mirror of
https://gitlab.com/sheaf/metabrush.git
synced 2024-11-23 15:34:06 +00:00
optimise root-finding functions
* use PrimArray to represent polynomials * add some strictness annotations * turn on some optimisation flags * use quadratic formula for quadratic polynomials
This commit is contained in:
parent
eb8e7012aa
commit
58ca70c1bd
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@ -40,6 +40,8 @@ common common
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>= 1.2.0.1 && < 2.0
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, groups
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^>= 0.4.1.0
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, primitive
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^>= 0.7.1.0
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, transformers
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^>= 0.5.6.2
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@ -49,7 +51,11 @@ common common
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ghc-options:
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-O1
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-fexpose-all-unfoldings
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-funfolding-use-threshold=16
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-fexcess-precision
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-fspecialise-aggressively
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-optc-O3
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-optc-ffast-math
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-Wall
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-Wcompat
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-fwarn-missing-local-signatures
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@ -79,12 +85,14 @@ library
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, Math.Vector2D
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build-depends:
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groups-generic
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groups-generic
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^>= 0.1.0.0
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, hmatrix
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^>= 0.20.0.0
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, monad-par
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^>= 0.3.5
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, prim-instances
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^>= 0.2
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, vector
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^>= 0.12.1.2
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@ -214,7 +214,7 @@ main = do
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maxHistorySizeTVar <- STM.newTVarIO @Int 1000
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fitParametersTVar <- STM.newTVarIO @FitParameters
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( FitParameters
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{ maxSubdiv = 10
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{ maxSubdiv = 6
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, nbSegments = 12
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, dist_tol = 5e-3
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, t_tol = 1e-4
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@ -1,3 +1,4 @@
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{-# LANGUAGE BangPatterns #-}
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{-# LANGUAGE DerivingStrategies #-}
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{-# LANGUAGE FlexibleContexts #-}
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{-# LANGUAGE GeneralizedNewtypeDeriving #-}
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@ -72,7 +73,7 @@ newtype UniqueSupply = UniqueSupply { uniqueSupplyTVar :: STM.TVar Unique }
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freshUnique :: UniqueSupply -> STM Unique
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freshUnique ( UniqueSupply { uniqueSupplyTVar } ) = do
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uniq@( Unique i ) <- STM.readTVar uniqueSupplyTVar
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uniq@( Unique !i ) <- STM.readTVar uniqueSupplyTVar
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STM.writeTVar uniqueSupplyTVar ( Unique ( succ i ) )
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pure uniq
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@ -1,4 +1,5 @@
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{-# LANGUAGE AllowAmbiguousTypes #-}
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{-# LANGUAGE BangPatterns #-}
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{-# LANGUAGE DeriveAnyClass #-}
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{-# LANGUAGE DeriveGeneric #-}
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{-# LANGUAGE DeriveTraversable #-}
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@ -57,6 +58,10 @@ import Data.Group
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import Data.Group.Generics
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()
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-- primitive
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import Data.Primitive.Types
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( Prim )
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-- MetaBrush
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import qualified Math.Bezier.Quadratic as Quadratic
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( Bezier(..), bezier )
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@ -154,28 +159,30 @@ subdivide ( Bezier {..} ) t = ( Bezier p0 q1 q2 pt, Bezier pt r1 r2 p3 )
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-- | Polynomial coefficients of the derivative of the distance to a cubic Bézier curve.
