ideas-0.6: src/Domain/Math/Polynomial/Views.hs
-----------------------------------------------------------------------------
-- Copyright 2010, Open Universiteit Nederland. This file is distributed
-- under the terms of the GNU General Public License. For more information,
-- see the file "LICENSE.txt", which is included in the distribution.
-----------------------------------------------------------------------------
-- |
-- Maintainer : bastiaan.heeren@ou.nl
-- Stability : provisional
-- Portability : portable (depends on ghc)
--
-----------------------------------------------------------------------------
module Domain.Math.Polynomial.Views
( polyView, polyViewFor, polyViewWith, polyViewForWith
, quadraticView, quadraticViewFor, quadraticViewWith, quadraticViewForWith
, linearView, linearViewFor, linearViewWith, linearViewForWith
, constantPolyView, linearPolyView, quadraticPolyView, cubicPolyView
, monomialPolyView, binomialPolyView, trinomialPolyView
, polyNormalForm
, linearEquationView, quadraticEquationView, quadraticEquationsView
, higherDegreeEquationsView
) where
import Prelude hiding ((^))
import Control.Monad
import Common.Apply
import Common.View
import Common.Traversable
import Common.Uniplate (transform, uniplate, children)
import Common.Utils (distinct)
import Domain.Math.Data.Polynomial
import Domain.Math.Data.Relation
import Domain.Math.Data.OrList
import Domain.Math.Expr
import Domain.Math.Numeric.Views
import Domain.Math.Equation.CoverUpRules
import Domain.Math.Polynomial.CleanUp
import Data.Maybe
import qualified Domain.Math.Data.SquareRoot as SQ
import Domain.Math.SquareRoot.Views
import Domain.Math.Power.Views (powerFactorViewForWith)
-------------------------------------------------------------------
-- Polynomial view
polyView :: View Expr (String, Polynomial Expr)
polyView = polyViewWith identity
polyViewFor :: String -> View Expr (Polynomial Expr)
polyViewFor v = polyViewForWith v identity
polyViewWith :: Fractional a => View Expr a -> View Expr (String, Polynomial a)
polyViewWith v = makeView f (uncurry g)
where
f expr = do
pv <- selectVar expr
p <- match (polyViewForWith pv v) expr
return (pv, p)
g pv = build (polyViewForWith pv v)
polyViewForWith :: Fractional a => String -> View Expr a -> View Expr (Polynomial a)
polyViewForWith pv v = makeView f g
where
f expr =
case expr of
Var s | pv == s -> Just var
Nat n -> Just (fromIntegral n)
Negate a -> liftM negate (f a)
a :+: b -> liftM2 (+) (f a) (f b)
a :-: b -> liftM2 (-) (f a) (f b)
a :*: b -> liftM2 (*) (f a) (f b)
a :/: b -> do
c <- match v b
guard (c /= 0)
guard (pv `notElem` collectVars b)
p <- f a
return (fmap (/c) p)
Sym s [a, n] | s == powerSymbol ->
liftM2 power (f a) (matchNat n)
_ -> do
guard (pv `notElem` collectVars expr)
liftM con (match v expr)
g = build sumView . map h . reverse . terms
h (a, n) = build v a .*. (Var pv .^. fromIntegral n)
matchNat expr = do
n <- match integralView expr
guard (n >= 0)
return n
-------------------------------------------------------------------
-- Quadratic view
quadraticView :: View Expr (String, Expr, Expr, Expr)
quadraticView = quadraticViewWith identity
quadraticViewFor :: String -> View Expr (Expr, Expr, Expr)
quadraticViewFor v = quadraticViewForWith v identity
quadraticViewWith :: Fractional a => View Expr a -> View Expr (String, a, a, a)
quadraticViewWith v = polyViewWith v >>> second quadraticPolyView >>> makeView f g
where
f (s, (a, b, c)) = return (s, a, b, c)
g (s, a, b, c) = (s, (a, b, c))
quadraticViewForWith :: Fractional a => String -> View Expr a -> View Expr (a, a, a)
quadraticViewForWith pv v = polyViewForWith pv v >>> quadraticPolyView
