mirror of
https://gitlab.com/sheaf/metabrush.git
synced 2024-11-23 15:34:06 +00:00
Documentation in root isolation module
This commit is contained in:
sheaf
parent
8009983b37
commit
0e6b9a822b
|
|
@ -97,12 +97,12 @@ import qualified Math.Ring as Ring
|
|||
|
||||
--------------------------------------------------------------------------------
|
||||
|
||||
-- | A tree recording the steps taken when doing cusp finding.
|
||||
-- | A tree recording the steps taken when isolating roots.
|
||||
data RootIsolationTree d
|
||||
= RootIsolationLeaf String d
|
||||
| RootIsolationStep RootIsolationStep [(d, RootIsolationTree d)]
|
||||
|
||||
-- | A description of a step taken in cusp-finding.
|
||||
-- | A description of a step taken when isolating roots.
|
||||
data RootIsolationStep
|
||||
= GaussSeidelStep
|
||||
| BisectionStep String Double
|
||||
|
|
@ -155,17 +155,19 @@ type BoxCt n d = ( n ~ N, d ~ 3 )
|
|||
)
|
||||
-}
|
||||
|
||||
-- | Options for the root isolation methods in 'findCusps' and 'isolateRootsIn'.
|
||||
-- | Options for the root isolation methods in 'isolateRootsIn'.
|
||||
type RootIsolationOptions :: Nat -> Nat -> Type
|
||||
data RootIsolationOptions n d
|
||||
newtype RootIsolationOptions n d
|
||||
= RootIsolationOptions
|
||||
{ -- | Minimum width of a box.
|
||||
minWidth :: !Double
|
||||
-- | Given a box and its history, return a round of cusp-finding strategies
|
||||
{ -- | Given a box and its history, return a round of root isolation strategies
|
||||
-- to run, in sequence, on this box.
|
||||
--
|
||||
-- Returning an empty list means: give up on this box.
|
||||
, cuspFindingAlgorithms :: !( Box n -> BoxHistory n -> Either String ( NE.NonEmpty ( RootIsolationAlgorithm n d ) ) )
|
||||
-- A return value of @Left whyStop@ indicates stopping any further iterations
|
||||
-- on this box, e.g. because it is too small.
|
||||
rootIsolationAlgorithms
|
||||
:: Box n
|
||||
-> BoxHistory n
|
||||
-> Either String ( NE.NonEmpty ( RootIsolationAlgorithm n d ) )
|
||||
}
|
||||
|
||||
type RootIsolationAlgorithm :: Nat -> Nat -> Type
|
||||
|
|
@ -184,16 +186,16 @@ type BisectionOptions :: Nat -> Nat -> Type
|
|||
data BisectionOptions n d =
|
||||
BisectionOptions
|
||||
{ canHaveSols :: !( ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) ) -> Box n -> Bool )
|
||||
, fallbackBisectionDim :: !( BisectionDimPicker n d )
|
||||
, fallbackBisectionCoord :: !( BisectionCoordPicker n d )
|
||||
}
|
||||
|
||||
type BisectionDimPicker n d
|
||||
= forall r
|
||||
. [ ( RootIsolationStep, Box n ) ]
|
||||
-- | A function to choose along which coordination dimension we should
|
||||
-- perform a bisection.
|
||||
type BisectionCoordPicker n d
|
||||
= [ ( RootIsolationStep, Box n ) ]
|
||||
-> BoxHistory n
|
||||
-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
|
||||
-> NE.NonEmpty ( Fin n, r )
|
||||
-> ( r, String )
|
||||
-> forall r. ( NE.NonEmpty ( Fin n, r ) -> ( r, String ) )
|
||||
|
||||
-- | Options for the interval Gauss–Seidel method.
|
||||
type GaussSeidelOptions :: Nat -> Nat -> Type
|
||||
|
|
@ -204,12 +206,13 @@ data GaussSeidelOptions n d =
|
|||
-- | Function that projects over the equations we will consider
|
||||
-- (the identity for a well-determined problem, or a projection for
|
||||
-- an overdetermined system).
