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Documentation in root isolation module
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@ -97,12 +97,12 @@ import qualified Math.Ring as Ring
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--------------------------------------------------------------------------------
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-- | A tree recording the steps taken when doing cusp finding.
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-- | A tree recording the steps taken when isolating roots.
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data RootIsolationTree d
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= RootIsolationLeaf String 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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-- | A description of a step taken when isolating roots.
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data RootIsolationStep
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= GaussSeidelStep
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| BisectionStep String Double
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@ -155,17 +155,19 @@ type BoxCt n d = ( n ~ N, d ~ 3 )
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)
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-}
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-- | Options for the root isolation methods in 'findCusps' and 'isolateRootsIn'.
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-- | Options for the root isolation methods in 'isolateRootsIn'.
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type RootIsolationOptions :: Nat -> Nat -> Type
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data RootIsolationOptions n d
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newtype RootIsolationOptions n d
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= RootIsolationOptions
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{ -- | Minimum width of a box.
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minWidth :: !Double
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-- | Given a box and its history, return a round of cusp-finding strategies
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{ -- | Given a box and its history, return a round of root isolation strategies
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-- to run, in sequence, on this box.
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--
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-- Returning an empty list means: give up on this box.
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, cuspFindingAlgorithms :: !( Box n -> BoxHistory n -> Either String ( NE.NonEmpty ( RootIsolationAlgorithm n d ) ) )
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-- A return value of @Left whyStop@ indicates stopping any further iterations
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-- on this box, e.g. because it is too small.
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rootIsolationAlgorithms
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:: Box n
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-> BoxHistory n
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-> Either String ( NE.NonEmpty ( RootIsolationAlgorithm n d ) )
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}
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type RootIsolationAlgorithm :: Nat -> Nat -> Type
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@ -184,16 +186,16 @@ type BisectionOptions :: Nat -> Nat -> Type
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data BisectionOptions n d =
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BisectionOptions
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{ canHaveSols :: !( ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) ) -> Box n -> Bool )
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, fallbackBisectionDim :: !( BisectionDimPicker n d )
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, fallbackBisectionCoord :: !( BisectionCoordPicker n d )
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}
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type BisectionDimPicker n d
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= forall r
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. [ ( RootIsolationStep, Box n ) ]
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-- | A function to choose along which coordination dimension we should
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-- perform a bisection.
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type BisectionCoordPicker n d
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= [ ( RootIsolationStep, Box n ) ]
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-> BoxHistory n
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-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
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-> NE.NonEmpty ( Fin n, r )
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-> ( r, String )
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-> forall r. ( NE.NonEmpty ( Fin n, r ) -> ( r, String ) )
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-- | Options for the interval Gauss–Seidel method.
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type GaussSeidelOptions :: Nat -> Nat -> Type
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@ -204,12 +206,13 @@ data GaussSeidelOptions n d =
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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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, gsDims :: ( 𝕀ℝ d -> 𝕀ℝ n ) }
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, gsPickEqs :: ( 𝕀ℝ d -> 𝕀ℝ n ) }
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-- | Options for the @box(1)@-consistency method.
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data Box1Options n d =
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Box1Options
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{ box1EpsEq :: !Double
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, box1EpsBis :: !Double
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, box1CoordsToNarrow :: [ Fin n ]
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, box1EqsToUse :: [ Fin d ]
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}
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@ -218,6 +221,7 @@ data Box1Options n d =
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data Box2Options n d =
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Box2Options
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{ box2EpsEq :: !Double
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, box2EpsBis :: !Double
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, box2LambdaMin :: !Double
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, box2CoordsToNarrow :: [ Fin n ]
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, box2EqsToUse :: [ Fin d ]
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@ -226,24 +230,20 @@ data Box2Options n d =
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defaultRootIsolationOptions :: BoxCt n d => RootIsolationOptions n d
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defaultRootIsolationOptions =
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RootIsolationOptions
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{ minWidth
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, cuspFindingAlgorithms = defaultRootIsolationAlgorithms minWidth narrowAbs
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{ rootIsolationAlgorithms = defaultRootIsolationAlgorithms minWidth ε_eq
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}
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where
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minWidth = 1e-5
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narrowAbs = 5e-3
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ε_eq = 5e-3
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defaultRootIsolationAlgorithms
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:: forall n d
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. BoxCt n d
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=> Double -> Double -> Box n -> BoxHistory n -> Either String ( NE.NonEmpty ( RootIsolationAlgorithm n d ) )
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defaultRootIsolationAlgorithms minWidth narrowAbs box history
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defaultRootIsolationAlgorithms minWidth ε_eq box history
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-- All the box widths are lower than the minimum width: give up.
