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--{-# OPTIONS_GHC #-}
--module Main where
import Data.Packed.Internal
import Data.Packed.Internal.Vector
import Data.Packed.Internal.Matrix
import Data.Packed.Internal.Tensor
import Data.Packed.Matrix
import GSL.Vector
import LAPACK
import Complex
import Numeric(showGFloat)
import Data.List(transpose,intersperse,sort,elemIndex,nub,foldl',foldl1')
import Foreign.Storable
vr = fromList [1..15::Double]
vc = fromList (map (\x->x :+ (x+1)) [1..15::Double])
mi = (2 >< 3) [1 .. 6::Int]
mz = (2 >< 3) [1,2,3,4,5,6:+(1::Double)]
ac = (2><3) [1 .. 6::Double]
bc = (3><4) [7 .. 18::Double]
af = (2>|<3) [1,4,2,5,3,6::Double]
bf = (3>|<4) [7,11,15,8,12,16,9,13,17,10,14,18::Double]
a |=| b = rows a == rows b &&
cols a == cols b &&
toList (cdat a) == toList (cdat b)
mulC a b = multiply RowMajor a b
mulF a b = multiply ColumnMajor a b
cc = mulC ac bf
cf = mulF af bc
r = mulC cc (trans cf)
rd = (2><2)
[ 27736.0, 65356.0
, 65356.0, 154006.0 ]
main = do
print $ r |=| rd
print $ foldl part t [("p",1),("q",0),("r",2)]
print $ foldl part t [("p",1),("r",2),("q",0)]
print $ foldl part t $ reverse [("p",1),("r",2),("q",0)]
t = T [(4,(Covariant,"p")),(2,(Covariant,"q")),(3,(Contravariant,"r"))] $ fromList [1..24::Double]
t1 = T [(4,(Covariant,"p")),(4,(Contravariant,"q")),(2,(Covariant,"r"))] $ fromList [1..32::Double]
t2 = T [(4,(Covariant,"p")),(4,(Contravariant,"q"))] $ fromList [1..16::Double]
addT ts = T (dims (head ts)) (fromList $ sumT ts)
delta i j | i==j = 1
| otherwise = 0
e i n = fromList [ delta k i | k <- [1..n]]
diagl = diag.fromList
scalar x = T [] (fromList [x])
tensorFromVector idx v = T {dims = [(dim v,idx)], ten = v}
tensorFromMatrix idxr idxc m = T {dims = [(rows m,idxr),(cols m,idxc)], ten = cdat m}
td = tensorFromMatrix (Contravariant,"i") (Covariant,"j") $ diagl [1..4] :: Tensor Double
tn = tensorFromMatrix (Contravariant,"i") (Covariant,"j") $ (2><3) [1..6] :: Tensor Double
tt = tensorFromMatrix (Contravariant,"i") (Covariant,"j") $ (2><3) [1..6] :: Tensor Double
tq = T [(3,(Covariant,"p")),(2,(Covariant,"q")),(2,(Covariant,"r"))] $ fromList [11 .. 22] :: Tensor Double
r1 = contraction tt "j" tq "p"
r1' = contraction' tt "j" tq "p"
pru = do
mapM_ (putStrLn.shdims.dims.normal) (contractions t1 t2)
let t1 = contraction tt "i" tq "q"
print $ normal t1
print $ foldl part t1 [("j",0),("p'",1),("r'",1)]
let t2 = contraction' tt "i" tq "q"
print $ normal t2
print $ foldl part t2 [("j",0),("p'",1),("r'",1)]
let t1 = contraction tq "q" tt "i"
print $ normal t1
print $ foldl part t1 [("j'",0),("p",1),("r",1)]
let t2 = contraction' tq "q" tt "i"
print $ normal t2
print $ foldl part t2 [("j'",0),("p",1),("r",1)]
scsig t = scalar (signature (nms t)) `prod` t
where nms = map (snd.snd) . dims
antisym' t = addT $ map (scsig . flip tridx t) (perms (names t))
{-
where T d v = t
t' = T d' v
fixdim (T _ v) = T d v
d' = [(n,(c,show (pos q))) | (n,(c,q)) <- d]
pos n = i where Just i = elemIndex n nms
nms = map (snd.snd) d
-}