|
||||
, gsDims :: ( 𝕀ℝ d -> 𝕀ℝ n ) }
|
||||
, gsPickEqs :: ( 𝕀ℝ d -> 𝕀ℝ n ) }
|
||||
|
||||
-- | Options for the @box(1)@-consistency method.
|
||||
data Box1Options n d =
|
||||
Box1Options
|
||||
{ box1EpsEq :: !Double
|
||||
, box1EpsBis :: !Double
|
||||
, box1CoordsToNarrow :: [ Fin n ]
|
||||
, box1EqsToUse :: [ Fin d ]
|
||||
}
|
||||
|
|
@ -218,6 +221,7 @@ data Box1Options n d =
|
|||
data Box2Options n d =
|
||||
Box2Options
|
||||
{ box2EpsEq :: !Double
|
||||
, box2EpsBis :: !Double
|
||||
, box2LambdaMin :: !Double
|
||||
, box2CoordsToNarrow :: [ Fin n ]
|
||||
, box2EqsToUse :: [ Fin d ]
|
||||
|
|
@ -226,24 +230,20 @@ data Box2Options n d =
|
|||
defaultRootIsolationOptions :: BoxCt n d => RootIsolationOptions n d
|
||||
defaultRootIsolationOptions =
|
||||
RootIsolationOptions
|
||||
{ minWidth
|
||||
, cuspFindingAlgorithms = defaultRootIsolationAlgorithms minWidth narrowAbs
|
||||
{ rootIsolationAlgorithms = defaultRootIsolationAlgorithms minWidth ε_eq
|
||||
}
|
||||
where
|
||||
minWidth = 1e-5
|
||||
narrowAbs = 5e-3
|
||||
ε_eq = 5e-3
|
||||
|
||||
defaultRootIsolationAlgorithms
|
||||
:: forall n d
|
||||
. BoxCt n d
|
||||
=> Double -> Double -> Box n -> BoxHistory n -> Either String ( NE.NonEmpty ( RootIsolationAlgorithm n d ) )
|
||||
defaultRootIsolationAlgorithms minWidth narrowAbs box history
|
||||
defaultRootIsolationAlgorithms minWidth ε_eq box history
|
||||
-- All the box widths are lower than the minimum width: give up.
|
||||
| and $ ( \ cd -> width cd <= minWidth ) <$> coordinates box
|
||||
= Left $ "sizes <= " ++ show minWidth
|
||||
-- Give up after depth 50
|
||||
-- | length history >= 50
|
||||
-- = Left "depth >= 50"
|
||||
= Left $ "widths <= " ++ show minWidth
|
||||
| otherwise
|
||||
= Right $ case history of
|
||||
lastRoundBoxes : _
|
||||
|
|
@ -253,7 +253,7 @@ defaultRootIsolationAlgorithms minWidth narrowAbs box history
|
|||
-- ...and the last round didn't sufficiently reduce the size of the box...
|
||||
, not $ box `sufficientlySmallerThan` lastRoundFirstBox
|
||||
-- ...then try bisecting the box.
|
||||
-> NE.singleton $ Bisection ( defaultBisectionOptions minWidth narrowAbs box )
|
||||
-> NE.singleton $ Bisection ( defaultBisectionOptions minWidth ε_eq box )
|
||||
-- Otherwise, do a normal round.
|
||||
-- Currently: we try an interval Gauss–Seidel step followed by box(1)-consistency.
