{-# OPTIONS -fno-implicit-prelude #-} {- | Copyright : (c) Henning Thielemann 2006 License : GPL Maintainer : numericprelude@henning-thielemann.de Stability : provisional Exact Real Arithmetic - Computable reals. Inspired by ''The most unreliable technique for computing pi.'' See also <http://www.haskell.org/haskellwiki/Exact_real_arithmetic> . -} module Number.Positional where import qualified MathObj.LaurentPolynomial as LPoly import qualified MathObj.Polynomial as Poly import qualified Algebra.IntegralDomain as Integral import qualified Algebra.Ring as Ring import qualified Algebra.Additive as Additive import qualified Algebra.ToInteger as ToInteger import qualified Prelude as P98 import qualified PreludeBase as P import qualified NumericPrelude as NP import PreludeBase import NumericPrelude hiding (sqrt, tan, one, zero, ) import qualified Data.List as List import Data.Char (intToDigit) import NumericPrelude.Condition (toMaybe, select, if') import NumericPrelude.List (replicateMatch, sliceVert, zipNeighborsWith, padLeft, padRight, mapLast) {- bugs: defltBase = 10 defltExp = 4 (sqrt 0.5) -- wrong result, probably due to undetected overflows -} {- * types -} type T = (Exponent, Mantissa) type FixedPoint = (Integer, Mantissa) type Mantissa = [Digit] type Digit = Int type Exponent = Int type Basis = Int {- * basic helpers -} moveToZero :: Basis -> Digit -> (Digit,Digit) moveToZero b n = let b2 = NP.div b 2 (q,r) = divMod (n+b2) b in (q,r-b2) checkPosDigit :: Basis -> Digit -> Digit checkPosDigit b d = if d>=0 && d<b then d else error ("digit " ++ show d ++ " out of range [0," ++ show b ++ ")") checkDigit :: Basis -> Digit -> Digit checkDigit b d = if abs d < b then d else error ("digit " ++ show d ++ " out of range (" ++ show (-b) ++ "," ++ show b ++ ")") {- | 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!) -} nonNegative :: Basis -> T -> T nonNegative b x = let (xe,xm) = compress b x in (xe, nonNegativeMant b xm) {- | Requires, that no digit is @(basis-1)@ or @(1-basis)@. The leading digit might be negative and might be @-basis@ or @basis@. -} nonNegativeMant :: Basis -> Mantissa -> Mantissa nonNegativeMant b = let recurse (x0:x1:xs) = select (replaceZeroChain x0 (x1:xs)) [(x1 >= 1, x0 : recurse (x1:xs)), (x1 <= -1, (x0-1) : recurse ((x1+b):xs))] recurse xs = xs replaceZeroChain x xs = let (xZeros,xRem) = span (0==) xs in case xRem of [] -> (x:xs) -- keep trailing zeros, because they show precision in 'show' functions (y:ys) -> if y>=0 -- equivalent to y>0 then x : replicateMatch xZeros 0 ++ recurse xRem else (x-1) : replicateMatch xZeros (b-1) ++ recurse ((y+b) : ys) in recurse {- | May prepend a digit. -} compress :: Basis -> T -> T compress _ x@(_, []) = x compress b (xe, xm) = let (hi:his,los) = unzip (map (moveToZero b) xm) in prependDigit hi (xe, Poly.add his los) {- | Compress first digit. May prepend a digit. -} compressFirst :: Basis -> T -> T compressFirst _ x@(_, []) = x compressFirst b (xe, x:xs) = let (hi,lo) = moveToZero b x in prependDigit hi (xe, lo:xs) {- | Does not prepend a digit. -} compressMant :: Basis -> Mantissa -> Mantissa compressMant _ [] = [] compressMant b (x:xs) = let (his,los) = unzip (map (moveToZero b) xs) in Poly.add his (x:los) {- | Compress second digit. Sometimes this is enough to keep the digits in the admissible range. Does not prepend a digit. -} compressSecondMant :: Basis -> Mantissa -> Mantissa compressSecondMant b (x0:x1:xs) = let (hi,lo) = moveToZero b x1 in x0+hi : lo : xs compressSecondMant _ xm = xm prependDigit :: Basis -> T -> T prependDigit 0 x = x prependDigit x (xe, xs) = (xe+1, x:xs) {- | Eliminate leading zero digits. This will fail for zero. -} trim :: T -> T trim (xe,xm) = let (xZero, xNonZero) = span (0 ==) xm in (xe - length xZero, xNonZero) {- | Trim until a minimum exponent is reached. Safe for zeros. -} trimUntil :: Exponent -> T -> T trimUntil e = until (\(xe,xm) -> xe<=e || not (null xm || head xm == 0)) trimOnce