diff --git a/Algebra/Polynomials/Bernstein.lhs b/Algebra/Polynomials/Bernstein.lhs new file mode 100644 --- /dev/null +++ b/Algebra/Polynomials/Bernstein.lhs @@ -0,0 +1,1009 @@ +\begin{code} +{-# OPTIONS -XUnboxedTuples -XScopedTypeVariables -XFlexibleContexts -XBangPatterns -XUndecidableInstances -XMultiParamTypeClasses -XFunctionalDependencies -XFlexibleInstances -XMagicHash -XTypeFamilies #-} +-- | Various functions for manipulating polynomials, essentially when +-- represented in the Bernstein basis, in one or two variables. +module Algebra.Polynomials.Bernstein (Bernsteinp(..),solve,Bernstein(..), + derivate,reorient) where + + +import Control.Monad.ST +import Algebra.Polynomials.Numerical + +import qualified Data.Vector.Unboxed as UV +import qualified Data.Vector.Unboxed.Mutable as MUV +import qualified Data.Vector as V +import Data.Vector.Generic as GV hiding ((++)) + +-- | The type for Bernstein polynomials with an arbitrary number of variables +data Bernsteinp a b=Bernsteinp { bounds::a, coefs::UV.Vector b } deriving (Eq,Show) + +type family Param a b +type instance Param Int a=a +type instance Param (Int,Int) a=(a,a) +type instance Param (Int,Int,Int) a=(a,a,a) +type instance Param (Int,Int,Int,Int) a=(a,a,a,a) + +type family Inter a b +type instance Inter Int a=(a,a) +type instance Inter (Int,Int) a=(a,a,a,a) +type instance Inter (Int,Int,Int) a=(a,a,a,a,a,a) +type instance Inter (Int,Int,Int,Int) a=(a,a,a,a,a,a,a,a) + + +class Bernstein a where + -- Constructeurs + (?)::UV.Unbox b=>Bernsteinp a b->a->b + constant::(UV.Unbox b,Num b, Fractional b)=>b->Bernsteinp a b + scale::(Num b, Fractional b,UV.Unbox b)=>b->Bernsteinp a b->Bernsteinp a b + scale a (Bernsteinp i b)=Bernsteinp i $ UV.map (*a) b + promote::(UV.Unbox b,Num b, Fractional b)=>Int->Bernsteinp Int b->Bernsteinp a b + elevate::(UV.Unbox b,Num b, Fractional b)=>a->Bernsteinp a b->Bernsteinp a b + eval::(UV.Unbox b,Num b,Fractional b)=>Bernsteinp a b->Param a b->b + restriction::(UV.Unbox b,Fractional b,Num b)=>Bernsteinp a b->Param a b->Param a b->Bernsteinp a b + + +instance Num (Bernsteinp a Interval)=>Intervalize (Bernsteinp a) where + intervalize (Bernsteinp i x)=Bernsteinp i $! UV.map interval x + intersects bxf bxg= + let (Bernsteinp _ xx)=bxf-bxg in + UV.all (\(Interval a b)->a<=0 && b>=0) xx + +binomials::(Num a, MUV.Unbox a)=>Int->UV.Vector a +binomials m= + UV.create $ do + v<-MUV.replicate ((m+1)*(m+1)) 0 + MUV.write v 0 1 + let fill i + | i>=(m+1)*(m+1) = return v + | i`mod`(m+1) == 0 = do + MUV.write v i 1 + fill (i+1) + | otherwise = do + a<-MUV.read v (i-m-1) + b<-MUV.read v (i-m-2) + MUV.write v i (a+b) + fill (i+1) + fill (m+1) + + +instance Bernstein Int where + (?) (Bernsteinp _ a) b=a!b + constant x=Bernsteinp 1 $ GV.singleton x + promote _=id + elevate r (Bernsteinp n f)= + if r<=0 then Bernsteinp n f else + let coef j= + let sumAll i result + | (i>j) || (i>=n) = result + | otherwise = + sumAll (i+1) $ result+(f!i)*((bin i (n-1))*((bin (j-i) r))/(bin j (n+r-1))) + in + (sumAll 0 0) + binomial=binomials $ n+r-1 + bin a b=binomial!(b*(n+r)+a) + in + Bernsteinp (n+r) $ UV.generate (n+r) coef + eval (Bernsteinp n points) t= + if n==0 then 0 else runST $ do + arr<-thaw points + let fill i s + | s>=n = MUV.read arr 0 + | i>=n-s = fill 0 (s+1) + | otherwise = do + a<-MUV.read arr i + b<-MUV.read arr (i+1) + MUV.write arr i $ a*(1-t)+b*t + fill (i+1) s + fill 0 1 + + restriction (Bernsteinp n0 points) a b= + runST $ do + pf<-thaw points + let casteljau bef aft nv u v i k s j + | i>=bef = return () + | k>=aft= casteljau bef aft nv u v (i+1) 0 1 0 + | s>=nv = casteljau bef aft nv u v i (k+1) 1 0 + | j>=nv = casteljau bef aft nv u v i k (s+1) 0 + | otherwise= + let idx i_ j_ k_=(i_*nv+j_)*aft + k_ + v'=(v-u)/(1-u) + in + if s+j0 = ra_ + | otherwise = 0 + rb + | rb_>0 = rb_ + | otherwise = 0 + in + if a<=0 || b<=0 then Bernsteinp (a+ra,b+rb) $ GV.replicate ((a+ra)*(b+rb)) 0 else + let idx i j=(i*b)+j + idx' i j=(i*(b+rb))+j + vect=create $ do + v<-MUV.new ((a+ra)*(b+rb)) + let coef i j + | i>=(a+ra) = return v + | j>=(b+rb) = coef (i+1) 0 + | otherwise=do + let sumAll i' j' !result + | i'>=a || i'>i = result + | j'>=b || j'>j = sumAll (i'+1) 0 result + | otherwise = + let x0=(bin i' (a-1))*((bin (i-i') ra)/(bin i (a+ra-1))) + x1=(bin j' (b-1))*((bin (j-j') rb)/(bin j (b+rb-1))) + in + sumAll i' (j'+1) $! result+x0*x1*(f!(idx i' j')) + MUV.write v (idx' i j) $ sumAll 0 0 0 + coef i (j+1) + coef 0 0 + + m=max (a+ra-1) (b+rb-1) + bin i j=binomial!