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| and $ ( \ cd -> width cd <= minWidth ) <$> coordinates box
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= Left $ "sizes <= " ++ show minWidth
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-- Give up after depth 50
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-- | length history >= 50
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-- = Left "depth >= 50"
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= Left $ "widths <= " ++ show minWidth
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| otherwise
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= Right $ case history of
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lastRoundBoxes : _
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@ -253,7 +253,7 @@ defaultRootIsolationAlgorithms minWidth narrowAbs box history
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-- ...and the last round didn't sufficiently reduce the size of the box...
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, not $ box `sufficientlySmallerThan` lastRoundFirstBox
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-- ...then try bisecting the box.
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-> NE.singleton $ Bisection ( defaultBisectionOptions minWidth narrowAbs box )
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-> NE.singleton $ Bisection ( defaultBisectionOptions minWidth ε_eq box )
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-- Otherwise, do a normal round.
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-- Currently: we try an interval Gauss–Seidel step followed by box(1)-consistency.
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_ -> GaussSeidel _gaussSeidelOptions
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@ -263,7 +263,8 @@ defaultRootIsolationAlgorithms minWidth narrowAbs box history
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_box1Options :: Box1Options n d
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_box1Options =
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Box1Options
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{ box1EpsEq = narrowAbs
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{ box1EpsEq = ε_eq
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, box1EpsBis = minWidth
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, box1CoordsToNarrow = toList $ universe @n
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, box1EqsToUse = toList $ universe @d
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}
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@ -271,7 +272,8 @@ defaultRootIsolationAlgorithms minWidth narrowAbs box history
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_box2Options :: Box2Options n d
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_box2Options =
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Box2Options
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{ box2EpsEq = narrowAbs
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{ box2EpsEq = ε_eq
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, box2EpsBis = minWidth
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, box2LambdaMin = 0.001
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, box2CoordsToNarrow = toList $ universe @n
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, box2EqsToUse = toList $ universe @d
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@ -281,7 +283,7 @@ defaultRootIsolationAlgorithms minWidth narrowAbs box history
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_gaussSeidelOptions =
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GaussSeidelOptions
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{ gsPreconditioner = InverseMidJacobian
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, gsDims =
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, gsPickEqs =
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\ ( 𝕀 ( ℝ3 a_lo b_lo c_lo ) ( ℝ3 a_hi b_hi c_hi ) ) ->
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case length history `mod` 3 of
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0 -> 𝕀 ( ℝ2 a_lo b_lo ) ( ℝ2 a_hi b_hi )
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@ -289,11 +291,12 @@ defaultRootIsolationAlgorithms minWidth narrowAbs box history
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_ -> 𝕀 ( ℝ2 a_lo c_lo ) ( ℝ2 a_hi c_hi )
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}
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-- Did we reduce the box width by at least "narrowAbs" in at least one of the dimensions?
