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evilactually/polyfit.hs

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Polynomial fitter for PWTIR table and the resulting polynomial in separate file.
Raw
polyfit.hs
import Data.List
import Data.Array
import Data.Ratio
-- Uncertainty
-- (130, 62, 486/488)
-- row 70, may contain errors
-- | Input data
pwtir :: [(Ratio Integer, Ratio Integer, Ratio Integer)]
pwtir = [(120, 55, 123), (121, 55, 546), (122, 55, 844), (123, 55, 765), (124, 55, 645), (125, 55, 645), (126, 55, 324), (127, 55, 435), (128, 55, 545), (129, 55, 535), (130, 55, 768), (131, 55, 932), (132, 55, 434),
(120, 56, 345), (121, 56, 432), (122, 56, 345), (123, 56, 532), (124, 56, 534), (125, 56, 675), (126, 56, 785), (127, 56, 875), (128, 56, 876), (129, 56, 634), (130, 56, 543), (131, 56, 876), (132, 56, 986),
(120, 57, 432), (121, 57, 567), (122, 57, 789), (123, 57, 874), (124, 57, 546), (125, 57, 876), (126, 57, 096), (127, 57, 087), (128, 57, 023), (129, 57, 053), (130, 57, 921), (131, 57, 109), (132, 57, 012),
(120, 58, 213), (121, 58, 132), (122, 58, 345), (123, 58, 243), (124, 58, 234), (125, 58, 569), (126, 58, 927), (127, 58, 019), (128, 58, 893), (129, 58, 423), (130, 58, 656), (131, 58, 746), (132, 58, 737),
(120, 59, 536), (121, 59, 488), (122, 59, 985), (123, 59, 748), (124, 59, 847), (125, 59, 899), (126, 59, 000), (127, 59, 001), (128, 59, 002), (129, 59, 928), (130, 59, 764), (131, 59, 830), (132, 59, 100),
(120, 60, 011), (121, 60, 022), (122, 60, 033), (123, 60, 044), (124, 60, 055), (125, 60, 012), (126, 60, 011), (127, 60, 033), (128, 60, 122), (129, 60, 111), (130, 60, 133), (131, 60, 333), (132, 60, 444),
(120, 61, 456), (121, 61, 767), (122, 61, 874), (123, 61, 988), (124, 61, 774), (125, 61, 874), (126, 61, 847), (127, 61, 746), (128, 61, 928), (129, 61, 626), (130, 61, 887), (131, 61, 736), (132, 61, 111),
(120, 62, 122), (121, 62, 345), (122, 62, 546), (123, 62, 756), (124, 62, 444), (125, 62, 335), (126, 62, 647), (127, 62, 737), (128, 62, 987), (129, 62, 635), (130, 62, 486), (131, 62, 045), (132, 62, 395),
(120, 63, 391), (121, 63, 305), (122, 63, 793), (123, 63, 539), (124, 63, 026), (125, 63, 729), (126, 63, 593), (127, 63, 719), (128, 63, 291), (129, 63, 374), (130, 63, 532), (131, 63, 115), (132, 63, 732),
(120, 64, 229), (121, 64, 394), (122, 64, 836), (123, 64, 811), (124, 64, 437), (125, 64, 801), (126, 64, 729), (127, 64, 964), (128, 64, 692), (129, 64, 401), (130, 64, 329), (131, 64, 495), (132, 64, 012),
(120, 65, 301), (121, 65, 903), (122, 65, 406), (123, 65, 593), (124, 65, 602), (125, 65, 167), (126, 65, 876), (127, 65, 027), (128, 65, 923), (129, 65, 024), (130, 65, 674), (131, 65, 934), (132, 65, 023),
(120, 66, 930), (121, 66, 304), (122, 66, 555), (123, 66, 223), (124, 66, 945), (125, 66, 730), (126, 66, 821), (127, 66, 556), (128, 66, 038), (129, 66, 120), (130, 66, 543), (131, 66, 736), (132, 66, 736),
