mirror of
https://gitlab.com/sheaf/metabrush.git
synced 2024-11-05 14:53:37 +00:00
Split off root-isolation algorithms into Math.Root.Isolation
This commit is contained in:
sheaf
parent
bd468fcf82
commit
131753da82
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@ -124,6 +124,7 @@ library
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, Math.Orientation
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, Math.Ring
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, Math.Roots
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, Math.Root.Isolation
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, Debug.Utils
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other-modules:
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@ -40,6 +40,8 @@ import Numeric
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( showFFloat )
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-- containers
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import qualified Data.IntMap.Strict as IntMap
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( fromList, toList )
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import Data.Sequence
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( Seq )
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import qualified Data.Sequence as Seq
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@ -72,6 +74,7 @@ import Math.Linear
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import Math.Module
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import Math.Ring
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( Transcendental )
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import Math.Root.Isolation
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--------------------------------------------------------------------------------
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@ -120,7 +123,12 @@ benchTestCase ( TestCase { testName, testBrushStroke, testCuspOptions, testStart
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let ( _, testStrokeFnI ) = brushStrokeFunctions testBrushStroke
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( dunno, sols ) =
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foldMap ( \ ( i, ( _trees, ( mbCusps, defCusps ) ) ) -> ( map ( i , ) mbCusps, map ( i, ) defCusps ) ) $
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computeCusps testCuspOptions testStrokeFnI testStartBoxes
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IntMap.toList $
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findCuspsIn testCuspOptions testStrokeFnI $
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IntMap.fromList
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[ ( i, [ box ] )
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| ( i, box ) <- testStartBoxes
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]
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rnf dunno `seq` rnf sols `seq` return ()
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after <- getMonotonicTime
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let dt = after - before
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@ -175,9 +183,7 @@ benchCases :: [ TestCase ]
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benchCases =
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[ ellipseTestCase opts "full" ( 0, 1 ) pi $ defaultStartBoxes [ 0 .. 3 ] ]
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where
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opts = defaultCuspFindingOptions
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opts = defaultRootIsolationOptions
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--------------------------------------------------------------------------------
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@ -193,7 +199,7 @@ data TestCase =
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TestCase
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{ testName :: String
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, testBrushStroke :: BrushStroke
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, testCuspOptions :: CuspFindingOptions
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, testCuspOptions :: RootIsolationOptions
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, testStartBoxes :: [ ( Int, Box ) ]
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}
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@ -250,7 +256,7 @@ take 10 $ Data.List.sortOn ( \ ( _, ℝ1 e, v) -> abs e + norm v ) [ let { v = m
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potentialCusp $ eval fI $ mkBox (0.5798, 0.5799) 3 (0.26798, 0.26799)
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> True
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let nbPotentialSols b = let ( _newtTrees, ( dunno, sols ) ) = findCuspsFrom NoPreconditioning 1e-7 fI b in length dunno + length sols
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let nbPotentialSols b = let ( _newtTrees, ( dunno, sols ) ) = isolateRootsIn NoPreconditioning 1e-7 fI b in length dunno + length sols
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nbPotentialSols $ mkBox (0.5798, 0.5799) 3 (0.26798, 0.26799)
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1
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@ -258,7 +264,7 @@ nbPotentialSols $ mkBox (0.5798, 0.5799) 3 (0.26798, 0.26799)
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nbPotentialSols $ mkBox (0.5798, 0.675) 3 (0.26798, 0.26799)
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0
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let showTrees b = map ( uncurry showIntervalNewtonTree ) $ fst $ findCuspsFrom NoPreconditioning 1e-7 fI b
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let showTrees b = map ( uncurry showIntervalNewtonTree ) $ fst $ isolateRootsIn NoPreconditioning 1e-7 fI b
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putStrLn $ unlines $ map Data.Tree.View.showTree $ showTrees $ mkBox (0.5798, 0.675) 3 (0.26798, 0.26799)
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@ -464,7 +470,7 @@ maximum [ _y $ sup $ unT $ _D12_v $ dEdsdcdt $ eval fI $ mkBox (t, t) 2 (s, s) |
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let showTrees b = map ( uncurry showIntervalNewtonTree ) $ fst $ findCuspsFrom NoPreconditioning 1e-7 fI b
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let showTrees b = map ( uncurry showIntervalNewtonTree ) $ fst $ isolateRootsIn NoPreconditioning 1e-7 fI b
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putStrLn $ unlines $ map Data.Tree.View.showTree $ showTrees $ mkBox (0.5486101933245248, 0.5486102071493622) 2 (0.548095036738487, 0.5480952)
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@ -482,7 +488,7 @@ showD float = showFFloat (Just 6) float ""
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--------------------------------------------------------------------------------
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ellipseTestCase :: CuspFindingOptions -> String -> ( Double, Double ) -> Double -> [ ( Int, Box ) ] -> TestCase
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ellipseTestCase :: RootIsolationOptions -> String -> ( Double, Double ) -> Double -> [ ( Int, Box ) ] -> TestCase
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ellipseTestCase opts str k0k1 rot startBoxes =
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TestCase
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{ testName = "ellipse (" ++ str ++ ")"
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@ -512,7 +518,7 @@ trickyCusp2TestCase =
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TestCase
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{ testName = "trickyCusp2"
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, testBrushStroke = trickyCusp2BrushStroke
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, testCuspOptions = defaultCuspFindingOptions
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, testCuspOptions = defaultRootIsolationOptions
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, testStartBoxes = defaultStartBoxes [ 0 .. 3 ]
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}
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@ -601,19 +607,6 @@ getStrokeFunctions brush sp0 crv =
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brush @3 @𝕀 proxy# singleton )
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{-# INLINEABLE getStrokeFunctions #-}
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computeCusps
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:: CuspFindingOptions
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-> ( 𝕀ℝ 1 -> Seq ( 𝕀ℝ 1 -> StrokeDatum 3 𝕀 ) )
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-> [ ( Int, Box ) ]
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-> [ ( Int, ( [ ( Box, RootFindingTree Box ) ], ( [ Box ], [ Box ] ) ) ) ]
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computeCusps cuspOpts eqs startBoxes =
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map ( \ ( i, box ) -> ( i , ) $ findCuspsFrom cuspOpts ( mkEqn i ) box ) startBoxes
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where
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mkEqn i ( 𝕀 ( ℝ2 t_lo s_lo ) ( ℝ2 t_hi s_hi ) ) =
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let t = 𝕀 ( ℝ1 t_lo ) ( ℝ1 t_hi )
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s = 𝕀 ( ℝ1 s_lo ) ( ℝ1 s_hi )
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in ( eqs t `Seq.index` i ) s
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defaultStartBoxes :: [ Int ] -> [ ( Int, Box ) ]
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defaultStartBoxes is =
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[ ( i, mkBox (zero, one) (zero, one) ) | i <- is ]
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@ -623,8 +616,8 @@ getR1 (ℝ1 u) = u
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{-
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(f, fI) = brushStrokeFunctions $ ellipseBrushStroke (0,1) pi
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nbPotentialSols box = let ( _newtTrees, ( dunno, sols ) ) = findCuspsFrom NoPreconditioning 1e-7 fI box in length dunno + length sols
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showTrees box = putStrLn $ unlines $ map Data.Tree.View.showTree $ map ( uncurry showIntervalNewtonTree ) $ fst $ findCuspsFrom NoPreconditioning 1e-7 fI box
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nbPotentialSols box = let ( _newtTrees, ( dunno, sols ) ) = isolateRootsIn NoPreconditioning 1e-7 fI box in length dunno + length sols
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showTrees box = putStrLn $ unlines $ map Data.Tree.View.showTree $ map ( uncurry showIntervalNewtonTree ) $ fst $ isolateRootsIn NoPreconditioning 1e-7 fI box
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sol_t = 0.5486100729150693796677845183880669025324233347060776339185 :: Double
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sol_s = 0.5480950141859386853197594577293968665598143630958601978245 :: Double
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File diff suppressed because it is too large
Load diff
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@ -317,9 +317,9 @@ evaluateCubic bez t =
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let inf_bez = fmap inf bez
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sup_bez = fmap sup bez
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mins = fmap (Cubic.bezier @( T Double ) inf_bez)
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$ inf t :| ( sup t : filter ( inside t ) ( Cubic.extrema inf_bez ) )
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$ inf t :| ( sup t : filter ( `inside` t ) ( Cubic.extrema inf_bez ) )