(j*(m+1)+i) + binomial=binomials m + in + Bernsteinp (a+ra,b+rb) vect + eval (Bernsteinp (n0,n1) points) (a,b)= + if n0<=0 || n1<=0 then 0 else + runST $ do + pf<-thaw points + let casteljau p0 p1 u i j s + | i>=p0 = return () + | s>=p1 = casteljau p0 p1 u (i+1) 0 1 + | j>=(p1-s) = casteljau p0 p1 u i 0 (s+1) + | otherwise = do + x0<-MUV.read pf $ i+p0*j + x1<-MUV.read pf $ i+p0*(j+1) + MUV.write pf (i+p0*j) $ (1-u)*x0+u*x1 + casteljau p0 p1 u i (j+1) s + casteljau n1 n0 a 0 0 1 + casteljau 1 n1 b 0 0 1 + MUV.read pf 0 + + restriction (Bernsteinp (n0,n1) points) (a,c) (b,d)= + runST $ do + pf<-thaw points + let casteljau bef aft nv u v i k s j + | i>=bef = return () + | k>=aft= casteljau bef aft nv u v (i+1) 0 1 0 + | s>=nv = casteljau bef aft nv u v i (k+1) 1 0 + | j>=nv = casteljau bef aft nv u v i k (s+1) 0 + | otherwise= + let idx i_ j_ k_=(i_*nv+j_)*aft + k_ + v'=(v-u)/(1-u) + in + if s+j0 = ra_ + | otherwise = 0 + rb + | rb_>0 = rb_ + | otherwise = 0 + rc + | rc_>0 = rc_ + | otherwise = 0 + in + if a<=0 || b<=0 || c<=0 then + Bernsteinp (a+ra,b+rb,c+rc) $ GV.replicate ((a+ra)*(b+rb)*(c+rc)) 0 + else + let idx i j k=((i*b)+j)*c+k + idx' i j k=((i*(b+rb))+j)*(c+rc)+k + vect=create $ do + v<-MUV.new ((a+ra)*(b+rb)*(c+rc)) + let coef i j k + | i>=a+ra = return v + | j>=b+rb = coef (i+1) 0 0 + | k>=c+rc = coef i (j+1) 0 + | otherwise=do + let sumAll i' j' k' !result + | i'>=a || i'>i = result + | j'>=b || j'>j = sumAll (i'+1) 0 0 result + | k'>=c || k'>k = sumAll i' (j'+1) 0 result + | otherwise = + let x0=(bin i' (a-1))*((bin (i-i') ra)/(bin i (a+ra-1))) + x1=(bin j' (b-1))*((bin (j-j') rb)/(bin j (b+rb-1))) + x2=(bin k' (c-1))*((bin (k-k') rc)/(bin k (c+rc-1))) + in + sumAll i' j' (k'+1) $! result+x0*x1*x2*(f!(idx i' j' k')) + MUV.write v (idx' i j k) $ sumAll 0 0 0 0 + coef i j (k+1) + coef 0 0 0 + + m=max (max (a+ra-1) (b+rb-1)) (c+rc-1) + bin i j=binomial!(j*(m+1)+i) + binomial=binomials m + in + Bernsteinp (a+ra,b+rb,c+rc) vect + eval (Bernsteinp (n0,n1,n2) points) (a,b,c)= + if n0<=0 || n1<=0 || n2<=0 then 0 else + runST $ do + pf<-thaw points + let casteljau p0 p1 u i j s + | i>=p0 = return () + | s>=p1 = casteljau p0 p1 u (i+1) 0 1 + | j>=(p1-s) = casteljau p0 p1 u i 0 (s+1) + | otherwise = do + x0<-MUV.read pf $ i+p0*j + x1<-MUV.read pf $ i+p0*(j+1) + MUV.write pf (i+p0*j) $ (1-u)*x0+u*x1 + casteljau p0 p1 u i (j+1) s + casteljau (n1*n2) n0 a 0 0 1 + casteljau n2 n1 b 0 0 1 + casteljau 1 n2 c 0 0 1 + MUV.read pf 0 + restriction (Bernsteinp (n0,n1,n2) points) (a,c,e) (b,d,f)= + runST $ do + pf<-thaw points + let casteljau bef aft nv u v i k s j + | i>=bef = return () + | k>=aft= casteljau bef aft nv u v (i+1) 0 1 0 + | s>=nv = casteljau bef aft nv u v i (k+1) 1 0 + | j>=nv = casteljau bef aft nv u v i k (s+1) 0 + | otherwise= + let idx i_ j_ k_=(i_*nv+j_)*aft + k_ + v'=(v-u)/(1-u) + in + if s+j0 = ra_ + | otherwise = 0 + rb + | rb_>0 = rb_ + | otherwise = 0 + rc + | rc_>0 = rc_ + | otherwise = 0 + rd + | rd_>0 = rd_ + | otherwise = 0 + in + if a<=0 || b<=0 || c<=0 || d<=0 then + Bernsteinp (a+ra,b+rb,c+rc,d+rd) $ GV.replicate ((a+ra)*(b+rb)*(c+rc)*(d+rd)) 0 + else + let idx i j k l=(((i*b)+j)*c+k)*d+l + idx' i j k l=(((i*(b+rb))+j)*(c+rc)+k)*(d+rd)+l + vect=create $ do + v<-MUV.new ((a+ra)*(b+rb)*(c+rc)*(d+rd)) + let coef i j k l + | i>=a+ra = return v + | j>=b+rb = coef (i+1) 0 0 0 + | k>=c+rc = coef i (j+1) 0 0 + | l>=d+rd = coef i j (k+1) 0 + | otherwise=do + let sumAll i' j' k' l' !result + | i'>=a || i'>i = result + | j'>=b || j'>j = sumAll (i'+1) 0 0 0 result + | k'>=c || k'>k = sumAll i' (j'+1) 0 0 result + | l'>=d || l'>l = sumAll i' j' (k'+1) 0 result + | otherwise = + let x0=(bin i' (a-1))*((bin (i-i') ra)/(bin i (a+ra-1))) + x1=(bin j' (b-1))*((bin (j-j') rb)/(bin j (b+rb-1))) + x2=(bin k' (c-1))*((bin (k-k') rc)/(bin k (c+rc-1))) + x3=(bin l' (d-1))*((bin (l-l') rd)/(bin l (d+rd-1))) + in + sumAll i' j' k' (l'+1) $! result+x0*x1*x2*x3*(f!(idx i' j' k' l')) + MUV.write v (idx' i j k l) $ sumAll 0 0 0 0 0 + coef i j k (l+1) + coef 0 0 0 0 + + m=max (max (a+ra) (b+rb)) (max (c+rc) (d+rd)) + bin i j=binomial!(j*(m+1)+i) + binomial=binomials m + in + Bernsteinp (a+ra,b+rb,c+rc,d+rd) vect + + eval (Bernsteinp (n0,n1,n2,n3) points) (a,b,c,d)= + if n0<=0 || n1<=0 || n2<=0 || n3<=0 then 0 else + runST $ do + pf<-thaw points + let casteljau p0 p1 u i j s + | i>=p0 = return () + | s>=p1 = casteljau p0 p1 u (i+1) 0 1 + | j>=(p1-s) = casteljau p0 p1 u i 0 (s+1) + | otherwise = do + x0<-MUV.read pf $ i+p0*j + x1<-MUV.read pf $ i+p0*(j+1) + MUV.write pf (i+p0*j) $ (1-u)*x0+u*x1 + casteljau p0 p1 u i (j+1) s + casteljau (n1*n2*n3) n0 a 0 0 1 + casteljau (n2*n3) n1 b 0 0 1 + casteljau n3 n2 c 0 0 1 + casteljau 1 n3 d 0 0 1 + MUV.read pf 0 + + restriction (Bernsteinp (n0,n1,n2,n3) points) (a,c,e,g) (b,d,f,h)= + runST $ do + pf<-thaw points + let casteljau bef aft nv u v i k s j + | i>=bef = return () + | k>=aft= casteljau bef aft nv u v (i+1) 0 1 0 + | s>=nv = casteljau bef aft nv u v i (k+1) 1 0 + | j>=nv = casteljau bef aft nv u v i k (s+1) 0 + | otherwise= + let idx i_ j_ k_=(i_*nv+j_)*aft + k_ + v'=(v-u)/(1-u) + in + if s+jNum (Bernsteinp Int a) where + + (+) bf@(Bernsteinp m _) bg@(Bernsteinp n _)= + let (Bernsteinp m' f')=elevate (n-m) bf + (Bernsteinp _ g')=elevate (m-n) bg + in + Bernsteinp m' $ UV.generate m' $ \i->f'!i+g'!i + + + + (*) ff@(Bernsteinp (af) _) gg@(Bernsteinp (ag) _)= + if af<=0 || ag<=0 then + Bernsteinp 0 UV.empty + else + let mm=(af+ag)-2 + binomial=binomials mm + bin a b=binomial!(b*(mm+1)+a) + in + Bernsteinp (af+ag-1) $ create $ do + v<-MUV.new $ af+ag-1 + let fill i + | i>=af+ag-1 = return v + | otherwise = do + let mCoef' i' result + | i'>i || i'>=af = + result + | otherwise = + let a=((bin i' (af-1))*(bin (i-i') (ag-1)))/(bin i (af+ag-2)) in + mCoef' (i'+1) $ + result+a*(ff?i')*(gg?