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ddist :: forall v r p. ( Torsor v p, Inner r v, RealFloat r ) => Bezier p -> p -> [ r ]
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ddist ( Bezier {..} ) c = [ a0, a1, a2, a3, a4, a5 ]
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ddist ( Bezier {..} ) c = [ a5, a4, a3, a2, a1, a0 ]
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where
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v, v', v'', v''' :: v
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v = c --> p0
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v' = p0 --> p1
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v'' = p1 --> p0 ^+^ p1 --> p2
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v''' = p0 --> p3 ^+^ 3 *^ ( p2 --> p1 )
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!v = c --> p0
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!v' = p0 --> p1
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!v'' = p1 --> p0 ^+^ p1 --> p2
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!v''' = p0 --> p3 ^+^ 3 *^ ( p2 --> p1 )
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a0, a1, a2, a3, a4, a5 :: r
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a0 = v ^.^ v'
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a1 = 3 * squaredNorm v' + 2 * v ^.^ v''
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a2 = 9 * v' ^.^ v'' + v ^.^ v'''
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a3 = 6 * squaredNorm v'' + 4 * v' ^.^ v'''
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a4 = 5 * v'' ^.^ v'''
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a5 = squaredNorm v'''
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!a0 = v ^.^ v'
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!a1 = 3 * squaredNorm v' + 2 * v ^.^ v''
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!a2 = 9 * v' ^.^ v'' + v ^.^ v'''
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!a3 = 6 * squaredNorm v'' + 4 * v' ^.^ v'''
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!a4 = 5 * v'' ^.^ v'''
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!a5 = squaredNorm v'''
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-- | Finds the closest point to a given point on a cubic Bézier curve.
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closestPoint :: forall v r p. ( Torsor v p, Inner r v, RealFloat r ) => Bezier p -> p -> ArgMin r ( r, p )
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closestPoint
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:: forall v r p. ( Torsor v p, Inner r v, RealFloat r, Prim r )
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=> Bezier p -> p -> ArgMin r ( r, p )
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closestPoint pts@( Bezier {..} ) c = pickClosest ( 0 :| 1 : roots ) -- todo: also include the self-intersection point if one exists
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where
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roots :: [ r ]
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roots = filter ( \ r -> r > 0 && r < 1 ) ( realRoots $ ddist @v pts c )
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roots = filter ( \ r -> r > 0 && r < 1 ) ( realRoots 2000 $ ddist @v pts c )
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pickClosest :: NonEmpty r -> ArgMin r ( r, p )
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pickClosest ( s :| ss ) = go s q nm0 ss
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@ -46,6 +46,10 @@ import qualified Data.Sequence as Seq
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import Control.DeepSeq
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( NFData )
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-- primitive
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import Data.Primitive.PrimArray
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( primArrayFromListN, unsafeThawPrimArray )
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-- transformers
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import Control.Monad.Trans.State.Strict
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( execStateT, modify' )
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@ -241,9 +245,12 @@ fitPiece dist_tol t_tol maxIters p tp qs r tr =
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( dts_changed, argmax_sq_dist ) <- ( `execStateT` ( False, Max ( Arg 0 0 ) ) ) $ for_ ( zip qs [ 0 .. ] ) \( q, i ) -> do
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ti <- lift ( Unboxed.MVector.unsafeRead ts i )
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let
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poly :: [ Complex Double ]
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poly = map (:+ 0) $ Cubic.ddist @( Vector2D Double ) bez q
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ti' <- case laguerre epsilon 1 poly ( ti :+ 0 ) of
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laguerreStepResult :: Complex Double
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laguerreStepResult = runST do
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coeffs <- unsafeThawPrimArray . primArrayFromListN 6 . map (:+ 0)
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$ Cubic.ddist @( Vector2D Double ) bez q
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laguerre epsilon 1 coeffs ( ti :+ 0 )
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ti' <- case laguerreStepResult of
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x :+ y
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| isNaN x
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|| isNaN y
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@ -1,4 +1,5 @@
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{-# LANGUAGE AllowAmbiguousTypes #-}
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{-# LANGUAGE BangPatterns #-}
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{-# LANGUAGE DeriveAnyClass #-}
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{-# LANGUAGE DeriveGeneric #-}
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{-# LANGUAGE DeriveTraversable #-}
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@ -54,6 +55,10 @@ import Data.Group
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import Data.Group.Generics
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()
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-- primitive
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import Data.Primitive.Types
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( Prim )
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-- MetaBrush
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import Math.Epsilon
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( epsilon )
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@ -125,25 +130,27 @@ subdivide ( Bezier {..} ) t = ( Bezier p0 q1 pt, Bezier pt r1 p2 )
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-- | Polynomial coefficients of the derivative of the distance to a quadratic Bézier curve.