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maxs = fmap (Cubic.bezier @( T Double ) sup_bez)
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$ inf t :| ( sup t : filter ( inside t ) ( Cubic.extrema sup_bez ) )
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$ inf t :| ( sup t : filter ( `inside` t ) ( Cubic.extrema sup_bez ) )
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in 𝕀 ( minimum mins ) ( maximum maxs )
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-- | Evaluate a quadratic Bézier curve, when both the coefficients and the
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@ -330,9 +330,9 @@ evaluateQuadratic bez t =
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let inf_bez = fmap inf bez
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sup_bez = fmap sup bez
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mins = fmap (Quadratic.bezier @( T Double ) inf_bez)
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$ inf t :| ( sup t : filter ( inside t ) ( Quadratic.extrema inf_bez ) )
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$ inf t :| ( sup t : filter ( `inside` t ) ( Quadratic.extrema inf_bez ) )
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maxs = fmap (Quadratic.bezier @( T Double ) sup_bez)
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$ inf t :| ( sup t : filter ( inside t ) ( Quadratic.extrema sup_bez ) )
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$ inf t :| ( sup t : filter ( `inside` t ) ( Quadratic.extrema sup_bez ) )
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in 𝕀 ( minimum mins ) ( maximum maxs )
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{-
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@ -59,8 +59,8 @@ width ( 𝕀 lo hi ) = hi - lo
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nonDecreasing :: ( a -> b ) -> 𝕀 a -> 𝕀 b
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nonDecreasing f ( 𝕀 lo hi ) = 𝕀 ( f lo ) ( f hi )
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inside :: Ord a => 𝕀 a -> a -> Bool
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inside ( 𝕀 lo hi ) x = x >= lo && x <= hi
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inside :: Ord a => a -> 𝕀 a -> Bool
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inside x ( 𝕀 lo hi ) = x >= lo && x <= hi
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deriving via ViaAbelianGroup ( T ( 𝕀 Double ) )
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instance Semigroup ( T ( 𝕀 Double ) )
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@ -12,7 +12,7 @@ module Math.Linear
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, ℝ(..), T(.., V2, V3, V4)
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, Fin(..), MFin(..)
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, RepDim, RepresentableQ(..)
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, Representable(..), injection, projection
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, Representable(..), set, injection, projection
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, Vec(..), (!), find, zipIndices, unzip
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) where
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@ -6,7 +6,9 @@ module Math.Linear.Internal
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, Fin(..), MFin(..)
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, RepDim
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, RepresentableQ(..)
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, Representable(..), projection, injection
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, Representable(..)
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, set
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, projection, injection
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)
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where
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@ -111,6 +113,12 @@ class Representable r v | v -> r where
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tabulate :: ( Fin ( RepDim v ) -> r ) -> v
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index :: v -> Fin ( RepDim v ) -> r
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{-# INLINEABLE set #-}
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set :: Representable r v => Fin ( RepDim v ) -> r -> v -> v
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set i r u = tabulate \ j ->
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if i == j
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then r
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else index u j
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projection :: ( Representable r u, Representable r v )
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=> ( Fin ( RepDim v ) -> Fin ( RepDim u ) )
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@ -138,20 +146,24 @@ instance RepresentableQ Double ( ℝ 0 ) where
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instance RepresentableQ Double ( ℝ 1 ) where
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tabulateQ f = [|| ℝ1 $$( f ( Fin 1 ) ) ||]
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indexQ p _ = [|| unℝ1 $$p ||]
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indexQ p = \ case
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Fin 1 -> [|| unℝ1 $$p ||]
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Fin i -> error $ "invalid index for ℝ 1: " ++ show i
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instance RepresentableQ Double ( ℝ 2 ) where
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tabulateQ f = [|| ℝ2 $$( f ( Fin 1 ) ) $$( f ( Fin 2 ) ) ||]
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indexQ p = \ case
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Fin 1 -> [|| _ℝ2_x $$p ||]
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_ -> [|| _ℝ2_y $$p ||]
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Fin 2 -> [|| _ℝ2_y $$p ||]
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Fin i -> error $ "invalid index for ℝ 2: " ++ show i
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instance RepresentableQ Double ( ℝ 3 ) where
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tabulateQ f = [|| ℝ3 $$( f ( Fin 1 ) ) $$( f ( Fin 2 ) ) $$( f ( Fin 3 ) ) ||]
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indexQ p = \ case
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Fin 1 -> [|| _ℝ3_x $$p ||]
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Fin 2 -> [|| _ℝ3_y $$p ||]
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_ -> [|| _ℝ3_z $$p ||]
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Fin 3 -> [|| _ℝ3_z $$p ||]
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Fin i -> error $ "invalid index for ℝ 3: " ++ show i
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instance RepresentableQ Double ( ℝ 4 ) where
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tabulateQ f = [|| ℝ4 $$( f ( Fin 1 ) ) $$( f ( Fin 2 ) ) $$( f ( Fin 3 ) ) $$( f ( Fin 4 ) ) ||]
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@ -159,7 +171,8 @@ instance RepresentableQ Double ( ℝ 4 ) where
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Fin 1 -> [|| _ℝ4_x $$p ||]
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Fin 2 -> [|| _ℝ4_y $$p ||]
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Fin 3 -> [|| _ℝ4_z $$p ||]
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_ -> [|| _ℝ4_w $$p ||]
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Fin 4 -> [|| _ℝ4_w $$p ||]
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Fin i -> error $ "invalid index for ℝ 4: " ++ show i
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instance Representable Double ( ℝ 0 ) where
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tabulate _ = ℝ0
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@ -170,7 +183,9 @@ instance Representable Double ( ℝ 0 ) where
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instance Representable Double ( ℝ 1 ) where
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tabulate f = ℝ1 ( f ( Fin 1 ) )
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{-# INLINE tabulate #-}
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index p _ = unℝ1 p
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index p = \ case
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Fin 1 -> unℝ1 p
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Fin i -> error $ "invalid index for ℝ 1: " ++ show i
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{-# INLINE index #-}
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instance Representable Double ( ℝ 2 ) where
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@ -178,7 +193,8 @@ instance Representable Double ( ℝ 2 ) where
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{-# INLINE tabulate #-}
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index p = \ case
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Fin 1 -> _ℝ2_x p
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_ -> _ℝ2_y p
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Fin 2 -> _ℝ2_y p
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Fin i -> error $ "invalid index for ℝ 2: " ++ show i
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{-# INLINE index #-}
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instance Representable Double ( ℝ 3 ) where
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@ -187,7 +203,8 @@ instance Representable Double ( ℝ 3 ) where
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index p = \ case
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Fin 1 -> _ℝ3_x p
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Fin 2 -> _ℝ3_y p
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_ -> _ℝ3_z p
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Fin 3 -> _ℝ3_z p
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Fin i -> error $ "invalid index for ℝ 3: " ++ show i
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{-# INLINE index #-}
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instance Representable Double ( ℝ 4 ) where
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@ -197,5 +214,6 @@ instance Representable Double ( ℝ 4 ) where
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Fin 1 -> _ℝ4_x p
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Fin 2 -> _ℝ4_y p
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Fin 3 -> _ℝ4_z p
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_ -> _ℝ4_w p
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Fin 4 -> _ℝ4_w p
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Fin i -> error $ "invalid index for ℝ 4: " ++ show i
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{-# INLINE index #-}
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746
brush-strokes/src/lib/Math/Root/Isolation.hs
Normal file
746
brush-strokes/src/lib/Math/Root/Isolation.hs
Normal file
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@ -0,0 +1,746 @@
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module Math.Root.Isolation
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( -- * Root isolation using interval arithmetic
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Box, BoxHistory
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, isolateRootsIn
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-- ** Configuration of the root isolation methods
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, RootIsolationOptions(..), defaultRootIsolationOptions
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, RootIsolationAlgorithm(..), defaultRootIsolationAlgorithms
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-- *** Options for the bisection method
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, BisectionOptions(..)
