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Sprinkle in a bit of unicode

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sheaf 2024-04-24 01:03:46 +02:00
parent edba0416aa
commit a59f1695fb
7 changed files with 58 additions and 54 deletions
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6
brush-strokes/src/lib/Math/Bezier/Stroke.hs
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@ -1234,10 +1234,10 @@ findCuspsIn opts boxStrokeData initBoxes =
, _D22_dy = T ( 𝕀 ( ℝ1 ee_s_lo ) ( ℝ1 ee_s_hi ) ) }
} = ( boxStrokeData t `Seq.index` i ) s
-- λ = ∂E/∂t / ∂E/∂s
λ1 = 𝕀 ee_t_lo ee_t_hi `extendedDivide` 𝕀 ee_s_lo ee_s_hi
λ1 = 𝕀 ee_t_lo ee_t_hi ⊘ 𝕀 ee_s_lo ee_s_hi
-- λ = u / v
λ2 = 𝕀 ux_lo ux_hi `extendedDivide` 𝕀 vx_lo vx_hi
λ3 = 𝕀 uy_lo uy_hi `extendedDivide` 𝕀 vy_lo vy_hi
λ2 = 𝕀 ux_lo ux_hi ⊘ 𝕀 vx_lo vx_hi
λ3 = 𝕀 uy_lo uy_hi ⊘ 𝕀 vy_lo vy_hi
λ = [ 𝕀 ( recip -0 ) ( recip 0 ) ]
`intersectMany` λ1
`intersectMany` λ2

8
brush-strokes/src/lib/Math/Bezier/Stroke/EnvelopeEquation.hs
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@ -451,9 +451,9 @@ evaluateCubic bez t =
let inf_bez = fmap inf bez
sup_bez = fmap sup bez
mins = fmap (Cubic.bezier @( T Double ) inf_bez)
$ inf t :| ( sup t : filter ( `inside` t ) ( Cubic.extrema inf_bez ) )
$ inf t :| ( sup t : filter ( ∈ t ) ( Cubic.extrema inf_bez ) )
maxs = fmap (Cubic.bezier @( T Double ) sup_bez)
$ inf t :| ( sup t : filter ( `inside` t ) ( Cubic.extrema sup_bez ) )
$ inf t :| ( sup t : filter ( ∈ t ) ( Cubic.extrema sup_bez ) )
in 𝕀 ( minimum mins ) ( maximum maxs )
-- | Evaluate a quadratic Bézier curve, when both the coefficients and the
@ -464,7 +464,7 @@ evaluateQuadratic bez t =
let inf_bez = fmap inf bez
sup_bez = fmap sup bez
mins = fmap (Quadratic.bezier @( T Double ) inf_bez)
$ inf t :| ( sup t : filter ( `inside` t ) ( Quadratic.extrema inf_bez ) )
$ inf t :| ( sup t : filter ( ∈ t ) ( Quadratic.extrema inf_bez ) )
maxs = fmap (Quadratic.bezier @( T Double ) sup_bez)
$ inf t :| ( sup t : filter ( `inside` t ) ( Quadratic.extrema sup_bez ) )
$ inf t :| ( sup t : filter ( ∈ t ) ( Quadratic.extrema sup_bez ) )
in 𝕀 ( minimum mins ) ( maximum maxs )