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ddist :: forall v r p. ( Torsor v p, Inner r v, RealFloat r ) => Bezier p -> p -> [ r ]
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ddist ( Bezier {..} ) c = [ a0, a1, a2, a3 ]
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ddist ( Bezier {..} ) c = [ a3, a2, a1, a0 ]
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where
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v, v', v'' :: v
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v = c --> p0
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v' = p0 --> p1
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v'' = p1 --> p0 ^+^ p1 --> p2
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!v = c --> p0
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!v' = p0 --> p1
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!v'' = p1 --> p0 ^+^ p1 --> p2
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a0, a1, a2, a3 :: r
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a0 = v ^.^ v'
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a1 = v ^.^ v'' + 2 * squaredNorm v'
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a2 = 3 * v' ^.^ v''
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a3 = squaredNorm v''
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!a0 = v ^.^ v'
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!a1 = v ^.^ v'' + 2 * squaredNorm v'
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!a2 = 3 * v' ^.^ v''
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!a3 = squaredNorm v''
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-- | Finds the closest point to a given point on a quadratic Bézier curve.
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closestPoint :: forall v r p. ( Torsor v p, Inner r v, RealFloat r ) => Bezier p -> p -> ArgMin r ( r, p )
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closestPoint
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:: forall v r p. ( Torsor v p, Inner r v, RealFloat r, Prim r )
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=> Bezier p -> p -> ArgMin r ( r, p )
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closestPoint pts@( Bezier {..} ) c = pickClosest ( 0 :| 1 : roots )
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where
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roots :: [ r ]
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roots = filter ( \ r -> r > 0 && r < 1 ) ( realRoots $ ddist @v pts c )
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roots = filter ( \ r -> r > 0 && r < 1 ) ( realRoots 2000 $ ddist @v pts c )
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pickClosest :: NonEmpty r -> ArgMin r ( r, p )
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pickClosest ( s :| ss ) = go s q nm0 ss
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@ -78,7 +78,7 @@ import Math.Module
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, lerp, squaredNorm
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)
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import Math.Roots
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( realRoots )
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( solveQuadratic )
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import Math.Vector2D
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( Point2D(..), Vector2D(..), cross )
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@ -580,14 +580,14 @@ withTangent tgt ( spt0 :<| spt1 :<| spts ) =
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| otherwise
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= Nothing
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in
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case mapMaybe correctTangentParam $ realRoots [ c01, 2 * ( c12 - c01 ), c01 + c23 - 2 * c12 ] of
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case mapMaybe correctTangentParam $ solveQuadratic c01 ( 2 * ( c12 - c01 ) ) ( c01 + c23 - 2 * c12 ) of
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( t : _ )
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-> Offset i ( Just t ) ( MkVector2D $ Cubic.bezier @( Vector2D Double ) bez t )
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-- Fallback in case we couldn't solve the quadratic for some reason.
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_
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| Just s <- between tgt tgt0 tgt2
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-- Fallback in case we couldn't solve the quadratic for some reason.
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-> Offset i ( Just s ) ( MkVector2D $ Cubic.bezier @( Vector2D Double ) bez s )
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-- Otherwise: go to next piece of the curve.
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-- Otherwise: go to next piece of the curve.