(i-i')) + + MUV.write v i $! mCoef' (max 0 (i-ag+1)) 0 + fill (i+1) + fill 0 + + (-) bf (Bernsteinp i g)= bf+(Bernsteinp i $ UV.map negate g) + + signum _=error "No signum operation for Bernstein1" + abs _=error "No abs operation for Bernstein1" + + fromInteger x=Bernsteinp 1 $ UV.singleton $ fromIntegral x + +instance (Fractional a, Num a,UV.Unbox a)=>Num (Bernsteinp (Int,Int) a) where + + (+) bff@(Bernsteinp (af,bf) _) bgg@(Bernsteinp (ag,bg) _)= + let (Bernsteinp (a,b) f')=elevate (ag-af,bg-bf) bff + (Bernsteinp _ g')=elevate (af-ag,bf-bg) bgg + in + Bernsteinp (a,b) $ UV.generate (a*b) $ \i->f'!i+g'!i + + + (*) ff@(Bernsteinp (af,bf) _) gg@(Bernsteinp (ag,bg) _)= + if af<=0 || bf<=0 || ag<=0 || bg<=0 then + Bernsteinp (0,0) UV.empty + else + let mm=max (af+ag) (bf+bg)-2 + binomial=binomials mm + bin a b=binomial!(b*(mm+1)+a) + in + Bernsteinp (af+ag-1,bf+bg-1) $ create $ do + v<-MUV.new $ (af+ag-1)*(bf+bg-1) + let idx i j=i*(bf+bg-1)+j + fill i j + | i>=af+ag-1 = return v + | j>=bf+bg-1 = fill (i+1) 0 + | otherwise = + do + let mCoef' i' j' result + | i'>i || i'>=af = + let b=(bin i (af+ag-2))*(bin j (bf+bg-2)) in + result/b + | j'>j || j'>=bf = mCoef' (i'+1) (max 0 (j-bg+1)) result + | otherwise = + let a=(bin i' (af-1))*(bin (i-i') (ag-1))* + (bin j' (bf-1))*(bin (j-j') (bg-1)) + in + mCoef' i' (j'+1) $ + result+a*(ff?(i',j'))*(gg?(i-i',j-j')) + + MUV.write v (idx i j) $! + mCoef' (max 0 (i-ag+1)) (max 0 (j-bg+1)) 0 + fill i (j+1) + fill 0 0 + + (-) bf bg=bf+(bg { coefs=UV.map negate $ coefs bg }) + + signum _=error "No signum operation for Bernstein1" + abs _=error "No abs operation for Bernstein1" + + fromInteger x=Bernsteinp (1,1) $ UV.singleton $ fromIntegral x +instance (Fractional a, Num a, UV.Unbox a)=>Num (Bernsteinp (Int,Int,Int) a) where + + (+) bff@(Bernsteinp (af,bf,cf) _) bgg@(Bernsteinp (ag,bg,cg) _)= + let (Bernsteinp (a,b,c) f')=elevate (ag-af,bg-bf,cg-cf) bff + (Bernsteinp _ g')=elevate (af-ag,bf-bg,cf-cg) bgg + in + Bernsteinp (a,b,c) $ UV.generate (a*b*c) $ \i->f'!i+g'!i + + (*) ff@(Bernsteinp (af,bf,cf) _) gg@(Bernsteinp (ag,bg,cg) _)= + if af<=0 || bf<=0 || cf<=0 || ag<=0 || bg<=0 || cg<=0 then + Bernsteinp (0,0,0) UV.empty + else + let mm=(max (max (af+ag) (bf+bg)) (cf+cg))-2 + binomial=binomials mm + bin a b=binomial!(b*(mm+1)+a) + in + Bernsteinp (af+ag-1,bf+bg-1,cf+cg-1) $ create $ do + v<-MUV.new $ (af+ag-1)*(bf+bg-1)*(cf+cg-1) + let idx i j k=(i*(bf+bg-1)+j)*(cf+cg-1)+k + fill i j k + | i>=af+ag-1 = return v + | j>=bf+bg-1 = fill (i+1) 0 0 + | k>=cf+cg-1 = fill i (j+1) 0 + | otherwise = + do + let mCoef' i' j' k' result + | i'>i || i'>=af = + let b=(bin i (af+ag-2))*(bin j (bf+bg-2))* + (bin k (cf+cg-2)) + in + result/b + | j'>j || j'>=bf = mCoef' (i'+1) (max 0 (j-bg+1)) (max 0 (k-cg+1)) result + | k'>k || k'>=cf = mCoef' i' (j'+1) (max 0 (k-cg+1)) result + | otherwise = + let a=(bin i' (af-1))*(bin (i-i') (ag-1))* + (bin j' (bf-1))*(bin (j-j') (bg-1))* + (bin k' (cf-1))*(bin (k-k') (cg-1)) + in + mCoef' i' j' (k'+1) $ + result+a*(ff?(i',j',k'))*(gg?(i-i',j-j',k-k')) + + MUV.write v (idx i j k) $! + mCoef' (max 0 (i-ag+1)) (max 0 (j-bg+1)) (max 0 (k-cg+1)) 0 + fill i j (k+1) + fill 0 0 0 + + (-) bf bg=bf+(bg { coefs=UV.map negate $ coefs bg }) + + signum _=error "No signum operation for Bernstein1" + abs _=error "No abs operation for Bernstein1" + + fromInteger x=Bernsteinp (1,1,1) $ UV.singleton $ fromIntegral x +instance (Fractional a, Num a,UV.Unbox a)=>Num (Bernsteinp (Int,Int,Int,Int) a) where + + (+) bff@(Bernsteinp (af,bf,cf,df) _) bgg@(Bernsteinp (ag,bg,cg,dg) _)= + let (Bernsteinp (a,b,c,d) f')=elevate (ag-af,bg-bf,cg-cf,dg-df) bff + (Bernsteinp _ g')=elevate (af-ag,bf-bg,cf-cg,df-dg) bgg + in + Bernsteinp (a,b,c,d) $ UV.generate (a*b*c*d) $ \i->f'!i+g'!i + + (*) ff@(Bernsteinp (af,bf,cf,df) _) gg@(Bernsteinp (ag,bg,cg,dg) _)= + if af<=0 || bf<=0 || cf<=0 || df<=0 || ag<=0 || bg<=0 || cg<=0 || dg<=0 then + Bernsteinp (0,0,0,0) UV.empty + else + let mm=(max (max (af+ag) (bf+bg)) (max (cf+cg) (df+dg)))-2 + binomial=binomials mm + bin a b=binomial!(b*(mm+1)+a) + in + Bernsteinp (af+ag-1,bf+bg-1,cf+cg-1,df+dg-1) $ create $ do + v<-MUV.new $ (af+ag-1)*(bf+bg-1)*(cf+cg-1)*(df+dg-1) + let idx i j k l=((i*(bf+bg-1)+j)*(cf+cg-1)+k)*(df+dg-1)+l + fill i j k l + | i>=af+ag-1 = return v + | j>=bf+bg-1 = fill (i+1) 0 0 0 + | k>=cf+cg-1 = fill i (j+1) 0 0 + | l>=df+dg-1 = fill i j (k+1) 0 + | otherwise = + do + let mCoef' i' j' k' l' result + | i'>i || i'>=af = + let b=(bin i (af+ag-2))*(bin j (bf+bg-2))* + (bin k (cf+cg-2))*(bin l (df+dg-2)) + in + result/b + | j'>j || j'>=bf = mCoef' (i'+1) (max 0 (j-bg+1)) (max 0 (k-cg+1)) (max 0 (l-dg+1)) result + | k'>k || k'>=cf = mCoef' i' (j'+1) (max 0 (k-cg+1)) (max 0 (l-dg+1)) result + | l'>l || l'>=df = mCoef' i' j' (k'+1) (max 0 (l-dg+1)) result + | otherwise = + let a=(bin i' (af-1))*(bin (i-i') (ag-1))* + (bin j' (bf-1))*(bin (j-j') (bg-1))* + (bin k' (cf-1))*(bin (k-k') (cg-1))* + (bin l' (df-1))*(bin (l-l') (dg-1)) + in + mCoef' i' j' k' (l'+1) $ + result+a*(ff?(i',j',k',l'))*(gg?