trimOnce :: T -> T trimOnce (xe,[]) = (xe-1,[]) trimOnce (xe,0:xm) = (xe-1,xm) trimOnce x = x {- | Accept a high leading digit for the sake of a reduced exponent. This eliminates one leading digit. Like 'pumpFirst' but with exponent management. -} decreaseExp :: Basis -> T -> T decreaseExp b (xe,xm) = (xe-1, pumpFirst b xm) {- | Merge leading and second digit. This is somehow an inverse of 'compressMant'. -} pumpFirst :: Basis -> Mantissa -> Mantissa pumpFirst b xm = case xm of (x0:x1:xs) -> x0*b+x1:xs (x0:[]) -> x0*b:[] [] -> [] decreaseExpFP :: Basis -> (Exponent, FixedPoint) -> (Exponent, FixedPoint) decreaseExpFP b (xe,xm) = (xe-1, pumpFirstFP b xm) pumpFirstFP :: Basis -> FixedPoint -> FixedPoint pumpFirstFP b (x,xm) = let xb = x * fromIntegral b in case xm of (x0:xs) -> (xb + fromIntegral x0, xs) [] -> (xb, []) {- | 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. -} negativeExp :: Basis -> T -> T negativeExp b x = let tx = trimUntil (-10) x in if fst tx >=0 then decreaseExp b tx else tx {- * conversions -} {- ** integer -} fromBaseCardinal :: Basis -> Integer -> T fromBaseCardinal b n = let mant = mantissaFromCard b n in (length mant - 1, mant) fromBaseInteger :: Basis -> Integer -> T fromBaseInteger b n = if n<0 then neg b (fromBaseCardinal b (negate n)) else fromBaseCardinal b n mantissaToNum :: Ring.C a => Basis -> Mantissa -> a mantissaToNum bInt = let b = fromIntegral bInt in foldl (\x d -> x*b + fromIntegral d) 0 mantissaFromCard :: (ToInteger.C a) => Basis -> a -> Mantissa mantissaFromCard bInt n = let b = NP.fromIntegral bInt in reverse (map NP.fromIntegral (Integral.decomposeVarPositional (repeat b) n)) mantissaFromInt :: (ToInteger.C a) => Basis -> a -> Mantissa mantissaFromInt b n = if n<0 then negate (mantissaFromCard b (negate n)) else mantissaFromCard b n mantissaFromFixedInt :: Basis -> Exponent -> Integer -> Mantissa mantissaFromFixedInt bInt e n = let b = NP.fromIntegral bInt in map NP.fromIntegral (uncurry (:) (List.mapAccumR (\x () -> divMod x b) n (replicate (pred e) ()))) {- ** rational -} fromBaseRational :: Basis -> Rational -> T fromBaseRational bInt x = let b = NP.fromIntegral bInt xDen = denominator x (xInt,xNum) = divMod (numerator x) xDen (xe,xm) = fromBaseInteger bInt xInt xFrac = List.unfoldr (\num -> toMaybe (num/=0) (divMod (num*b) xDen)) xNum in (xe, xm ++ map NP.fromInteger xFrac) {- ** fixed point -} {- | Split into integer and fractional part. -} toFixedPoint :: Basis -> T -> FixedPoint toFixedPoint b (xe,xm) = if xe>=0 then let (x0,x1) = splitAtPadZero (xe+1) xm in (mantissaToNum b x0, x1) else (0, replicate (- succ xe) 0 ++ xm) fromFixedPoint :: Basis -> FixedPoint -> T fromFixedPoint b (xInt,xFrac) = let (xe,xm) = fromBaseInteger b xInt in (xe, xm++xFrac) {- ** floating point -} toDouble :: Basis -> T -> Double toDouble b (xe,xm) = let txm = take (mantLengthDouble b) xm bf = fromIntegral b br = recip bf in fieldPower xe bf * foldr (\xi y -> fromIntegral xi + y*br) 0 txm {- | cf. 'Numeric.floatToDigits' -} fromDouble :: Basis -> Double -> T fromDouble b x = let (n,frac) = splitFraction x (mant,e) = P98.decodeFloat frac fracR = alignMant b (-1) (fromBaseRational b (mant % ringPower (-e) 2)) in fromFixedPoint b (n, fracR) {- | Only return as much digits as are contained in Double. This will speedup further computations. -} fromDoubleApprox :: Basis -> Double -> T fromDoubleApprox b x = let (xe,xm) = fromDouble b x in (xe, take (mantLengthDouble b) xm) fromDoubleRough :: Basis -> Double -> T fromDoubleRough b x = let (xe,xm) = fromDouble b x in (xe, take 2 xm) mantLengthDouble :: Basis -> Exponent mantLengthDouble b = let fi = fromIntegral :: Int -> Double x = undefined :: Double in ceiling (logBase (fi b) (fromInteger (P98.floatRadix x)) * fi (P98.floatDigits x)) liftDoubleApprox :: Basis -> (Double -> Double) -> T -> T liftDoubleApprox b f = fromDoubleApprox b . f . toDouble b liftDoubleRough :: Basis -> (Double -> Double) -> T -> T