-------------------------------------------------------------------
-- Linear view
linearView :: View Expr (String, Expr, Expr)
linearView = linearViewWith identity
linearViewFor :: String -> View Expr (Expr, Expr)
linearViewFor v = linearViewForWith v identity
linearViewWith :: Fractional a => View Expr a -> View Expr (String, a, a)
linearViewWith v = polyViewWith v >>> second linearPolyView >>> makeView f g
where
f (s, (a, b)) = return (s, a, b)
g (s, a, b) = (s, (a, b))
linearViewForWith :: Fractional a => String -> View Expr a -> View Expr (a, a)
linearViewForWith pv v = polyViewForWith pv v >>> linearPolyView
-------------------------------------------------------------------
-- Views on polynomials (degree)
constantPolyView :: Num a => View (Polynomial a) a
constantPolyView = makeView (isList1 . polynomialList) (buildList . list1)
linearPolyView :: Num a => View (Polynomial a) (a, a)
linearPolyView = makeView (isList2 . polynomialList) (buildList . list2)
quadraticPolyView :: Num a => View (Polynomial a) (a, a, a)
quadraticPolyView = makeView (isList3 . polynomialList) (buildList . list3)
cubicPolyView :: Num a => View (Polynomial a) (a, a, a, a)
cubicPolyView = makeView (isList4 . polynomialList) (buildList . list4)
-------------------------------------------------------------------
-- Views on polynomials (number of terms)
monomialPolyView :: Num a => View (Polynomial a) (a, Int)
monomialPolyView = makeView (isList1. terms) (buildPairs . list1)
binomialPolyView :: Num a => View (Polynomial a) ((a, Int), (a, Int))
binomialPolyView = makeView (isList2 . terms) (buildPairs . list2)
trinomialPolyView :: Num a => View (Polynomial a) ((a, Int), (a, Int), (a, Int))
trinomialPolyView = makeView (isList3 . terms) (buildPairs . list3)
-- helpers
buildList :: Num a => [a] -> Polynomial a
buildList = buildPairs . flip zip [0..] . reverse
buildPairs :: Num a => [(a, Int)] -> Polynomial a
buildPairs as
| null as = 0
| otherwise = foldl1 (+) (map f as)
where
f (a, n) = con a * power var n
polynomialList :: Num a => Polynomial a -> [a]
polynomialList p = map (`coefficient` p) [d, d-1 .. 0]
where d = degree p
list1 (a) = [a]
list2 (a, b) = [a, b]
list3 (a, b, c) = [a, b, c]
list4 (a, b, c, d) = [a, b, c, d]
isList1 [a] = Just a
isList1 _ = Nothing
isList2 [a, b] = Just (a, b)
isList2 _ = Nothing
isList3 [a, b, c] = Just (a, b, c)
isList3 _ = Nothing
isList4 [a, b, c, d] = Just (a, b, c, d)
isList4 _ = Nothing
-------------------------------------------------------------------
-- Normal form, and list of power factors
listOfPowerFactors :: Num a => String -> View Expr a -> View Expr [(a, Int)]
listOfPowerFactors pv v = sumView >>> listView (powerFactorViewForWith pv v)
polyNormalForm :: Num a => View Expr a -> View Expr (String, Polynomial a)
polyNormalForm v = makeView f (uncurry g)
where
f e = do
pv <- selectVar e
xs <- match (listOfPowerFactors pv v) e
guard (distinct (map snd xs))
return (pv, buildPairs xs)
g pv = build (listOfPowerFactors pv v) . reverse . terms
-------------------------------------------------------------------
-- Normal forms for equations
-- Excludes equations such as 1==1 or 0==1
linearEquationViewWith :: Fractional a => View Expr a -> View (Equation Expr) (String, a)
linearEquationViewWith v = makeView f g
where
f (lhs :==: rhs) = do
(x, a, b) <- match (linearViewWith v) (lhs - rhs)
return (x, -b/a)
g (x, r) = Var x :==: build v r
linearEquationView :: View (Equation Expr) (String, Rational)
linearEquationView = linearEquationViewWith rationalView
quadraticEquationsView:: View (OrList (Equation Expr)) (OrList (String, SQ.SquareRoot Rational))
