exp-pairs (empty) → 0.1.0.0
raw patch · 13 files changed
+1595/−0 lines, 13 filesdep +QuickCheckdep +basedep +exp-pairssetup-changed
Dependencies added: QuickCheck, base, exp-pairs, matrix, memoize, smallcheck
Files
- LICENSE +674/−0
- Math/ExpPairs.hs +89/−0
- Math/ExpPairs/Ivic.hs +185/−0
- Math/ExpPairs/Kratzel.hs +64/−0
- Math/ExpPairs/LinearForm.hs +85/−0
- Math/ExpPairs/Matrix3.hs +166/−0
- Math/ExpPairs/MenzerNowak.hs +15/−0
- Math/ExpPairs/Pair.hs +41/−0
- Math/ExpPairs/Process.hs +117/−0
- Math/ExpPairs/RatioInf.hs +97/−0
- Math/ExpPairs/Tests.hs +19/−0
- Setup.hs +2/−0
- exp-pairs.cabal +41/−0
+ LICENSE view
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Of course, your program's commands+might be different; for a GUI interface, you would use an "about box".++ You should also get your employer (if you work as a programmer) or school,+if any, to sign a "copyright disclaimer" for the program, if necessary.+For more information on this, and how to apply and follow the GNU GPL, see+<http://www.gnu.org/licenses/>.++ The GNU General Public License does not permit incorporating your program+into proprietary programs. If your program is a subroutine library, you+may consider it more useful to permit linking proprietary applications with+the library. If this is what you want to do, use the GNU Lesser General+Public License instead of this License. But first, please read+<http://www.gnu.org/philosophy/why-not-lgpl.html>.
+ Math/ExpPairs.hs view
@@ -0,0 +1,89 @@+module Math.ExpPairs (optimize, LinearForm (..), RationalForm (..), IneqType (..), Constraint (..), InitPair, Path, simulateOptimize,simulateOptimize', RatioInf (..), RationalInf, OptimizeResult, optimalValue, optimalPair, optimalPath) where++import Data.Ratio+import Data.Ord+import Data.List+import Data.Monoid++import Math.ExpPairs.LinearForm+import Math.ExpPairs.Process+import Math.ExpPairs.Pair+import Math.ExpPairs.RatioInf++fracs2proj :: (Rational, Rational) -> (Integer, Integer, Integer)+fracs2proj (q, r) = (k, l, m) where+ dq = denominator q+ dr = denominator r+ m = lcm dq dr+ k = numerator q * (m `div` dq)+ l = numerator r * (m `div` dr)++proj2fracs :: (Integer, Integer, Integer) -> (Rational, Rational)+proj2fracs (k, l, m) = (k%m, l%m)+++evalFunctional :: [InitPair] -> [InitPair] -> [RationalForm Rational] -> [Constraint Rational] -> Path -> (RationalInf, InitPair)+evalFunctional corners interiors rfs cons path = if null rs then (InfPlus, undefined) else minimumBy (comparing fst) rs where+ applyPath ips = map (evalPath path . fracs2proj . initPairToValue) ips `zip` ips+ corners' = applyPath corners+ interiors' = applyPath interiors++ predicate (p, _) = all (checkConstraint p) cons+ qs = if all predicate corners' then corners' else filter predicate interiors'++ rs = map (\(p, ip) -> (maximum $ map (evalRF p) rfs, ip)) qs++checkMConstraints :: Path -> [Constraint Rational] -> Bool+checkMConstraints path = all (\con -> any (\p -> checkConstraint (evalPath path p) con ) triangleT) where+ triangleT = map fracs2proj [ (0%1,1%1), (0%1,1%2), (1%2,1%2)]++data OptimizeResult = OptimizeResult {+ optimalValue :: RationalInf,+ optimalPair :: InitPair,+ optimalPath :: Path+ }++instance Show OptimizeResult where+ show (OptimizeResult r' ip p) = show' r' ++ "\n" ++ show ip ++ "\t" ++ show p where+ show' (Finite r) = show (fromRational r :: Double) ++ " = " ++ show r+ show' r = show r++instance Eq OptimizeResult where+ a==b = optimalValue a == optimalValue b++instance Ord OptimizeResult where+ compare a b = compare (optimalValue a) (optimalValue b)++simulateOptimize :: Rational -> OptimizeResult+simulateOptimize r = OptimizeResult (Finite r) Corput01 mempty++simulateOptimize' :: RationalInf -> OptimizeResult+simulateOptimize' r = OptimizeResult