(i-i',j-j',k-k',l-l')) + + MUV.write v (idx i j k l) $! + mCoef' (max 0 (i-ag+1)) (max 0 (j-bg+1)) (max 0 (k-cg+1)) (max 0 (l-dg+1)) 0 + fill i j k (l+1) + fill 0 0 0 0 + (-) bf bg=bf+(bg { coefs=UV.map negate $ coefs bg }) + + signum _=error "No signum operation for Bernstein1" + abs _=error "No abs operation for Bernstein1" + + fromInteger x=Bernsteinp (1,1,1,1) $ UV.singleton $ fromIntegral x + +-- | Computes the derivative of a univariate Bernstein polynomial. +derivate::(UV.Unbox a,Num a)=>Bernsteinp Int a->Bernsteinp Int a +derivate (Bernsteinp n f) + | n<=1 = Bernsteinp 0 $ UV.empty + | otherwise=Bernsteinp (n-1) $ UV.generate (n-1) (\i->(f!(i+1)-f!i)*(fromIntegral $ n-1)) + +-- | Computes @f(1-x)@ (useful when used with Bezier curves). +reorient::(UV.Unbox a)=>Bernsteinp Int a->Bernsteinp Int a +reorient (Bernsteinp n f)=Bernsteinp n (UV.reverse f) + +\end{code} + +\begin{code} +{- +restrict::Int->Int->Int->Bernsteinp i Interval->Double->Double->Bernsteinp i Interval +restrict bef aft nv (Bernsteinp ix poly) a b= + --traceShow "Restrict" $ + runST $ do + poly'<-thaw poly :: ST s (MUV.STVector s Interval) + casteljau poly' 0 0 1 0 + unsafeFreeze poly' >>= return.(Bernsteinp ix) + + where + + + (# bl,bu #)= + let (# b0,b1 #)=minus b b a a + (# b2,b3 #)=minus 1 1 a a + in + over b0 b1 b2 b3 + + idx i j k= --traceShow (i,j,k,d) $ + (i*nv+j)*aft + k + + casteljau::MUV.STVector s Interval-> + Int->Int->Int->Int->ST s () + + casteljau pf i k s j + | i>=bef = return () + | k>=aft= casteljau pf (i+1) 0 1 0 + | s>=nv = casteljau pf i (k+1) 1 0 + | j>=nv = casteljau pf i k (s+1) 0 + + -- Au boulot + | s+jInt->Int->Bernsteinp i a->a->a->Bernsteinp i a +restrict bef aft nv (Bernsteinp ix poly) a b= + --traceShow "Restrict" $ + runST $ do + poly'<-thaw poly + casteljau poly' 0 0 1 0 + unsafeFreeze poly' >>= return.(Bernsteinp ix) + + where + + + casteljau pf bef aft nv a b i k s j + | i>=bef = return () + | k>=aft= casteljau pf bef aft nv a b (i+1) 0 1 0 + | s>=nv = casteljau pf bef aft nv a b i (k+1) 1 0 + | j>=nv = casteljau pf bef aft nv a b i k (s+1) 0 + | otherwise= + let idx i j k=(i*nv+j)*aft + k + b'=(b-a)/(1-a) + in + if s+jInt->Int->Bernsteinp i Interval->Double->Double->(Bool,Double,Double) +convexHull bef aft nv (Bernsteinp _ points) a b= + let (allzero,pointsl,pointsu)=runST $ do + let idx i j k=(i*nv+j)*aft + k + pl<-MUV.replicate nv (1/0) + pu<-MUV.replicate nv (-1/0) + let fill i j k allzero_ -- a avant, b courant, c apres + | i>=bef = return allzero_ + | j>=nv = fill (i+1) 0 0 allzero_ + | k>=aft = fill i (j+1) 0 allzero_ + | otherwise = do + pl0<-MUV.read pl j + pu0<-MUV.read pu j + MUV.write pl j (min pl0 $ ilow $ points!(idx i j k)) + MUV.write pu j (max pu0 $ iup $ points!(idx i j k)) + fill i j (k+1) (allzero_ && pl0<=0 && pu0>=0) + allzero_<-fill 0 0 0 True + pl'<-UV.unsafeFreeze pl + pu'<-UV.unsafeFreeze pu + return (allzero_,pl',pu') + inter::Int->Int->(Double,Double) + inter i j + | i>j = inter j i + | otherwise = + let yli=pointsl!i + yui=pointsu!i + ylj=pointsl!j + yuj=pointsu!j + fi=fromIntegral i + fj=fromIntegral j + inter0 yi yj= + let k=fromIntegral $ i-j in + Interval fi fi + + (Interval yi yi)*(Interval k k)/ + (Interval yj yj-Interval yi yi) + in + if yli<=0 then + if yui>=0 then + if ylj<=0 then + if yuj>=0 then + -- 1 les deux sont sur la ligne + --traceShow "1" $ + (fi,fj) + else + -- 2 M est sur la ligne, M' est en-dessous + --traceShow "2" $ + (fi, iup $ inter0 yui yuj) + else + -- 3 M est sur la ligne, M' est au-dessus + --traceShow "3" $ + (fi,iup $ inter0 yli ylj) + else + -- M est en-dessous de la ligne + if ylj<=0 then + if yuj>=0 then + -- 4 M' est sur la ligne + --traceShow "4" $ + (ilow $ inter0 yui yuj, fj) + else + -- 5 M' est en-dessous (comme M) + --traceShow "5" $ + (1/0,-1/0) + else + -- 6 M' est au-dessus + --traceShow "6" $ + (ilow $ inter0 yui yuj, iup $ inter0 yli ylj) + else + -- M est au-dessus de la ligne + if ylj<=0 then + if yuj>=0 then + -- 7 M' est sur la ligne + --traceShow "7" $ + (ilow $ inter0 yli ylj,fj) + else + -- 8 M' est en-dessous (comme M) + --traceShow "8" $ + (ilow $ inter0 yli ylj, iup $ inter0 yui yuj) + else + -- 9 M' est au-dessus + --traceShow "9" $ + (1/0,-1/0) + + testAll i j m0 m1 + | i>=nv = + let fn=fromIntegral (nv-1) + (# a0,b0 #)=over m0 m0 fn fn + (# a1,b1 #)=minus b b a a + (# a2,b2 #)=times a0 b0 a1 b1 + (# a3,_ #)=plus a a a2 b2 + + (# c0,d0 #)=over m1 m1 fn fn + (# c2,d2 #)=times c0 d0 a1 b1 + (# _,d3 #)=plus a a c2 d2 + in + (False,a3,d3) + | j>=nv = testAll (i+1) (i+1) m0 m1 + | otherwise = + let (m0',m1')=inter i j in + testAll i (j+1) + (min m0 m0') (max m1 m1') + in + --traceShow allzero $ + if allzero then (True,a,b) else + testAll 0 0 (1/0) (-1/0) +--traceShow ("convexHull",pointsl,pointsu,m) $ m + +class Box a i | a->i where + cut::Int->a->[a] + size::Int->a->Double + restriction#::a->Bernsteinp i Interval->a + variables::a->Int + +instance Box (Double,Double) Int where + cut _ (a,b)= + let m=(a+b)/2 in + if a1 = convexHull 1 n1 n0 restr a b + | otherwise = (False,a,b) + (allz1,c',d') + | n1>1 = convexHull n0 1 n1 restr c d + | otherwise = (False,c,d) + in + (max a a', min b b', max c c', min d d') + variables _=2 + + +instance Box (Double,Double,Double,Double,Double,Double) (Int,Int,Int) where + cut 0 x@(a,b,c,d,e,f)= + let m=(a+b)/2 in + if