(120, 67, 125), (121, 67, 873), (122, 67, 747), (123, 67, 435), (124, 67, 827), (125, 67, 873), (126, 67, 123), (127, 67, 837), (128, 67, 983), (129, 67, 833), (130, 67, 837), (131, 67, 928), (132, 67, 826),
(120, 68, 937), (121, 68, 837), (122, 68, 837), (123, 68, 826), (124, 68, 927), (125, 68, 726), (126, 68, 927), (127, 68, 277), (128, 68, 928), (129, 68, 289), (130, 68, 726), (131, 68, 625), (132, 68, 872),
(120, 69, 123), (121, 69, 762), (122, 69, 781), (123, 69, 871), (124, 69, 871), (125, 69, 871), (126, 69, 122), (127, 69, 124), (128, 69, 827), (129, 69, 762), (130, 69, 872), (131, 69, 872), (132, 69, 812),
(120, 70, 123), (121, 70, 776), (122, 70, 827), (123, 70, 019), (124, 70, 928), (125, 70, 190), (126, 70, 827), (127, 70, 982), (128, 70, 982), (129, 70, 921), (130, 70, 123), (131, 70, 368), (132, 70, 000),
(120, 71, 001), (121, 71, 002), (122, 71, 003), (123, 71, 123), (124, 71, 123), (125, 71, 432), (126, 71, 523), (127, 71, 543), (128, 71, 522), (129, 71, 245), (130, 71, 255), (131, 71, 235), (132, 71, 234),
(120, 72, 678), (121, 72, 478), (122, 72, 879), (123, 72, 984), (124, 72, 743), (125, 72, 383), (126, 72, 838), (127, 72, 333), (128, 72, 777), (129, 72, 322), (130, 72, 873), (131, 72, 823), (132, 72, 209),
(120, 73, 489), (121, 73, 029), (122, 73, 432), (123, 73, 237), (124, 73, 238), (125, 73, 238), (126, 73, 387), (127, 73, 287), (128, 73, 328), (129, 73, 287), (130, 73, 274), (131, 73, 234), (132, 73, 222),
(120, 74, 001), (121, 74, 002), (122, 74, 111), (123, 74, 223), (124, 74, 442), (125, 74, 433), (126, 74, 663), (127, 74, 666), (128, 74, 222), (129, 74, 298), (130, 74, 238), (131, 74, 273), (132, 74, 287),
(120, 75, 839), (121, 75, 236), (122, 75, 277), (123, 75, 231), (124, 75, 345), (125, 75, 784), (126, 75, 989), (127, 75, 908), (128, 75, 908), (129, 75, 349), (130, 75, 904), (131, 75, 349), (132, 75, 493),
(120, 76, 546), (121, 76, 356), (122, 76, 635), (123, 76, 632), (124, 76, 736), (125, 76, 267), (126, 76, 873), (127, 76, 872), (128, 76, 256), (129, 76, 738), (130, 76, 001), (131, 76, 732), (132, 76, 276),
(120, 77, 673), (121, 77, 736), (122, 77, 782), (123, 77, 728), (124, 77, 287), (125, 77, 269), (126, 77, 293), (127, 77, 938), (128, 77, 362), (129, 77, 287), (130, 77, 286), (131, 77, 286), (132, 77, 266),
(120, 78, 536), (121, 78, 242), (122, 78, 423), (123, 78, 672), (124, 78, 276), (125, 78, 632), (126, 78, 786), (127, 78, 786), (128, 78, 782), (129, 78, 123), (130, 78, 823), (131, 78, 673), (132, 78, 376),
(120, 79, 453), (121, 79, 534), (122, 79, 355), (123, 79, 256), (124, 79, 726), (125, 79, 728), (126, 79, 988), (127, 79, 837), (128, 79, 662), (129, 79, 255), (130, 79, 536), (131, 79, 767), (132, 79, 637),
(120, 80, 983), (121, 80, 938), (122, 80, 273), (123, 80, 267), (124, 80, 847), (125, 80, 462), (126, 80, 727), (127, 80, 267), (128, 80, 256), (129, 80, 894), (130, 80, 963), (131, 80, 927), (132, 80, 278),