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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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sufficientlySmallerThan :: Box n -> Box n -> Bool
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b1 `sufficientlySmallerThan` b2 =
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or $
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( \ cd1 cd2 -> width cd2 - width cd1 >= narrowAbs )
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( \ cd1 cd2 -> width cd2 - width cd1 >= ε_eq )
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<$> coordinates b1
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<*> coordinates b2
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{-# INLINEABLE defaultRootIsolationAlgorithms #-}
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@ -303,7 +306,7 @@ defaultBisectionOptions
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. ( 1 <= n, BoxCt n d )
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=> Double -> Double
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-> Box n -> BisectionOptions n d
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defaultBisectionOptions minWidth _narrowAbs box =
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defaultBisectionOptions minWidth _ε_eq box =
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BisectionOptions
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{ canHaveSols =
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\ eqs box' ->
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@ -313,16 +316,16 @@ defaultBisectionOptions minWidth _narrowAbs box =
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in unT ( origin @Double ) `inside` iRange'
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-- box(1)-consistency
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--let box1Options = Box1Options _narrowAbs ( toList $ universe @n ) ( toList $ universe @d )
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--let box1Options = Box1Options _ε_eq ( toList $ universe @n ) ( toList $ universe @d )
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--in not $ null $ makeBox1Consistent _minWidth box1Options eqs box'
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-- box(2)-consistency
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--let box2Options = Box2Options _narrowAbs 0.001 ( toList $ universe @n ) ( toList $ universe @d )
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--let box2Options = Box2Options _ε_eq 0.001 ( toList $ universe @n ) ( toList $ universe @d )
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-- box'' = makeBox2Consistent _minWidth box2Options eqs box'
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-- iRange'' :: Box d
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-- iRange'' = eqs box'' `monIndex` zeroMonomial
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--in unT ( origin @Double ) `inside` iRange''
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, fallbackBisectionDim =
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, fallbackBisectionCoord =
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\ _roundHist _prevRoundsHist eqs possibleCoordChoices ->
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let datPerCoord =
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possibleCoordChoices <&> \ ( i, r ) ->
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@ -334,14 +337,15 @@ defaultBisectionOptions minWidth _narrowAbs box =
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}
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-- First, check if the largest dimension is over 20 times larger
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-- than the smallest dimension; if so bisect along that coordinate.
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-- than the smallest dimension; if so bisect that dimension.
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in case sortOnArgNE ( width . coordInterval ) datPerCoord of
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Arg _ d NE.:| [] -> ( coordBisectionData d, "" )
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Arg w0 _ NE.:| ds ->
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let Arg w1 d1 = last ds
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in if w1 >= 20 * w0
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then ( coordBisectionData d1, "tooWide" )
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-- Otherwise, pick the dimension with the largest spread = width * Jacobian column norm
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-- Otherwise, pick the coordination dimension with maximum spread
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-- (spread = width * Jacobian column norm).
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else
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let isTooSmall ( Arg ( Dual w ) _ ) = w < minWidth
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in case NE.filter ( not . isTooSmall ) $ sortOnArgNE ( Dual . spread ) datPerCoord of
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@ -391,12 +395,7 @@ isolateRootsIn
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-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
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-> Box n
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-> ( [ ( Box n, RootIsolationTree ( Box n ) ) ], ( [ Box n ], [ Box n ] ) )
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isolateRootsIn
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( RootIsolationOptions
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{ minWidth
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, cuspFindingAlgorithms
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}
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)
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isolateRootsIn ( RootIsolationOptions { rootIsolationAlgorithms } )
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eqs initBox = runWriter $ map ( initBox , ) <$> go [] initBox
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where
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@ -411,7 +410,7 @@ isolateRootsIn
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-- Box doesn't contain a solution: discard it.
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= return []
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| otherwise
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= case cuspFindingAlgorithms cand history of
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= case rootIsolationAlgorithms cand history of
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Right strats -> doStrategies history strats cand
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Left whyStop -> do
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-- We are giving up on this box (e.g. because it is too small).
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@ -421,7 +420,7 @@ isolateRootsIn
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iRange :: Box d
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iRange = eqs cand `monIndex` zeroMonomial
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-- Run a round of cusp finding strategies, then recur.
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-- Run a round of root isolation strategies, then recur.
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doStrategies
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:: BoxHistory n
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-> NonEmpty ( RootIsolationAlgorithm n d )
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@ -437,7 +436,7 @@ isolateRootsIn
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[ RootIsolationTree ( Box n )]
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do_strats thisRoundHist ( algo NE.:| algos ) box = do
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-- Run one strategy in the round.
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( step, boxes ) <- doStrategy thisRoundHist prevRoundsHist eqs minWidth algo box
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( step, boxes ) <- doStrategy thisRoundHist prevRoundsHist eqs algo box
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case algos of
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-- If there are other algorithms to run in this round, run them next.