|
||||
_ -> GaussSeidel _gaussSeidelOptions
|
||||
|
|
@ -263,7 +263,8 @@ defaultRootIsolationAlgorithms minWidth narrowAbs box history
|
|||
_box1Options :: Box1Options n d
|
||||
_box1Options =
|
||||
Box1Options
|
||||
{ box1EpsEq = narrowAbs
|
||||
{ box1EpsEq = ε_eq
|
||||
, box1EpsBis = minWidth
|
||||
, box1CoordsToNarrow = toList $ universe @n
|
||||
, box1EqsToUse = toList $ universe @d
|
||||
}
|
||||
|
|
@ -271,7 +272,8 @@ defaultRootIsolationAlgorithms minWidth narrowAbs box history
|
|||
_box2Options :: Box2Options n d
|
||||
_box2Options =
|
||||
Box2Options
|
||||
{ box2EpsEq = narrowAbs
|
||||
{ box2EpsEq = ε_eq
|
||||
, box2EpsBis = minWidth
|
||||
, box2LambdaMin = 0.001
|
||||
, box2CoordsToNarrow = toList $ universe @n
|
||||
, box2EqsToUse = toList $ universe @d
|
||||
|
|
@ -281,7 +283,7 @@ defaultRootIsolationAlgorithms minWidth narrowAbs box history
|
|||
_gaussSeidelOptions =
|
||||
GaussSeidelOptions
|
||||
{ gsPreconditioner = InverseMidJacobian
|
||||
, gsDims =
|
||||
, gsPickEqs =
|
||||
\ ( 𝕀 ( ℝ3 a_lo b_lo c_lo ) ( ℝ3 a_hi b_hi c_hi ) ) ->
|
||||
case length history `mod` 3 of
|
||||
0 -> 𝕀 ( ℝ2 a_lo b_lo ) ( ℝ2 a_hi b_hi )
|
||||
|
|
@ -289,11 +291,12 @@ defaultRootIsolationAlgorithms minWidth narrowAbs box history
|
|||
_ -> 𝕀 ( ℝ2 a_lo c_lo ) ( ℝ2 a_hi c_hi )
|
||||
}
|
||||
|
||||
-- Did we reduce the box width by at least "narrowAbs" in at least one of the dimensions?
|
||||
-- Did we reduce the box width by at least ε_eq
|
||||
-- in at least one of the coordinates?
|
||||
sufficientlySmallerThan :: Box n -> Box n -> Bool
|
||||
b1 `sufficientlySmallerThan` b2 =
|
||||
or $
|
||||
( \ cd1 cd2 -> width cd2 - width cd1 >= narrowAbs )
|
||||
( \ cd1 cd2 -> width cd2 - width cd1 >= ε_eq )
|
||||
<$> coordinates b1
|
||||
<*> coordinates b2
|
||||
{-# INLINEABLE defaultRootIsolationAlgorithms #-}
|
||||
|
|
@ -303,7 +306,7 @@ defaultBisectionOptions
|
|||
. ( 1 <= n, BoxCt n d )
|
||||
=> Double -> Double
|
||||
-> Box n -> BisectionOptions n d
|
||||
defaultBisectionOptions minWidth _narrowAbs box =
|
||||
defaultBisectionOptions minWidth _ε_eq box =
|
||||
BisectionOptions
|
||||
{ canHaveSols =
|
||||
\ eqs box' ->
|
||||
|
|
@ -313,16 +316,16 @@ defaultBisectionOptions minWidth _narrowAbs box =
|
|||
in unT ( origin @Double ) `inside` iRange'
|
||||
|
||||
-- box(1)-consistency
|
||||
--let box1Options = Box1Options _narrowAbs ( toList $ universe @n ) ( toList $ universe @d )
|
||||
--let box1Options = Box1Options _ε_eq ( toList $ universe @n ) ( toList $ universe @d )
|
||||
--in not $ null $ makeBox1Consistent _minWidth box1Options eqs box'
|
||||
|
||||
-- box(2)-consistency
|
||||
--let box2Options = Box2Options _narrowAbs 0.001 ( toList $ universe @n ) ( toList $ universe @d )
|
||||
--let box2Options = Box2Options _ε_eq 0.001 ( toList $ universe @n ) ( toList $ universe @d )
|
||||
-- box'' = makeBox2Consistent _minWidth box2Options eqs box'
|
||||
-- iRange'' :: Box d
|
||||
-- iRange'' = eqs box'' `monIndex` zeroMonomial
|
||||
--in unT ( origin @Double ) `inside` iRange''
|
||||
, fallbackBisectionDim =
|
||||
, fallbackBisectionCoord =
|
||||
\ _roundHist _prevRoundsHist eqs possibleCoordChoices ->
|
||||
let datPerCoord =
|
||||
possibleCoordChoices <&> \ ( i, r ) ->
|
||||
|
|
@ -334,14 +337,15 @@ defaultBisectionOptions minWidth _narrowAbs box =
|
|||
}
|
||||
|
||||
-- First, check if the largest dimension is over 20 times larger
|
||||
-- than the smallest dimension; if so bisect along that coordinate.
