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Interval Newton method for cusp isolation
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sheaf
parent
eb68c27941
commit
d2a485f71e
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@ -31,11 +31,13 @@ import Control.Monad.ST
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import Data.Bifunctor
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( Bifunctor(bimap) )
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import Data.Coerce
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( Coercible )
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( Coercible, coerce )
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import Data.Foldable
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( for_, toList )
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import Data.Functor.Identity
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( Identity(..) )
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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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( unzip )
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import Data.Maybe
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@ -548,14 +550,20 @@ outlineFunction ptParams toBrushParams brushFromParams sp0 crv =
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$ runD ( brushFromParams @Point proxy# id )
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$ toBrushParams params_t
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bisSols = bisection 0.0001 curvesI
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( newtDunno, newtSols ) = intervalNewtonGS 0.0001 curvesI
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in --trace
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-- ( unlines $
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-- ( "bisectionMethod: #(possible zeroes) = " ++ show ( length bisSols ) ) :
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-- "" :
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-- map show bisSols )
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fwdBwd
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-- ( unlines $
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-- [ "newtonMethod: #(definite zeroes) = " ++ show ( length newtSols )
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-- , "newtonMethod: #(unknown) = " ++ show ( length newtDunno )
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-- , ""
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-- , "definite solutions:"
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-- , if null newtSols then "[]" else unlines $ map show newtSols
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-- , ""
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-- , "unknown:"
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-- , if null newtDunno then "[]" else unlines $ map show newtDunno ]
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-- ) $
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fwdBwd
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-----------------------------------
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-- Various utility functions
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@ -1056,75 +1064,154 @@ brushStrokeData path params brush =
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--------------------------------------------------------------------------------
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bisection :: Double
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-> ( 𝕀ℝ 1 -> Seq ( 𝕀ℝ 1 -> StrokeDatum 'Interval ) )
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-> [ ( 𝕀ℝ 1, Int, 𝕀ℝ 1, 𝕀ℝ 1, 𝕀ℝ 2 ) ]
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bisection minWidth eqs =
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bisect initialCands [] []
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-- Take one interval Gauss–Seidel step for the equation \( A X = B \),
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-- refining the initial guess box for \( X \) into up to four (disjoint) new boxes.
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--
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-- The boolean indicates whether the Gauss–Seidel step was a contraction.
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gaussSeidel :: ( 𝕀ℝ 2, 𝕀ℝ 2 ) -- ^ columns of \( A \)
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-> 𝕀ℝ 2 -- ^ \( B \)
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-> ( 𝕀ℝ 1, 𝕀ℝ 1 ) -- ^ initial box \( X \)
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-> [ ( ( 𝕀ℝ 1, 𝕀ℝ 1 ), Bool ) ]
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gaussSeidel
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( 𝕀 ( ℝ2 a11_lo a21_lo ) ( ℝ2 a11_hi a21_hi )
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, 𝕀 ( ℝ2 a12_lo a22_lo ) ( ℝ2 a12_hi a22_hi ) )
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( 𝕀 ( ℝ2 b1_lo b2_lo ) ( ℝ2 b1_hi b2_hi ) )
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( 𝕀 ( ℝ1 t0_lo ) ( ℝ1 t0_hi ), 𝕀 ( ℝ1 s0_lo ) ( ℝ1 s0_hi ) )
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= let !a11 = 𝕀 a11_lo a11_hi
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!a12 = 𝕀 a12_lo a12_hi
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!a21 = 𝕀 a21_lo a21_hi
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!a22 = 𝕀 a22_lo a22_hi
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!b1 = 𝕀 b1_lo b1_hi
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!b2 = 𝕀 b2_lo b2_hi
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!t0 = 𝕀 t0_lo t0_hi
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!s0 = 𝕀 s0_lo s0_hi
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in nub $ do
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t' <- ( b1 - a12 * s0 ) `extendedDivide` a11
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( t@( 𝕀 t_lo t_hi ), sub_t ) <- t' `intersect` t0
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s' <- ( b2 - a21 * t ) `extendedDivide` a22
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( 𝕀 s_lo s_hi, sub_s ) <- s' `intersect` s0
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return ( ( 𝕀 ( ℝ1 t_lo ) ( ℝ1 t_hi ), 𝕀 ( ℝ1 s_lo ) ( ℝ1 s_hi ) )
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, sub_t && sub_s )
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intersect :: 𝕀 Double -> 𝕀 Double -> [ ( 𝕀 Double, Bool ) ]
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intersect ( 𝕀 lo1 hi1 ) ( 𝕀 lo2 hi2 )
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| lo > hi
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= [ ]
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| otherwise
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= [ ( 𝕀 lo hi, lo == lo1 && hi == hi1 ) ]
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where
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lo = max lo1 lo2
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hi = min hi1 hi2
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extendedDivide :: 𝕀 Double -> 𝕀 Double -> [ 𝕀 Double ]
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extendedDivide x y = map ( x * ) ( extendedRecip y )
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extendedRecip :: 𝕀 Double -> [ 𝕀 Double ]
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extendedRecip x@( 𝕀 lo hi )
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| lo == 0 && hi == 0
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= [ 𝕀 ( -1 / 0 ) ( 1 / 0 ) ]
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| lo >= 0 || hi <= 0
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= [ recip x ]
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| otherwise
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= [ recip ( 𝕀 lo 0 ), recip ( 𝕀 0 hi ) ]
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-- | Interval Newton method with Gauss–Seidel step for inversion
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-- of the interval Jacobian.
