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Add complete interval-union Gauss–Seidel step method
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@ -69,7 +69,7 @@ benchTestCase testName ( TestCase { testDescription, testBrushStroke, testCuspOp
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( dunno, sols ) =
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foldMap
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( \ ( i, ( _trees, DoneBoxes { doneSolBoxes = defCusps, doneGiveUpBoxes = mbCusps } ) ) ->
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( map ( ( i , ) . snd ) mbCusps, map ( i, ) defCusps ) ) $
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( map ( ( i , ) . fst ) mbCusps, map ( i, ) defCusps ) ) $
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IntMap.toList $
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findCuspsIn testCuspOptions testStrokeFnI $
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IntMap.fromList
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@ -128,8 +128,17 @@ pattern V4 x y z w = T ( ℝ4 x y z w )
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type Vec :: Nat -> Type -> Type
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newtype Vec n a = Vec { vecList :: [ a ] }
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deriving newtype ( Show, Eq, Ord, Functor, Foldable )
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deriving Applicative via ZipList
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type role Vec nominal representational
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deriving newtype instance Show a => Show ( Vec n a )
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deriving newtype instance Eq a => Eq ( Vec n a )
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deriving newtype instance Ord a => Ord ( Vec n a )
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deriving newtype instance Functor ( Vec n )
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deriving newtype instance Foldable ( Vec n )
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deriving via ZipList
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instance Applicative ( Vec n )
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instance Traversable ( Vec n ) where
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traverse f ( Vec as ) = Vec <$> traverse f as
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universe :: forall n. KnownNat n => Vec n ( Fin n )
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universe = Vec [ Fin i | i <- [ 1 .. fromIntegral ( natVal' @n proxy# ) ] ]
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@ -138,9 +138,9 @@ defaultRootIsolationAlgorithms minWidth ε_eq history box =
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where
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verySmall = and $ ( \ cd -> width cd <= minWidth ) <$> coordinates box
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_bisOptions = defaultBisectionOptions minWidth ε_eq box
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_gsOptions = defaultGaussSeidelOptions history
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_box1Options = defaultBox1Options minWidth ε_eq
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_bisOptions = defaultBisectionOptions @n @d minWidth ε_eq box
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_gsOptions = defaultGaussSeidelOptions @n @d history
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_box1Options = defaultBox1Options @n @d minWidth ε_eq
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-- Did we reduce the box width by at least ε_eq
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-- in at least one of the coordinates?
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@ -89,7 +89,6 @@ type BoxCt n d =
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, Vars ( D 1 ( ℝ n ) ) ~ n
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, Module Double ( T ( ℝ n ) )
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, Module ( 𝕀 Double ) ( T ( 𝕀ℝ n ) )
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, Applicative ( Vec n )
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, Ord ( ℝ d )
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, Module Double ( T ( ℝ d ) )
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@ -12,6 +12,7 @@ module Math.Root.Isolation.GaussSeidel
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where
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-- base
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import Prelude hiding ( unzip )
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import Control.Arrow
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( first )
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import Data.Bifunctor
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@ -24,6 +25,8 @@ import Data.Foldable
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( toList )
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import Data.List
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( partition )
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import Data.List.NonEmpty
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( unzip )
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import Data.Proxy
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( Proxy(..) )
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import Data.Type.Ord
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@ -71,12 +74,21 @@ instance RootIsolationAlgorithm GaussSeidel where
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type GaussSeidelOptions :: Nat -> Nat -> Type
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data GaussSeidelOptions n d =
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GaussSeidelOptions
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{ -- | Which preconditioner to user?
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{ -- | Which preconditioner to use?
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gsPreconditioner :: !Preconditioner
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-- | Function that projects over the equations we will consider
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-- (the identity for a well-determined problem, or a projection for
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-- an overdetermined system).
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, gsPickEqs :: ( 𝕀ℝ d -> 𝕀ℝ n ) }
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, gsPickEqs :: !( 𝕀ℝ d -> 𝕀ℝ n )
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-- | Whether to use a partial or a complete Gauss–Seidel update
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, gsUpdate :: !GaussSeidelUpdateMethod
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}
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-- | Whether to use a partial or a complete Gauss–Seidel update.
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data GaussSeidelUpdateMethod
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= GS_Partial
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| GS_Complete
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deriving stock Show
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-- | Default options for the interval Gauss–Seidel method.
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defaultGaussSeidelOptions
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@ -90,7 +102,7 @@ defaultGaussSeidelOptions
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-> GaussSeidelOptions n d
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defaultGaussSeidelOptions history =
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GaussSeidelOptions
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{ gsPreconditioner = InverseMidJacobian
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{ gsPreconditioner = InverseMidpoint
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, gsPickEqs =
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case cmpNat @n @d Proxy Proxy of
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EQI -> id
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@ -103,13 +115,14 @@ defaultGaussSeidelOptions history =
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choice :: Vec n ( Fin d )
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choice = choices !! ( length history `mod` length choices )
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in \ u -> tabulate \ i -> index u ( choice ! i )
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, gsUpdate = GS_Complete
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}
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{-# INLINEABLE defaultGaussSeidelOptions #-}
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-- | Preconditioner to use with the interval Gauss–Seidel method.