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-- *** Options for the interval Newton method with Gauss–Seidel step
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, GaussSeidelOptions(..), Preconditioner(..)
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-- *** Options for the @box(1)@-consistency algorithm
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, Box1Options(..)
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-- *** Options for the @box(2)@-consistency algorithm
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, Box2Options(..)
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-- ** Trees recording search space of root isolation algorithms
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, RootIsolationTree(..), showRootIsolationTree
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, RootIsolationStep(..)
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, RootIsolationLeaf(..)
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)
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where
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-- base
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import Control.Arrow
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( first )
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import Control.Monad
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( when )
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import Data.Bifunctor
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( Bifunctor(bimap) )
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import Data.List
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( nub, partition)
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import Data.List.NonEmpty
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( NonEmpty )
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import qualified Data.List.NonEmpty as NE
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( NonEmpty(..), last )
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import Numeric
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( showFFloat )
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-- containers
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import Data.Tree
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( Tree(..) )
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-- transformers
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import Control.Monad.Trans.State.Strict as State
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( State, evalState, get, put )
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import Control.Monad.Trans.Writer.CPS
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( Writer, runWriter, tell )
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-- MetaBrush
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import Math.Algebra.Dual
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( D, D1𝔸2(..) )
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import Math.Epsilon
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( nearZero )
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import Math.Float.Utils
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( succFP, prevFP )
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import Math.Interval
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import Math.Linear
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import Math.Module
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( Module(..) )
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import qualified Math.Ring as Ring
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--import Debug.Utils
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--------------------------------------------------------------------------------
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-- | A tree recording the steps taken when doing cusp finding.
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data RootIsolationTree d
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= RootIsolationLeaf (RootIsolationLeaf d)
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| RootIsolationStep RootIsolationStep [(d, RootIsolationTree d)]
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-- | A description of a step taken in cusp-finding.
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data RootIsolationStep
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= GaussSeidelStep
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| BisectionStep (String, Double)
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| Box1Step
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| Box2Step
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instance Show RootIsolationStep where
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showsPrec _ GaussSeidelStep = showString "GS"
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showsPrec _ ( BisectionStep (coord, w) )
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= showString "bis "
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. showParen True
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( showString coord
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. showString " = "
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. showsPrec 0 w
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)
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showsPrec _ Box1Step = showString "box(1)"
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showsPrec _ Box2Step = showString "box(2)"
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data RootIsolationLeaf d
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= TooSmall d
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deriving stock Show
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showRootIsolationTree :: Box -> RootIsolationTree Box -> Tree String
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showRootIsolationTree cand (RootIsolationLeaf l) = Node (show cand ++ " " ++ showArea (boxArea cand) ++ " " ++ show l) []
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showRootIsolationTree cand (RootIsolationStep s ts)
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= Node (show cand ++ " abc " ++ showArea (boxArea cand) ++ " " ++ show s) $ map (\ (c,t) -> showRootIsolationTree c t) ts
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boxArea :: Box -> Double
|
||||
boxArea ( 𝕀 ( ℝ2 t_lo s_lo ) ( ℝ2 t_hi s_hi ) )
|
||||
= abs ( t_hi - t_lo ) * abs ( s_hi - s_lo )
|
||||
|
||||
showArea :: Double -> String
|
||||
showArea area = "(area " ++ showFFloat (Just 6) area "" ++ ")"
|
||||
|
||||
type Box = 𝕀ℝ 2
|
||||
type BoxHistory = [ NE.NonEmpty ( RootIsolationStep, Box ) ]
|
||||
|
||||
-- | Options for the root isolation methods in 'findCusps' and 'isolateRootsIn'.
|
||||
data RootIsolationOptions
|
||||
= RootIsolationOptions
|
||||
{ -- | Minimum width of a box.
|
||||
minWidth :: !Double
|
||||
-- | Given a box and its history, return a round of cusp-finding strategies
|
||||
-- to run, in sequence, on this box.
|
||||
, cuspFindingAlgorithms :: !( Box -> BoxHistory -> NonEmpty RootIsolationAlgorithm )
|
||||
}
|
||||
|
||||
data RootIsolationAlgorithm
|
||||
-- | Bisection step.
|
||||
= Bisection !BisectionOptions
|
||||
-- | Gauss–Seidel step with the given preconditioning method.
|
||||
| GaussSeidel !GaussSeidelOptions
|
||||
-- | @box(1)@-consistency.
|
||||
| Box1 !Box1Options
|
||||
-- | @box(2)@-consistency.
|
||||
| Box2 !Box2Options
|
||||
|
||||
-- | Options for the bisection method.
|
||||
data BisectionOptions =
|
||||
BisectionOptions
|
||||
{ canHaveSols :: !( ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) ) -> Box -> Bool )
|
||||
, fallbackBisectionDim :: !( [ ( RootIsolationStep, Box ) ] -> BoxHistory -> ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) ) -> ( Int, String ) )
|
||||
}
|
||||
|
||||
-- | Options for the interval Gauss–Seidel method.
|
||||
data GaussSeidelOptions =
|
||||
GaussSeidelOptions
|
||||
{ gsPreconditioner :: !Preconditioner }
|
||||
|
||||
-- | Options for the @box(1)@-consistency method.