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nextAlgo : otherAlgos ->
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@ -459,35 +458,39 @@ isolateRootsIn
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return [ RootIsolationStep step $ concat rest ]
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{-# INLINEABLE isolateRootsIn #-}
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-- | Execute a cusp-finding strategy, replacing the input box with
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-- | Execute a root isolation strategy, replacing the input box with
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-- (hopefully smaller) output boxes.
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doStrategy
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:: BoxCt n d
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=> [ ( RootIsolationStep, Box n ) ]
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-> BoxHistory n
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-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
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-> Double
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-> RootIsolationAlgorithm n d
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-> Box n
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-> Writer ( [ Box n ], [ Box n ] )
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( RootIsolationStep, [ Box n ] )
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doStrategy roundHistory previousRoundsHistory eqs minWidth algo box =
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doStrategy roundHistory previousRoundsHistory eqs algo box =
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case algo of
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GaussSeidel gsOptions -> do
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boxes <- intervalGaussSeidel gsOptions eqs box
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return ( GaussSeidelStep, boxes )
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Box1 box1Options ->
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return ( Box1Step, makeBox1Consistent minWidth box1Options eqs box )
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return ( Box1Step, makeBox1Consistent box1Options eqs box )
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Box2 box2Options ->
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return ( Box2Step, [ makeBox2Consistent minWidth box2Options eqs box ] )
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Bisection ( BisectionOptions { canHaveSols, fallbackBisectionDim } ) -> do
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let ( boxes, ( whatBis, mid ) ) = bisect ( canHaveSols eqs ) ( fallbackBisectionDim roundHistory previousRoundsHistory eqs ) box
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return ( Box2Step, [ makeBox2Consistent box2Options eqs box ] )
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Bisection ( BisectionOptions { canHaveSols, fallbackBisectionCoord } ) -> do
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let ( boxes, ( whatBis, mid ) ) =
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bisect
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( canHaveSols eqs )
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( fallbackBisectionCoord roundHistory previousRoundsHistory eqs )
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box
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return ( BisectionStep whatBis mid, boxes )
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{-# INLINEABLE doStrategy #-}
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-- | Bisect the given box.
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--
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-- (The difficult part lies in determining along which dimension to bisect.)
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-- (The difficult part lies in determining along which coordinate
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-- dimension to bisect.)
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bisect
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:: forall n
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. ( 1 <= n, KnownNat n, Representable Double ( ℝ n ) )
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@ -564,14 +567,13 @@ intervalGaussSeidel
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-- ^ box
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-> Writer ( [ 𝕀ℝ n ], [ 𝕀ℝ n ] ) [ 𝕀ℝ n ]
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intervalGaussSeidel
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( GaussSeidelOptions { gsPreconditioner = precondMeth, gsDims = projectDims } )
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( GaussSeidelOptions { gsPreconditioner = precondMeth, gsPickEqs = pickEqs } )
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eqs
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box
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| let boxMid = singleton $ midpoint box
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df = eqs box
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f' :: Vec n ( 𝕀ℝ n )
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f' = fmap ( \ i -> projectDims $ df `monIndex` linearMonomial i ) ( universe @n )
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f_mid = projectDims $ eqs boxMid `monIndex` zeroMonomial
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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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= 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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@ -622,7 +624,7 @@ gaussSeidelStep
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-> T ( 𝕀ℝ n ) -- ^ initial box \( X \)
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-> [ ( T ( 𝕀ℝ n ), Bool ) ]
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gaussSeidelStep as b ( T x0 ) = coerce $
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forEachDim @n ( x0, True ) $ \ i ( x, contraction ) -> do
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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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@ -635,14 +637,14 @@ gaussSeidelStep as b ( T x0 ) = coerce $
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-- | Run an effectful computation several times in sequence, piping its output
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-- into the next input, once for each coordinate dimension.
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forEachDim :: forall n a m. ( KnownNat n, Monad m ) => a -> ( Fin n -> a -> m a ) -> m a
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forEachDim a0 f = go ( toList $ universe @n ) a0
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forEachCoord :: forall n a m. ( KnownNat n, Monad m ) => a -> ( Fin n -> a -> m a ) -> m a
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forEachCoord a0 f = go ( toList $ universe @n ) a0
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where
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go [] a = return a
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go ( i : is ) a = do
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a' <- f i a
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go is a'
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{-# INLINEABLE forEachDim #-}
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{-# INLINEABLE forEachCoord #-}
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-- | Compute the intersection of two intervals.