(120, 81, 456), (121, 81, 526), (122, 81, 636), (123, 81, 672), (124, 81, 290), (125, 81, 873), (126, 81, 362), (127, 81, 827), (128, 81, 666), (129, 81, 237), (130, 81, 222), (131, 81, 088), (132, 81, 087),
(120, 82, 536), (121, 82, 367), (122, 82, 928), (123, 82, 273), (124, 82, 978), (125, 82, 773), (126, 82, 928), (127, 82, 233), (128, 82, 298), (129, 82, 298), (130, 82, 298), (131, 82, 111), (132, 82, 111),
(120, 83, 467), (121, 83, 732), (122, 83, 827), (123, 83, 352), (124, 83, 762), (125, 83, 287), (126, 83, 278), (127, 83, 289), (128, 83, 902), (129, 83, 092), (130, 83, 272), (131, 83, 222), (132, 83, 244),
(120, 84, 647), (121, 84, 764), (122, 84, 748), (123, 84, 373), (124, 84, 222), (125, 84, 477), (126, 84, 748), (127, 84, 837), (128, 84, 278), (129, 84, 873), (130, 84, 782), (131, 84, 356), (132, 84, 654),
(120, 85, 867), (121, 85, 958), (122, 85, 598), (123, 85, 938), (124, 85, 930), (125, 85, 276), (126, 85, 622), (127, 85, 626), (128, 85, 262), (129, 85, 747), (130, 85, 646), (131, 85, 545), (132, 85, 522),
(120, 86, 635), (121, 86, 526), (122, 86, 872), (123, 86, 928), (124, 86, 647), (125, 86, 545), (126, 86, 244), (127, 86, 222), (128, 86, 111), (129, 86, 111), (130, 86, 737), (131, 86, 526), (132, 86, 625),
(120, 87, 452), (121, 87, 564), (122, 87, 352), (123, 87, 762), (124, 87, 702), (125, 87, 888), (126, 87, 999), (127, 87, 933), (128, 87, 555), (129, 87, 333), (130, 87, 902), (131, 87, 737), (132, 87, 363),
(120, 88, 333), (121, 88, 892), (122, 88, 989), (123, 88, 090), (124, 88, 909), (125, 88, 383), (126, 88, 822), (127, 88, 121), (128, 88, 232), (129, 88, 299), (130, 88, 283), (131, 88, 828), (132, 88, 273),
(120, 89, 647), (121, 89, 637), (122, 89, 522), (123, 89, 536), (124, 89, 267), (125, 89, 366), (126, 89, 266), (127, 89, 726), (128, 89, 992), (129, 89, 833), (130, 89, 636), (131, 89, 993), (132, 89, 333),
(120, 90, 545), (121, 90, 535), (122, 90, 727), (123, 90, 222), (124, 90, 888), (125, 90, 575), (126, 90, 757), (127, 90, 444), (128, 90, 332), (129, 90, 282), (130, 90, 092), (131, 90, 881), (132, 90, 882)]
variable_names = [(0,"x"), (1, "y")]
-- | Ordered arrays of values
weights :: Array Int (Ratio Integer)
weights =
let
unique_weights = (nub.sort $ map (\(x,_,_) -> x) pwtir)
(_, indexed_weights) = mapAccumL (\i x -> (i+1,(i,x))) 0 unique_weights in array (0, length unique_weights - 1) indexed_weights
weights_count = b + 1 where (_,b) = bounds weights
mphs :: Array Int (Ratio Integer)
mphs =
let
unique_mphs = (nub.sort $ map (\(_,x,_) -> x) pwtir)
(_, indexed_mphs) = mapAccumL (\i x -> (i+1,(i,x))) 0 unique_mphs in array (0, length unique_mphs - 1) indexed_mphs
mphs_count = b + 1 where (_,b) = bounds mphs
-- | Arrays of base polynomials
weight_bases :: Array Int [(Ratio Integer, [(Integer, Integer)])]
weight_bases = array (bounds weights) $ map (\idx -> (idx, polySimplify $ basisPoly 0 (elems weights) idx)) (indices weights)