|
||||
data Box1Options =
|
||||
Box1Options
|
||||
{ box1EpsEq :: !Double }
|
||||
|
||||
-- | Options for the @box(2)@-consistency method.
|
||||
data Box2Options =
|
||||
Box2Options
|
||||
{ box2EpsEq :: !Double
|
||||
, box2LambdaMin :: !Double
|
||||
}
|
||||
|
||||
defaultRootIsolationOptions :: RootIsolationOptions
|
||||
defaultRootIsolationOptions =
|
||||
RootIsolationOptions
|
||||
{ minWidth
|
||||
, cuspFindingAlgorithms = defaultRootIsolationAlgorithms minWidth narrowAbs
|
||||
}
|
||||
where
|
||||
minWidth = 1e-5
|
||||
narrowAbs = 5e-3
|
||||
|
||||
defaultRootIsolationAlgorithms :: Double -> Double -> Box -> BoxHistory -> NonEmpty RootIsolationAlgorithm
|
||||
defaultRootIsolationAlgorithms minWidth narrowAbs box history =
|
||||
case history of
|
||||
lastRoundBoxes : _
|
||||
-- If, in the last round of strategies, we didn't try bisection...
|
||||
| any ( \case { BisectionStep {} -> False; _ -> True } . fst ) lastRoundBoxes
|
||||
, (_, lastRoundFirstBox ) <- NE.last lastRoundBoxes
|
||||
-- ...and the last round didn't sufficiently reduce the size of the box...
|
||||
, not $ box `sufficientlySmallerThan` lastRoundFirstBox
|
||||
-- ...then try bisecting the box.
|
||||
-> Bisection ( defaultBisectionOptions minWidth narrowAbs box ) NE.:| []
|
||||
|
||||
-- Otherwise, do a normal round.
|
||||
-- Currently: we try an interval Gauss–Seidel step followed by box(1)-consistency.
|
||||
_ -> GaussSeidel defaultGaussSeidelOptions
|
||||
NE.:| [ Box1 ( Box1Options { box1EpsEq = narrowAbs } ) ]
|
||||
where
|
||||
sufficientlySmallerThan :: Box -> Box -> Bool
|
||||
sufficientlySmallerThan
|
||||
( 𝕀 ( ℝ2 t1_lo s1_lo ) ( ℝ2 t1_hi s1_hi ) )
|
||||
( 𝕀 ( ℝ2 t2_lo s2_lo ) ( ℝ2 t2_hi s2_hi ) ) =
|
||||
( ( t1_hi - t1_lo ) - ( t2_hi - t2_lo ) > narrowAbs )
|
||||
||
|
||||
( ( s1_hi - s1_lo ) - ( s2_hi - s2_lo ) > narrowAbs )
|
||||
|
||||
defaultGaussSeidelOptions :: GaussSeidelOptions
|
||||
defaultGaussSeidelOptions = GaussSeidelOptions { gsPreconditioner = InverseMidJacobian }
|
||||
|
||||
defaultBisectionOptions :: Double -> Double -> Box -> BisectionOptions
|
||||
defaultBisectionOptions
|
||||
_minWidth
|
||||
_narrowAbs
|
||||
box@( 𝕀 ( ℝ2 t_lo s_lo ) ( ℝ2 t_hi s_hi ) )
|
||||
= BisectionOptions
|
||||
{ canHaveSols =
|
||||
\ eqs box' ->
|
||||
-- box(0)-consistency
|
||||
let D12 iRange _ _ = eqs box'
|
||||
in ℝ3 0 0 0 `inside` iRange
|
||||
|
||||
-- box(1)-consistency
|
||||
--not $ null $ makeBox1Consistent eqs _minWidth _narrowAbs box'
|
||||
|
||||
-- box(2)-consistency
|
||||
--let box'' = makeBox2Consistent eqs _minWidth _narrowAbs 0.2 box'
|
||||
-- D12 iRange'' = eqs box''
|
||||
--in ℝ3 0 0 0 `inside` iRange''
|
||||
, fallbackBisectionDim =
|
||||
\ _roundHist _prevRoundsHist eqs ->
|
||||
let D12 _ ( T f_t ) ( T f_s ) = eqs box
|
||||
wd_t = t_hi - t_lo
|
||||
wd_s = s_hi - s_lo
|
||||
tJWidth = wd_t * normVI3 f_t
|
||||
sJWidth = wd_s * normVI3 f_s
|
||||
in if tJWidth >= sJWidth
|
||||
-- bisect along dimension that maximises width(x_j) * || J_j || ...
|
||||
-- ... but don't allow thin boxes
|
||||
|| ( wd_t >= 10 * wd_s )
|
||||
&& not ( wd_s >= 10 * wd_t )
|
||||
then (0, "")
|
||||
else (1, "")
|
||||
}
|
||||
|
||||
-- | Use the following algorithms using interval arithmetic in order
|
||||
-- to isolate roots:
|
||||
--
|
||||
-- - interval Newton method with Gauss–Seidel step for inversion
|
||||
-- of the interval Jacobian,
|
||||
-- - coordinate-wise Newton method (@box(1)@-consistency algorithm),
|
||||
-- - @box(2)@-consistency algorithm,
|
||||
-- - bisection.
|
||||
--
|
||||
-- Returns @(tree, (dunno, sols))@ where @tree@ is the full tree (useful for debugging),
|
||||
-- @sols@ are boxes that contain a unique solution (and to which Newton's method
|
||||
-- will converge starting from anywhere inside the box), and @dunno@ are small
|
||||
-- boxes which might or might not contain solutions.
|
||||
isolateRootsIn
|
||||
:: RootIsolationOptions
|
||||
-> ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) )
|
||||
-> Box
|
||||
-> ( [ ( Box, RootIsolationTree Box ) ], ( [ Box ], [ Box ] ) )
|
||||
isolateRootsIn
|
||||
( RootIsolationOptions
|
||||
{ minWidth
|
||||
, cuspFindingAlgorithms
|
||||
}
|
||||
)
|
||||
eqs initBox = runWriter $ map ( initBox , ) <$> go [] initBox
|
||||
|
||||
where
|
||||
|
||||
go :: BoxHistory
|
||||
-> Box -- box to work on
|
||||
-> Writer ( [ Box ], [ Box ] )
|
||||
[ RootIsolationTree Box ]
|
||||
go history
|
||||
cand@( 𝕀 ( ℝ2 t_lo s_lo ) ( ℝ2 t_hi s_hi ) )
|
||||
| -- Check the envelope equation interval contains zero.
|
||||
not $ ( ℝ3 0 0 0 `inside` iRange )
|
||||
-- Box doesn't contain a solution: discard it.
|
||||
= return []
|
||||
-- Box is small: stop processing it.
|
||||
| t_hi - t_lo < minWidth && s_hi - s_lo < minWidth
|
||||
= do let res = TooSmall cand
|
||||
tell ( [ cand ], [] )
|
||||
return [ RootIsolationLeaf res ]
|
||||
| otherwise
|
||||
= doStrategies history ( cuspFindingAlgorithms cand history ) cand
|
||||
where
|
||||
D12 iRange _ _ = eqs $ 𝕀 ( ℝ2 t_lo s_lo ) ( ℝ2 t_hi s_hi )
|
||||
|
||||
-- Run a round of cusp finding strategies, then recur.