liftDoubleRough b f = fromDoubleRough b . f . toDouble b {- ** text -} {- | Show a number with respect to basis @10^e@. -} showDec :: Exponent -> T -> String showDec = showBasis 10 showHex :: Exponent -> T -> String showHex = showBasis 16 showBin :: Exponent -> T -> String showBin = showBasis 2 showBasis :: Basis -> Exponent -> T -> String showBasis b e xBig = let x = rootBasis b e xBig (sign,absX) = case cmp b x (fst x,[]) of LT -> ("-", neg b x) _ -> ("", x) (int, frac) = toFixedPoint b (nonNegative b absX) checkedFrac = map (checkPosDigit b) frac intStr = if b==10 then show int else map intToDigit (mantissaFromInt b int) in sign ++ intStr ++ '.' : map intToDigit checkedFrac {- ** basis -} {- | Convert from a @b@ basis representation to a @b^e@ basis. Works well with every exponent. -} powerBasis :: Basis -> Exponent -> T -> T powerBasis b e (xe,xm) = let (ye,r) = divMod xe e (y0,y1) = splitAtPadZero (r+1) xm y1pad = mapLast (padRight 0 e) (sliceVert e y1) in (ye, map (mantissaToNum b) (y0 : y1pad)) {- | Convert from a @b^e@ basis representation to a @b@ basis. Works well with every exponent. -} rootBasis :: Basis -> Exponent -> T -> T rootBasis b e (xe,xm) = let splitDigit d = padLeft 0 e (mantissaFromInt b d) in nest (e-1) trimOnce ((xe+1)*e-1, concatMap splitDigit (map (checkDigit (ringPower e b)) xm)) {- | Convert between arbitrary bases. This conversion is expensive (quadratic time). -} fromBasis :: Basis -> Basis -> T -> T fromBasis bDst bSrc x = let (int,frac) = toFixedPoint bSrc x in fromFixedPoint bDst (int, fromBasisMant bDst bSrc frac) fromBasisMant :: Basis -> Basis -> Mantissa -> Mantissa fromBasisMant _ _ [] = [] fromBasisMant bDst bSrc xm = let {- We use a counter that alerts us, when the digits are grown too much by Poly.scale. Then it is time to do some carry/compression. 'inc' is essentially the fractional number digits needed to represent the destination base in the source base. It is multiplied by 'unit' in order to allow integer computation. -} inc = ceiling (logBase (fromIntegral bSrc) (fromIntegral bDst) * unit * 1.1 :: Double) -- Without the correction factor, invalid digits are emitted - why? unit :: Ring.C a => a unit = 10000 {- This would create finite representations in some cases (input is finite, and the result is finite) but will cause infinite loop otherwise. dropWhileRev (0==) . compressMant bDst -} cmpr (mag,xs) = (mag - unit, compressMant bSrc xs) scl (_,[]) = Nothing scl (mag,xs) = let (newMag,y:ys) = until ((<unit) . fst) cmpr (mag + inc, Poly.scale bDst xs) (d,y0) = moveToZero bSrc y in Just (d, (newMag, y0:ys)) in List.unfoldr scl (0::Int,xm) {- * 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. -} cmp :: Basis -> T -> T -> Ordering cmp b x y = let (_,dm) = sub b x y {- Only differences above 2 allow a safe decision, because 1(-9)(-9)(-9)(-9)... and (-1)9999... represent the same number, namely zero. -} recurse [] = EQ recurse (d:[]) = compare d 0 recurse (d0:d1:ds) = select (recurse (d0*b+d1 : ds)) [(d0 < -2, LT), (d0 > 2, GT)] in recurse dm {- Compare two numbers approximately. This circumvents the infinite loop if both numbers are equal. If @lessApprox bnd b x y@ then @x@ is definitely smaller than @y@. The converse is not true. You should use this one instead of 'cmp' for checking for bounds. -} lessApprox :: Basis -> Exponent -> T -> T -> Bool lessApprox b bnd x y = let tx = trunc bnd x ty = trunc bnd y in LT == cmp b (liftLaurent2 LPoly.add (bnd,[2]) tx) ty trunc :: Exponent -> T -> T trunc bnd (xe, xm) = if bnd > xe then (bnd, []) else (xe, take (1+xe-bnd) xm) equalApprox :: Basis -> Exponent -> T -> T -> Bool equalApprox b bnd x y = fst (trimUntil bnd (sub b x y)) == bnd align :: Basis -> Exponent -> T -> T align b ye x = (ye, alignMant b ye x) {- | Get the mantissa in such a form that it fits an expected exponent. @x@ and @(e, alignMant b e x)@ represent the same number. -} alignMant :: Basis -> Exponent -> T -> Mantissa alignMant b e (xe,xm) = if e>=xe then replicate (e-xe) 0 ++ xm else let (xm0,xm1) = splitAtPadZero (xe-e+1) xm in mantissaToNum b xm0 : xm1 absolute :: T -> T absolute (xe,xm) = (xe, absMant xm) absMant :: Mantissa -> Mantissa absMant xa@(x:xs) = case compare x 0 of EQ -> x : absMant xs LT -> Poly.negate xa GT -> xa absMant [] = [] {- * arithmetic -} fromLaurent :: LPoly.T Int -> T fromLaurent (LPoly.Cons nxe xm) = (NP.negate nxe, xm) toLaurent :: T -> LPoly.T Int toLaurent (xe, xm) = LPoly.Cons (NP.negate xe) xm liftLaurent2 :: (LPoly.T Int -> LPoly.T Int -> LPoly.T Int) -> (T -> T -> T) liftLaurent2 f x y = fromLaurent (f (toLaurent x) (toLaurent y)) liftLaurentMany :: ([LPoly.T Int] -> LPoly.T Int) -> ([T] -> T) liftLaurentMany f = fromLaurent . f . map toLaurent {- | Add two numbers but do not eliminate leading zeros. -} add :: Basis -> T -> T -> T add b x y = compress b (liftLaurent2 LPoly.add x y) sub :: Basis -> T -> T -> T sub b x y = compress b (liftLaurent2 LPoly.sub x y) neg :: Basis -> T -> T neg _ (xe, xm) = (xe, Poly.negate xm) {- | Add at most @basis@ summands. More summands will violate the allowed digit range. -} addSome :: Basis -> [T] -> T addSome b = compress b . liftLaurentMany sum {- | Add many numbers efficiently by computing sums of sub lists with only little carry propagation. -} addMany :: Basis -> [T] -> T addMany _ [] = zero addMany b ys = let recurse xs = case map (addSome b) (sliceVert b xs) of [s] -> s sums -> recurse sums in recurse ys type Series = [(Exponent, T)] {- | 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. -} series :: Basis -> Series -> T series _ [] = error "empty series: don't know a good exponent" -- series _ [] = (0,[]) -- unfortunate choice of exponent series b summands = {- Some pre-processing that asserts decreasing exponents. Increasing coefficients can appear legally due to non-unique number representation. -} let (es,xs) = unzip summands safeSeries = zip (scanl1 min es) xs in seriesPlain b safeSeries seriesPlain :: Basis -> Series -> T seriesPlain _ [] = error "empty series: don't know a good exponent" seriesPlain b summands = let (es,m:ms) = unzip (map (uncurry (align b)) summands) eDifs = zipNeighborsWith (-) es eDifLists = sliceVert (pred b) eDifs mLists = sliceVert (pred b) ms accum sumM (eDifList,mList) = let subM = LPoly.addShiftedMany eDifList (sumM:mList) -- lazy unary sum len = concatMap (flip replicate ()) eDifList (high,low) = splitAtMatchPadZero len subM {- 'compressMant' looks unsafe when the residue doesn't decrease for many summands. Then there is a leading digit of a chunk which is not compressed for long time. However, if the residue is estimated correctly there can be no overflow. -} in (compressMant b low, high) (trailingDigits, chunks) = List.mapAccumL accum m (zip eDifLists mLists) in compress b (head es, concat chunks ++ trailingDigits) {- An alternative series implementation could reduce carries by do the following cycle (split, add sub-lists). This would reduce carries to the minimum but we must work hard in order to find out lazily how many digits can be emitted. -} {- | Like 'splitAt', but it pads with zeros if the list is too short. This way it preserves @ length (fst (splitAtPadZero n xs)) == n @ -} splitAtPadZero :: Int -> Mantissa -> (Mantissa, Mantissa) splitAtPadZero n [] = (replicate n 0, []) splitAtPadZero 0 xs = ([], xs) splitAtPadZero n (x:xs) = let (ys, zs) = splitAtPadZero (n-1) xs in (x:ys, zs) -- must get a case for negative index splitAtMatchPadZero :: [()] -> Mantissa -> (Mantissa, Mantissa) splitAtMatchPadZero n [] = (replicateMatch n 0, []) splitAtMatchPadZero [] xs = ([], xs) splitAtMatchPadZero n (x:xs) = let (ys, zs) = splitAtMatchPadZero (tail n) xs in (x:ys, zs) {- | help showing series summands -} truncSeriesSummands :: Series -> Series truncSeriesSummands = map (\(e,x) -> (e,trunc (-20) x)) scale :: Basis -> Digit -> T -> T scale b y x = compress b (scaleSimple y x) {- scaleSimple :: ToInteger.C