a1 = convexHull 1 (n1*n2) n0 restr a b + | otherwise = (False,a,b) + (allz1,c',d') + | n1>1 = convexHull n0 n2 n1 restr c d + | otherwise = (False,c,d) + (allz2,e',f') + | n2>1 = convexHull (n0*n1) 1 n2 restr e f + | otherwise = (False,e,f) + in + (max a a', min b b', max c c', min d d', max e e', min f f') + + variables _=3 + +instance Box (Double,Double,Double,Double,Double,Double,Double,Double) (Int,Int,Int,Int) where + cut 0 x@(a,b,c,d,e,f,g,h)= + let m=(a+b)/2 in + if a1 = convexHull 1 (n1*n2*n3) n0 restr a b + | otherwise = (False,a,b) + (allz1,c',d') + | n1>1 = convexHull n0 (n2*n3) n1 restr c d + | otherwise = (False,c,d) + (allz2,e',f') + | n2>1 = convexHull (n0*n1) n3 n2 restr e f + | otherwise = (False,e,f) + (allz3,g',h') + | n3>1 = convexHull (n0*n1*n2) 1 n3 restr g h + | otherwise = (False,g,h) + in + --traceShow restr $ + (max a a', min b b', max c c', min d d', + max e e', min f f', max g g', min h h') + + variables _=4 + +-- | Computes the intersection of a given Bezier hypersurface, given +-- by its graph, with plane @z=0@. +solve::(Show a,Show i,Eq a,Box a i)=>Double->V.Vector (Bernsteinp i Interval)->a->[a] +solve sizeThreshold equations h= -- traceShow h $ + let splitThreshold=0.95 + restrictAll neq box + | neq>=V.length equations = box + | not (check 0 box) = box + | otherwise = + let next=restriction# box (equations!neq) in + restrictAll (neq+1) next + check v box= + (v>=(variables box)) || + (let s=size v box in + 0<=s && s<(1/0) && check (v+1) box) + + h'=restrictAll 0 h + + isSmall v box= + (v>=variables box) || + ((size v box <= sizeThreshold) && (isSmall (v+1) box)) + + in + if isSmall 0 h' then + if check 0 (restrictAll 0 h') then + [h'] + else + [] + else + if check 0 h' then + let cutAll v boxes + | v>=(variables h) = boxes + | otherwise = + cutAll (v+1) $ + Prelude.concatMap (\b->if (size v b)>=splitThreshold*(size v h) + && (size v b)>sizeThreshold + then + cut v b + else [b]) boxes + cc=cutAll 0 [h'] + in + case cc of + [h''] + | h''==h -> + [h] + | otherwise -> Prelude.concatMap (solve sizeThreshold equations) cc + _-> + Prelude.concatMap (solve sizeThreshold equations) cc + else + [] +\end{code} diff --git a/Algebra/Polynomials/Numerical.hs b/Algebra/Polynomials/Numerical.hs new file mode 100644 --- /dev/null +++ b/Algebra/Polynomials/Numerical.hs @@ -0,0 +1,263 @@ +{-# CFILES cnumerical.c #-} +{-# OPTIONS -XUnboxedTuples -XMagicHash -XScopedTypeVariables -XBangPatterns -cpp -XTypeFamilies -XMultiParamTypeClasses #-} +{-# LANGUAGE ForeignFunctionInterface #-} +-- | This module contains the definition of the main arithmetic tools +-- used in Metafont'. +module Algebra.Polynomials.Numerical( + -- * Raw operations + fromIntegral#,plus,minus,over,times, + sqrt#,cos#,sin#,acos#,asin#, + -- * The 'Interval' type + Interval(..),Intervalize(..), + interval,intersectsd, union, + fpred,fsucc + ) where + + +import Data.Vector.Unboxed as UV +import qualified Data.Vector.Generic.Mutable as GMV +import qualified Data.Vector.Generic as GV +import Foreign.C.Types +foreign import ccall unsafe "c_succ" c_fsucc::CDouble->CDouble +foreign import ccall unsafe "c_pred" c_fpred::CDouble->CDouble + +fsucc,fpred::Double->Double +fpred=realToFrac.c_fpred.realToFrac +fsucc=realToFrac.c_fsucc.realToFrac + +{-# INLINE plus #-} +-- | Interval addition +plus::Double->Double->Double->Double->(# Double, Double #) +plus !a !b !c !d= + let !x=a+c + !y=b+d + in + (# fpred x,fsucc y #) + +-- | Interval substraction +{-# INLINE minus #-} +minus::Double->Double->Double->Double->(# Double, Double #) +minus !a !b !c !d= + let !x=a-d + !y=b-c + in + (# fpred x, fsucc y #) +-- | Interval multiplication +{-# INLINE times #-} +times::Double->Double->Double->Double->(# Double, Double #) +times !a !b !c !d= + let !w=a*c + !x=a*d + !y=b*c + !z=b*d + + (# !aa,!bb #)=if wDouble->Double->Double->(# Double, Double #) +over !a !b !c !d= + if c*d<=0 then + if a>0 then (# 1/0,1/0 #) else + if b<0 then (# (-1/0), (-1/0) #) else + (# 0/0, 0/0 #) + + else + let !w=a/c + !x=a/d + !y=b/c + !z=b/d + + !(aa,bb)=if wx->(# Double,Double #) +fromIntegral# n= + let !n_=fromIntegral n in + (# fpred n_,fsucc n_ #) + +-- | Interval cosine +cos#::Double->Double->(# Double,Double #) +cos# !a !b= + let (# !_m0,!_m0' #)=if cos a<=cos b then (# cos a, cos b #) else (# cos b, cos a #) + !m0=fpred _m0 + !m0'=fsucc _m0' + checkUp !(k::Int) !m !m'= + let (# !ka,!kb #)=fromIntegral# k + (# !ka0,!kb0 #)=times ka kb (fpred pi) (fsucc pi) + in + if ka0>b then (# m,m' #) else + if kb0Double->(# Double,Double #) +sin# !a !b= + let (# _m0,_m0' #) + | sin ab then + (# m,m' #) + else + if kb2Double->(# Double,Double #) +sqrt# !a !b= + let sa=sqrt a + sb=sqrt b + sa_=max 0 (fpred sa) + sb_=fsucc sb + in + (# sa_, sb_ #) + +acos#::Double->Double->(# Double,Double #) +acos# !a !b= + let aca=acos $ max (-1) a + acb=acos $ min 1 b + in + (# fpred (min aca acb), fsucc (max aca acb) #) + +asin#::Double->Double->(# Double,Double #) +asin# !a !b= + let aca=asin $ max (-1) a + acb=asin $ min 1 b + in + (# fpred (min aca acb), fsucc (max aca acb) #) + +-- | The interval type (most of its operations are calls to the raw functions) +data Interval=Interval {ilow::Double,iup::Double} deriving (Eq, Show) + +instance Floating Interval where + cos (Interval a b)= + let (# c,d #)=cos# a b in + Interval c d + sin (Interval a b)= + let (# c,d #)=sin# a b in + Interval c d + sqrt (Interval a b)= + let (# a#,b# #)=sqrt# a b in + Interval a# b# + acos (Interval a b)= + let (# a#,b# #)=acos# a b in + Interval a# b# + asin (Interval a b)= + let (# a#,b# #)=asin# a b in + Interval a# b# + pi=Interval (fpred pi) (fsucc pi) + + +-- | Intersection of two 'Interval's. +{-# INLINE intersectsd #-} +intersectsd::Interval->Interval->Bool +intersectsd (Interval a b) (Interval c d) = b>=c && a<=d + +-- | Union of two intersecting intervals (undefined behaviour if they do not intersect). +{-# INLINE union #-} +union::Interval->Interval->Interval +union (Interval a b) (Interval c d) = Interval (min a c) (max b d) + +-- | Two common operations on types defined with intervals. +class Intervalize a where + intervalize::a Double->a Interval + intersects::a Interval->a Interval->Bool + +-- | Converts an optimal IEEE-754 representation of a number into an +-- optimal interval containing this number. +interval::Double->Interval +interval x=Interval (fpred x) (fsucc x) + +instance Num Interval where + (+) (Interval a b) (Interval c d)= + let (# a',b' #)=plus a b c d in + Interval a' b' + (-) (Interval a b) (Interval c d)= + let (# a',b' #)=minus a b c d in + Interval a' b' + (*) (Interval a b) (Interval c d)= + let (# a',b' #)=times a b c d in + Interval a' b' + abs x@(Interval a b)= + if b<=0 then Interval (negate b) (negate a) else + if a<=0 then + Interval 0 (max b $ negate a) + else + x + + signum _=undefined + + fromInteger=interval.fromInteger + +instance Fractional Interval where + + (/) (Interval a b) (Interval c d)= + let (# a',b' #)=over a b c d in + Interval a' b' + + fromRational r= + let r'=fromRational r in + Interval (fpred r') (fsucc r') + + +newtype instance UV.MVector s Interval = MV_Interval (UV.MVector s (Double,Double)) +newtype instance UV.Vector Interval = V_Interval (UV.Vector (Double,Double)) +instance Unbox Interval + +instance GMV.MVector UV.MVector Interval where + basicLength (MV_Interval a)=GMV.basicLength a + basicUnsafeSlice a b (MV_Interval c)=MV_Interval $ GMV.basicUnsafeSlice a b c + basicOverlaps (MV_Interval a) (MV_Interval b)=GMV.basicOverlaps a b + basicUnsafeNew a=GMV.basicUnsafeNew a >>= return.MV_Interval + basicUnsafeReplicate a (Interval b c)=GMV.basicUnsafeReplicate a (b,c)>>=return.MV_Interval + basicUnsafeRead (MV_Interval a) b=GMV.basicUnsafeRead a b >>= (\(u,v)->return $ Interval u v) + basicUnsafeWrite (MV_Interval a) b (Interval c d)=GMV.basicUnsafeWrite a b (c,d) + basicClear (MV_Interval a)=GMV.basicClear a + basicSet (MV_Interval a) (Interval b c)=GMV.basicSet a (b,c) + basicUnsafeCopy (MV_Interval a) (MV_Interval b)=GMV.basicUnsafeCopy a b + basicUnsafeGrow (MV_Interval a) b=GMV.basicUnsafeGrow a b >>= return.MV_Interval + +instance GV.Vector UV.Vector Interval where + basicUnsafeFreeze (MV_Interval a)=GV.basicUnsafeFreeze a >>= return.V_Interval + basicUnsafeThaw (V_Interval a)=GV.basicUnsafeThaw a >>= return.MV_Interval + basicLength (V_Interval a)=GV.basicLength a + basicUnsafeSlice a b (V_Interval c)=V_Interval (GV.basicUnsafeSlice a b c) + basicUnsafeIndexM (V_Interval a) b=GV.basicUnsafeIndexM a b >>= (\(u,v)->return $ Interval u v) + +(!#)::UV.Vector Interval->Int->(# Double,Double #) +(!#) a b= + let Interval u v=a!b in (# u,v #) + diff --git a/Algebra/Polynomials/cnumerical.c b/Algebra/Polynomials/cnumerical.c new file mode 100644 --- /dev/null +++ b/Algebra/Polynomials/cnumerical.c @@ -0,0 +1,143 @@ +#include +#include + +union stg_ieee754_dbl +{ + double d; + struct { + +#if WORDS_BIGENDIAN + unsigned int negative:1; + unsigned int exponent:11; + unsigned int mantissa0:20; + unsigned int mantissa1:32; +#else +#if FLOAT_WORDS_BIGENDIAN + unsigned int mantissa0:20; + unsigned int exponent:11; + unsigned int negative:1; + unsigned int mantissa1:32; +#else + unsigned int mantissa1:32; + unsigned int mantissa0:20; + unsigned int exponent:11; + unsigned int negative:1; +#endif +#endif + } ieee; + /* This format makes it easier to see if a NaN is a signalling NaN. */ + struct { + +#if WORDS_BIGENDIAN + unsigned int negative:1; + unsigned int exponent:11; + unsigned int quiet_nan:1; + unsigned int mantissa0:19; + unsigned int mantissa1:32; +#else +#if FLOAT_WORDS_BIGENDIAN + unsigned int mantissa0:19; + unsigned int quiet_nan:1; + unsigned int exponent:11; + unsigned int negative:1; + unsigned int mantissa1:32; +#else + unsigned int mantissa1:32; + unsigned int mantissa0:19; + unsigned int quiet_nan:1; + unsigned int exponent:11; + unsigned int negative:1; +#endif +#endif + } ieee_nan; +}; + + + +double c_succ(double y) +{ + union stg_ieee754_dbl su; + + su.d=y; + if (su.ieee.negative==0) { /* y >= 0 */ + if (su.ieee.exponent!=2047 || su.ieee.mantissa0!=0 || su.ieee.mantissa1!=0) + if (su.ieee.mantissa1==0xffffffff) { + su.ieee.mantissa1=0; + if (su.ieee.mantissa0==1048575) { + su.ieee.mantissa0=0; + su.ieee.exponent++; + } else { + su.ieee.mantissa0++; + } + } else { + su.ieee.mantissa1++; + } + } + else { /* y < 0 */ + if (su.ieee.exponent!