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--
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-- Returns @(dunno, sols)@ where @sols@ are boxes that contain a unique solution
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-- (and to which Newton's method will converge starting from anywhere inside
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-- the box), and @dunno@ which are small boxes which might or might not
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-- contain solutions.
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intervalNewtonGS :: Double
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-> ( 𝕀ℝ 1 -> Seq ( 𝕀ℝ 1 -> StrokeDatum 'Interval ) )
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-> ( [ ( 𝕀ℝ 1, Int, 𝕀ℝ 1 ) ], [ ( 𝕀ℝ 1, Int, 𝕀ℝ 1 ) ] )
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intervalNewtonGS minWidth eqs =
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go [ ( 𝕀 ( ℝ1 0 ) ( ℝ1 1 ), i, 𝕀 ( ℝ1 0 ) ( ℝ1 1 ) )
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| i <- [ 0 .. length ( eqs ( 𝕀 ( ℝ1 0 ) ( ℝ1 1 ) ) ) - 1 ]
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]
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[]
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[]
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where
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bisect :: [ ( 𝕀ℝ 1, Int, 𝕀ℝ 1, 𝕀ℝ 1, 𝕀ℝ 2 ) ] -- have solutions, need bisection to refine
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-> [ ( 𝕀ℝ 1, Int, 𝕀ℝ 1 ) ] -- have been bisected, don't know if they contain solutions yet
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-> [ ( 𝕀ℝ 1, Int, 𝕀ℝ 1, 𝕀ℝ 1, 𝕀ℝ 2 ) ] -- have solutions, don't bisect further
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-> [ ( 𝕀ℝ 1, Int, 𝕀ℝ 1, 𝕀ℝ 1, 𝕀ℝ 2 ) ]
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bisect [] [] sols = sols
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bisect cands ( ( t, i, s ) : toTry ) sols
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| Just ( ee, 𝛿E𝛿sdcdt ) <- isCand t i s
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= bisect ( ( t, i, s, ee, 𝛿E𝛿sdcdt ) : cands ) toTry sols
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| otherwise
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= bisect cands toTry sols
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bisect ( cand@( t@( 𝕀 ( ℝ1 t_lo ) ( ℝ1 t_hi ) )
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, i
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, s@( 𝕀 ( ℝ1 s_lo ) ( ℝ1 s_hi ) )
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, _, _
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) : cands )
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toTry
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sols
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-- If the box is small, don't bisect it further, and store it as a candidate solution.
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go :: [ ( 𝕀ℝ 1, Int, 𝕀ℝ 1 ) ] -- boxes to work on
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-> [ ( 𝕀ℝ 1, Int, 𝕀ℝ 1 ) ] -- too small: don't shrink further
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-> [ ( 𝕀ℝ 1, Int, 𝕀ℝ 1 ) ] -- found solutions
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-> ( [ ( 𝕀ℝ 1, Int, 𝕀ℝ 1 ) ], [ ( 𝕀ℝ 1, Int, 𝕀ℝ 1 ) ] )
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go [] giveUp sols = ( giveUp, sols )
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go ( cand@( t@( 𝕀 ( ℝ1 t_lo ) ( ℝ1 t_hi ) )
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, i
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, s@( 𝕀 ( ℝ1 s_lo ) ( ℝ1 s_hi ) )
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) : cands ) giveUp sols
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-- Box is small: stop processing it.