|
||||
doStrategies
|
||||
:: BoxHistory
|
||||
-> NonEmpty RootIsolationAlgorithm
|
||||
-> Box
|
||||
-> Writer ( [ Box ], [ Box ] )
|
||||
[ RootIsolationTree Box ]
|
||||
doStrategies prevRoundsHist = do_strats []
|
||||
where
|
||||
do_strats :: [ ( RootIsolationStep, Box ) ]
|
||||
-> NE.NonEmpty RootIsolationAlgorithm
|
||||
-> Box
|
||||
-> Writer ( [ Box ], [ Box ] )
|
||||
[ RootIsolationTree Box ]
|
||||
do_strats thisRoundHist ( algo NE.:| algos ) box = do
|
||||
-- Run one strategy in the round.
|
||||
( step, boxes ) <- doStrategy thisRoundHist prevRoundsHist eqs minWidth algo box
|
||||
case algos of
|
||||
-- If there are other algorithms to run in this round, run them next.
|
||||
nextAlgo : otherAlgos ->
|
||||
let thisRoundHist' = ( step, box ) : thisRoundHist
|
||||
in recur step ( do_strats thisRoundHist' ( nextAlgo NE.:| otherAlgos ) ) boxes
|
||||
-- Otherwise, we have done one full round of strategies.
|
||||
-- Recur back to the top (calling 'go').
|
||||
[] ->
|
||||
let thisRoundHist' = ( step, box ) NE.:| thisRoundHist
|
||||
history = thisRoundHist' : prevRoundsHist
|
||||
in recur step ( go history ) boxes
|
||||
|
||||
recur :: RootIsolationStep
|
||||
-> ( Box -> Writer ( [ Box ], [ Box ] ) [ RootIsolationTree Box ] )
|
||||
-> [ Box ]
|
||||
-> Writer ( [Box], [Box] ) [ RootIsolationTree Box ]
|
||||
recur step r cands = do
|
||||
rest <- traverse ( \ c -> do { trees <- r c; return [ (c, tree) | tree <- trees ] } ) cands
|
||||
return [ RootIsolationStep step $ concat rest ]
|
||||
|
||||
|
||||
-- | Execute a cusp-finding strategy, replacing the input box with
|
||||
-- (hopefully smaller) output boxes.
|
||||
doStrategy :: [ ( RootIsolationStep, Box ) ]
|
||||
-> BoxHistory
|
||||
-> ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) )
|
||||
-> Double
|
||||
-> RootIsolationAlgorithm
|
||||
-> Box
|
||||
-> Writer ( [ Box ], [ Box ] )
|
||||
( RootIsolationStep, [ Box ] )
|
||||
doStrategy roundHistory previousRoundsHistory eqs minWidth algo box =
|
||||
case algo of
|
||||
GaussSeidel ( GaussSeidelOptions { gsPreconditioner } ) -> do
|
||||
boxes <- intervalGaussSeidel eqs gsPreconditioner box
|
||||
return ( GaussSeidelStep, boxes )
|
||||
Box1 ( Box1Options { box1EpsEq } ) ->
|
||||
return ( Box1Step, makeBox1Consistent eqs minWidth box1EpsEq box )
|
||||
Box2 ( Box2Options { box2LambdaMin, box2EpsEq } ) ->
|
||||
return ( Box2Step, [ makeBox2Consistent eqs minWidth box2EpsEq box2LambdaMin box ] )
|
||||
Bisection ( BisectionOptions { canHaveSols, fallbackBisectionDim } ) -> do
|
||||
let ( boxes, whatBis ) = bisect ( canHaveSols eqs ) ( fallbackBisectionDim roundHistory previousRoundsHistory eqs ) box
|
||||
return ( BisectionStep whatBis, boxes )
|
||||
|
||||
-- | Bisect the given box.
|
||||
--
|
||||
-- (The difficult part lies in determining along which dimension to bisect.)
|
||||
bisect :: ( Box -> Bool )
|
||||
-- ^ how to check whether a box contains solutions
|
||||
-> ( Int, String )
|
||||
-- ^ fall-back choice of dimension (and "why" we chose it)
|
||||
-> Box
|
||||
-> ( [ Box ], ( String, Double ) )
|
||||
bisect canHaveSols ( dim, why )
|
||||
( 𝕀 ( ℝ2 t_lo s_lo ) ( ℝ2 t_hi s_hi ) )
|
||||
-- We try to bisect along a dimension which eliminates zeros from one of the
|
||||
-- sub-regions.
|
||||
--
|
||||
-- If this fails, we fall-back to the provided dimension choice.
|
||||
= let t_mid = 0.5 * ( t_lo + t_hi )
|
||||
s_mid = 0.5 * ( s_lo + s_hi )
|
||||
l = 𝕀 ( ℝ2 t_lo s_lo ) ( ℝ2 t_mid s_hi )
|
||||
r = 𝕀 ( ℝ2 t_mid s_lo ) ( ℝ2 t_hi s_hi )
|
||||
d = 𝕀 ( ℝ2 t_lo s_lo ) ( ℝ2 t_hi s_mid )
|
||||
u = 𝕀 ( ℝ2 t_lo s_mid ) ( ℝ2 t_hi s_hi )
|
||||
l_skip = not $ canHaveSols l
|
||||
r_skip = not $ canHaveSols r
|
||||
d_skip = not $ canHaveSols d
|
||||
u_skip = not $ canHaveSols u
|
||||
|
||||
( bisGuesses, whatBis )
|
||||
| l_skip && r_skip
|
||||
= ( [], ( "lr", t_mid ) )
|
||||
| d_skip && u_skip
|
||||
= ( [], ( "du", s_mid ) )
|
||||
| l_skip
|
||||
= ( [ r ], ( "r", t_mid ) )
|
||||
| r_skip
|
||||
= ( [ l ], ( "l", t_mid ) )
|
||||
| d_skip
|
||||
= ( [ u ], ( "u", s_mid ) )
|
||||
| u_skip
|
||||
= ( [ d ], ( "d", s_mid ) )
|
||||
| otherwise
|
||||
= let why' = case why of
|
||||
"" -> ""
|
||||
_ -> " (" ++ why ++ ")"
|
||||
in case dim of
|
||||
0 -> ( [ l, r ], ( "t" ++ why', t_mid ) )
|
||||
1 -> ( [ d, u ], ( "s" ++ why', s_mid ) )
|
||||
_ -> error "bisect: fall-back bisection dimension should be either 0 or 1"
|
||||
in ( bisGuesses, whatBis )
|
||||
|
||||
-- | Interval Newton method with Gauss–Seidel step.