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| otherwise
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-> continue ( i + 3 ) tgt2 sp3 ps
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go _ _ _ _ _
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@ -1,7 +1,7 @@
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{-# LANGUAGE ScopedTypeVariables #-}
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module Math.Epsilon
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( epsilon )
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( epsilon, nearZero )
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where
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--------------------------------------------------------------------------------
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@ -10,3 +10,6 @@ module Math.Epsilon
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{-# SPECIALISE epsilon :: Double #-}
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epsilon :: forall r. RealFloat r => r
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epsilon = encodeFloat 1 ( 5 - floatDigits ( 0 :: r ) )
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nearZero :: RealFloat r => r -> Bool
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nearZero x = abs x < epsilon
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@ -1,120 +1,215 @@
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{-# LANGUAGE ScopedTypeVariables #-}
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{-# LANGUAGE BangPatterns #-}
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{-# LANGUAGE BlockArguments #-}
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{-# LANGUAGE GADTs #-}
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{-# LANGUAGE NamedWildCards #-}
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{-# LANGUAGE PartialTypeSignatures #-}
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{-# LANGUAGE ScopedTypeVariables #-}
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{-# LANGUAGE TypeApplications #-}
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module Math.Roots
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{-# OPTIONS_GHC -fno-warn-partial-type-signatures #-}
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where
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module Math.Roots where
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-- base
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import Control.Monad
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( unless )
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import Control.Monad.ST
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( ST, runST )
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import Data.Complex
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( Complex(..), magnitude )
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import Data.List.NonEmpty
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( NonEmpty(..), toList )
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import Data.Maybe
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( mapMaybe )
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-- primitive
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import Control.Monad.Primitive
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( PrimMonad(PrimState) )
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import Data.Primitive.PrimArray
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( PrimArray, MutablePrimArray
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, primArrayFromList
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, getSizeofMutablePrimArray, sizeofPrimArray
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, unsafeThawPrimArray, cloneMutablePrimArray
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, shrinkMutablePrimArray, readPrimArray, writePrimArray
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)
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import Data.Primitive.Types
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( Prim )
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-- prim-instances
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import Data.Primitive.Instances
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() -- instance Prim a => Prim ( Complex a )
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-- MetaBrush
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import Math.Epsilon
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( epsilon )
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( epsilon, nearZero )
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--------------------------------------------------------------------------------
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-- | Find real roots of a polynomial.
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-- Coefficients are given in order of increasing degree, e.g.:
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-- x² + 7 is given by [ 7, 0, 1 ].
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realRoots :: forall r. RealFloat r => [ r ] -> [ r ]
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realRoots p = mapMaybe isReal ( roots epsilon 10000 ( map (:+ 0) p ) )
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-- | Real solutions to a quadratic equation.
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solveQuadratic :: forall a. RealFloat a => a -> a -> a -> [ a ]
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solveQuadratic a0 a1 a2
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| nearZero a1 && nearZero a2
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= if nearZero a0
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then [ 0, 0.5, 1 ] -- convention
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else []
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| nearZero ( a0 * a0 * a2 / ( a1 * a1 ) )
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= [ - a0 / a1 ]
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| disc < 0
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= [] -- non-real solutions
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| otherwise
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= let
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r :: a
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r =
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if a1 >= 0
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then 2 * a0 / ( - a1 - sqrt disc )
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else 0.5 * ( - a1 + sqrt disc) / a2
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in [ r, -r - a1 ]
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where
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isReal :: Complex r -> Maybe r
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disc :: a
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disc = a1 * a1 - 4 * a0 * a2
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-- | Find real roots of a polynomial.
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--
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-- Coefficients are given in order of decreasing degree, e.g.:
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-- x² + 7 is given by [ 1, 0, 7 ].
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realRoots :: forall a. ( RealFloat a, Prim a ) => Int -> [ a ] -> [ a ]
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realRoots maxIters coeffs = mapMaybe isReal ( roots epsilon maxIters ( map (:+ 0) coeffs ) )
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where
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isReal :: Complex a -> Maybe a
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isReal ( a :+ b )
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| abs b < epsilon = Just a
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| otherwise = Nothing
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-- | Compute all roots of a polynomial using Laguerre's method and (forward) deflation.
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--
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-- Polynomial coefficients are given in order of ascending degree (e.g. constant coefficient first).
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-- Polynomial coefficients are given in order of descending degree (e.g. constant coefficient last).
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--
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-- N.B. The forward deflation process is only guaranteed to be numerically stable
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-- if Laguerre's method finds roots in increasing order of magnitude.
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roots :: forall a. RealFloat a => a -> Int -> [ Complex a ] -> [ Complex a ]
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roots eps maxIters p = go p []
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where
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go :: [ Complex a ] -> [ Complex a ] -> [ Complex a ]
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go q rs
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| length q <= 2 = r : rs
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| otherwise = go ( deflate r q ) ( r : rs )
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where
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r :: Complex a
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r = laguerre eps maxIters q 0
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-- Start the iteration at 0 for best chance of numerical stability.