=2047 || su.ieee.mantissa0!=0 || su.ieee.mantissa1==0){ + if (su.ieee.negative==1 && su.ieee.exponent==0 && su.ieee.mantissa0==0 && su.ieee.mantissa1==0) { + su.ieee.negative=0; + su.ieee.mantissa1=1; + } else { + if (su.ieee.mantissa1==0) { + su.ieee.mantissa1=0xffffffff; + if (su.ieee.mantissa0==0) { + su.ieee.mantissa0=1048575; + su.ieee.exponent--; + } else { + su.ieee.mantissa0--; + } + } else { + su.ieee.mantissa1--; + } + } + } + } + return su.d; +} /* end function q_succ */ + + + + + + +double c_pred(double y) +{ + union stg_ieee754_dbl su; + + su.d=y; + if (su.ieee.negative==1) { /* y < 0 */ + if (su.ieee.exponent!=2047 || su.ieee.mantissa0!=0 || su.ieee.mantissa1!=0 ) + if (su.ieee.mantissa1==0xffffffff) { + su.ieee.mantissa1=0; + if (su.ieee.mantissa0==1048575) { + su.ieee.mantissa0=0; + su.ieee.exponent++; + } else { + su.ieee.mantissa0++; + } + } else + su.ieee.mantissa1++; + } else { /* y >= 0 */ + if (su.ieee.exponent!=2047 || su.ieee.mantissa0!=0 || su.ieee.mantissa1!=0) + if (su.ieee.exponent==0 && su.ieee.mantissa0==0 && su.ieee.mantissa1==0) { + su.ieee.negative=1; + su.ieee.mantissa1=1; + } else { + if (su.ieee.mantissa1==0) { + su.ieee.mantissa1=0xffffffff; + if (su.ieee.mantissa0==0) { + su.ieee.mantissa0=1048575; + su.ieee.exponent--; + } else { + su.ieee.mantissa0--; + } + } else { + su.ieee.mantissa1--; + } + } + } + + return su.d; +} /* end function q_pred */ + diff --git a/LICENSE b/LICENSE new file mode 100644 --- /dev/null +++ b/LICENSE @@ -0,0 +1,340 @@ + GNU GENERAL PUBLIC LICENSE + Version 2, June 1991 + + Copyright (C) 1989, 1991 Free Software Foundation, Inc. + 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + Preamble + + The licenses for most software are designed to take away your +freedom to share and change it. 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You may copy and distribute the Program (or a work based on it, +under Section 2) in object code or executable form under the terms of +Sections 1 and 2 above provided that you also do one of the following: + + a) Accompany it with the complete corresponding machine-readable + source code, which must be distributed under the terms of Sections + 1 and 2 above on a medium customarily used for software interchange; or, + + b) Accompany it with a written offer, valid for at least three + years, to give any third party, for a charge no more than your + cost of physically performing source distribution, a complete + machine-readable copy of the corresponding source code, to be + distributed under the terms of Sections 1 and 2 above on a medium + customarily used for software interchange; or, + + c) Accompany it with the information you received as to the offer + to distribute corresponding source code. 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Each time you redistribute the Program (or any work based on the +Program), the recipient automatically receives a license from the +original licensor to copy, distribute or modify the Program subject to +these terms and conditions. You may not impose any further +restrictions on the recipients' exercise of the rights granted herein. +You are not responsible for enforcing compliance by third parties to +this License. + + 7. If, as a consequence of a court judgment or allegation of patent +infringement or for any other reason (not limited to patent issues), +conditions are imposed on you (whether by court order, agreement or +otherwise) that contradict the conditions of this License, they do not +excuse you from the conditions of this License. If you cannot +distribute so as to satisfy simultaneously your obligations under this +License and any other pertinent obligations, then as a consequence you +may not distribute the Program at all. For example, if a patent +license would not permit royalty-free redistribution of the Program by +all those who receive copies directly or indirectly through you, then +the only way you could satisfy both it and this License would be to +refrain entirely from distribution of the Program. + +If any portion of this section is held invalid or unenforceable under +any particular circumstance, the balance of the section is intended to +apply and the section as a whole is intended to apply in other +circumstances. + +It is not the purpose of this section to induce you to infringe any +patents or other property right claims or to contest validity of any +such claims; this section has the sole purpose of protecting the +integrity of the free software distribution system, which is +implemented by public license practices. 