a => a -> T -> T scaleSimple y (xe, xm) = (xe, Poly.scale (fromIntegral y) xm) -} scaleSimple :: Basis -> T -> T scaleSimple y (xe, xm) = (xe, Poly.scale y xm) scaleMant :: Basis -> Digit -> Mantissa -> Mantissa scaleMant b y xm = compressMant b (Poly.scale y xm) mulSeries :: Basis -> T -> T -> Series mulSeries _ (xe,[]) (ye,_) = [(xe+ye, zero)] mulSeries b (xe,xm) (ye,ym) = let zes = iterate pred (xe+ye+1) zs = zipWith (\xd e -> scale b xd (e,ym)) xm (tail zes) in zip zes zs {- | For obtaining n result digits it is mathematically sufficient to know the first (n+1) digits of the operands. However this implementation needs (n+2) digits, because of calls to 'compress' in both 'scale' and 'series'. We should fix that. -} mul :: Basis -> T -> T -> T mul b x y = trimOnce (seriesPlain b (mulSeries b x y)) {- | 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. -} divide :: Basis -> T -> T -> T divide b (xe,xm) (ye',ym') = let (ye,ym) = until ((>=b) . abs . head . snd) (decreaseExp b) (ye',ym') in nest 3 trimOnce (compress b (xe-ye, divMant b ym xm)) divMant :: Basis -> Mantissa -> Mantissa -> Mantissa divMant _ [] _ = error "Number.Positional: division by zero" divMant b ym xm0 = snd $ List.mapAccumL (\xm fullCompress -> let z = div (head xm) (head ym) {- 'scaleMant' contains compression, which is not much of a problem, because it is always applied to @ym@. That is, there is no nested call. -} dif = pumpFirst b (Poly.sub xm (scaleMant b z ym)) cDif = if fullCompress then compressMant b dif else compressSecondMant b dif in (cDif, z)) xm0 (cycle (replicate (b-1) False ++ [True])) divMantSlow :: Basis -> Mantissa -> Mantissa -> Mantissa divMantSlow _ [] = error "Number.Positional: division by zero" divMantSlow b ym = List.unfoldr (\xm -> let z = div (head xm) (head ym) d = compressMant b (pumpFirst b (Poly.sub xm (Poly.scale z ym))) in Just (z, d)) reciprocal :: Basis -> T -> T reciprocal b = divide b one {- | Fast division for small integral divisors, which occur for instance in summands of power series. -} divIntMant :: Basis -> Int -> Mantissa -> Mantissa divIntMant b y xInit = List.unfoldr (\(r,rxs) -> let rb = r*b (rbx, xs', run) = case rxs of [] -> (rb, [], r/=0) (x:xs) -> (rb+x, xs, True) (d,m) = divMod rbx y in toMaybe run (d, (m, xs'))) (0,xInit) -- this version is simple but ignores the possibility of a terminating result divIntMantInf :: Basis -> Int -> Mantissa -> Mantissa divIntMantInf b y = map fst . tail . scanl (\(_,r) x -> divMod (r*b+x) y) (undefined,0) . (++ repeat 0) divInt :: Basis -> Digit -> T -> T divInt b y (xe,xm) = -- (xe, divIntMant b y xm) let z = (xe, divIntMant b y xm) {- Division by big integers may cause leading zeros. Eliminate as many as we can expect from the division. If the input number has leading zeros (it might be equal to zero), then the result may have, too. -} tz = until ((<=1) . fst) (\(yi,zi) -> (div yi b, trimOnce zi)) (y,z) in snd tz {- * algebraic functions -} sqrt :: Basis -> T -> T sqrt b = sqrtDriver b sqrtFP sqrtDriver :: Basis -> (Basis -> FixedPoint -> Mantissa) -> T -> T sqrtDriver _ _ (xe,[]) = (div xe 2, []) sqrtDriver b sqrtFPworker x = let (exe,ex0:exm) = if odd (fst x) then decreaseExp b x else x (nxe,(nx0,nxm)) = until (\xi -> fst (snd xi) >= fromIntegral b ^ 2) (nest 2 (decreaseExpFP b)) (exe, (fromIntegral ex0, exm)) in compress b (div nxe 2, sqrtFPworker b (nx0,nxm)) sqrtMant :: Basis -> Mantissa -> Mantissa sqrtMant _ [] = [] sqrtMant b (x:xs) = sqrtFP b (fromIntegral x, xs) {- | 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) `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 @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)@ ? -} sqrtFP :: Basis -> FixedPoint -> Mantissa sqrtFP b (x0,xs) = let y0 = round (NP.sqrt (fromInteger x0 :: Double)) dyx0 = fromInteger (x0 - fromIntegral y0 ^ 2) accum dif (e,ty) = -- (e,ty) == xm - (trunc j y)^2 let yj = div (head dif + y0) (2*y0) newDif = pumpFirst b $ LPoly.addShifted e (Poly.sub dif (scaleMant b (2*yj) ty)) [yj*yj] {- We could always compress the full difference number, but it is not necessary, and we save dependencies on less significant digits. -} cNewDif = if mod e b == 0 then compressMant b newDif else compressSecondMant b newDif in (cNewDif, yj) truncs = lazyInits y y = y0 : snd (List.mapAccumL accum (pumpFirst b (dyx0 : xs)) (zip [1..] (drop 2 truncs))) in y sqrtNewton :: Basis -> T -> T sqrtNewton b = sqrtDriver b sqrtFPNewton {- | 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.