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data Preconditioner
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= NoPreconditioning
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| InverseMidJacobian
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| InverseMidpoint
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deriving stock Show
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-- | Interval Newton method with Gauss–Seidel step.
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@ -123,44 +136,66 @@ intervalGaussSeidel
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-- ^ box
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-> Writer ( DoneBoxes n ) [ 𝕀ℝ n ]
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intervalGaussSeidel
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( GaussSeidelOptions { gsPreconditioner = precondMeth, gsPickEqs = pickEqs } )
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( GaussSeidelOptions
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{ gsPreconditioner = precondMeth
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, gsPickEqs = pickEqs
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, gsUpdate
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} )
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eqs
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box
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| let boxMid = singleton $ boxMidpoint box
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f' :: Vec n ( 𝕀ℝ n )
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f' = fmap ( \ i -> pickEqs $ eqs box `monIndex` linearMonomial i ) ( universe @n )
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f_mid = pickEqs $ eqs boxMid `monIndex` zeroMonomial
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x
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| let x_mid = singleton $ boxMidpoint x
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f'_x :: Vec n ( 𝕀ℝ n )
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f'_x = fmap ( \ i -> pickEqs $ eqs x `monIndex` linearMonomial i ) ( universe @n )
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f_x_mid = pickEqs $ eqs x_mid `monIndex` zeroMonomial
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= let -- Interval Newton method: take one Gauss–Seidel step
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-- for the system of equations f'(x) ( x - x_mid ) = - f(x_mid).
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minus_f_x_mid = unT $ -1 *^ T ( boxMidpoint f_x_mid )
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-- Precondition the above linear system into A ( x - x_mid ) = B.
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( a, b ) = precondition precondMeth
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( fmap boxMidpoint f' )
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f' ( singleton $ unT $ -1 *^ T ( boxMidpoint f_mid ) )
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f'_x ( singleton minus_f_x_mid )
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-- NB: we have to change coordinates, putting the midpoint of the box
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-- at the origin, in order to take a Gauss–Seidel step.
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gsGuesses = map ( first ( \ box' -> unT $ box' ^+^ T boxMid ) )
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$ gaussSeidelStep a b ( T box ^-^ T boxMid )
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gsGuesses = map ( first ( \ x' -> unT $ x' ^+^ T x_mid ) )
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$ gaussSeidelUpdate gsUpdate a b ( T x ^-^ T x_mid )
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in
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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 fst ) ( map fst )
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$ partition snd gsGuesses
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let ( done, todo ) = bimap ( map fst ) ( map fst )
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$ partition snd gsGuesses
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in do tell $ noDoneBoxes { doneSolBoxes = done }
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return todo
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where
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{-# INLINEABLE intervalGaussSeidel #-}
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-- | A partial or complete Gauss–Seidel step for the equation \( A X = B \),
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-- refining the initial guess box for \( X \) into up to \( 2^n \) (disjoint) new boxes.
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gaussSeidelUpdate
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:: forall n
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. ( Representable Double ( ℝ n ), n ~ RepDim ( ℝ n ), Eq ( ℝ n ) )
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=> GaussSeidelUpdateMethod -- ^ which step method to use
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-> Vec n ( 𝕀ℝ n ) -- ^ columns of \( A \)
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-> 𝕀ℝ n -- ^ \( B \)
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-> T ( 𝕀ℝ n ) -- ^ initial box \( X \)
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-> [ ( T ( 𝕀ℝ n ), Bool ) ]
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gaussSeidelUpdate upd as b x =
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case upd of
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GS_Partial -> gaussSeidelStep as b x
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GS_Complete -> gaussSeidelStep_Complete as b x
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{-# INLINEABLE gaussSeidelUpdate #-}
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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 \( 2^n \) (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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gaussSeidelStep
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:: forall n
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. ( Representable Double ( ℝ n ), Eq ( ℝ n ) )
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. ( Representable Double ( ℝ n ), n ~ RepDim ( ℝ n ), Eq ( ℝ n ) )
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=> Vec n ( 𝕀ℝ n ) -- ^ columns of \( A \)
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-> 𝕀ℝ n -- ^ \( B \)
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-> T ( 𝕀ℝ n ) -- ^ initial box \( X \)
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@ -168,15 +203,80 @@ gaussSeidelStep
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gaussSeidelStep as b ( T x0 ) = coerce $
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forEachCoord @n ( x0, True ) $ \ i ( x, contraction ) -> do
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-- x_i' = ( b_i - sum { j /= i } a_ij * x_j ) / a_ii
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x_i'0 <- ( b `index` i - sum [ ( as ! j ) `index` i * x `index` j | j <- toList ( universe @n ), j /= i ] )
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`extendedDivide` ( ( as ! i ) `index` i )
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( x_i', sub_i ) <- x_i'0 `intersect` ( x `index` i )
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return $ ( set i x_i' x, sub_i && contraction )
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-- TODO: try implementing the complete interval union Gauss–Seidel algorithm.