auxrename (T d v) = T d' v
where d' = [(n,(c,show (pos q))) | (n,(c,q)) <- d]
pos n = i where Just i = elemIndex n nms
nms = map (snd.snd) d
antisym t = T (dims t) (ten (antisym' (auxrename t)))
norper t = prod t (scalar (recip $ fromIntegral $ product [1 .. length (dims t)]))
antinorper t = prod t (scalar (fromIntegral $ product [1 .. length (dims t)]))
tvector n v = tensorFromVector (Contravariant,n) v
tcovector n v = tensorFromVector (Covariant,n) v
vector n v = tvector n (fromList v) :: Tensor Double
wedge a b = antisym (prod (norper a) (norper b))
a /\ b = wedge a b
a <*> b = normal $ prod a b
u = vector "p" [1,1,0]
v = vector "q" [0,1,1]
w = vector "r" [1,0,1]
uv = u /\ v
uw = u /\ w
normAT t = sqrt $ innerAT t t
innerAT t1 t2 = dot (ten t1) (ten t2) / fromIntegral (fact $ length $ dims t1)
det m = product $ toList s where (_,s,_) = svdR' m
fact n = product [1..n]
l1 = vector "p" [0,0,0,1]
l2 = vector "q" [1,0,0,1]
l3 = vector "r" [0,1,0,1]
leviCivita n = antisym $ foldl1 prod $ zipWith tcovector (map show [1..]) (toRows (ident n))
contractionF t1 t2 = contraction t1 n1 t2 n2
where n1 = fn t1
n2 = fn t2
fn = snd . snd . head . dims
dualV vs = foldl' contractionF (leviCivita n) vs
where n = fst . head . dims . head $ vs
dual1 = foldl' contractionF (leviCivita 3) [u,v]
dual2 = foldl' contractionF (leviCivita 3) [u,v,w]
contract1b t (n1,n2) = contract1 t n1 n2
dual1' = prod (foldl' contract1b ((leviCivita 3) <*> (u /\ v)) [("1","p'"),("2'","q''")]) (scalar (recip $ fact 2))
dual2' = prod (foldl' contract1b ((leviCivita 3) <*> (u /\ v /\ w)) [("1","p'"),("2'","q''"),("3'","r''")]) (scalar (recip $ fact 3))
x1 = vector "p" [0,0,1]
x2 = vector "q" [2,2,2]
x3 = vector "r" [-3,-1,-1]
x4 = vector "s" [12,0,3]
raise (T d v) = T (map raise' d) v
where raise' (n,(Covariant,s)) = (n,(Contravariant,s))
raise' (n,(Contravariant,s)) = (n,(Covariant,s))
dualMV t = prod (foldl' contract1b (lc <*> t) ds) (scalar (recip $ fromIntegral $ fact (length ds)))
where
lc = leviCivita n
nms1 = map (snd.snd) (dims lc)
nms2 = map ((++"'").snd.snd) (dims t)
ds = zip nms1 nms2
n = fst . head . dims $ t
-- intersection of two lines :-)
-- > raise $ dualMV $ raise $ dualMV (x1/\x2) /\ dualV [x3,x4]
--(3'^[3]) [24.0,24.0,12.0]
y1 = vector "p" [0,0,0,1]
y2 = vector "q" [2,2,0,2]
y3 = vector "r" [-3,-1,0,-1]
y4 = vector "s" [12,0,0,3]
-- why not in R^4?
-- > raise $ dualMV $ raise $ dualMV (y1/\y2) /\ dualV [y3,y4]
-- scalar 0.0
-- it seems that the sum of ranks must be greater than n :(
asBase r n = filter (\x-> (x==nub x && x==sort x)) $ sequence $ replicate r [1..n]
partF t i = part t (name,i) where name = snd . snd . head . dims $ t
--partL = foldl' partF
niceAS t = filter ((/=0.0).fst) $ zip vals base
where vals = map ((`at` 0).ten.foldl' partF t) (map (map pred) base)
base = asBase r n
r = length (dims t)
n = fst . head . dims $ t
z1 = vector "p" [0,0,0,1]
z2 = vector "q" [1,0,0,1]
z3 = vector "r" [0,1,0,1]
z4 = vector "s" [0,0,1,1]
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