Modulenumeric-prelude-0.4.4Haskell98
Number.Positional
Exact Real Arithmetic - Computable reals. Inspired by ''The most unreliable technique for computing pi.'' See also http://www.haskell.org/haskellwiki/Exact_real_arithmetic .
Contents
Synopsis
- type T
- type FixedPoint
- type Mantissa
- type Digit
- type Exponent
- type Basis
- value moveToZero
- value checkPosDigit
- value checkDigit
- value nonNegative
- value nonNegativeMant
- value compress
- value compressFirst
- value compressMant
- value compressSecondMant
- value prependDigit
- value trim
- value trimUntil
- value trimOnce
- value decreaseExp
- value pumpFirst
- value decreaseExpFP
- value pumpFirstFP
- value negativeExp
- value fromBaseCardinal
- value fromBaseInteger
- value mantissaToNum
- value mantissaFromCard
- value mantissaFromInt
- value mantissaFromFixedInt
- value fromBaseRational
- value toFixedPoint
- value fromFixedPoint
- value toDouble
- value fromDouble
- value fromDoubleApprox
- value fromDoubleRough
- value mantLengthDouble
- value liftDoubleApprox
- value liftDoubleRough
- value showDec
- value showHex
- value showBin
- value showBasis
- value powerBasis
- value rootBasis
- value fromBasis
- value fromBasisMant
- value cmp
- value lessApprox
- value trunc
- value equalApprox
- value ifLazy
- value mean2
- value withTwoMantissas
- value align
- value alignMant
- value absolute
- value absMant
- value fromLaurent
- value toLaurent
- value liftLaurent2
- value liftLaurentMany
- value add
- value sub
- value neg
- value addSome
- value addMany
- type Series
- value series
- value seriesPlain
- value splitAtPadZero
- value splitAtMatchPadZero
- value truncSeriesSummands
- value scale
- value scaleSimple
- value scaleMant
- value mulSeries
- value mul
- value divide
- value divMant
- value divMantSlow
- value reciprocal
- value divIntMant
- value divIntMantInf
- value divInt
- value sqrt
- value sqrtDriver
- value sqrtMant
- value sqrtFP
- value sqrtNewton
- value sqrtFPNewton
- value lazyInits
- value expSeries
- value expSmall
- value expSeriesLazy
- value expSmallLazy
- value exp
- value intPower
- value cardPower
- value powerSeries
- value powerSmall
- value power
- value root
- value cosSinhSmall
- value cosSinSmall
- value cosSinFourth
- value cosSin
- value tan
- value cot
- value lnSeries
- value lnSmall
- value lnNewton
- value lnNewton'
- value ln
- value angle
- value arctanSeries
- value arctanSmall
- value arctanStem
- value arctan
- value arctanClassic
- value zero
- value one
- value minusOne
- value eConst
- value recipEConst
- value piConst
- value sliceVertPair
- Packagenumeric-prelude-0.4.4
- Exports128
- LanguageHaskell98
- LicenceBSD-3-Clause
- SourcePositional.hs
types
basic helpers
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!)
Requires, that no digit is (basis-1) or (1-basis).
The leading digit might be negative and might be -basis or basis.
May prepend a digit.
Compress first digit. May prepend a digit.
Does not prepend a digit.
Compress second digit. Sometimes this is enough to keep the digits in the admissible range. Does not prepend a digit.
Eliminate leading zero digits. This will fail for zero.
Trim until a minimum exponent is reached. Safe for zeros.
Accept a high leading digit for the sake of a reduced exponent. This eliminates one leading digit. Like pumpFirst but with exponent management.
Merge leading and second digit. This is somehow an inverse of compressMant.
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
rational
fixed point
Split into integer and fractional part.
floating point
cf. floatToDigits
Only return as much digits as are contained in Double. This will speedup further computations.
text
Show a number with respect to basis 10^e.
basis
Convert from a b basis representation to a b^e basis.
Works well with every exponent.
Convert from a b^e basis representation to a b basis.
Works well with every exponent.
Convert between arbitrary bases. This conversion is expensive (quadratic time).
comparison
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.
If all values are completely defined, then it holds
if b then x else y == ifLazy b x yHowever if b is undefined,
then it is at least known that the result is between x and y.
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
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
Add two numbers but do not eliminate leading zeros.
Add at most basis summands.
More summands will violate the allowed digit range.
Add many numbers efficiently by computing sums of sub lists with only little carry propagation.
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.
Like splitAt,
but it pads with zeros if the list is too short.
This way it preserves
length (fst (splitAtPadZero n xs)) == n
help showing series summands
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.
Fast division for small integral divisors, which occur for instance in summands of power series.
algebraic functions
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 nth digit of the square root
depends roughly only on the first n digits of the input.
This is because sqrt (1+eps) equalApprox 1 + eps/2.
However this implementation requires 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 basisth 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) ?
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.
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.undefinedThe following routine is more lazy than inits
and even lazier than 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.undefinedtranscendent functions
exponential functions
Absolute value of argument should be below 1.
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.
Absolute value of argument should be below 1.
Absolute value of argument should be below 1.
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
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.
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.
Arcus tangens of arguments with absolute value less than 1 / sqrt 3.
Efficient computation of Arcus tangens of an argument of the form 1/n.
This implementation gets the first decimal place for free by calling the arcus tangens implementation for Doubles.
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.