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| t_hi - t_lo < minWidth && s_hi - s_lo < minWidth
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= trace ( "bisection sol: " ++ show cand ++ "\nnbCands = " ++ show ( length cands ) ++ "\nnbToTry = " ++ show ( length toTry ) )
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$ bisect cands toTry ( cand : sols )
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-- Otherwise, bisect in its longest direction and add the two resulting
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-- boxes to the list of boxes to try.
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| otherwise
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= let newToTry
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| t_hi - t_lo > s_hi - s_lo
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, let t_mid = 0.5 * ( t_lo + t_hi )
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= ( 𝕀 ( ℝ1 t_lo ) ( ℝ1 t_mid ), i, s )
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: ( 𝕀 ( ℝ1 t_mid ) ( ℝ1 t_hi ), i, s )
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: toTry
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| let s_mid = 0.5 * ( s_lo + s_hi )
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= ( t, i, 𝕀 ( ℝ1 s_lo ) ( ℝ1 s_mid ) )
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: ( t, i, 𝕀 ( ℝ1 s_mid ) ( ℝ1 s_hi ) )
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: toTry
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in bisect cands newToTry sols
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= go cands ( cand : giveUp ) sols
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initialCands =
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getCands
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( 𝕀 ( ℝ1 0 ) ( ℝ1 1 ) )
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( 𝕀 ( ℝ1 0 ) ( ℝ1 1 ) )
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| StrokeDatum { ee = D22 ee _ _ _ _ _
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, 𝛿E𝛿sdcdt = D12 ( T f ) ( T ( T f_t ) ) ( T ( T f_s ) ) }
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<- ( eqs t `Seq.index` i ) s
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getCands t s =
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[ (t, i, s, ee, 𝛿E𝛿sdcdt )
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| let !eqs_t = eqs t
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, ( eq_t, i ) <- zip ( toList eqs_t ) ( [0,1..] :: [Int] )
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, let !( StrokeDatum { ee = D22 ee _ _ _ _ _, 𝛿E𝛿sdcdt = D12 ( T 𝛿E𝛿sdcdt ) _ _ } ) = eq_t s
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, Interval.inf ( ival ee ) < Rounded ( ℝ1 0 )
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, Interval.sup ( ival ee ) > Rounded ( ℝ1 0 )
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, cmpℝ2 (<) ( getRounded ( Interval.inf $ ival 𝛿E𝛿sdcdt ) ) ( ℝ2 0 0 )
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, cmpℝ2 (>) ( getRounded ( Interval.sup $ ival 𝛿E𝛿sdcdt ) ) ( ℝ2 0 0 )
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]
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, StrokeDatum { 𝛿E𝛿sdcdt = D12 ( T f_mid ) ( T ( T f_t_mid ) ) ( T ( T f_s_mid ) ) }
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<- ( eqs i_t_mid `Seq.index` i ) i_s_mid
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= if | Interval.inf ( ival ee ) < Rounded ( ℝ1 0 )
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, Interval.sup ( ival ee ) > Rounded ( ℝ1 0 )
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, cmpℝ2 (<) ( getRounded ( Interval.inf $ ival f ) ) ( ℝ2 0 0 )
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, cmpℝ2 (>) ( getRounded ( Interval.sup $ ival f ) ) ( ℝ2 0 0 )
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-> let -- Interval Newton method: take one Gauss–Seidel step
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-- for the equation f'(X) v = - f(x_mid).
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-- !precond = matInverse ( f_t_mid, f_s_mid )
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-- !a = matMul precond ( f_t, f_s )
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-- !b = matMulVec precond ( neg f_mid )
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-- !gsGuesses = gaussSeidel a b ( t, s )
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!gsGuesses = gaussSeidel
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( f_t, f_s )
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( neg f_mid )
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( coerce ( (-) @( 𝕀 Double ) ) t i_t_mid
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, coerce ( (-) @( 𝕀 Double ) ) s i_s_mid )
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in if all ( smaller . fst ) gsGuesses
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then
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-- If the Gauss–Seidel step was a contraction, then the box
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-- contains a unique solution (by the Banach fixed point theorem).
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--
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-- These boxes can thus be directly added to the solution set:
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-- Newton's method is guaranteed to converge to the unique solution.