|
||||
-- than the smallest dimension; if so bisect that dimension.
|
||||
in case sortOnArgNE ( width . coordInterval ) datPerCoord of
|
||||
Arg _ d NE.:| [] -> ( coordBisectionData d, "" )
|
||||
Arg w0 _ NE.:| ds ->
|
||||
let Arg w1 d1 = last ds
|
||||
in if w1 >= 20 * w0
|
||||
then ( coordBisectionData d1, "tooWide" )
|
||||
-- Otherwise, pick the dimension with the largest spread = width * Jacobian column norm
|
||||
-- Otherwise, pick the coordination dimension with maximum spread
|
||||
-- (spread = width * Jacobian column norm).
|
||||
else
|
||||
let isTooSmall ( Arg ( Dual w ) _ ) = w < minWidth
|
||||
in case NE.filter ( not . isTooSmall ) $ sortOnArgNE ( Dual . spread ) datPerCoord of
|
||||
|
|
@ -391,12 +395,7 @@ isolateRootsIn
|
|||
-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
|
||||
-> Box n
|
||||
-> ( [ ( Box n, RootIsolationTree ( Box n ) ) ], ( [ Box n ], [ Box n ] ) )
|
||||
isolateRootsIn
|
||||
( RootIsolationOptions
|
||||
{ minWidth
|
||||
, cuspFindingAlgorithms
|
||||
}
|
||||
)
|
||||
isolateRootsIn ( RootIsolationOptions { rootIsolationAlgorithms } )
|
||||
eqs initBox = runWriter $ map ( initBox , ) <$> go [] initBox
|
||||
|
||||
where
|
||||
|
|
@ -411,7 +410,7 @@ isolateRootsIn
|
|||
-- Box doesn't contain a solution: discard it.
|
||||
= return []
|
||||
| otherwise
|
||||
= case cuspFindingAlgorithms cand history of
|
||||
= case rootIsolationAlgorithms cand history of
|
||||
Right strats -> doStrategies history strats cand
|
||||
Left whyStop -> do
|
||||
-- We are giving up on this box (e.g. because it is too small).
|
||||
|
|
@ -421,7 +420,7 @@ isolateRootsIn
|
|||
iRange :: Box d
|
||||
iRange = eqs cand `monIndex` zeroMonomial
|
||||
|
||||
-- Run a round of cusp finding strategies, then recur.
|
||||
-- Run a round of root isolation strategies, then recur.
|
||||
doStrategies
|
||||
:: BoxHistory n
|
||||
-> NonEmpty ( RootIsolationAlgorithm n d )
|
||||
|
|
@ -437,7 +436,7 @@ isolateRootsIn
|
|||
[ RootIsolationTree ( Box n )]
|
||||
do_strats thisRoundHist ( algo NE.:| algos ) box = do
|
||||
-- Run one strategy in the round.
|
||||
( step, boxes ) <- doStrategy thisRoundHist prevRoundsHist eqs minWidth algo box
|
||||
( step, boxes ) <- doStrategy thisRoundHist prevRoundsHist eqs algo box
|
||||
case algos of
|
||||
-- If there are other algorithms to run in this round, run them next.