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roots :: forall a. ( RealFloat a, Prim a ) => a -> Int -> [ Complex a ] -> [ Complex a ]
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roots eps maxIters coeffs = runST do
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let
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coeffPrimArray :: PrimArray ( Complex a )
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coeffPrimArray = primArrayFromList coeffs
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sz :: Int
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sz = sizeofPrimArray coeffPrimArray
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p <- unsafeThawPrimArray coeffPrimArray
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let
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go :: Int -> [ Complex a ] -> ST _s [ Complex a ]
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go i rs = do
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!r <- laguerre eps maxIters p 0 -- Start Laguerre's method at 0 for best chance of numerical stability.
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if i <= 2
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then pure ( r : rs )
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else do
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deflate r p
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go ( i - 1 ) ( r : rs )
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go sz []
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-- | Deflate a polynomial: factor out a root of the polynomial.
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--
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-- The polynomial must have degree at least 2.
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deflate :: forall a. Num a => a -> [ a ] -> [ a ]
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deflate r ( _ : c : cs ) = toList $ go ( c :| cs )
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where
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go :: NonEmpty a -> NonEmpty a
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go ( a :| [] ) = a :| []
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go ( a :| a' : as ) = case go ( a' :| as ) of
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( b' :| bs ) -> ( a + r * b' ) :| ( b' : bs )
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deflate _ _ = error "deflate: polynomial of degree < 2"
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deflate :: forall a m s. ( Num a, Prim a, PrimMonad m, s ~ PrimState m ) => a -> MutablePrimArray s a -> m ()
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deflate r p = do
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deg <- subtract 1 <$> getSizeofMutablePrimArray p
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case compare deg 2 of
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LT -> pure ()
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EQ -> shrinkMutablePrimArray p deg
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GT -> do
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shrinkMutablePrimArray p deg
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let
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go :: a -> Int -> m ()
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go b i = unless ( i >= deg ) do
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ai <- readPrimArray p i
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writePrimArray p i ( ai + r * b )
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go ai ( i + 1 )
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a0 <- readPrimArray p 0
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go a0 1
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-- | Laguerre's method.
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laguerre
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:: forall a. RealFloat a
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=> a -- ^ error tolerance
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-> Int -- ^ max number of iterations
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-> [ Complex a ] -- ^ polynomial
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-> Complex a -- ^ initial point
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-> Complex a
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laguerre eps maxIters p = go maxIters
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where
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p', p'' :: [ Complex a ]
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p' = derivative p
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p'' = derivative p'
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go :: Int -> Complex a -> Complex a
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go iterationsLeft x
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| iterationsLeft <= 1
|
||||
|| magnitude ( x' - x ) < eps = x'
|
||||
| otherwise = go ( iterationsLeft - 1 ) x'
|
||||
where
|
||||
x' :: Complex a
|
||||
x' = laguerreStep eps p p' p'' x
|
||||
:: forall a m s
|
||||
. ( RealFloat a, Prim a, PrimMonad m, s ~ PrimState m )
|
||||
=> a -- ^ error tolerance
|
||||
-> Int -- ^ max number of iterations
|
||||
-> MutablePrimArray s ( Complex a ) -- ^ polynomial
|
||||
-> Complex a -- ^ initial point
|
||||
-> m ( Complex a )
|
||||
laguerre eps maxIters p x0 = do
|
||||
p' <- derivative p
|
||||
p'' <- derivative p'
|
||||
let
|
||||
go :: Int -> Complex a -> m ( Complex a )
|
||||
go iterationsLeft x = do
|
||||
x' <- laguerreStep eps p p' p'' x
|
||||
if iterationsLeft <= 1 || magnitude ( x' - x ) < eps
|
||||
then pure x'
|
||||
else go ( iterationsLeft - 1 ) x'
|
||||
go maxIters x0
|
||||
|
||||
-- | Take a single step in Laguerre's method.
