * 2 == 1.4142137288854335
`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