mphs_bases :: Array Int [(Ratio Integer, [(Integer, Integer)])]
mphs_bases = array (bounds mphs) $ map (\idx -> (idx, polySimplify $ basisPoly 1 (elems mphs) idx)) (indices mphs)
-- | Function value look-up array
pwtir_values :: Array (Int, Int) (Ratio Integer)
pwtir_values =
array ((0,0),(weights_count-1,mphs_count-1)) $
map
(\(w,m,v) ->
let
(Just w_idx) = elemIndex w $ elems $ weights
(Just m_idx) = elemIndex m $ elems $ mphs
in ((w_idx,m_idx),v))
pwtir
pwtirMap f = [ f w_idx m_idx | w_idx <- indices weights, m_idx <- indices mphs]
-- | The answer to life universe and everything
pwtir_polynomial :: [(Ratio Integer, [(Integer, Integer)])]
pwtir_polynomial = polyAddAll [ polyMultiplyAll [(weight_bases ! w_idx), (mphs_bases ! m_idx), numberToPoly (pwtir_values ! (w_idx, m_idx))] | w_idx <- indices weights, m_idx <- indices mphs]
-- | Compute basis polynomial
basisPoly :: Integral a => a -> [Ratio Integer] -> Int -> [(Ratio Integer, [(a,Integer)])]
basisPoly b ts i =
let
n = polyMultiplyAll $ map (\t -> [(1, [(b,1)]), (-t, [])] ) ts /!! i
d = foldl (\d t -> d*((ts !! i)-t)) 1 (ts /!! i)
in polyMultiply n [( (1/d),[])]
-- HELPTER FUNCTIONS
-- | Infix operator to delete element from a list
xs /!! n = ys ++ (tail zs) where (ys,zs) = splitAt n xs
-- Polynomial Display Functions
baseTokens ns [] = ""
baseTokens ns bs =
concat $
intersperse "*" $
map
(\(b,e) ->
let Just bn = lookup b ns in
bn ++ if e > 1 then "^" ++ (show e) else "") bs
polyTokens ns =
map
(\(k, bs) ->
--((if k < 0 then "-" else "+"), (show (abs k)),"*", baseTokens ns bs))
((if k < 0 then "-" else "+"), "(" ++ (show (abs (k))) ++ ")","*", baseTokens ns bs))
ruleNoLooseStars =
map
(\(s,k,m,bs) ->
if null bs || null k then
(s,k,"",bs)
else
(s,k,m,bs))
ruleNoLeadingPlus (("+",k,m,bs):ms) = ("",k,m,bs) : ms
ruleNoLeadingPlus ms = ms
processTokens = ruleNoLooseStars . ruleNoLeadingPlus
concatTokens = concatMap (\(t1,t2,t3,t4) -> t1 ++ t2 ++ t3 ++ t4)
--showPoly ns = concatTokens . processTokens . polyTokens ns . polyFloatify
showPoly ns = concatTokens . processTokens . polyTokens ns
-- Functions for symbolic manipulations on polynomials
polyProductIdentity = [(1,[])]
--polyAdditionIdentity = [(0,[])]
membersAdd p1@(k1,bs) p2@(k2,_)
| (polyHash p1 == polyHash p2) = (k1+k2, bs)
membersAddAll a@((_,bs):_) = foldl membersAdd (0, bs) a
memberMultiply (k1,bs1) (k2,bs2) = memberSimplify (k1*k2, bs1 ++ bs2)
groupBases bs = groupBy (\(b1,_) (b2,_) -> b1 == b2) . sortBy (\(b1,_) (b2,_) -> compare b1 b2) $ bs
memberSimplify (k,bs) = (k, filter ((/=0).snd) . map (foldl (\(_,e1) (b,e2) -> (b, e1 + e2)) (0,0)) . groupBases $ bs)
hashBase = toInteger $ max mphs_count weights_count
polyHash (_,[]) = 0
polyHash (_,bs) = foldl (\h (b,e) -> h + e*hashBase^b) 0 bs
polyGroupByHash p =
map (map fst) .
groupBy (\(_,h1) (_,h2) -> h1 == h2) .
sortBy (\(_,h1) (_,h2) -> compare h1 h2) .