|
||||
nextAlgo : otherAlgos ->
|
||||
|
|
@ -459,35 +458,39 @@ isolateRootsIn
|
|||
return [ RootIsolationStep step $ concat rest ]
|
||||
{-# INLINEABLE isolateRootsIn #-}
|
||||
|
||||
-- | Execute a cusp-finding strategy, replacing the input box with
|
||||
-- | Execute a root isolation strategy, replacing the input box with
|
||||
-- (hopefully smaller) output boxes.
|
||||
doStrategy
|
||||
:: BoxCt n d
|
||||
=> [ ( RootIsolationStep, Box n ) ]
|
||||
-> BoxHistory n
|
||||
-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
|
||||
-> Double
|
||||
-> RootIsolationAlgorithm n d
|
||||
-> Box n
|
||||
-> Writer ( [ Box n ], [ Box n ] )
|
||||
( RootIsolationStep, [ Box n ] )
|
||||
doStrategy roundHistory previousRoundsHistory eqs minWidth algo box =
|
||||
doStrategy roundHistory previousRoundsHistory eqs algo box =
|
||||
case algo of
|
||||
GaussSeidel gsOptions -> do
|
||||
boxes <- intervalGaussSeidel gsOptions eqs box
|
||||
return ( GaussSeidelStep, boxes )
|
||||
Box1 box1Options ->
|
||||
return ( Box1Step, makeBox1Consistent minWidth box1Options eqs box )
|
||||
return ( Box1Step, makeBox1Consistent box1Options eqs box )
|
||||
Box2 box2Options ->
|
||||
return ( Box2Step, [ makeBox2Consistent minWidth box2Options eqs box ] )
|
||||
Bisection ( BisectionOptions { canHaveSols, fallbackBisectionDim } ) -> do
|
||||
let ( boxes, ( whatBis, mid ) ) = bisect ( canHaveSols eqs ) ( fallbackBisectionDim roundHistory previousRoundsHistory eqs ) box
|
||||
return ( Box2Step, [ makeBox2Consistent box2Options eqs box ] )
|
||||
Bisection ( BisectionOptions { canHaveSols, fallbackBisectionCoord } ) -> do
|
||||
let ( boxes, ( whatBis, mid ) ) =
|
||||
bisect
|
||||
( canHaveSols eqs )
|
||||
( fallbackBisectionCoord roundHistory previousRoundsHistory eqs )
|
||||
box
|
||||
return ( BisectionStep whatBis mid, boxes )
|
||||
{-# INLINEABLE doStrategy #-}
|
||||
|
||||
-- | Bisect the given box.
|
||||
--
|
||||
-- (The difficult part lies in determining along which dimension to bisect.)
|
||||
-- (The difficult part lies in determining along which coordinate
|
||||
-- dimension to bisect.)
|
||||
bisect
|
||||
:: forall n
|
||||
. ( 1 <= n, KnownNat n, Representable Double ( ℝ n ) )
|
||||
|
|
@ -564,14 +567,13 @@ intervalGaussSeidel
|
|||
-- ^ box
|
||||
-> Writer ( [ 𝕀ℝ n ], [ 𝕀ℝ n ] ) [ 𝕀ℝ n ]
|
||||
intervalGaussSeidel
|
||||
( GaussSeidelOptions { gsPreconditioner = precondMeth, gsDims = projectDims } )
|
||||
( GaussSeidelOptions { gsPreconditioner = precondMeth, gsPickEqs = pickEqs } )
|
||||
eqs
|
||||
box
|
||||
| let boxMid = singleton $ midpoint box
|
||||
df = eqs box
|
||||
f' :: Vec n ( 𝕀ℝ n )
|
||||
f' = fmap ( \ i -> projectDims $ df `monIndex` linearMonomial i ) ( universe @n )
|
||||
f_mid = projectDims $ eqs boxMid `monIndex` zeroMonomial
|
||||
f' = fmap ( \ i -> pickEqs $ eqs box `monIndex` linearMonomial i ) ( universe @n )
|
||||
f_mid = pickEqs $ eqs boxMid `monIndex` zeroMonomial
|
||||
|
||||
= let -- Interval Newton method: take one Gauss–Seidel step
|
||||
-- for the system of equations f'(x) ( x - x_mid ) = - f(x_mid).
