np-linear 0.3 → 0.3.0.1
raw patch · 2 files changed
+22/−22 lines, 2 files
Files
- np-linear.cabal +1/−1
- src/Algebra/Linear.hs +21/−21
np-linear.cabal view
@@ -1,5 +1,5 @@ name: np-linear-version: 0.3+version: 0.3.0.1 synopsis: Linear algebra for the numeric-prelude framework -- description: license: BSD3
src/Algebra/Linear.hs view
@@ -69,7 +69,7 @@ -- | Calculate all dependencies among the given vectors of degree d. dependencies :: (Algebra.Field.C k,Eq k) => Integer -> [Vector k] -> [Relation k]-dependencies d = map (Relation . genericDrop d) . filter (all (== zero) . genericTake d) . (\ (r,_,_) -> r) . reduce' . adorn+dependencies d = map (Relation . genericDrop d) . filter (all (== zero) . genericTake d) . (\ (r,_,_) -> r) . reduce . adorn -- | Calculate the equations satisfied by the subspace spanned by the given vectors of degree d. equations :: (Algebra.Field.C k,Eq k) => Integer -> [Vector k] -> [Relation k]@@ -81,10 +81,10 @@ inverseImage :: (Algebra.Field.C k,Eq k) => Matrix k -> Vector k -> Vector k inverseImage a = solveUpperTriangular u . matrixVector b where- (u,b,_) = reduce' a+ (u,b,_) = reduce a invert :: (Algebra.Field.C k,Eq k) => Matrix k -> Maybe (Matrix k)-invert m = fmap strip . process . (\ (r,_,_) -> r) . reduce' . adorn $ m where+invert m = fmap strip . process . (\ (r,_,_) -> r) . reduce . adorn $ m where n = length m process x = go x where go [v] = Just [v]@@ -101,10 +101,10 @@ v₀ = v !! i strip = map (drop n) -determinant :: (Algebra.Field.C k,Eq k,DebugDeterminant k) => Matrix k -> k+determinant :: (Algebra.Field.C k,Eq k) => Matrix k -> k determinant x = case reduce x of (_,_,σ) -> σ -adjoint :: (Algebra.Lattice.C k,Algebra.Field.C k,Eq k,DebugDeterminant k) => Matrix k -> Maybe (Matrix k)+adjoint :: (Algebra.Lattice.C k,Algebra.Field.C k,Eq k) => Matrix k -> Maybe (Matrix k) adjoint a = (abs (determinant a) *>) <$> invert a diagonal :: Matrix k -> [k]@@ -142,22 +142,17 @@ | lᵢᵢ == 0 = error "Linear.solveLowerTriangular: zero on diagonal" | otherwise = y / lᵢᵢ -class DebugDeterminant a where--reduce :: (Algebra.Field.C k,Eq k,DebugDeterminant k) => Matrix k -> (Matrix k,Matrix k,k)-reduce = reduce'- -- Compute the row echelon form of the matrix, -- together with the basis transformation matrix, -- and its determinant.-reduce' :: (Algebra.Field.C k,Eq k) => Matrix k -> (Matrix k,Matrix k,k)-reduce' [] = ([],[],1)-reduce' xs@([] : _) = (xs,identity (length xs),1)-reduce' vs = case nonZero of- [] -> (\ (x,u,_σ) -> (map (0 :) x,u,0)) $ reduce' (map tail vs)+reduce :: (Algebra.Field.C k,Eq k) => Matrix k -> (Matrix k,Matrix k,k)+reduce [] = ([],[],1)+reduce xs@([] : _) = (xs,identity (length xs),1)+reduce vs = case nonZero of+ [] -> (\ (x,u,_σ) -> (map (0 :) x,u,0)) $ reduce (map tail vs) (v@(v₀ : _),i) : [] -> let subMatrix = map (tail . fst) startZero- (h,u,σ) = reduce' subMatrix+ (h,u,σ) = reduce subMatrix sign = if odd i then 1 else -1 in ( (map (/ v₀) v :) . map (0 :) $ h@@ -168,7 +163,7 @@ (reduced,translates) = unzip . flip map rest $ \ (x@(x₀ : _),j) -> let c = x₀ / v₀ in ((zipWith (\ vᵢ xᵢ -> xᵢ - c * vᵢ) v x,j),(j,c)) subMatrix = v : map fst reduced ++ map fst startZero- (h,u,σ) = reduce' subMatrix+ (h,u,σ) = reduce subMatrix permutation = i : map snd reduced ++ map snd startZero in ( h@@ -274,11 +269,16 @@ test1,test2,test3,test4,test5,test7,test8 :: Matrix Rational test1 =- [[-169096784620000 % 453365147813277,-838720051715200 % 4987016625946047,6763871384800 % 151121715937759,216443884313600 % 4987016625946047,392304540318400 % 4987016625946047,27055485539200 % 1662338875315349,351721312009600 % 453365147813277,1975050444361600 % 4987016625946047,101458070772000 % 1662338875315349,5221708709065600 % 4987016625946047,1102511035722400 % 4987016625946047,1 % 1],[0 % 1,-135277427696 % 525510452511,-270554855392 % 1732881574809,4328877686272 % 209678670551889,7846090806368 % 209678670551889,5546374535536 % 29954095793127,135277427696 % 247554510687,39501008887232 % 209678670551889,676387138480 % 23297630061321,9604697366416 % 29954095793127,22050220714448 % 209678670551889,66973810188487 % 45384993625950],[0 % 1,16774401034304 % 399540911,270554855392 % 3301991,-4328877686272 % 399540911,-7846090806368 % 399540911,-1623329132352 % 399540911,-135277427696 % 471713,-39501008887232 % 399540911,-6087484246320 % 399540911,-104434174181312 % 399540911,-22050220714448 % 399540911,-64766190245461 % 259442150],[0 % 1,30302143803904 % 101277,-4328877686272 % 174933,-138524085960704 % 5772789,-251074905803776 % 5772789,-17315510745088 % 1924263,-225101639686144 % 524799,-1264032284391424 % 5772789,-21644388431360 % 641421,-3341893573801984 % 5772789,-705607062862336 % 5772789,-2417947450630819 % 4373325],[0 % 1,-536780833097728 % 11,12986633058816 % 1,138524085960704 % 11,251074905803776 % 11,51946532235264 % 11,225101639686144 % 1,1264032284391424 % 11,194799495882240 % 11,3341893573801984 % 11,705607062862336 % 11,7253842365012457 % 25],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,-108221942156800 % 524799,0 % 1,0 % 1,108221942156800 % 524799,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,-541109710784 % 95367421875,0 % 1,0 % 1,0 % 1,541109710784 % 95367421875,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,6763871384800 % 16663069252269,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,211370980775 % 536346624,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,-211370980775 % 537394176,0 % 1,211370980775 % 537394176,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,-211370980775 % 537394176,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,108221942156800 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1]]+ [[-169096784620000 % 453365147813277,-838720051715200 % 4987016625946047,6763871384800 % 151121715937759,216443884313600 % 4987016625946047,392304540318400 % 4987016625946047,27055485539200 % 1662338875315349,351721312009600 % 453365147813277,1975050444361600 % 4987016625946047,101458070772000 % 1662338875315349,5221708709065600 % 4987016625946047,1102511035722400 % 4987016625946047,1 % 1],[0 % 1,-135277427696 % 525510452511,-270554855392 % 1732881574809,4328877686272 % 209678670551889,7846090806368 % 209678670551889,5546374535536 % 29954095793127,135277427696 % 247554510687,39501008887232 % 209678670551889,676387138480 % 23297630061321,9604697366416 % 29954095793127,22050220714448 % 209678670551889,66973810188487 % 45384993625950],[0 % 1,16774401034304 % 399540911,270554855392 % 3301991,-4328877686272 % 399540911,-7846090806368 % 399540911,-1623329132352 % 399540911,-135277427696 % 471713,-39501008887232 % 399540911,-6087484246320 % 399540911,-104434174181312 % 399540911,-22050220714448 % 399540911,-+64766190245461 % 259442150],[0 % 1,30302143803904 % 101277,-4328877686272 % 174933,-138524085960704 % 5772789,-251074905803776 % 5772789,-17315510745088 % 1924263,-225101639686144 % 524799,-1264032284391424 % 5772789,-21644388431360 % 641421,-3341893573801984 % 5772789,-705607062862336 % 5772789,-2417947450630819 % 4373325],[0 % 1,-536780833097728 % 11,12986633058816 % 1,138524085960704 % 11,251074905803776 % 11,51946532235264 % 11,225101639686144 % 1,1264032284391424 % 11,194799495882240 % 11,3341893573801984 % 11,705607062862336 % 11,7253842365012457 % 25],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,-108221942156800 % 524799,0 % 1,0 % 1,108221942156800 % 524799,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,-541109710784 % 95367421875,0 % 1,0 % 1,0 % 1,541109710784 % 95367421875,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,6763871384800 % 16663069252269,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,211370980775 % 536346624,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,-211370980775 % 537394176,0 % +1,211370980775 % 537394176,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,-211370980775 % 537394176,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,108221942156800 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1]] test2 =- [[-135277427696 % 525510452511,-270554855392 % 1732881574809,4328877686272 % 209678670551889,7846090806368 % 209678670551889,5546374535536 % 29954095793127,135277427696 % 247554510687,39501008887232 % 209678670551889,676387138480 % 23297630061321,9604697366416 % 29954095793127,22050220714448 % 209678670551889,66973810188487 % 45384993625950],[0 % 1,6763871384800 % 119772219,-108221942156800 % 14492438499,-196152270159200 % 14492438499,54110971078400 % 2070348357,-3381935692400 % 17110317,-987525222180800 % 14492438499,-50729035386000 % 4830812833,-432887768627200 % 2070348357,-551255517861200 % 14492438499,-1124184870087 % 125475658],[0 % 1,-108221942156800 % 524799,0 % 1,0 % 1,108221942156800 % 524799,108221942156800 % 524799,0 % 1,0 % 1,-108221942156800 % 524799,0 % 1,1162300833 % 1],[0 % 1,16990844918617600 % 399,3463102149017600 % 399,6276872645094400 % 399,-1731551074508800 % 57,6926204298035200 % 57,31600807109785600 % 399,1623329132352000 % 133,13852408596070400 % 57,17640176571558400 % 399,197878620851139 % 19],[0 % 1,2325080788525 % 9746376192,77573149944425 % 214420276224,-6129758442475 % 107210138112,-8666210211775 % 30631468032,-2325080788525 % 2784678912,-15430081596575 % 53605069056,-1056854903875 % 23824475136,-211370980775 % 239308344,-34453469866325 % 214420276224,-66944106946759 % 29703241728],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,-108221942156800 % 524799,0 % 1,0 % 1,108221942156800 % 524799,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,-541109710784 % 95367421875,0 % 1,0 % 1,0 % 1,541109710784 % 95367421875,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,6763871384800 % 16663069252269,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,211370980775 % 536346624,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,108221942156800 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1]]-test3 = [[6763871384800 % 119772219,-108221942156800 % 14492438499,-196152270159200 % 14492438499,54110971078400 % 2070348357,-3381935692400 % 17110317,-987525222180800 % 14492438499,-50729035386000 % 4830812833,-432887768627200 % 2070348357,-551255517861200 % 14492438499,-1124184870087 % 125475658],[0 % 1,-1731551074508800 % 63500679,-3138436322547200 % 63500679,6385094587251200 % 21166893,-270554855392000 % 524799,-15800403554892800 % 63500679,-270554855392000 % 7055631,-61578285087219200 % 63500679,-8820088285779200 % 63500679,724539378362069 % 641421],[0 % 1,1731551074508800 % 121,3138436322547200 % 121,-6060428760780800 % 121,270554855392000 % 1,15800403554892800 % 121,2434993698528000 % 121,48483430086246400 % 121,8820088285779200 % 121,188875843483779 % 11],[0 % 1,211370980775 % 537394176,0 % 1,-211370980775 % 537394176,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,-178053373 % 80352],[0 % 1,0 % 1,0 % 1,0 % 1,-108221942156800 % 524799,0 % 1,0 % 1,108221942156800 % 524799,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,-541109710784 % 95367421875,0 % 1,0 % 1,0 % 1,541109710784 % 95367421875,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,6763871384800 % 16663069252269,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,211370980775 % 536346624,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,108221942156800 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1]]-test4 = [[-1731551074508800 % 63500679,-3138436322547200 % 63500679,6385094587251200 % 21166893,-270554855392000 % 524799,-15800403554892800 % 63500679,-270554855392000 % 7055631,-61578285087219200 % 63500679,-8820088285779200 % 63500679,724539378362069 % 641421],[0 % 1,0 % 1,108221942156800 % 1,0 % 1,0 % 1,0 % 1,-108221942156800 % 1,0 % 1,609975477158400 % 1],[0 % 1,-6129758442475 % 8598306816,34030727904775 % 8598306816,-11625403942625 % 1563328512,-15430081596575 % 4299153408,-1056854903875 % 1910734848,-120270088060975 % 8598306816,-34453469866325 % 17196613632,5634017560436093 % 400212099072],[0 % 1,0 % 1,0 % 1,-108221942156800 % 524799,0 % 1,0 % 1,108221942156800 % 524799,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,-541109710784 % 95367421875,0 % 1,0 % 1,0 % 1,541109710784 % 95367421875,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,6763871384800 % 16663069252269,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,211370980775 % 536346624,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,108221942156800 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1]]+ [[-135277427696 % 525510452511,-270554855392 % 1732881574809,4328877686272 % 209678670551889,7846090806368 % 209678670551889,5546374535536 % 29954095793127,135277427696 % 247554510687,39501008887232 % 209678670551889,676387138480 % 23297630061321,9604697366416 % 29954095793127,22050220714448 % 209678670551889,66973810188487 % 45384993625950],[0 % 1,6763871384800 % 119772219,-108221942156800 % 14492438499,-196152270159200 % 14492438499,54110971078400 % 2070348357,-3381935692400 % 17110317,-987525222180800 % 14492438499,-50729035386000 % 4830812833,-432887768627200 % 2070348357,-551255517861200 % 14492438499,-1124184870087 % 125475658],[0 % 1,-108221942156800 % 524799,0 % 1,0 % 1,108221942156800 % 524799,108221942156800 % 524799,0 % 1,0 % 1,-108221942156800 % 524799,0 % 1,1162300833 % 1],[0 % 1,16990844918617600 % 399,3463102149017600 % 399,6276872645094400 % 399,-1731551074508800 % 57,6926204298035200 % 57,31600807109785600 % 399,1623329132352000 % 133,13852408596070400 % 57,17640176571558400 % 399,+197878620851139 % 19],[0 % 1,2325080788525 % 9746376192,77573149944425 % 214420276224,-6129758442475 % 107210138112,-8666210211775 % 30631468032,-2325080788525 % 2784678912,-15430081596575 % 53605069056,-1056854903875 % 23824475136,-211370980775 % 239308344,-34453469866325 % 214420276224,-66944106946759 % 29703241728],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,-108221942156800 % 524799,0 % 1,0 % 1,108221942156800 % 524799,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,-541109710784 % 95367421875,0 % 1,0 % 1,0 % 1,541109710784 % 95367421875,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,6763871384800 % 16663069252269,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,211370980775 % 536346624,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,108221942156800 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1]]+test3 = [[6763871384800 % 119772219,-108221942156800 % 14492438499,-196152270159200 % 14492438499,54110971078400 % 2070348357,-3381935692400 % 17110317,-987525222180800 % 14492438499,-50729035386000 % 4830812833,-432887768627200 % 2070348357,-551255517861200 % 14492438499,-1124184870087 % 125475658],[0 % 1,-1731551074508800 % 63500679,-3138436322547200 % 63500679,6385094587251200 % 21166893,-270554855392000 % 524799,-15800403554892800 % 63500679,-270554855392000 % 7055631,-61578285087219200 % 63500679,-8820088285779200 % 63500679,724539378362069 % 641421],[0 % 1,1731551074508800 % 121,3138436322547200 % 121,-6060428760780800 % 121,270554855392000 % 1,15800403554892800 % 121,2434993698528000 % 121,48483430086246400 % 121,8820088285779200 % 121,188875843483779 % 11],[0 % 1,211370980775 % 537394176,0 % 1,-211370980775 % 537394176,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,-178053373 % 80352],[0 % 1,0 % 1,0 % 1,0 % 1,-108221942156800 % 524799,0 % 1,0 % 1,108221942156800 % 524799,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,-+541109710784 % 95367421875,0 % 1,0 % 1,0 % 1,541109710784 % 95367421875,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,6763871384800 % 16663069252269,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,211370980775 % 536346624,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,108221942156800 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1]]+test4 = [[-1731551074508800 % 63500679,-3138436322547200 % 63500679,6385094587251200 % 21166893,-270554855392000 % 524799,-15800403554892800 % 63500679,-270554855392000 % 7055631,-61578285087219200 % 63500679,-8820088285779200 % 63500679,724539378362069 % 641421],[0 % 1,0 % 1,108221942156800 % 1,0 % 1,0 % 1,0 % 1,-108221942156800 % 1,0 % 1,609975477158400 % 1],[0 % 1,-6129758442475 % 8598306816,34030727904775 % 8598306816,-11625403942625 % 1563328512,-15430081596575 % 4299153408,-1056854903875 % 1910734848,-120270088060975 % 8598306816,-34453469866325 % 17196613632,5634017560436093 % 400212099072],[0 % 1,0 % 1,0 % 1,-108221942156800 % 524799,0 % 1,0 % 1,108221942156800 % 524799,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,-541109710784 % 95367421875,0 % 1,0 % 1,0 % 1,541109710784 % 95367421875,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,6763871384800 % 16663069252269,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,211370980775 % 536346624,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,+0 % 1,108221942156800 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1]] test5 = [[-108221942156800 % 524799,0 % 1,0 % 1,108221942156800 % 524799,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,-541109710784 % 95367421875,541109710784 % 95367421875,577984359097 % 577984375000],[0 % 1,0 % 1,0 % 1,108221942156800 % 1,0 % 1,524800 % 1],[0 % 1,0 % 1,6763871384800 % 16663069252269,0 % 1,0 % 1,1 % 1],[0 % 1,211370980775 % 536346624,0 % 1,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,0 % 1,0 % 1,1 % 1]] test7 = [[0 % 1,-541109710784 % 95367421875,541109710784 % 95367421875,577984359097 % 577984375000],[0 % 1,108221942156800 % 1,0 % 1,524800 % 1],[6763871384800 % 16663069252269,0 % 1,0 % 1,1 % 1],[0 % 1,0 % 1,0 % 1,1 % 1]] test8 = [[-541109710784 % 95367421875,541109710784 % 95367421875,577984359097 % 577984375000],[108221942156800 % 1,0 % 1,524800 % 1],[0 % 1,0 % 1,1 % 1]]