quadraticEquationsView = makeView f (fmap g)
where
f eq = do
ors <- switch (fmap (match quadraticEquationView) eq)
return (normalize (join ors))
g (x, a) = Var x :==: build (squareRootViewWith rationalView) a
quadraticEquationView :: View (Equation Expr) (OrList (String, SQ.SquareRoot Rational))
quadraticEquationView = makeView f g
where
f (lhs :==: rhs) = do
(s, p) <- match (polyViewWith (squareRootViewWith rationalView)) (lhs - rhs)
guard (degree p <= 2)
liftM (fmap ((,) s)) $
case polynomialList p of
[a, b, c] -> do
discr <- SQ.fromSquareRoot (b*b - SQ.scale 4 (a*c))
let sdiscr = SQ.sqrtRational discr
twoA = SQ.scale 2 a
case compare discr 0 of
LT -> return false
EQ -> return $ orList [-b/twoA]
GT -> return $ orList [(-b+sdiscr)/twoA, (-b-sdiscr)/twoA]
[a, b] -> return $ orList [-b/a]
[a] | a==0 -> return true
_ -> return false
g ors =
case disjunctions ors of
Nothing -> 0 :==: 0
Just xs ->
let make (x, a) = Var x .-. build (squareRootViewWith rationalView) a
in build productView (False, map make xs) :==: 0
higherDegreeEquationsView :: View (OrList (Equation Expr)) (OrList Expr)
higherDegreeEquationsView = makeView f (fmap g)
where
f = let make (a :==: b) = orList (filter (not . hasNegSqrt) $ map cleanUpExpr2 (normHDE (a-b)))
in Just . normalize . join . fmap make . cuRules
g = (:==: 0)
cuRules :: OrList (Equation Expr) -> OrList (Equation Expr)
cuRules xs =
let new = fmap (fmap (cleanUpExpr2 . distr)) xs in
case msum (map (`apply` new) coverUpRulesOr) of
Just ys -> cuRules ys
Nothing -> new
hasNegSqrt :: Expr -> Bool
hasNegSqrt (Sqrt a) =
case match rationalView a of
Just r | r < 0 -> True
_ -> hasNegSqrt a
hasNegSqrt (Sym s [a, b]) | s == rootSymbol =
case (match rationalView a, match integerView b) of
(Just r, Just n) | r < 0 && even n -> True
_ -> hasNegSqrt a || hasNegSqrt b
hasNegSqrt a =
any hasNegSqrt (children a)
distr :: Expr -> Expr
distr = transform f
where
f ((a :+: b) :/: c) = (a ./. c) .+. (b ./. c)
f ((a :-: b) :/: c) = (a ./. c) .-. (b ./. c)
f a = a
normHDE :: Expr -> [Expr]
normHDE e =
case match (polyViewWith rationalView) e of
Just (x, p) -> g x p
Nothing -> fromMaybe [e] $ do
(x, a) <- match (linearEquationViewWith (squareRootViewWith rationalView)) (e :==: 0)
return [ Var x .+. build (squareRootViewWith rationalView) (-a) ]
where
g :: String -> Polynomial Rational -> [Expr]
g x p
| d==0 = []
| length (terms p) <= 1 = [Var x]
| d==1 = [Var x .+. fromRational (coefficient 0 p / coefficient 1 p)]
| d==2 = let [a,b,c] = [ coefficient n p | n <- [2,1,0] ]
discr = b*b - 4*a*c
sdiscr = SQ.sqrtRational discr
in if discr < 0 then [] else
map ((Var x .+.) . build (squareRootViewWith rationalView))
[ SQ.scale (1/(2*a)) (SQ.con b + sdiscr)
, SQ.scale (1/(2*a)) (SQ.con b - sdiscr)
]
| otherwise =
case terms p of
[(c, 0), (b, e1), (a, e2)] | e1 > 1 && e2 `mod` e1 == 0 ->
let list = [(c, 0), (b, 1), (a, e2 `div` e1)]
newp = sum (map (\(x, y) -> scale x (power var y)) list)
sub = map (substitute (x, Var x^fromIntegral e1))
in concatMap normHDE (sub (g x newp))
[(c, 0), (a, n)]
| odd n -> if c/a >= 0
then [Var x + root (fromRational (c/a)) (fromIntegral n)]
else [Var x - root (fromRational (abs (c/a))) (fromIntegral n)]
| even n -> if c/a > 0
then []
else [ Var x + root (fromRational (abs (c/a))) (fromIntegral n)
, Var x - root (fromRational (abs (c/a))) (fromIntegral n)
]
_ ->
case factorize p of
ps | length ps > 1 -> concatMap (g x) ps
_ -> [build (polyViewWith rationalView) (x, p)]
where
d = degree p
substitute :: (String, Expr) -> Expr -> Expr
substitute (s, a) (Var b) | s==b = a
substitute pair expr = f (map (substitute pair) cs)
where
(cs, f) = uniplate expr