map (\x -> (x, polyHash x)) $ p
polySimplify p = filter ((/=0).fst) . map membersAddAll . polyGroupByHash . map memberSimplify $ p
polyMultiply p1 p2 = [memberMultiply m1 m2 | m1 <- p1, m2 <- p2]
polyMultiplyAll ps = foldl (\acc p -> polySimplify $ polyMultiply acc p) polyProductIdentity ps
polyAddAll ps = concat ps
numberToPoly x = [(x,[])]
polyFloatify p = map (\(k,bs) -> ((fromRational k :: Double) , bs)) p
polyCompute vs =
sum . map (\(k,bs) ->
(k*).product.map (\(b,e) -> (vs !! (fromInteger b :: Int) )^e) $ bs)
pwtirTest =
and $
pwtirMap (\w_idx m_idx ->
let
w = toRational $ weights ! w_idx
m = toRational $ mphs ! m_idx
in polyCompute [w, m] pwtir_polynomial == pwtir_values ! (w_idx, m_idx))
main = do
putStrLn $ showPoly variable_names $ polySimplify $ pwtir_polynomial
Raw
pwtir.hs
import Data.Ratio
pwtir x y = -(2152178128125761793289982828071119159109613935591743069592 % 1)+(631385761950148339077463635446494874165335240134770323733873 % 3080)*x-(4961085170038969660477002288033232137238519507247934127388399 % 554400)*x^2+(4772341930227294895004569076935109540229645737322527999557 % 20160)*x^3-(159766710543670207959467351079105789059314300528987853431 % 37800)*x^4+(2596266970565166163511187927605918329908706227052007347 % 48384)*x^5-(3604812421770479651329062827112145351189862270285894983 % 7257600)*x^6+(9078823700978064102874009099351600369247040575081 % 2688)*x^7-(40510890307284967923893919467516735857239917669027 % 2419200)*x^8+(1586823679260395510396244190138751011214053807 % 26880)*x^9-(113270117376636025071018311153724333785401169 % 806400)*x^10+(134745123553372047174793730633675576085719 % 665280)*x^11-(10685179108588858950315452439390345693037 % 79833600)*x^12+(25536896079871415772901402204651044073834020625431468131318170499825293 % 24067258815600)*y-(9194446720997780020927863210353461524705485966436915977486282761896804559 % 90974238322968000)*x*y+(529796106012365877623549629846222862332096297673838811888697789285387945903 % 120085994586317760000)*x^2*y-(402347331485909795772093228998847034542452799521554768053738814432818157 % 3447444820659840000)*x^3*y+(4379430852982484200957684049576767273820725034077110756859908588977393 % 2101650853236480000)*x^4*y-(1386279929786569286333719785371610322077066006058518453762018186707497 % 52401161274029568000)*x^5*y+(192479326828086422637653214996610048289497330968669341460530552052143 % 786017419110443520000)*x^6*y-(15308353568252515710387818555183711380866063121545229519388476411 % 9193186188426240000)*x^7*y+(4326162880355724049175107332810643676905610150650582064057648029 % 524011612740295680000)*x^8*y-(762556619451261554346646527379214004021561482681262609357557 % 26200580637014784000)*x^9*y+(234623018437414603918294925225551421416735359403098056393 % 3388006116855360000)*x^10*y-(52325203950207431000704681917935393098612756890700718759 % 524011612740295680000)*x^11*y+(39347233502992331780538087191958831682285312208470841 % 596289076566543360000)*x^12*y-(24958783531442163815690087241239026932033425814379143342851835270334008229559 % 98292610383615648000)*y^2+(12356157650463250001054480457536196502849772665044972293733990638290057545734323 % 510875842468842330480000)*x*y^2-(9414582171970017707656788240609854143600914303843534714979267021739021404118119 % 8917105614001611586560000)*x^2*y^2+(93394123832393968654185800843025831534541608234674137332698380678374025371797 % 3343914605250604344960000)*x^3*y^2-(3163686242781763044553998950282059754904995887461513953466346364103830792059 % 6344185812333162393600000)*x^4*y^2+(169361787540711098780435823497757230341417905015315260319629236320510759343 % 26751316842004834759680000)*x^5*y^2-(9406074361575933857041551921580795996211445373473326217547056700061504747 % 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