|
||||
intervalGaussSeidel
|
||||
:: ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) )
|
||||
-- ^ equations
|
||||
-> Preconditioner
|
||||
-- ^ preconditioner to use for the Gauss–Seidel step
|
||||
-> 𝕀ℝ 2
|
||||
-> Writer ( [ 𝕀ℝ 2 ], [ 𝕀ℝ 2 ] ) [ 𝕀ℝ 2 ]
|
||||
intervalGaussSeidel
|
||||
eqs
|
||||
precondMethod
|
||||
ts@( 𝕀 ( ℝ2 t_lo s_lo ) ( ℝ2 t_hi s_hi ) )
|
||||
| D12 _f ( T ee_f_t ) ( T ee_f_s )
|
||||
<- eqs ts
|
||||
, let f_t = take2 ee_f_t
|
||||
f_s = take2 ee_f_s
|
||||
, D12 ee_f_mid ( T _ee_f_t_mid ) ( T _ee_f_s_mid )
|
||||
<- eqs ts_mid
|
||||
, let f_mid = take2 ee_f_mid
|
||||
|
||||
= let -- Interval Newton method: take one Gauss–Seidel step
|
||||
-- for the equation f'(x) v = - f(x_mid),
|
||||
-- where f = 𝛿E/𝛿s * dc/dt
|
||||
( a, b ) = precondition precondMethod
|
||||
( midI f_t, midI f_s )
|
||||
( f_t, f_s ) ( neg f_mid )
|
||||
|
||||
-- NB: we have to re-center around zero to take a Gauss–Seidel step.
|
||||
gsGuesses = map ( first ( \ ts' -> unT $ T ts' ^+^ T ts_mid ) )
|
||||
$ gaussSeidelStep a b ( unT $ T ts ^-^ T ts_mid )
|
||||
in
|
||||
-- If the Gauss–Seidel step was a contraction, then the box
|
||||
-- contains a unique solution (by the Banach fixed point theorem).
|
||||
--
|
||||
-- These boxes can thus be directly added to the solution set:
|
||||
-- Newton's method is guaranteed to converge to the unique solution.
|
||||
let !(done, todo) = bimap ( map fst ) ( map fst )
|
||||
$ partition snd gsGuesses
|
||||
in do tell ([], done)
|
||||
return todo
|
||||
where
|
||||
t_mid = 0.5 * ( t_lo + t_hi )
|
||||
s_mid = 0.5 * ( s_lo + s_hi )
|
||||
ts_mid = singleton ( ℝ2 t_mid s_mid )
|
||||
neg ( 𝕀 ( ℝ2 x_lo y_lo ) ( ℝ2 x_hi y_hi ) )
|
||||
= let !( 𝕀 x'_lo x'_hi ) = negate $ 𝕀 x_lo x_hi
|
||||
!( 𝕀 y'_lo y'_hi ) = negate $ 𝕀 y_lo y_hi
|
||||
in 𝕀 ( ℝ2 x'_lo y'_lo ) ( ℝ2 x'_hi y'_hi )
|
||||
take2 :: 𝕀ℝ 3 -> 𝕀ℝ 2
|
||||
take2 ( 𝕀 ( ℝ3 _a_lo b_lo c_lo ) ( ℝ3 _a_hi b_hi c_hi ) )
|
||||
= 𝕀 ( ℝ2 b_lo c_lo ) ( ℝ2 b_hi c_hi )
|
||||
-- TODO: always using the last 2 coordinates, instead of any
|
||||
-- pair of two coordinates.
|
||||
|
||||
-- Take one interval Gauss–Seidel step for the equation \( A X = B \),
|
||||
-- refining the initial guess box for \( X \) into up to four (disjoint) new boxes.
|
||||
--
|
||||
-- The boolean indicates whether the Gauss–Seidel step was a contraction.
|
||||
gaussSeidelStep :: ( 𝕀ℝ 2, 𝕀ℝ 2 ) -- ^ columns of \( A \)
|
||||
-> 𝕀ℝ 2 -- ^ \( B \)
|
||||
-> 𝕀ℝ 2 -- ^ initial box \( X \)
|
||||
-> [ ( 𝕀ℝ 2, Bool ) ]
|
||||
gaussSeidelStep
|
||||
( 𝕀 ( ℝ2 a11_lo a21_lo ) ( ℝ2 a11_hi a21_hi )
|
||||
, 𝕀 ( ℝ2 a12_lo a22_lo ) ( ℝ2 a12_hi a22_hi ) )
|
||||
( 𝕀 ( ℝ2 b1_lo b2_lo ) ( ℝ2 b1_hi b2_hi ) )
|
||||
( 𝕀 ( ℝ2 x1_lo x2_lo ) ( ℝ2 x1_hi x2_hi ) )
|
||||
= let !a11 = 𝕀 a11_lo a11_hi
|
||||
!a12 = 𝕀 a12_lo a12_hi
|
||||
!a21 = 𝕀 a21_lo a21_hi
|
||||
!a22 = 𝕀 a22_lo a22_hi
|
||||
!b1 = 𝕀 b1_lo b1_hi
|
||||
!b2 = 𝕀 b2_lo b2_hi
|
||||
!x1 = 𝕀 x1_lo x1_hi
|
||||
!x2 = 𝕀 x2_lo x2_hi
|
||||
in nub $ do
|
||||
|
||||
-- x1' = ( ( b1 - a12 * x2 ) / a11 ) ∩ x1
|
||||
x1'_0 <- ( b1 - a12 * x2 ) `extendedDivide` a11
|
||||
( x1'@( 𝕀 x1'_lo x1'_hi ), sub_x1 ) <- x1'_0 `intersect` x1
|
||||
-- x2' = ( ( b2 - a21 * x1' ) / a22 ) ∩ x2
|
||||
x2'_0 <- ( b2 - a21 * x1' ) `extendedDivide` a22
|
||||
( 𝕀 x2'_lo x2'_hi, sub_x2 ) <- x2'_0 `intersect` x2
|
||||
|
||||
return ( ( 𝕀 ( ℝ2 x1'_lo x2'_lo) ( ℝ2 x1'_hi x2'_hi ) )
|
||||
, sub_x1 && sub_x2 )
|
||||
-- TODO: try implementing the complete interval union Gauss–Seidel algorithm.
|
||||
-- See "Algorithm 2" in
|
||||
-- "Using interval unions to solve linear systems of equations with uncertainties"
|
||||
|
||||
-- | Compute the intersection of two intervals.
|
||||
--
|
||||
-- Returns whether the first interval is a strict subset of the second interval,
|
||||
-- or the intersection is a single point.
|
||||
intersect :: 𝕀 Double -> 𝕀 Double -> [ ( 𝕀 Double, Bool ) ]
|
||||
intersect ( 𝕀 lo1 hi1 ) ( 𝕀 lo2 hi2 )
|
||||
| lo > hi
|
||||
= [ ]
|
||||
| otherwise
|
||||
= [ ( 𝕀 lo hi, ( lo1 > lo2 && hi1 < hi2 ) || ( lo == hi ) ) ]
|
||||
where
|
||||
lo = max lo1 lo2
|
||||
hi = min hi1 hi2
|
||||
|
||||
-- | Preconditioner to use with the interval Gauss–Seidel method.