@ -} sqrtFPNewton :: Basis -> FixedPoint -> Mantissa sqrtFPNewton bInt (x0,xs) = let b = fromIntegral bInt chunkLengths = iterate (2*) 1 xChunks = map (mantissaToNum bInt) $ snd $ List.mapAccumL (\x cl -> flipPair (splitAtPadZero cl x)) xs chunkLengths basisPowers = iterate (^2) b truncXs = scanl (\acc (bp,frac) -> acc*bp+frac) x0 (zip basisPowers xChunks) accum y (bp, x) = let ybp = y * bp newY = div (ybp + div (x * div bp b) y) 2 in (newY, newY-ybp) y0 = round (NP.sqrt (fromInteger x0 :: Double)) yChunks = snd $ List.mapAccumL accum y0 (zip basisPowers (tail truncXs)) yFrac = concat $ zipWith (mantissaFromFixedInt bInt) chunkLengths yChunks in fromInteger y0 : yFrac {- | 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 but restricted to infinite lists. -} lazyInits :: [a] -> [[a]] lazyInits ~(x:xs) = [] : map (x:) (lazyInits xs) -- this pattern is never reached, GHC does not complain about it lazyInits [] = [[]] {- * transcendent functions -} {- ** exponential functions -} expSeries :: Basis -> T -> Series expSeries b xOrig = let x = negativeExp b xOrig xps = scanl (\p n -> divInt b n (mul b x p)) one [1..] in map (\xp -> (fst xp, xp)) xps {- | Absolute value of argument should be below 1. -} expSmall :: Basis -> T -> T expSmall b x = series b (expSeries b x) expSeriesLazy :: Basis -> T -> Series expSeriesLazy b x@(xe,_) = let xps = scanl (\p n -> divInt b n (mul b x p)) one [1..] {- much effort for computing the residue exponents without touching the arguments mantissa -} es :: [Double] es = zipWith (-) (map fromIntegral (iterate ((1+xe)+) 0)) (scanl (+) 0 (map (logBase (fromIntegral b) . fromInteger) [1..])) in zip (map ceiling es) xps expSmallLazy :: Basis -> T -> T expSmallLazy b x = series b (expSeriesLazy b x) exp :: Basis -> T -> T exp b x = let (xInt,xFrac) = toFixedPoint b (compress b x) yFrac = expSmall b (-1,xFrac) {- (xFrac0,xFrac1) = splitAt 2 xFrac yFrac = mul b -- slow convergence but simple argument (expSmall b (-1, xFrac0)) -- fast convergence but big argument (expSmall b (-3, xFrac1)) -} in intPower b xInt yFrac (recipEConst b) (eConst b) intPower :: Basis -> Integer -> T -> T -> T -> T intPower b expon neutral recipX x = if expon >= 0 then cardPower b expon neutral x else cardPower b (-expon) neutral recipX cardPower :: Basis -> Integer -> T -> T -> T cardPower b expon neutral x = if expon >= 0 then reduceRepeated (mul b) neutral x expon else error "negative exponent - use intPower" {- | 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. -} powerSeries :: Basis -> Rational -> T -> Series powerSeries b expon xOrig = let scaleRat ni yi = divInt b (fromInteger (denominator yi) * ni) . scaleSimple (fromInteger (numerator yi)) x = negativeExp b (sub b xOrig one) xps = scanl (\p fac -> uncurry scaleRat fac (mul b x p)) one (zip [1..] (iterate (subtract 1) expon)) in map (\xp -> (fst xp, xp)) xps powerSmall :: Basis -> Rational -> T -> T powerSmall b y x = series b (powerSeries b y x) power :: Basis -> Rational -> T -> T power b expon x = let num = numerator expon den = denominator expon rootX = root b den x in intPower b num one (reciprocal b rootX) rootX root :: Basis -> Integer -> T -> T root b expon x = let estimate = liftDoubleApprox b (** (1 / fromInteger expon)) x estPower = cardPower b expon one estimate residue = divide b x estPower in mul b estimate (powerSmall b (1 % fromIntegral expon) residue) {- | Absolute value of argument