r Corput01 mempty++optimize :: [RationalForm Rational] -> [Constraint Rational] -> OptimizeResult+optimize rfs cons = optimize' rfs cons (OptimizeResult r0 ip0 mempty) where+ (r0, ip0) = evalFunctional [Corput01, Corput12] [Corput01, Corput12] rfs cons mempty++optimize' :: [RationalForm Rational] -> [Constraint Rational] -> OptimizeResult -> OptimizeResult+optimize' rfs cons ret@(OptimizeResult r _ path)+ | lengthPath path > 100 = ret+ | otherwise = retBA where+ ret0@(OptimizeResult r0 ip0 _) = if r0' < r then OptimizeResult r0' ip0' path else ret where+ (r0', ip0') = evalFunctional corners interiors rfs cons path+ corners = [Mix 1 0, Mix 0 1, Mix 0 0]+ interiors = initPairs++ cons0 = if r0==InfPlus then cons else cons ++ map (consBuilder r0) rfs++ retA@(OptimizeResult r1 ip1 _) = if checkMConstraints patha cons0 && r1' < r0 then branchA else ret0 where+ patha = path `mappend` aPath+ branchA@(OptimizeResult r1' _ _) = optimize' rfs cons (OptimizeResult r0 ip0 patha)++ cons1 = if r1==r0 then cons0 else cons ++ map (consBuilder r1) rfs++ retBA = if checkMConstraints pathba cons1 && r2' < r1 then branchB else retA where+ pathba = path `mappend` baPath+ branchB@(OptimizeResult r2' _ _) = optimize' rfs cons (OptimizeResult r1 ip1 pathba)++ consBuilder rr (RationalForm num den) = Constraint (substituteLF (num, den, 1) (LinearForm (-1) (toRational rr) 0)) Strict+
+ Math/ExpPairs/Ivic.hs view
@@ -0,0 +1,185 @@+module Math.ExpPairs.Ivic where++import Data.Ratio+import Data.List+import Data.Ord++import Math.ExpPairs++zetaOnS :: Rational -> OptimizeResult+zetaOnS s+ | s >= 1 = simulateOptimize 0+ | s >= 1%2 = optimize+ [RationalForm (LinearForm 1 1 (-s)) 2]+ [Constraint (LinearForm (-1) 1 (-s)) NonStrict]+ | otherwise = optRes {optimalValue = r} where+ optRes = zetaOnS (1-s)+ r = Finite (1%2 - s) + optimalValue optRes++zetaOnHalf :: Rational+zetaOnHalf = 32%205++reverseZetaOnS :: Rational -> OptimizeResult+reverseZetaOnS mu+ | mu >= 1%2 = simulateOptimize 0+ | mu > zetaOnHalf = optimize [RationalForm (LinearForm 1 (-1) 1) 1] [Constraint (LinearForm 0 (-2) (1+2*mu)) NonStrict]+ | otherwise = optRes {optimalValue = negate $ optimalValue optRes} where+ optRes = optimize [RationalForm (LinearForm 1 (-1) 0) 1] [Constraint (LinearForm 1 0 (-mu)) NonStrict, Constraint (LinearForm (-1) 1 (-1%2)) NonStrict]++lemma82_f :: Rational -> Rational+lemma82_f s+ | s < 1%2 = undefined+ | s<= 2%3 = 2/(3-4*s)+ | s<=11%14 = 10/(7-8*s)+ | s<=13%15 = 34/(15-16*s)+ | s<=57%62 = 98/(31-32*s)+ | otherwise = 5/(1-s)++-- Ivic, (8.97)+-- R << T V^{-2f(sigma)} + T^alpha1 V^beta1 + T^alpha2 V^beta2+--+-- If a<1 then T^a V^b << T V^{b+(a-1)/muS}+--+-- (8.97) implies that alpha1 <= 1 for S >= 1/2+-- and that alpha2 <= 1 for S >= 2/3 or S >= 5/8 and+-- (4S-2)k + (8S-6)l + 2S-1 >=0++mOnS :: Rational -> OptimizeResult+mOnS s+ | s < 1%2 = simulateOptimize 0+ | s < 5%8 = simulateOptimize $ 4/(3-4*s)+ | s>= 1 = simulateOptimize' InfPlus+ | otherwise = minimumBy (comparing optimalValue) [x1, x2, simulateOptimize (lemma82_f s * 2)] where++ optRes = zetaOnS s+ muS = toRational $ optimalValue optRes+ alpha1 = (4-4*s)/(1+2*s)+ beta1 = -12/(1+2*s)+ x1 = optRes {optimalValue = Finite $ (1-alpha1)/muS - beta1}++ --alpha2 = 4*(1-s)*(k+l)/((2*m+4*l)*s-m+2*k-2*l)+ --beta2 = -4*(m+2*k+2*l)/((2*m+4*l)*s-m+2*k-2*l)+ --ratio = (1-alpha2)/muS - beta2+ --numer = numerator ratio+ --denom = denominator ratio+ numer = LinearForm+ (-4*s + (-8*muS + 2))+ (-8*s + (-8*muS + 6))+ (-2*s + (-4*muS + 1))+ denom = LinearForm+ (2*muS)+ (4*muS*s - 2*muS)+ (2*muS*s - muS)++ cons = if s >= 2%3 then [] else [Constraint+ (LinearForm (4*s-2) (8*s-6) (2*s-1)) NonStrict+ ]++ x2' = optimize [RationalForm numer denom] cons+ x2 = x2' {optimalValue = negate $ optimalValue x2'}++reverseMOnS