|
||||
data Preconditioner
|
||||
= NoPreconditioning
|
||||
| InverseMidJacobian
|
||||
deriving stock Show
|
||||
|
||||
-- | An implementation of "bc_enforce" from the paper
|
||||
-- "Parallelization of a bound-consistency enforcing procedure and its application in solving nonlinear systems"
|
||||
--
|
||||
-- See also
|
||||
-- "Presentation of a highly tuned multithreaded interval solver for underdetermined and well-determined nonlinear systems"
|
||||
makeBox1Consistent
|
||||
:: ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) )
|
||||
-> Double -> Double
|
||||
-> Box -> [ Box ]
|
||||
makeBox1Consistent eqs minWidth epsEq x =
|
||||
( `State.evalState` False ) $
|
||||
pipeFunctionsWhileTrue ( allNarrowingOperators epsEq minWidth eqs ) x
|
||||
|
||||
-- | An implementation of "bound-consistency" from the paper
|
||||
-- "Parallelization of a bound-consistency enforcing procedure and its application in solving nonlinear systems"
|
||||
makeBox2Consistent
|
||||
:: ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) )
|
||||
-> Double -> Double -> Double
|
||||
-> Box -> Box
|
||||
makeBox2Consistent eqs minWidth epsEq lambdaMin x0 = ( `State.evalState` False ) $ doLoop 0.25 x0
|
||||
where
|
||||
doLoop :: Double -> Box -> State Bool Box
|
||||
doLoop lambda x = do
|
||||
x' <- boundConsistency get_t set_t lambda x
|
||||
x'' <- boundConsistency get_s set_s lambda x'
|
||||
modified <- State.get
|
||||
let lambda' = if modified then lambda else 0.5 * lambda
|
||||
if lambda' < lambdaMin
|
||||
then return x''
|
||||
else do { State.put False ; doLoop lambda' x'' }
|
||||
|
||||
boundConsistency :: ( Box -> 𝕀 Double )
|
||||
-> ( 𝕀 Double -> Box -> Box )
|
||||
-> Double -> Box -> State Bool Box
|
||||
boundConsistency getter setter lambda box = do
|
||||
let x@( 𝕀 x_inf x_sup ) = getter box
|
||||
c1 = ( 1 - lambda ) * x_inf + lambda * x_sup
|
||||
c2 = lambda * x_inf + ( 1 - lambda ) * x_sup
|
||||
x'_inf =
|
||||
case makeBox1Consistent eqs minWidth epsEq ( setter ( 𝕀 x_inf c1 ) box ) of
|
||||
[] -> c1
|
||||
x's -> minimum $ map ( inf . getter ) x's
|
||||
x'_sup =
|
||||
case makeBox1Consistent eqs minWidth epsEq ( setter ( 𝕀 c2 x_sup ) box ) of
|
||||
[] -> c2
|
||||
x's -> maximum $ map ( sup . getter ) x's
|
||||
x' = 𝕀 x'_inf x'_sup
|
||||
when ( width x - width x' >= epsEq ) $
|
||||
State.put True
|
||||
return $ setter x' box
|
||||
|
||||
precondition :: Preconditioner
|
||||
-> ( 𝕀ℝ 2, 𝕀ℝ 2 )
|
||||
-> ( 𝕀ℝ 2, 𝕀ℝ 2 )
|
||||
-> 𝕀ℝ 2
|
||||
-> ( ( 𝕀ℝ 2, 𝕀ℝ 2 ), 𝕀ℝ 2 )
|
||||
precondition meth jac_mid a@( a1, a2 ) b =
|
||||
case meth of
|
||||
NoPreconditioning
|
||||
-> ( a, b )
|
||||
InverseMidJacobian
|
||||
| ( 𝕀 ( ℝ2 a11_lo a21_lo ) ( ℝ2 a11_hi a21_hi )
|
||||
, 𝕀 ( ℝ2 a12_lo a22_lo ) ( ℝ2 a12_hi a22_hi ) ) <- jac_mid
|
||||
, let !a11 = 0.5 * ( a11_lo + a11_hi )
|
||||
!a12 = 0.5 * ( a12_lo + a12_hi )
|
||||
!a21 = 0.5 * ( a21_lo + a21_hi )
|
||||
!a22 = 0.5 * ( a22_lo + a22_hi )
|
||||
!d = a11 * a22 - a12 * a21
|
||||
, not ( nearZero d )
|
||||
, let !precond = ( ℝ2 a22 -a12, ℝ2 -a21 a11 )
|
||||
!inv = recip d
|
||||
f x = scale inv $ matMulVec precond x
|
||||
-> ( ( f a1, f a2 ), f b )
|
||||
| otherwise
|
||||
-> ( a, b )
|
||||
|
||||
scale :: Double -> 𝕀ℝ 2 -> 𝕀ℝ 2
|
||||
scale s ( 𝕀 ( ℝ2 a1_lo a2_lo ) ( ℝ2 a1_hi a2_hi ) )
|
||||
= 𝕀 ( ℝ2 b1_lo b2_lo ) ( ℝ2 b1_hi b2_hi )
|
||||
where
|
||||
𝕀 b1_lo b1_hi = scaleInterval s ( 𝕀 a1_lo a1_hi )
|
||||
𝕀 b2_lo b2_hi = scaleInterval s ( 𝕀 a2_lo a2_hi )
|
||||
|
||||
matMulVec :: ( ℝ 2, ℝ 2 ) -> 𝕀ℝ 2 -> 𝕀ℝ 2
|
||||
matMulVec ( ℝ2 a11 a21, ℝ2 a12 a22 ) ( 𝕀 ( ℝ2 u_lo v_lo ) ( ℝ2 u_hi v_hi ) )
|
||||
= 𝕀 ( ℝ2 u'_lo v'_lo ) ( ℝ2 u'_hi v'_hi )
|
||||
where
|
||||
u = 𝕀 u_lo u_hi
|
||||
v = 𝕀 v_lo v_hi
|
||||
𝕀 u'_lo u'_hi = scaleInterval a11 u + scaleInterval a12 v
|
||||
𝕀 v'_lo v'_hi = scaleInterval a21 u + scaleInterval a22 v
|
||||
|
||||
midI :: 𝕀ℝ 2 -> 𝕀ℝ 2
|
||||
midI ( 𝕀 ( ℝ2 x_lo y_lo ) ( ℝ2 x_hi y_hi ) ) =
|
||||
singleton $ ℝ2 ( 0.5 * ( x_lo + x_hi ) ) ( 0.5 * ( y_lo + y_hi ) )
|
||||
--width :: 𝕀ℝ 1 -> Double
|
||||
--width ( 𝕀 ( ℝ1 lo ) ( ℝ1 hi ) ) = hi - lo
|
||||
--normI :: 𝕀ℝ 1 -> Double
|
||||
--normI ( 𝕀 ( ℝ1 lo ) ( ℝ1 hi ) ) = sqrt $ sup $ ( 𝕀 lo hi ) Ring.^ 2
|
||||
--normVI :: 𝕀ℝ 2 -> Double
|
||||
--normVI ( 𝕀 ( ℝ2 x_lo y_lo ) ( ℝ2 x_hi y_hi ) ) =
|
||||
-- sqrt $ sup $ ( 𝕀 x_lo x_hi ) Ring.^ 2 + ( 𝕀 y_lo y_hi ) Ring.^ 2
|
||||
normVI3 :: 𝕀ℝ 3 -> Double
|
||||
normVI3 ( 𝕀 ( ℝ3 lo x_lo y_lo ) ( ℝ3 hi x_hi y_hi ) ) =
|
||||
sqrt $ max ( abs lo ) ( abs hi ) Ring.^ 2
|
||||
+ max ( abs x_lo ) ( abs x_hi ) Ring.^ 2
|
||||
+ max ( abs y_lo ) ( abs y_hi ) Ring.^ 2
|
||||
|
||||
-- Use the univariate interval Newton method to narrow from the left
|
||||
-- a candidate interval.