should be below 1. -} cosSinhSmall :: Basis -> T -> (T, T) cosSinhSmall b x = let (coshXps, sinhXps) = unzip (sliceVertPair (expSeries b x)) in (series b coshXps, series b sinhXps) {- | Absolute value of argument should be below 1. -} cosSinSmall :: Basis -> T -> (T, T) cosSinSmall b x = let (coshXps, sinhXps) = unzip (sliceVertPair (expSeries b x)) alternate s = zipWith3 if' (cycle [True,False]) s (map (\(e,y) -> (e, neg b y)) s) in (series b (alternate coshXps), series b (alternate sinhXps)) {- | 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. -} cosSinFourth :: Basis -> T -> (T, T) cosSinFourth b x = let (cosx, sinx) = cosSinSmall b (divInt b 4 x) sinx2 = mul b sinx sinx cosx2 = mul b cosx cosx sincosx = mul b sinx cosx in (add b one (scale b 8 (mul b cosx2 (sub b cosx2 one))), scale b 4 (mul b sincosx (sub b one (scale b 2 sinx2)))) cosSin :: Basis -> T -> (T, T) cosSin b x = let pi2 = divInt b 2 (piConst b) {- @compress@ ensures that the leading digit of the fractional part is close to zero -} (quadrant, frac) = toFixedPoint b (compress b (divide b x pi2)) -- it's possibly faster if we subtract quadrant*pi/4 wrapped = if quadrant==0 then x else mul b pi2 (-1, frac) (cosW,sinW) = cosSinSmall b wrapped in case mod quadrant 4 of 0 -> ( cosW, sinW) 1 -> (neg b sinW, cosW) 2 -> (neg b cosW, neg b sinW) 3 -> ( sinW, neg b cosW) _ -> error "error in implementation of 'mod'" tan :: Basis -> T -> T tan b x = uncurry (flip (divide b)) (cosSin b x) cot :: Basis -> T -> T cot b x = uncurry (divide b) (cosSin b x) {- ** logarithmic functions -} lnSeries :: Basis -> T -> Series lnSeries b xOrig = let x = negativeExp b (sub b xOrig one) mx = neg b x xps = zipWith (divInt b) [1..] (iterate (mul b mx) x) in map (\xp -> (fst xp, xp)) xps lnSmall :: Basis -> T -> T lnSmall b x = series b (lnSeries b x) {- | @ 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 + 2/1.4140) / 2 == 1.414213578500707 (1.4149 + 2/1.4149) / 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. -} lnNewton :: Basis -> T -> T lnNewton b y = let estimate = liftDoubleApprox b log y expRes = mul b y (expSmall b (neg b estimate)) -- try to reduce dependencies by feeding expSmall with a small argument residue = sub b (mul b expRes (expSmallLazy b (neg b resTrim))) one resTrim = -- (-3, replicate 4 0 ++ alignMant b (-7) residue) align b (- mantLengthDouble b) residue lazyAdd (xe,xm) (ye,ym) = (xe, LPoly.addShifted (xe-ye) xm ym) x = lazyAdd estimate resTrim in x lnNewton' :: Basis -> T -> T lnNewton' b y = let estimate = liftDoubleApprox b log y residue = sub b (mul b y (expSmall b (neg b x))) one -- sub b (mul b y (expSmall b (neg b estimate))) one -- sub b (mul b y (expSmall b (neg b -- (fst estimate, snd estimate ++ [undefined])))) one resTrim = -- align b (-6) residue align b (- mantLengthDouble b) residue -- align returns the new exponent immediately -- nest (mantLengthDouble b) trimOnce residue -- negativeExp b residue lazyAdd (xe,xm) (ye,ym) = (xe, LPoly.addShifted (xe-ye) xm ym) x = lazyAdd estimate resTrim -- add b estimate resTrim -- LPoly.add checks for empty lists and is thus too strict in x ln :: Basis -> T -> T ln b x@(xe,_) = let e = round (log (fromIntegral b) * fromIntegral xe :: Double) ei = fromIntegral e y = trim $ if e<0 then reduceRepeated (mul b) x (eConst b) (-ei) else reduceRepeated (mul b) x (recipEConst b) ei estimate = liftDoubleApprox b log y residue = mul b (expSmall b (neg b estimate)) y in addSome b [(0,[e]), estimate, lnSmall b residue] {- | 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. -} angle :: Basis -> (T,T) -> T angle b (cosx, sinx) = let wd = atan2 (toDouble b sinx) (toDouble b cosx) wApprox = fromDoubleApprox b wd (cosApprox, sinApprox) = cosSin b wApprox (cosD,sinD) = (add b (mul b cosx cosApprox) (mul b sinx sinApprox), sub b (mul b sinx cosApprox) (mul b cosx sinApprox)) sinDSmall = negativeExp b sinD in add b wApprox (arctanSmall b (divide b sinDSmall cosD)) {- | Arcus