m = reverseMOnS' from to where+ from = 1 % 2+ to = 1 % 1+ reverseMOnS' a b+ | b-a < 1%1000000 = a+ | optimalValue (mOnS ((a+b)/2)) > m = reverseMOnS' a ((a+b)/2)+ | otherwise = reverseMOnS' ((a+b)/2) b++checkAbscissa :: [(Rational, Rational)] -> Rational -> Bool+checkAbscissa xs s = sum rs < Finite 1 where+ qs = map (\(n,m) -> optimalValue (mOnS (n*s)) / Finite m) xs+ rs = map (\q -> 1/q) qs++searchMinAbscissa :: [(Rational, Rational)] -> Rational+searchMinAbscissa xs = searchMinAbscissa' from to where+ from = 1 % 2 / minimum (map fst xs)+ to = 1 % 1+ searchMinAbscissa' a b+ | b-a < 1%1000000 = a+ | checkAbscissa xs ((a+b)/2) = searchMinAbscissa' a ((a+b)/2)+ | otherwise = searchMinAbscissa' ((a+b)/2) b++-- % \begin{lemma}\label{l:pointwise-moments-on-1/2}+-- % For $A\ge12$ let+-- % $$+-- % f(A) = 1+\inf \{ l/k \mid (4-A)k+4l+2 \ge 0 \}.+-- % $$+-- % Then+-- % $$+-- % R \ll T V^{-6} \log^8 T + T^{f(A)+\eps} V^{-A}+-- % \ll T^{\max \{ 1+32(A-6)/205, f(A) \}} V^{-A}+-- % $$+-- % and thus+-- % $$+-- % M(A) \le \max \{ 1+32(A-6)/205, f(A) \}.+-- % $$+-- % \end{lemma}++-- Constant+-- is produced by+-- optimize [RationalForm (LinearForm 4 4 2) (LinearForm 1 0 0)] [Constraint (LinearForm (-64) (-77) 64) Strict]++mBigOnHalf a+ | a < 4 = simulateOptimize 1+ | a < 12 = simulateOptimize $ 1+(a-4)/8+ | a > 41614060315296730740083860226662 % 2636743270445733804969041895717 = simulateOptimize $ 1 + 32*(a-6)/205+ | otherwise = if Finite x >= optimalValue optRes+ then simulateOptimize x+ else optRes where+ optRes = optimize [RationalForm (LinearForm 1 1 0) (LinearForm 1 0 0)]+ [Constraint (LinearForm (4-a) 4 2) NonStrict]+ x = 1 + 32*(a-6)/205++reverseMBigOnHalf m+ | m <= 2 = simulateOptimize $ (m-1)*8 + 4+ | otherwise = if Finite a <= optimalValue optRes+ then simulateOptimize a+ else optRes where+ a = (m-1)*205/32 + 6+ optRes = optimize [RationalForm (LinearForm 4 4 2) (LinearForm 1 0 0)] [Constraint (LinearForm (1-m) 1 0) NonStrict]+++f l lambda = (l-lambda)*32/205+ + ((toRational . optimalValue . mBigOnHalf) (4*l/(4-lambda)) - 1) * (4-lambda) /4++--bestLambda l = minimum $ map (f l) lambdas `zip` lambdas where+-- lambdas = [0,1%100..4-1%100]++heckeZetaByHalf a = 1 - xt where+ d = toRational 12.571624917200547+ ia 2 = 0+ ia 3 = 1%4+ ia a+ | a<=12 = 32%205 * ((1 + 4 / d) * a - 4) + (toRational (optimalValue (mBigOnHalf d)) - 1) * a / d+ | a<=15 = 32%205 * a + toRational (optimalValue (mBigOnHalf a)) - 1+ | otherwise = 32%205 * (2 * a - 6)+ xt = 1%2 / (1 + ia a)+++bestLambda l = (\x -> (x, fromRational $ f l x)) (bestLambda' 0 (3999%1000)) where+ bestLambda' a b+ | b-a < 1%1000000000 = a+ | otherwise = if mx1 > mx2 then bestLambda' x1 b else bestLambda' a x2 where+ x1 = (2*a+b)/3+ x2 = (a+2*b)/3+ ma = f l a+ mb = f l b+ mx1 = f l x1+ mx2 = f l x2++difur a = a * m' - m + (77%205) where+ h = 1%(10^100)+ m = toRational $ optimalValue $ mBigOnHalf a+ mh = toRational $ optimalValue $ mBigOnHalf (a+h)+ m' = (mh-m) / h++solveD a b+ | b-a < 1%(10^6) = a+ | otherwise = if difur c > 0 then solveD a c else solveD c b where+ c = (a+b)/2++{-+D = 12.571624917200547+-}+
+ Math/ExpPairs/Kratzel.hs view
@@ -0,0 +1,64 @@+module Math.ExpPairs.Kratzel where++import Data.Ratio+import Data.Ord+import Data.List++import Math.ExpPairs++data TauabTheorem = Kr511a | Kr511b | Kr512a | Kr512b+ deriving (Show)++tauab :: Integer -> Integer -> (TauabTheorem, OptimizeResult)+tauab a' b' = minimumBy (comparing (optimalValue . snd)) [kr511a, kr511b, kr512a, kr512b] where+ a = a'%1+ b = b'%1+ kr511a = (Kr511a, optimize+ [RationalForm (LinearForm 2 2 (-1)) (LinearForm 0 0 (a+b))]+ [Constraint (LinearForm (-2*b) (2*a) (-a)) NonStrict])+ kr511b = (Kr511b, optimize+ [RationalForm (LinearForm 1 0 0) (LinearForm b (-a) a)]+ [Constraint (LinearForm (2*b) (-2*a) a) Strict])+ kr512a = (Kr512a, simulateOptimize r) where+ r = if 11*a >= 8*b then 19/29/(a+b) else 1%1+ kr512b = if 11*a >= 8*b then kr512a else (Kr512b, optimize+ [+ RationalForm (LinearForm (-11) 8 (-4)) (LinearForm (-29*b) (29*a) (4*b-20*a))+ ]+ [+ Constraint (LinearForm (-2*b) (2*a) (-a)) NonStrict,+ Constraint (LinearForm (-29) 0 4) Strict,+ Constraint (LinearForm 29 29 (-24)) Strict+ ])++data TauabcTheorem = Kolesnik | Kr61 | Kr62 | Kr63 | Kr64 | Kr65 | Kr66 | Tauab TauabTheorem+ deriving (Show)++tauabc :: Integer -> Integer -> Integer -> (TauabcTheorem, OptimizeResult)+tauabc 1 1 1 = (Kolesnik, simulateOptimize $ 43%96)+tauabc a' b' c' = minimumBy (comparing (optimalValue . snd)) [kr61, kr62, kr63, kr64, kr65, kr66] where+ a = a'%1+ b = b'%1+ c = c'%1+ kr61+ | c<a+b = (Kr61, simulateOptimize $ 2/(a+b+c))+ | optimalValue optRes < Finite (recip c) = (Kr61, simulateOptimize $ 1/c)+ | otherwise = (Tauab th, optRes)+ where+ (th, optRes) = tauab a' b'+ kr62 = (Kr62, optimize+ [RationalForm (LinearForm 2 2 0) (LinearForm 0 0 (a+b+c))]+ [+ Constraint (LinearForm (-b-c) a 0) NonStrict,+ Constraint (LinearForm (-2*c) (-2*c) (a+b+c)) NonStrict+ ])+ kr63 = (Kr63, optimize+ [RationalForm (LinearForm 4 2 3) (LinearForm (2*(a+b+c)) 0 (3*(a+b+c)))]+ [Constraint (LinearForm (2*(a-b-c)) (2*a) (2*a-b-c)) NonStrict])+ kr64 = (Kr64, simulateOptimize r) where+ r = recip (a+b+c) * minimum ((a+b+c):[2-4*(k-1)%(3*2^k-4) | k<-[1..maxk], (3*2^k-2*k-4)%1 * a >= 2 * (b+c), (3*2^k-8)%1 * (a+b) >= (3*2^k-4*k+4)%1 * c])+ maxk = 4 `max` floor (logBase 2 (fromRational $ b+c))+ kr65 = (Kr65, simulateOptimize r) where+ r = if 7*a>=2*(b+c) && 4*(a+b)>=5*c then 3%2/(a+b+c) else 1%1+ kr66 = (Kr66, simulateOptimize r) where+ r = if 18*a>=7*(b+c) && 2*(a+b)>=3*c then 25%17/(a+b+c) else 1%1
+ Math/ExpPairs/LinearForm.hs view
@@ -0,0 +1,85 @@+module Math.ExpPairs.LinearForm (LinearForm (..), RationalForm (..), IneqType (..), Constraint (..), checkConstraint, evalLF, evalRF, substituteLF) where++import Data.List+import Data.Ratio+import Data.Monoid+import Math.ExpPairs.RatioInf++data LinearForm t = LinearForm t t t+ deriving (Eq)++instance (Num t, Eq t, Show t) => Show (LinearForm t) where+ show (LinearForm a b c) = if (a==0) && (b==0) && (c==0)+ then "0"+ else "(" ++ intercalate " + " (filter (/=[]) $+ [if a/= 0 then show a ++ "k" else []] +++ [if b/= 0 then show b ++ "l" else []] +++ [if c/= 0 then show c else []] ) ++ ")" -- where+ -- show' :: Rational -> String+ -- show' z = if denominator z==1 then show (numerator z) else show z++instance Num t => Num (LinearForm t) where+ (LinearForm a b c) + (LinearForm d e f) = LinearForm (a+d) (b+e) (c+f)+ (*) = undefined+ negate (LinearForm a b c) = LinearForm (negate a) (negate b) (negate c)+ abs = undefined+ signum = undefined+ fromInteger n = LinearForm 0 0 (fromInteger n)++instance Num t => Monoid (LinearForm t) where+ mempty = 0+ mappend = (+)++scaleLF :: (Num t, Eq t) => t -> LinearForm t -> LinearForm t+scaleLF 0 (LinearForm {}) = LinearForm 0 0 0+scaleLF s (LinearForm a b c) = LinearForm (a*s) (b*s) (c*s)++evalLF :: Num t => (t, t, t) -> LinearForm t -> t+evalLF (k, l, m) (LinearForm a b c) = a*k+l*b+m*c++substituteLF :: (Eq t, Num t) => (LinearForm t, LinearForm t, LinearForm t) -> LinearForm t -> LinearForm t+substituteLF (k, l, m) (LinearForm a b c) = scaleLF a k + scaleLF b l + scaleLF c m+++data RationalForm t = RationalForm (LinearForm t) (LinearForm t)+ deriving (Show)++instance Num t => Num (RationalForm t) where+ (+) = undefined+ (*) = undefined+ negate (RationalForm a b) = RationalForm (negate a) b+ abs = undefined+ signum = undefined+ fromInteger n = RationalForm (fromInteger n) 1++instance Num t => Fractional (RationalForm t) where+ fromRational r = RationalForm (fromInteger $ numerator r) (fromInteger $ denominator r)+ recip (RationalForm a b) = RationalForm b a++evalRF :: (Real t, Num t) => (Integer, Integer, Integer) -> RationalForm t -> RationalInf+evalRF (k', l', m') (RationalForm num den) = if denom==0 then InfPlus else Finite (numer / denom) where+ k = fromInteger k'+ l = fromInteger l'+ m = fromInteger m'+ numer = toRational $ evalLF (k, l, m) num+ denom = toRational $ evalLF (k, l, m) den++substituteRF :: (Eq t, Num t) => (LinearForm t, LinearForm t, LinearForm t) -> RationalForm t -> RationalForm t+substituteRF (k, l, m) (RationalForm num den) = RationalForm (substituteLF (k, l, m) num) (substituteLF (k, l, m) den)+++data IneqType = Strict | NonStrict+ deriving (Eq, Show)++data Constraint t = Constraint (LinearForm t) IneqType+ deriving (Show)++checkConstraint :: (Num t, Eq t) => (Integer, Integer, Integer) -> Constraint t -> Bool+checkConstraint (k', l', m') (Constraint lf ineq)+ = if ineq==NonStrict+ then signum numer /= -1+ else signum numer == 1 where+ k = fromInteger k'+ l = fromInteger l'+ m = fromInteger m'+ numer = evalLF (k, l, m) lf
+ Math/ExpPairs/Matrix3.hs view
@@ -0,0 +1,166 @@+module Math.ExpPairs.Matrix3 (Matrix3 (..), Vector3 (..), fromList, toList, normalize, prettyMatrix, multCol, det) where++import qualified Data.List as List+import Data.Monoid++data Vector3 t = Vector3 {+ a1 :: t,+ a2 :: t,+ a3 :: t+ }+ deriving (Eq, Show)++data Matrix3 t = Matrix3 {+ a11 :: t,+ a12 :: t,+ a13 :: t,+ a21 :: t,+ a22 :: t,+ a23 :: t,+ a31 :: t,+ a32 :: t,+ a33 :: t+ }+ deriving (Eq, Show)++instance Num t => Num (Matrix3 t) where+ a + b = Matrix3 {+ a11 = a11 a + a11 b,+ a12 = a12 a + a12 b,+ a13 = a13 a + a13 b,+ a21 = a21 a + a21 b,+ a22 = a22 a + a22 b,+ a23 = a23 a + a23 b,+ a31 = a31 a + a31 b,+ a32 = a32 a + a32 b,+ a33 = a33 a + a33 b+ }++ -- intercalate ",\n" [ "a"++(show i)++(show j)++" = "++( intercalate " + " ["a"++(show i)++(show k)++" a * "++"a"++(show k)++(show j)++" b" | k<-[1..3]] ) | i<-[1..3], j<-[1..3]]+ a * b = Matrix3 {+ a11 = a11 a * a11 b + a12 a * a21 b + a13 a * a31 b,+ a12 = a11 a * a12 b + a12 a * a22 b + a13 a * a32 b,+ a13 = a11 a * a13 b + a12 a * a23 b + a13 a * a33 b,+ a21 = a21 a * a11 b + a22 a * a21 b + a23 a * a31 b,+ a22 = a21 a * a12 b + a22 a * a22 b + a23 a * a32 b,+ a23 = a21 a * a13 b + a22 a * a23 b + a23 a * a33 b,+ a31 = a31 a * a11 b + a32 a * a21 b + a33 a * a31 b,+ a32 = a31 a * a12 b + a32 a * a22 b + a33 a * a32 b,+ a33 = a31 a * a13 b + a32 a * a23 b + a33 a * a33 b+ }++ negate a = Matrix3 {+ a11 = - a11 a,+ a12 = - a12 a,+ a13 = - a13 a,+ a21 = - a21 a,+ a22 = - a22 a,+ a23 = - a23 a,+ a31 = - a31 a,+ a32 = - a32 a,+ a33 = - a33 a+ }++ abs = undefined++ signum = undefined++ -- Multiplicative, not additive behaviour+ fromInteger n = Matrix3 {+ a11 = fromInteger n,+ a12 = 0,+ a13 = 0,+ a21 = 0,+ a22 = fromInteger n,+ a23 = 0,+ a31 = 0,+ a32 = 0,+ a33 = fromInteger n+ }++det :: (Num t) => Matrix3 t -> t+det a =+ a11 a * (a22 a * a33 a - a32 a * a23 a)+ - a12 a * (a21 a * a33 a - a23 a * a31 a)+ + a13 a * (a21 a * a32 a - a22 a * a31 a)++instance Fractional t => Fractional (Matrix3 t) where+ -- Multiplicative, not additive behaviour+ fromRational n = Matrix3 {+ a11 = fromRational n,+ a12 = 0,+ a13 = 0,+ a21 = 0,+ a22 = fromRational n,+ a23 = 0,+ a31 = 0,+ a32 = 0,+ a33 = fromRational n+ }++ recip a = Matrix3 {+ a11 = (a22 a * a33 a - a32 a * a23 a) / d,+ a12 = -(a21 a * a33 a - a23 a * a31 a) / d,+ a13 = (a21 a * a32 a - a22 a * a31 a) / d,+ a21 = -(a12 a * a33 a - a13 a * a32 a) / d,+ a22 = (a11 a * a33 a - a13 a * a31 a) / d,+ a23 = -(a11 a * a32 a - a12 a * a31 a) / d,+ a31 = (a12 a * a23 a - a13 a * a22 a) / d,+ a32 = -(a11 a * a23 a - a13 a * a21 a) / d,+ a33 = (a11 a * a22 a - a12 a * a21 a) / d+ } where d = det a+++instance Num t => Monoid (Matrix3 t) where+ mempty = 1+ mappend = (*)++toList :: Matrix3 t -> [t]+toList a = [a11 a, a12 a, a13 a, a21 a, a22 a, a23 a, a31 a, a32 a, a33 a]++fromList :: [t] -> Matrix3 t+fromList as = Matrix3 {+ a11 = as!!0,+ a12 = as!!1,+ a13 = as!!2,+ a21 = as!!3,+ a22 = as!!4,+ a23 = as!!5,+ a31 = as!!6,+ a32 = as!!7,+ a33 = as!!8+ }++instance Functor Matrix3 where+ fmap f = fromList . List.map f . toList++normalize :: Integral t => Matrix3 t -> Matrix3 t+normalize m = m' where+ l = toList m+ d = foldl1 gcd l+ m' = if d==0 then m else fromList $ List.map (`div`d) l++maximum :: Ord t => Matrix3 t -> t+maximum = List.maximum . toList++prettyMatrix :: (Show t) => Matrix3 t -> String+prettyMatrix m =+ show (a11 m) ++ " " +++ show (a12 m) ++ " " +++ show (a13 m) ++ "\n" +++ show (a21 m) ++ " " +++ show (a22 m) ++ " " +++ show (a23 m) ++ "\n" +++ show (a31 m) ++ " " +++ show (a32 m) ++ " " +++ show (a33 m)++multCol :: (Num t) => Matrix3 t -> Vector3 t -> Vector3 t+multCol m v = Vector3 {+ a1 = a11 m * a1 v + a12 m * a2 v + a13 m * a3 v,+ a2 = a21 m * a1 v + a22 m * a2 v + a23 m * a3 v,+ a3 = a31 m * a1 v + a32 m * a2 v + a33 m * a3 v+ }+++
+ Math/ExpPairs/MenzerNowak.hs view
@@ -0,0 +1,15 @@+module Math.ExpPairs.MenzerNowak where++import Data.Ratio++import Math.ExpPairs++menzerNowak :: Integer -> Integer -> OptimizeResult+menzerNowak a' b' = optimize+ [+ RationalForm (LinearForm 1 1 0) (LinearForm (a+b) 0 (a+b)),+ RationalForm (LinearForm 1 0 0) (LinearForm (a+b) (-a) a)+ ]+ [] where+ a = a'%1+ b = b'%1
+ Math/ExpPairs/Pair.hs view
@@ -0,0 +1,41 @@+module Math.ExpPairs.Pair (InitPair' (..), InitPair, initPairs, initPairToValue) where++import Data.Ratio++data Triangle = Corput16 | HuxW87b1 | Hux05+ deriving (Show, Bounded, Enum, Eq, Ord)++data InitPair' t = Corput01 | Corput12 | Mix t t+ deriving (Eq)+type InitPair = InitPair' Rational++instance (Show t, Num t, Eq t) => Show (InitPair' t) where+ show Corput01 = "(0, 1)"+ show Corput12 = "(1/2, 1/2)"+ show (Mix r1 r2) =+ s1 ++ (if s1/="" && (s2/=""||s3/="") then " + " else "")+ ++ s2 ++ (if s2/="" && s3/="" then " + " else "") ++ s3+ where+ r3 = 1 - r1 - r2+ f r t = if r==0 then "" else (if r==1 then "" else show r ++ " * ") ++ show t+ s1 = f r1 Corput16+ s2 = f r2 HuxW87b1+ s3 = f r3 Hux05++sect :: Integer+sect = 30++initPairs :: [InitPair' (Ratio Integer)]+initPairs = Corput01 : Corput12 : [Mix (r1%sect) (r2%sect) | r1<-[0..sect], r2<-[0..sect-r1]]++initPairToValue :: InitPair -> (Rational, Rational)+initPairToValue Corput01 = (0, 1)+initPairToValue Corput12 = (1%2, 1%2)+initPairToValue (Mix r1 r2) = (x, y) where+ r3 = 1 - r1 - r2+ (x1, y1) = (1%6, 2%3)+ (x2, y2) = ( 2 % 13, 35 % 52)+ (x3, y3) = (32 % 205, 269 % 410)+ x = x1*r1 + x2*r2 + x3*r3+ y = y1*r1 + y2*r2 + y3*r3+
+ Math/ExpPairs/Process.hs view
@@ -0,0 +1,117 @@+{-# LANGUAGE TemplateHaskell #-}+module Math.ExpPairs.Process (Process, Path (), evalPath, lengthPath, aPath, baPath) where++import Data.Monoid+import Data.List+import Data.Ord+import Data.Function.Memoize++import qualified Math.ExpPairs.Matrix3 as Mx++data Process = A | BA+ deriving (Eq, Show, Read, Ord, Enum)++deriveMemoizable ''Process++type ProcessMatrix = Mx.Matrix3 Integer++process2matrix :: Process -> ProcessMatrix+process2matrix A = Mx.Matrix3 1 0 0 1 1 1 2 0 2+process2matrix BA = Mx.Matrix3 0 1 0 2 0 1 2 0 2++data Path = Path ProcessMatrix [Process]++aPath :: Path+aPath = Path (process2matrix A) [ A]+baPath :: Path+baPath = Path (process2matrix BA) [BA]++instance Monoid Path where+ mempty = Path 1 []+ mappend (Path m1 l1) (Path m2 l2) = Path (Mx.normalize $ m1*m2) (l1++l2)++instance Show Path where+ show (Path m l) = prettyProcesses l -- ++ "\n" ++ Mx.prettyMatrix m++instance Read Path where+ readsPrec _ zs = [reads' zs] where+ reads' ('A':xs) = (aPath `mappend` path, ys) where+ (path, ys) = reads' xs+ reads' ('B':'A':xs) = (baPath `mappend` path, ys) where+ (path, ys) = reads' xs+ reads' ('B':xs) = (baPath, xs)+ reads' xs = (mempty, xs)++instance Eq Path where+ (Path m1 _) == (Path m2 _) = Mx.normalize m1 == Mx.normalize m2++instance Ord Path where+ (Path _ q1) <= (Path _ q2) = cmp q1 q2 where+ cmp (A:p1) (A:p2) = cmp p1 p2+ cmp (BA:p1) (BA:p2) = cmp p2 p1+ cmp (A:_) (BA:_) = True+ cmp (BA:_) (A:_) = False+ cmp [] _ = True+ cmp _ [] = False++evalPath :: (Num t) => Path -> (t, t, t) -> (t, t, t)+evalPath (Path m _) (a,b,c) = (a',b',c') where+ m' = fmap fromInteger m+ (Mx.Vector3 a' b' c') = Mx.multCol m' (Mx.Vector3 a b c)++lengthPath :: Path -> Int+lengthPath (Path _ xs) = length xs++symbolWidth :: Int+symbolWidth = 10+bracketWidth :: Int+bracketWidth = 4+subscriptWidth :: Int+subscriptWidth = 4++-- Пусть строка из n символов является повторением строки из l символов+-- Какова длина такой записи?+len0 :: [Process] -> Int -> (Int, String)+len0 xs 1 = (lxs, pxs) where+ lxs = length pxs * symbolWidth+ pxs = concatMap show xs+len0 [A] n = (symbolWidth + subscriptWidth, show A ++ "^" ++ show n)+len0 [BA] n = (symbolWidth + bracketWidth*2 + subscriptWidth, "(" ++ show BA ++ ")^" ++ show n)+len0 xs n = (lxs + bracketWidth*2 + subscriptWidth, "(" ++ pxs ++ ")^" ++ show n) where+ (lxs, pxs) = len2M xs++len0M :: [Process] -> Int -> (Int, String)+len0M = memoize len0++-- Простейшая оптимизация: строка as, целиком состоящая из n повторений подстроки bs, может быть записана как bs^n+len1 :: [Process] -> (Int, String)+len1 as = if null inner+ then len0M as 1+ else len0M as 1 `min` minimumBy (comparing fst) inner where+ l = length as+ bs n = take n as+ cs m xs = concat (replicate m xs)+ inner = [len0M (bs n) (l`div`n) | n<-[1..l-1], l`mod`n==0, cs (l`div`n) (bs n) == as]++len1M :: [Process] -> (Int, String)+len1M = memoize len1++-- Перебираем все способы разбить строку на две части и применить к каждой из них len1+len2 :: [Process] -> (Int, String)+len2 as = if null inner+ then len1M as+ else len1M as `min` minimumBy (comparing fst) inner where+ l = length as+ bs n = take n as+ cs n = drop n as+ add (x, xs) (y, ys) = (x+y, xs++ys)+ inner = [ len2M (bs n) `add` len2M (cs n) | n<-[1..l-1] ]++len2M :: [Process] -> (Int, String)+len2M = memoize len2++prettyProcesses :: [Process] -> String+prettyProcesses = snd . len2M+++
+ Math/ExpPairs/RatioInf.hs view
@@ -0,0 +1,97 @@+{-|+Module : ExpPairs.RatioInf+Description : Rational numbers with infinities+Copyright : (c) Andrew Lelechenko, 2014-2015+License : GPL-3+Maintainer : andrew.lelechenko@gmail.com+Stability : experimental+Portability : POSIX++Provides types and necessary instances for rational numbers, extended with infinite values. Just use @RationalInf@ instead of @Rational@.+-}+module Math.ExpPairs.RatioInf (RatioInf (..), RationalInf) where++import Data.Ratio++-- |Extends a rational type with positive and negative+-- infinities.+data RatioInf t = InfMinus | Finite (Ratio t) | InfPlus+ deriving (Ord, Eq)++-- |Arbitrary-precision rational numbers with positive and negative+-- infinities+type RationalInf = RatioInf Integer++instance (Integral t, Show t) => Show (RatioInf t) where+ show InfMinus = "-Inf"+ show (Finite x) = show x+ show InfPlus = "+Inf"++instance (Integral t) => Num (RatioInf t) where+ InfMinus + InfPlus = error "Cannot add up negative and positive infinities"+ InfPlus + InfMinus = error "Cannot add up negative and positive infinities"+ InfMinus + _ = InfMinus+ InfPlus + _ = InfPlus+ _ + InfMinus = InfMinus+ _ + InfPlus = InfPlus+ (Finite a) + (Finite b) = Finite (a+b)++ fromInteger n = Finite (fromInteger n)++ signum InfMinus = Finite (-1)+ signum InfPlus = Finite 1+ signum (Finite r) = Finite (signum r)++ abs InfMinus = InfPlus+ abs InfPlus = InfPlus+ abs (Finite r) = Finite (abs r)++ negate InfMinus = InfPlus+ negate InfPlus = InfMinus+ negate (Finite r) = Finite (negate r)++ InfMinus * a+ | signum a == Finite 0 = error "Cannot multiply infinity by zero"+ | signum a == Finite 1 = InfMinus+ | signum a == Finite (-1) = InfPlus+ InfPlus * a+ | signum a == Finite 0 = error "Cannot multiply infinity by zero"+ | signum a == Finite 1 = InfPlus+ | signum a == Finite (-1) = InfMinus+ a * InfMinus+ | signum a == Finite 0 = error "Cannot multiply infinity by zero"+ | signum a == Finite 1 = InfMinus+ | signum a == Finite (-1) = InfPlus+ a * InfPlus+ | signum a == Finite 0 = error "Cannot multiply infinity by zero"+ | signum a == Finite 1 = InfPlus+ | signum a == Finite (-1) = InfMinus+ (Finite a) * (Finite b) = Finite (a*b)++instance (Integral t) => Fractional (RatioInf t) where+ fromRational = Finite . fromRational++ InfMinus / InfMinus = error "Cannot divide infinity by infinity"+ InfMinus / InfPlus = error "Cannot divide infinity by infinity"+ InfMinus / (Finite a)+ | signum a == 0 = error "Cannot divide infinity by zero"+ | signum a == 1 = InfMinus+ | signum a == -1 = InfPlus++ InfPlus / InfMinus = error "Cannot divide infinity by infinity"+ InfPlus / InfPlus = error "Cannot divide infinity by infinity"+ InfPlus / (Finite a)+ | signum a == 0 = error "Cannot divide infinity by zero"+ | signum a == 1 = InfPlus+ | signum a == -1 = InfMinus++ (Finite _) / InfPlus = Finite 0+ (Finite _) / InfMinus = Finite 0++ (Finite _) / (Finite 0) = error "Cannot divide finite value by zero"+ (Finite a) / (Finite b) = Finite (a/b)++instance (Integral t) => Real (RatioInf t) where+ toRational (Finite r) = toRational r+ toRational InfPlus = error "Cannot map infinity into Rational"+ toRational InfMinus = error "Cannot map infinity into Rational"
+ Math/ExpPairs/Tests.hs view
@@ -0,0 +1,19 @@+--module Math.ExpPairs.Tests where++import qualified Math.ExpPairs.Tests.RatioInf as RatioInf (testSuite)+import qualified Math.ExpPairs.Tests.Pair as Pair (testSuite)+import qualified Math.ExpPairs.Tests.Ivic as Ivic (testSuite)+import qualified Math.ExpPairs.Tests.MenzerNowak as MenzerNowak (testSuite)+import qualified Math.ExpPairs.Tests.Matrix3 as Matrix3 (testSuite)+import qualified Math.ExpPairs.Tests.Kratzel as Kratzel (testSuite)+import qualified Math.ExpPairs.Tests.LinearForm as LinearForm (testSuite)++main :: IO ()+main = do+ RatioInf.testSuite+ Pair.testSuite+ Ivic.testSuite+ MenzerNowak.testSuite+ Matrix3.testSuite+ Kratzel.testSuite+ LinearForm.testSuite
+ Setup.hs view
@@ -0,0 +1,2 @@+import Distribution.Simple+main = defaultMain
+ exp-pairs.cabal view
@@ -0,0 +1,41 @@+name: exp-pairs+version: 0.1.0.0+synopsis: Linear programming over exponent pairs+description: Package implements an algorithm to minimize rational objective function over the set of exponent pairs+homepage: https://github.com/Bodigrim/exp-pairs+license: GPL-3+license-file: LICENSE+author: Andrew Lelechenko+maintainer: andrew.lelechenko@gmail.com+category: Math+build-type: Simple+cabal-version: >=1.8++source-repository head+ type: git+ location: git://github.com/Bodigrim/exp-pairs.git++library+ exposed-modules: Math.ExpPairs,+ Math.ExpPairs.Ivic,+ Math.ExpPairs.Kratzel,+ Math.ExpPairs.MenzerNowak+ other-modules: Math.ExpPairs.LinearForm,+ Math.ExpPairs.Matrix3,+ Math.ExpPairs.Pair,+ Math.ExpPairs.Process,+ Math.ExpPairs.RatioInf+ build-depends: base >=4 && <5,+ memoize >=0.1,+ matrix >=0.1++test-suite tests+ ghc-options: -Wall+ type: exitcode-stdio-1.0+ main-is: Math/ExpPairs/Tests.hs+ build-depends: base >=4 && <5,+ QuickCheck >=2.4.2,+ smallcheck >=0.2.1,+ exp-pairs,+ memoize >=0.1,+ matrix >=0.1