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-- See "Algorithm 2" in
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-- "Using interval unions to solve linear systems of equations with uncertainties"
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let s = b `index` i - sum [ ( as ! j ) `index` i * x `index` j
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| j <- toList ( universe @n ), j /= i ]
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x_i = x `index` i
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a_ii = ( as ! i ) `index` i
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-- Take a shortcut before performing the division if possible.
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if | not $ 0 `inside` ( s - a_ii * x_i )
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-- No solutions: don't bother performing a division.
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-> [ ]
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| 0 `inside` s && 0 `inside` a_ii
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-- The division would produce [-oo,+oo]: don't do anything.
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-> [ ( x, False ) ]
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-- Otherwise, perform the division.
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| otherwise
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-> do
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x_i'0 <- s `extendedDivide` a_ii
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( x_i', sub_i ) <- x_i'0 `intersect` x_i
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return $ ( set i x_i' x, sub_i && contraction )
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{-# INLINEABLE gaussSeidelStep #-}
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-- | The complete interval-union Gauss–Seidel step.
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--
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-- Algorithm 2 from:
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-- "Using interval unions to solve linear systems of equations with uncertainties"
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-- (Montanher, Domes, Schichl, Neumaier) (2017)
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gaussSeidelStep_Complete
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:: forall n
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. ( Representable Double ( ℝ n ), n ~ RepDim ( ℝ n ) )
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=> Vec n ( 𝕀ℝ n ) -- ^ columns of \( A \)
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-> 𝕀ℝ n -- ^ \( B \)
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-> T ( 𝕀ℝ n ) -- ^ initial box \( X \)
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-> [ ( T ( 𝕀ℝ n ), Bool ) ]
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gaussSeidelStep_Complete as b ( T x0 ) = coerce $ do
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( x', subs ) <-
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forEachCoord @n ( x0, pure False ) $ \ i ( x, contractions ) -> do
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let s = b `index` i - sum [ ( as ! k ) `index` i * x `index` k
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| k <- toList ( universe @n ) ]
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x_i = x `index` i
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a_ii = ( as ! i ) `index` i
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( x', subs ) <- fromComponents \ j -> do
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let x_j = x `index` j
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a_ij = ( as ! j ) `index` i
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s_j = s `ominus` ( a_ij * x_j )
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-- Shortcut division if possible (see gaussSeidelStep for commentary).
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if | not $ 0 `inside` ( s_j - a_ii * x_i )
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-> [ ]
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| 0 `inside` s_j && 0 `inside` a_ij
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-> [ ( x_j, False ) ]
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| otherwise
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-> do
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x_j'0 <- s_j `extendedDivide` a_ij
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( x_j', sub_j ) <- x_j'0 `intersect` x_j
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return $ ( x_j', sub_j )
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return ( x', (||) <$> contractions <*> subs )
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return ( x', and subs )
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{-# INLINEABLE gaussSeidelStep_Complete #-}
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fromComponents
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:: forall n
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. ( Representable Double ( ℝ n ), n ~ RepDim ( ℝ n ) )
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=> ( Fin n -> [ ( 𝕀 Double, Bool ) ] ) -> [ ( 𝕀ℝ n, Vec n Bool ) ]
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fromComponents f = do
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( xs, bs ) <- unzip <$> traverse f ( universe @n )
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return $ ( tabulate $ \ i -> xs ! i, bs )
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-- TODO: this could be more efficient.
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{-# INLINEABLE fromComponents #-}
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infixl 6 `ominus`
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ominus :: 𝕀 Double -> 𝕀 Double -> 𝕀 Double
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ominus a@( 𝕀 lo1 hi1 ) b@( 𝕀 lo2 hi2 )
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| width a >= width b
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= 𝕀 ( lo1 - lo2 ) ( hi1 - hi2 )
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| otherwise
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= 𝕀 ( hi1 - hi2 ) ( lo1 - lo2 )
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-- | The midpoint of a box.
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boxMidpoint :: Representable Double ( ℝ n ) => 𝕀ℝ n -> ℝ n
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boxMidpoint box =
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@ -191,16 +291,15 @@ precondition
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:: forall n
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. ( KnownNat n, Representable Double ( ℝ n ) )
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=> Preconditioner -- ^ pre-conditioning method to use
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-> Vec n ( ℝ n ) -- ^ entry-wise midpoint matrix of the interval Jacobian matrix
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-> Vec n ( 𝕀ℝ n ) -- ^ columns of \( A \)
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-> 𝕀ℝ n -- ^ \( B \)
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-> ( Vec n ( 𝕀ℝ n ), 𝕀ℝ n )
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precondition meth jac_mid as b =
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precondition meth as b =
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case meth of
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NoPreconditioning
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-> ( as, b )
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InverseMidJacobian
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| let mat = toEigen jac_mid
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InverseMidpoint
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| let mat = toEigen $ fmap boxMidpoint as
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det = Eigen.determinant mat
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, not $ nearZero det
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-- (TODO: a bit wasteful to compute determinant then inverse.)
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