tangens of arguments with absolute value less than @1 \/ sqrt 3@. -} arctanSeries :: Basis -> T -> Series arctanSeries b xOrig = let x = negativeExp b xOrig mx2 = neg b (mul b x x) xps = zipWith (divInt b) [1,3..] (iterate (mul b mx2) x) in map (\xp -> (fst xp, xp)) xps arctanSmall :: Basis -> T -> T arctanSmall b x = series b (arctanSeries b x) {- | Efficient computation of Arcus tangens of an argument of the form @1\/n@. -} arctanStem :: Basis -> Int -> T arctanStem b n = let x = (0, divIntMant b n [1]) divN2 = divInt b n . divInt b (-n) {- this one can cause overflows in piConst too easily mn2 = - n*n divN2 = divInt b mn2 -} xps = zipWith (divInt b) [1,3..] (iterate (trim . divN2) x) in series b (map (\xp -> (fst xp, xp)) xps) {- | This implementation gets the first decimal place for free by calling the arcus tangens implementation for 'Double's. -} arctan :: Basis -> T -> T arctan b x = let estimate = liftDoubleRough b atan x tanEst = tan b estimate residue = divide b (sub b x tanEst) (add b one (mul b x tanEst)) in addSome b [estimate, arctanSmall b residue] {- | 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. -} arctanClassic :: Basis -> T -> T arctanClassic b x = let absX = absolute x pi2 = divInt b 2 (piConst b) in select (divInt b 2 (sub b pi2 (arctanSmall b (divInt b 2 (sub b (reciprocal b x) x))))) [(lessApprox b (-5) absX (fromBaseRational b (11%19)), arctanSmall b x), (lessApprox b (-5) (fromBaseRational b (19%11)) absX, sub b pi2 (arctanSmall b (reciprocal b x)))] {- * constants -} {- ** elementary -} zero :: T zero = (0,[]) one :: T one = (0,[1]) minusOne :: T minusOne = (0,[-1]) {- ** transcendental -} eConst :: Basis -> T eConst b = expSmall b one recipEConst :: Basis -> T recipEConst b = expSmall b minusOne piConst :: Basis -> T piConst b = let numCompress = takeWhile (0/=) (iterate (flip div b) (4*(44+7+12+24))) stArcTan k den = scaleSimple k (arctanStem b den) sum' = addSome b [stArcTan 44 57, stArcTan 7 239, stArcTan (-12) 682, stArcTan 24 12943] in foldl (const . compress b) (scaleSimple 4 sum') numCompress {- * auxilary functions -} {- | Candidate for a Utility module. -} nest :: Int -> (a -> a) -> a -> a nest 0 _ x = x nest n f x = f (nest (n-1) f x) flipPair :: (a,b) -> (b,a) flipPair ~(x,y) = (y,x) sliceVertPair :: [a] -> [(a,a)] sliceVertPair (x0:x1:xs) = (x0,x1) : sliceVertPair xs sliceVertPair [] = [] sliceVertPair _ = error "odd number of elements" {- Pi as a zero of trigonometric functions. - Is a corresponding computation that bad? Newton converges quadratically, but the involved trigonometric series converge only slightly more than linearly. -- lift cos to higher frequencies, in order to shift the zero to smaller values, which let trigonometric series converge faster take 10 $ Numerics.Newton.zero 0.7 (\x -> (cos (2*x), -2 * sin (2*x))) (\x -> (2 * cos x ^ 2 - 1, -4 * cos x * sin x)) (\x -> (cos x ^ 2 - sin x ^ 2, -4 * cos x * sin x)) (\x -> (tan x ^ 2 - 1, 4 * tan x)) -- compute arctan as inverse of tan by Newton zero 0.7 (\x -> (tan x - 1, 1 + tan x ^ 2)) zero 0.7 (\x -> (tan x - 1, 1 / cos x ^ 2)) iterate (\x -> x + (cos x - sin x) * cos x) 0.7 iterate (\x -> x + (cos x - sin x) * sqrt 0.5) 0.7 iterate (\x -> x + cos x ^ 2 - sin x * cos x) 0.7 iterate (\x -> x + 0.5 - sin x * cos x) 0.7 iterate (\x -> x + cos x ^ 2 - 0.5) 0.7 -- compute section of tan and cot zero 0.7 (\x -> (tan x - 1 / tan x, (1 + tan x ^ 2) * (1 + 1 / tan x ^ 2)) zero 0.7 (\x -> ((tan x ^ 2 - 1) * tan x, (1 + tan x ^ 2) ^ 2) iterate (\x -> x - (sin x ^ 2 - cos x ^ 2) * sin x * cos x) 0.7 iterate (\x -> x - (sin x ^ 2 - cos x ^ 2) * 0.5) 0.7 iterate (\x -> x + 1/2 - sin x ^ 2) 0.7 For using the last formula, the n-th digit of (sin x) must depend only on the n-th digit of x. The same holds for (^2). This means that no interim carry compensation is possible. This will certainly force usage of Integer for digits, otherwise the multiplication will overflow sooner or later. -}