|
||||
laguerreStep
|
||||
:: forall a. RealFloat a
|
||||
=> a -- ^ error tolerance
|
||||
-> [ Complex a ] -- ^ polynomial
|
||||
-> [ Complex a ] -- ^ first derivative of polynomial
|
||||
-> [ Complex a ] -- ^ second derivative of polynomial
|
||||
-> Complex a -- ^ initial point
|
||||
-> Complex a
|
||||
laguerreStep eps p p' p'' x
|
||||
| magnitude px < eps = x
|
||||
| otherwise = x - n / denom
|
||||
:: forall a m s
|
||||
. ( RealFloat a, Prim a, PrimMonad m, s ~ PrimState m )
|
||||
=> a -- ^ error tolerance
|
||||
-> MutablePrimArray s ( Complex a ) -- ^ polynomial
|
||||
-> MutablePrimArray s ( Complex a ) -- ^ first derivative of polynomial
|
||||
-> MutablePrimArray s ( Complex a ) -- ^ second derivative of polynomial
|
||||
-> Complex a -- ^ initial point
|
||||
-> m ( Complex a )
|
||||
laguerreStep eps p p' p'' x = do
|
||||
n <- fromIntegral @_ @a <$> getSizeofMutablePrimArray p
|
||||
px <- eval p x
|
||||
if magnitude px < eps
|
||||
then pure x
|
||||
else do
|
||||
p'x <- eval p' x
|
||||
p''x <- eval p'' x
|
||||
let
|
||||
g = p'x / px
|
||||
g² = g * g
|
||||
h = g² - p''x / px
|
||||
delta = sqrt $ ( n - 1 ) *: ( n *: h - g² )
|
||||
gp = g + delta
|
||||
gm = g - delta
|
||||
denom
|
||||
| magnitude gm > magnitude gp
|
||||
= gm
|
||||
| otherwise
|
||||
= gp
|
||||
pure $ x - n *: ( recip denom )
|
||||
|
||||
where
|
||||
n = fromIntegral ( length p )
|
||||
px = eval p x
|
||||
p'x = eval p' x
|
||||
p''x = eval p'' x
|
||||
g = p'x / px
|
||||
g² = g * g
|
||||
h = g² - p''x / px
|
||||
delta = sqrt $ ( n - 1 ) * ( n * h - g² )
|
||||
gp = g + delta
|
||||
gm = g - delta
|
||||
denom
|
||||
| magnitude gm > magnitude gp
|
||||
= gm
|
||||
| otherwise
|
||||
= gp
|
||||
(*:) :: a -> Complex a -> Complex a
|
||||
r *: (u :+ v) = ( r * u ) :+ ( r * v )
|
||||
|
||||
-- | Evaluate a polynomial.
|
||||
eval :: Num a => [ a ] -> a -> a
|
||||
eval as x = foldr ( \ a b -> a + x * b ) 0 as
|
||||
eval
|
||||
:: forall a m s
|
||||
. ( Num a, Prim a, PrimMonad m, s ~ PrimState m )
|
||||
=> MutablePrimArray s a -> a -> m a
|
||||
eval p x = do
|
||||
n <- getSizeofMutablePrimArray p
|
||||
let
|
||||
go :: a -> Int -> m a
|
||||
go !a i =
|
||||
if i >= n
|
||||
then pure a
|
||||
else do
|
||||
!b <- readPrimArray p i
|
||||
go ( b + x * a ) ( i + 1 )
|
||||
an <- readPrimArray p 0
|
||||
go an 1
|
||||
|
||||
-- | Derivative of a polynomial.
|
||||
derivative :: Num a => [ a ] -> [ a ]
|
||||
derivative as = zipWith ( \ i a -> fromIntegral i * a ) [ ( 1 :: Int ) .. ] ( tail as )
|
||||
derivative
|
||||
:: forall a m s
|
||||
. ( Num a, Prim a, PrimMonad m, s ~ PrimState m )
|
||||
=> MutablePrimArray s a -> m ( MutablePrimArray s a )
|
||||
derivative p = do
|
||||
deg <- subtract 1 <$> getSizeofMutablePrimArray p
|
||||
p' <- cloneMutablePrimArray p 0 deg
|
||||
let
|
||||
go :: Int -> m ()
|
||||
go i = unless ( i >= deg ) do
|
||||
a <- readPrimArray p' i
|
||||
writePrimArray p' i ( a * fromIntegral ( deg - i ) )
|
||||
go 0
|
||||
pure p'
|
||||
|
|
Loading…
Reference in a new issue