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The Free Software Foundation may publish revised and/or new versions +of the General Public License from time to time. Such new versions will +be similar in spirit to the present version, but may differ in detail to +address new problems or concerns. + +Each version is given a distinguishing version number. If the Program +specifies a version number of this License which applies to it and "any +later version", you have the option of following the terms and conditions +either of that version or of any later version published by the Free +Software Foundation. If the Program does not specify a version number of +this License, you may choose any version ever published by the Free Software +Foundation. + + 10. If you wish to incorporate parts of the Program into other free +programs whose distribution conditions are different, write to the author +to ask for permission. For software which is copyrighted by the Free +Software Foundation, write to the Free Software Foundation; we sometimes +make exceptions for this. Our decision will be guided by the two goals +of preserving the free status of all derivatives of our free software and +of promoting the sharing and reuse of software generally. + + NO WARRANTY + + 11. BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY +FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN +OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES +PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED +OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF +MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS +TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE +PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, +REPAIR OR CORRECTION. + + 12. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING +WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR +REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, +INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING +OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED +TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY +YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER +PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE +POSSIBILITY OF SUCH DAMAGES. + + END OF TERMS AND CONDITIONS + + How to Apply These Terms to Your New Programs + + If you develop a new program, and you want it to be of the greatest +possible use to the public, the best way to achieve this is to make it +free software which everyone can redistribute and change under these terms. + + To do so, attach the following notices to the program. It is safest +to attach them to the start of each source file to most effectively +convey the exclusion of warranty; and each file should have at least +the "copyright" line and a pointer to where the full notice is found. + + + Copyright (C) + + This program is free software; you can redistribute it and/or modify + it under the terms of the GNU General Public License as published by + the Free Software Foundation; either version 2 of the License, or + (at your option) any later version. + + This program is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + GNU General Public License for more details. + + You should have received a copy of the GNU General Public License + along with this program; if not, write to the Free Software + Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + + +Also add information on how to contact you by electronic and paper mail. + +If the program is interactive, make it output a short notice like this +when it starts in an interactive mode: + + Gnomovision version 69, Copyright (C) year name of author + Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'. + This is free software, and you are welcome to redistribute it + under certain conditions; type `show c' for details. + +The hypothetical commands `show w' and `show c' should show the appropriate +parts of the General Public License. Of course, the commands you use may +be called something other than `show w' and `show c'; they could even be +mouse-clicks or menu items--whatever suits your program. + +You should also get your employer (if you work as a programmer) or your +school, if any, to sign a "copyright disclaimer" for the program, if +necessary. Here is a sample; alter the names: + + Yoyodyne, Inc., hereby disclaims all copyright interest in the program + `Gnomovision' (which makes passes at compilers) written by James Hacker. + + , 1 April 1989 + Ty Coon, President of Vice + +This 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 Library General +Public License instead of this License. diff --git a/Setup.hs b/Setup.hs new file mode 100644 --- /dev/null +++ b/Setup.hs @@ -0,0 +1,3 @@ +import Distribution.Simple + +main=defaultMain diff --git a/polynomials-bernstein.cabal b/polynomials-bernstein.cabal new file mode 100644 --- /dev/null +++ b/polynomials-bernstein.cabal @@ -0,0 +1,22 @@ +Name: polynomials-bernstein +Version: 1 +Synopsis: A solver for systems of polynomial equations in bernstein form +Description: This library defines an optimized type for representing polynomials + in Bernstein form, as well as instances of numeric classes and other + manipulation functions, and a solver of systems of polynomial + equations in this form. +Category: Math +Maintainer: Pierre-Etienne Meunier +License: GPL +License-file: LICENSE +Build-Type: Simple +Cabal-Version: >=1.6 +source-repository this + type: darcs + location: http://www.lama.univ-savoie.fr/~meunier/darcs/polynomials + tag: 1.0 +Library + Build-Depends: base<5, vector + Exposed-modules: Algebra.Polynomials.Bernstein, Algebra.Polynomials.Numerical + ghc-options: -O2 -Wall + c-sources: algebra/polynomials/cnumerical.c