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--
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@ -671,32 +673,31 @@ data Preconditioner
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-- "Presentation of a highly tuned multithreaded interval solver for underdetermined and well-determined nonlinear systems"
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makeBox1Consistent
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:: BoxCt n d
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=> Double -> Box1Options n d
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=> Box1Options n d
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-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
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-> Box n -> [ Box n ]
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makeBox1Consistent minWidth box1Options eqs x =
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makeBox1Consistent box1Options eqs x =
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( `State.evalState` False ) $
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pipeFunctionsWhileTrue ( allNarrowingOperators minWidth box1Options eqs ) x
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pipeFunctionsWhileTrue ( allNarrowingOperators box1Options eqs ) x
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-- | An implementation of "bound-consistency" from the paper
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-- "Parallelization of a bound-consistency enforcing procedure and its application in solving nonlinear systems"
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makeBox2Consistent
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:: forall n d
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. BoxCt n d
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=> Double
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-> Box2Options n d
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=> Box2Options n d
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-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
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-> Box n -> Box n
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makeBox2Consistent minWidth (Box2Options ε_eq λMin coordsToNarrow eqsToUse) eqs x0
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makeBox2Consistent (Box2Options ε_eq ε_bis λMin coordsToNarrow eqsToUse) eqs x0
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= ( `State.evalState` False ) $ doLoop 0.25 x0
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where
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box1Options :: Box1Options n d
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box1Options = Box1Options ε_eq coordsToNarrow eqsToUse
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box1Options = Box1Options ε_eq ε_bis coordsToNarrow eqsToUse
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doBox1 :: Box n -> [ Box n ]
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doBox1 = makeBox1Consistent minWidth box1Options eqs
|
||||
doBox1 = makeBox1Consistent box1Options eqs
|
||||
doLoop :: Double -> Box n -> State Bool ( Box n )
|
||||
doLoop λ x = do
|
||||
x'' <- forEachDim @n x $ boundConsistency λ
|
||||
x'' <- forEachCoord @n x $ boundConsistency λ
|
||||
modified <- State.get
|
||||
let λ' = if modified then λ else 0.5 * λ
|
||||
if λ' < λMin
|
||||
|
|
@ -741,6 +742,7 @@ precondition meth jac_mid as b =
|
|||
| let mat = toEigen jac_mid
|
||||
det = Eigen.determinant mat
|
||||
, not $ nearZero det
|
||||
-- (TODO: a bit wasteful to compute determinant then inverse.)
|
||||
, let precond = Eigen.inverse mat
|
||||
doPrecond = matMulVec ( fromEigen precond )
|
||||
-> ( fmap doPrecond as, doPrecond b )
|
||||
|
|
@ -788,7 +790,7 @@ maxVI ( 𝕀 los his ) =
|
|||
where
|
||||
maxAbs :: Double -> Double -> Double
|
||||
maxAbs lo hi = max ( abs lo ) ( abs hi )
|
||||
{-# INLINEABLE maxVI #-}
|
||||
{-# INLINEABLE maxVI #-}
|
||||
|
||||
-- Use the univariate interval Newton method to narrow from the left
|
||||
-- a candidate interval.
|
||||
|
|
@ -889,11 +891,10 @@ pipeFunctionsWhileTrue fns = go fns
|
|||
allNarrowingOperators
|
||||
:: forall n d
|
||||
. BoxCt n d
|
||||
=> Double
|
||||
-> Box1Options n d
|
||||
=> Box1Options n d
|
||||
-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
|
||||
-> [ Box n -> State Bool [ Box n ] ]
|
||||
allNarrowingOperators ε_bis ( Box1Options ε_eq coordsToNarrow eqsToUse ) eqs =
|
||||
allNarrowingOperators ( Box1Options ε_eq ε_bis coordsToNarrow eqsToUse ) eqs =
|
||||
[ \ cand ->
|
||||
let getter = ( `index` coordIndex )
|
||||
setter = set coordIndex
|
||||
|
|
|
|||
Loading…
Reference in a new issue