34
brush-strokes/src/lib/Math/Interval.hs
View file

@ -11,10 +11,9 @@ module Math.Interval
, 𝕀ℝ
, isCanonical
, singleton, nonDecreasing
, inside
, aabb
, extendedDivide, extendedRecip
, intersect, bisect
, (∈), (⊖), (∩), (⊘)
, extendedRecip
, aabb, bisect
)
where
@ -72,9 +71,10 @@ nonDecreasing :: ( a -> b ) -> 𝕀 a -> 𝕀 b
nonDecreasing f ( 𝕀 lo hi ) = 𝕀 ( f lo ) ( f hi )
-- | Does the given value lie inside the specified interval?
inside :: Ord a => a -> 𝕀 a -> Bool
inside x ( 𝕀 lo hi ) = x >= lo && x <= hi
{-# INLINEABLE inside #-}
(∈) :: Ord a => a -> 𝕀 a -> Bool
x ∈ 𝕀 lo hi = x >= lo && x <= hi
{-# INLINEABLE (∈) #-}
infix 4 ∈
-- | Is this interval canonical, i.e. it consists of either 1 or 2 floating point
-- values only?
@ -91,8 +91,8 @@ midpoint ( 𝕀 x_inf x_sup ) = 0.5 * ( x_inf + x_sup )
--
-- Returns whether the first interval is a strict subset of the second interval
-- (or the intersection is a single point).
intersect :: 𝕀 Double -> 𝕀 Double -> [ ( 𝕀 Double, Bool ) ]
intersect ( 𝕀 lo1 hi1 ) ( 𝕀 lo2 hi2 )
(∩) :: 𝕀 Double -> 𝕀 Double -> [ ( 𝕀 Double, Bool ) ]
𝕀 lo1 hi1 ∩ 𝕀 lo2 hi2
| lo > hi
= [ ]
| otherwise
@ -100,6 +100,7 @@ intersect ( 𝕀 lo1 hi1 ) ( 𝕀 lo2 hi2 )
where
lo = max lo1 lo2
hi = min hi1 hi2
infix 3 ∩
-- | Bisect an interval.
--
@ -113,6 +114,15 @@ bisect x@( 𝕀 x_inf x_sup )
= 𝕀 x_inf x_mid NE.:| [ 𝕀 x_mid x_sup ]
where x_mid = midpoint x
infixl 6 ⊖
(⊖) :: ( Ring a, Ord a ) => 𝕀 a -> 𝕀 a -> 𝕀 a
(⊖) a@( 𝕀 lo1 hi1 ) b@( 𝕀 lo2 hi2 )
| width a >= width b
= 𝕀 ( lo1 - lo2 ) ( hi1 - hi2 )
| otherwise
= 𝕀 ( hi1 - hi2 ) ( lo1 - lo2 )
{-# INLINEABLE (⊖) #-}
deriving via ViaAbelianGroup ( T ( 𝕀 Double ) )
instance Semigroup ( T ( 𝕀 Double ) )
deriving via ViaAbelianGroup ( T ( 𝕀 Double ) )
@ -128,8 +138,10 @@ instance Torsor ( T ( 𝕀 Double ) ) ( 𝕀 Double ) where
-------------------------------------------------------------------------------
-- Extended division
extendedDivide :: 𝕀 Double -> 𝕀 Double -> [ 𝕀 Double ]
extendedDivide x y = nub $ map ( x * ) ( extendedRecip y )
-- | Extended division.
(⊘) :: 𝕀 Double -> 𝕀 Double -> [ 𝕀 Double ]
x ⊘ y = nub $ map ( x * ) ( extendedRecip y )
infixl 7 ⊘
extendedRecip :: 𝕀 Double -> [ 𝕀 Double ]
extendedRecip x@( 𝕀 lo hi )

2
brush-strokes/src/lib/Math/Root/Isolation.hs
View file

@ -192,7 +192,7 @@ isolateRootsIn ( RootIsolationOptions { rootIsolationAlgorithms } )
[ RootIsolationTree ( Box n ) ]
go history cand
| -- Check the range of the equations contains zero.
not $ ( unT ( origin @Double ) `inside` iRange )
not $ ( unT ( origin @Double ) ∈ iRange )
-- Box doesn't contain a solution: discard it.
= return []
| otherwise