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let !(done, todo) = bimap ( map ( mkGuess . fst ) ) ( map ( mkGuess . fst ) )
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$ partition snd gsGuesses
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in go ( todo ++ cands ) giveUp ( done ++ sols )
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else
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-- Gauss–Seidel failed to shrink the boxes.
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-- Bisect along the widest dimension instead.
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let bisGuesses
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| t_hi - t_lo > s_hi - s_lo
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= [ ( 𝕀 ( ℝ1 t_lo ) ( ℝ1 t_mid ), i, s )
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, ( 𝕀 ( ℝ1 t_mid ) ( ℝ1 t_hi ), i, s ) ]
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| otherwise
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= [ ( t, i, 𝕀 ( ℝ1 s_lo ) ( ℝ1 s_mid ) )
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, ( t, i, 𝕀 ( ℝ1 s_mid ) ( ℝ1 s_hi ) ) ]
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in go ( bisGuesses ++ cands ) giveUp sols
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isCand :: 𝕀ℝ 1 -> Int -> 𝕀ℝ 1 -> Maybe ( 𝕀ℝ 1, 𝕀ℝ 2 )
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isCand t i s = case ( ( eqs t ) `Seq.index` i ) s of
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StrokeDatum { ee = D22 ee _ _ _ _ _, 𝛿E𝛿sdcdt = D12 ( T 𝛿E𝛿sdcdt ) _ _ } ->
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do guard $
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Interval.inf ( ival ee ) < Rounded ( ℝ1 0 )
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&& Interval.sup ( ival ee ) > Rounded ( ℝ1 0 )
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&& cmpℝ2 (<) ( getRounded ( Interval.inf $ ival 𝛿E𝛿sdcdt ) ) ( ℝ2 0 0 )
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&& cmpℝ2 (>) ( getRounded ( Interval.sup $ ival 𝛿E𝛿sdcdt ) ) ( ℝ2 0 0 )
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return ( ee, 𝛿E𝛿sdcdt )
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-- Box doesn't contain a solution: discard it.
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| otherwise
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-> go cands giveUp sols
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where
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t_mid = 0.5 * ( t_lo + t_hi )
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s_mid = 0.5 * ( s_lo + s_hi )
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i_t_mid = 𝕀 ( ℝ1 t_mid ) ( ℝ1 t_mid )
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i_s_mid = 𝕀 ( ℝ1 s_mid ) ( ℝ1 s_mid )
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mkGuess ( t0, s0 ) = ( coerce ( (+) @( 𝕀 Double ) ) t0 i_t_mid
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, i
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, coerce ( (+) @( 𝕀 Double ) ) s0 i_s_mid )
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smaller ( 𝕀 ( ℝ1 t0_lo ) ( ℝ1 t0_hi ), 𝕀 ( ℝ1 s0_lo ) ( ℝ1 s0_hi ) )
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= ( t0_lo + t_mid ) > t_lo + 0.25 * minWidth
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|| ( t0_hi + t_mid ) < t_hi - 0.25 * minWidth
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|| ( s0_lo + s_mid ) > s_lo + 0.25 * minWidth
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|| ( s0_hi + s_mid ) < s_hi - 0.25 * minWidth
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neg ( 𝕀 ( ℝ2 x_lo y_lo ) ( ℝ2 x_hi y_hi ) )
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= let !( 𝕀 x'_lo x'_hi ) = negate $ 𝕀 x_lo x_hi
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!( 𝕀 y'_lo y'_hi ) = negate $ 𝕀 y_lo y_hi
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in 𝕀 ( ℝ2 x'_lo y'_lo ) ( ℝ2 x'_hi y'_hi )
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cmpℝ2 :: ( Double -> Double -> Bool ) -> ℝ 2 -> ℝ 2 -> Bool
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cmpℝ2 cmp ( ℝ2 x1 y1 ) ( ℝ2 x2 y2 )
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@ -41,6 +41,10 @@ import Math.Ring
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newtype 𝕀 a = MkI { ival :: Interval a }
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deriving newtype ( Prelude.Num, Prelude.Fractional, Prelude.Floating )
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instance Eq a => Eq ( 𝕀 a ) where
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𝕀 a b == 𝕀 c d =
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a == c && b == d
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{-# COMPLETE 𝕀 #-}
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pattern 𝕀 :: a -> a -> 𝕀 a
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pattern 𝕀 x y = MkI ( Interval.I ( Rounded x ) ( Rounded y ) )
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