|
||||
|
|
@ -622,7 +624,7 @@ gaussSeidelStep
|
|||
-> T ( 𝕀ℝ n ) -- ^ initial box \( X \)
|
||||
-> [ ( T ( 𝕀ℝ n ), Bool ) ]
|
||||
gaussSeidelStep as b ( T x0 ) = coerce $
|
||||
forEachDim @n ( x0, True ) $ \ i ( x, contraction ) -> do
|
||||
forEachCoord @n ( x0, True ) $ \ i ( x, contraction ) -> do
|
||||
-- x_i' = ( b_i - sum { j /= i } a_ij * x_j ) / a_ii
|
||||
x_i'0 <- ( b `index` i - sum [ ( as ! j ) `index` i * x `index` j | j <- toList ( universe @n ), j /= i ] )
|
||||
`extendedDivide` ( ( as ! i ) `index` i )
|
||||
|
|
@ -635,14 +637,14 @@ gaussSeidelStep as b ( T x0 ) = coerce $
|
|||
|
||||
-- | Run an effectful computation several times in sequence, piping its output
|
||||
-- into the next input, once for each coordinate dimension.
|
||||
forEachDim :: forall n a m. ( KnownNat n, Monad m ) => a -> ( Fin n -> a -> m a ) -> m a
|
||||
forEachDim a0 f = go ( toList $ universe @n ) a0
|
||||
forEachCoord :: forall n a m. ( KnownNat n, Monad m ) => a -> ( Fin n -> a -> m a ) -> m a
|
||||
forEachCoord a0 f = go ( toList $ universe @n ) a0
|
||||
where
|
||||
go [] a = return a
|
||||
go ( i : is ) a = do
|
||||
a' <- f i a
|
||||
go is a'
|
||||
{-# INLINEABLE forEachDim #-}
|
||||
{-# INLINEABLE forEachCoord #-}
|
||||
|
||||
-- | Compute the intersection of two intervals.
|
||||
--
|
||||
|
|
@ -671,32 +673,31 @@ data Preconditioner
|
|||
-- "Presentation of a highly tuned multithreaded interval solver for underdetermined and well-determined nonlinear systems"
|
||||
makeBox1Consistent
|
||||
:: BoxCt n d
|
||||
=> Double -> Box1Options n d
|
||||
=> Box1Options n d
|
||||
-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
|
||||
-> Box n -> [ Box n ]
|
||||
makeBox1Consistent minWidth box1Options eqs x =
|
||||
makeBox1Consistent box1Options eqs x =
|
||||
( `State.evalState` False ) $
|
||||
pipeFunctionsWhileTrue ( allNarrowingOperators minWidth box1Options eqs ) x
|
||||
pipeFunctionsWhileTrue ( allNarrowingOperators box1Options eqs ) x
|
||||
|
||||
-- | An implementation of "bound-consistency" from the paper
|
||||
-- "Parallelization of a bound-consistency enforcing procedure and its application in solving nonlinear systems"
|
||||
makeBox2Consistent
|
||||
:: forall n d
|
||||
. BoxCt n d
|
||||
=> Double
|
||||
-> Box2Options n d
|
||||
=> Box2Options n d
|
||||
-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
|
||||
-> Box n -> Box n
|
||||
makeBox2Consistent minWidth (Box2Options ε_eq λMin coordsToNarrow eqsToUse) eqs x0
|
||||
makeBox2Consistent (Box2Options ε_eq ε_bis λMin coordsToNarrow eqsToUse) eqs x0
|
||||
= ( `State.evalState` False ) $ doLoop 0.25 x0
|
||||
where
|
||||
box1Options :: Box1Options n d
|
||||
box1Options = Box1Options ε_eq coordsToNarrow eqsToUse
|
||||
box1Options = Box1Options ε_eq ε_bis coordsToNarrow eqsToUse
|
||||
doBox1 :: Box n -> [ Box n ]
|
||||
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 )
|
||||
|
|
@ -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