|
||||
--
|
||||
-- See Algorithm 5 (Procedure left_narrow) in
|
||||
-- "Parallelization of a bound-consistency enforcing procedure and its application in solving nonlinear systems"
|
||||
-- by Bartłomiej Jacek Kubica, 2017
|
||||
leftNarrow :: Double
|
||||
-> Double
|
||||
-> ( 𝕀 Double -> ( 𝕀 Double, 𝕀 Double ) )
|
||||
-> 𝕀 Double
|
||||
-> [ 𝕀 Double ]
|
||||
leftNarrow eps_equal eps_bisection ff' = left_narrow
|
||||
where
|
||||
left_narrow ( 𝕀 x_inf x_sup ) =
|
||||
go x_sup ( 𝕀 x_inf ( if x_inf == x_sup then x_inf else succFP x_inf ) )
|
||||
go x_sup x_left =
|
||||
let ( f_x_left, _f'_x_left ) = ff' x_left
|
||||
in
|
||||
if inf f_x_left <= 0 && sup f_x_left >= 0
|
||||
then [ 𝕀 ( inf x_left ) x_sup ]
|
||||
else
|
||||
let x = 𝕀 ( sup x_left ) x_sup
|
||||
( _f_x, f'_x ) = ff' x
|
||||
x's = do delta <- f_x_left `extendedDivide` f'_x --f'_x_left
|
||||
let x_new = x_left - delta
|
||||
map fst $ x_new `intersect` x
|
||||
in
|
||||
if | null x's
|
||||
-> []
|
||||
| ( width x - sum ( map width x's ) ) < eps_equal
|
||||
-- TODO: do a check with the relative reduction in size?
|
||||
-> x's
|
||||
| otherwise
|
||||
-> do
|
||||
x' <- x's
|
||||
if sup x' - inf x' < eps_bisection
|
||||
then return x'
|
||||
else left_narrow =<< bisectI x'
|
||||
|
||||
-- TODO: de-duplicate with 'leftNarrow'?
|
||||
rightNarrow :: Double
|
||||
-> Double
|
||||
-> ( 𝕀 Double -> ( 𝕀 Double, 𝕀 Double ) )
|
||||
-> 𝕀 Double
|
||||
-> [ 𝕀 Double ]
|
||||
rightNarrow eps_equal eps_bisection ff' = right_narrow
|
||||
where
|
||||
right_narrow ( 𝕀 x_inf x_sup ) =
|
||||
go x_inf ( 𝕀 ( if x_inf == x_sup then x_sup else prevFP x_sup ) x_sup )
|
||||
go x_inf x_right =
|
||||
let ( f_x_right, _f'_x_right ) = ff' x_right
|
||||
in
|
||||
if inf f_x_right <= 0 && sup f_x_right >= 0
|
||||
then [ 𝕀 x_inf ( sup x_right ) ]
|
||||
else
|
||||
let x = 𝕀 x_inf ( inf x_right )
|
||||
( _f_x, f'_x ) = ff' x
|
||||
x's = do delta <- f_x_right `extendedDivide` f'_x --f'_x_right
|
||||
let x_new = x_right - delta
|
||||
map fst $ x_new `intersect` x
|
||||
in
|
||||
if | null x's
|
||||
-> []
|
||||
| ( width x - sum ( map width x's ) ) < eps_equal
|
||||
-- TODO: do a check with the relative reduction in size?
|
||||
-> x's
|
||||
| otherwise
|
||||
-> do
|
||||
x' <- x's
|
||||
if sup x' - inf x' < eps_bisection
|
||||
then return x'
|
||||
else right_narrow =<< bisectI x'
|
||||
|
||||
bisectI :: 𝕀 Double -> [ 𝕀 Double ]
|
||||
bisectI x@( 𝕀 x_inf x_sup )
|
||||
| x_inf == x_sup
|
||||
= [ x ]
|
||||
| otherwise
|
||||
= [ 𝕀 x_inf x_mid, 𝕀 x_mid x_sup ]
|
||||
where x_mid = 0.5 * ( x_inf + x_sup )
|
||||
|
||||
-- | Apply each function in turn, piping the output of one function into
|
||||
-- the next.
|
||||
--
|
||||
-- Once all the functions have been applied, check whether the Bool is True.
|
||||
-- If it is, go around again with all the functions; otherwise, stop.
|
||||
pipeFunctionsWhileTrue :: [ a -> State Bool [ a ] ] -> a -> State Bool [ a ]
|
||||
pipeFunctionsWhileTrue fns = go fns
|
||||
where
|
||||
go [] x = do
|
||||
doAnotherRound <- State.get
|
||||
if doAnotherRound
|
||||
then do { State.put False ; go fns x }
|
||||
else return [ x ]
|
||||
go ( f : fs ) x = do
|
||||
xs <- f x
|
||||
concat <$> traverse ( go fs ) xs
|
||||
|
||||
allNarrowingOperators
|
||||
:: Double
|
||||
-> Double
|
||||
-> ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) )
|
||||
-> [ Box -> State Bool [ Box ] ]
|
||||
allNarrowingOperators eps_eq eps_bis eqs =
|
||||
[ \ cand ->
|
||||
let getter = ( `index` coordIndex )
|
||||
setter = set coordIndex
|
||||
newCands = map ( `setter` cand ) $ narrowFn eps_eq eps_bis ( \ x0 -> ff' $ setter x0 cand ) ( getter cand )
|
||||
in do
|
||||
when ( ( width ( getter cand ) - sum ( map ( width . getter ) newCands ) ) > eps_eq ) $
|
||||
-- TODO: do a check with the relative reduction in size?
|
||||
State.put True
|
||||
return newCands
|
||||
| narrowFn <- [ leftNarrow, rightNarrow ]
|
||||
, ( coordIndex, ff' ) <-
|
||||
[ ( Fin 1, ff'_t i ) | i <- [ Fin 1, Fin 2, Fin 3 ] ]
|
||||
++
|
||||
[ ( Fin 2, ff'_s i ) | i <- [ Fin 1, Fin 2, Fin 3 ] ]
|
||||
]
|
||||
where
|
||||
|
||||
ff'_t i ts =
|
||||
let D12 { _D12_v = f
|
||||
, _D12_dx = T f_t
|
||||
} = eqs ts
|
||||
in ( f `index` i, f_t `index` i )
|
||||
ff'_s i ts =
|
||||
let D12 { _D12_v = f
|
||||
, _D12_dy = T f_t
|
||||
} = eqs ts
|
||||
in ( f `index` i, f_t `index` i )
|
||||
|
||||
get_t, get_s :: Box -> 𝕀 Double
|
||||
get_t = ( `index` ( Fin 1 ) )
|
||||
get_s = ( `index` ( Fin 2 ) )
|
||||
|
||||
set_t, set_s :: 𝕀 Double -> Box -> Box
|
||||
set_t = set ( Fin 1 )
|
||||
set_s = set ( Fin 2 )
|
||||
Loading…
Reference in a new issue