4
brush-strokes/src/lib/Math/Root/Isolation/Bisection.hs
View file

@ -106,7 +106,7 @@ defaultBisectionOptions minWidth _ε_eq box =
-- box(0)-consistency
let iRange' :: Box d
iRange' = eqs box' `monIndex` zeroMonomial
in unT ( origin @Double ) `inside` iRange'
in unT ( origin @Double ) ∈ iRange'
-- box(1)-consistency
--let box1Options = Box1Options _ε_eq ( toList $ universe @n ) ( toList $ universe @d )
@ -117,7 +117,7 @@ defaultBisectionOptions minWidth _ε_eq box =
-- box'' = makeBox2Consistent _minWidth box2Options eqs box'
-- iRange'' :: Box d
-- iRange'' = eqs box'' `monIndex` zeroMonomial
--in unT ( origin @Double ) `inside` iRange''
--in unT ( origin @Double ) ∈ iRange''
, fallbackBisectionCoord =
\ _thisRoundHist _prevRoundsHist eqs possibleCoordChoices ->
let datPerCoord =

26
brush-strokes/src/lib/Math/Root/Isolation/GaussSeidel.hs
View file

@ -208,17 +208,17 @@ gaussSeidelStep as b ( T x0 ) = coerce $
x_i = x `index` i
a_ii = ( as ! i ) `index` i
-- Take a shortcut before performing the division if possible.
if | not $ 0 `inside` ( s - a_ii * x_i )
if | not $ 0 ∈ ( s - a_ii * x_i )
-- No solutions: don't bother performing a division.
-> [ ]
| 0 `inside` s && 0 `inside` a_ii
| 0 ∈ s && 0 ∈ a_ii
-- The division would produce [-oo,+oo]: don't do anything.
-> [ ( x, False ) ]
-- Otherwise, perform the division.
| otherwise
-> do
x_i'0 <- s `extendedDivide` a_ii
( x_i', sub_i ) <- x_i'0 `intersect` x_i
x_i'0 <- s ⊘ a_ii
( x_i', sub_i ) <- x_i'0 ∩ x_i
return $ ( set i x_i' x, sub_i && contraction )
{-# INLINEABLE gaussSeidelStep #-}
@ -244,16 +244,16 @@ gaussSeidelStep_Complete as b ( T x0 ) = coerce $ do
( x', subs ) <- fromComponents \ j -> do
let x_j = x `index` j
a_ij = ( as ! j ) `index` i
s_j = s `ominus` ( a_ij * x_j )
s_j = s ⊖ ( a_ij * x_j )
-- Shortcut division if possible (see gaussSeidelStep for commentary).
if | not $ 0 `inside` ( s_j - a_ii * x_i )
if | not $ 0 ∈ ( s_j - a_ii * x_i )
-> [ ]
| 0 `inside` s_j && 0 `inside` a_ij
| 0 ∈ s_j && 0 ∈ a_ij
-> [ ( x_j, False ) ]
| otherwise
-> do
x_j'0 <- s_j `extendedDivide` a_ij
( x_j', sub_j ) <- x_j'0 `intersect` x_j
x_j'0 <- s_j ⊘ a_ij
( x_j', sub_j ) <- x_j'0 ∩ x_j
return $ ( x_j', sub_j )
return ( x', (||) <$> contractions <*> subs )
return ( x', and subs )
@ -269,14 +269,6 @@ fromComponents f = do
-- TODO: this could be more efficient.
{-# INLINEABLE fromComponents #-}
infixl 6 `ominus`
ominus :: 𝕀 Double -> 𝕀 Double -> 𝕀 Double
ominus a@( 𝕀 lo1 hi1 ) b@( 𝕀 lo2 hi2 )
| width a >= width b
= 𝕀 ( lo1 - lo2 ) ( hi1 - hi2 )
| otherwise
= 𝕀 ( hi1 - hi2 ) ( lo1 - lo2 )
-- | The midpoint of a box.
boxMidpoint :: Representable Double ( ℝ n ) => 𝕀ℝ n -> ℝ n
boxMidpoint box =

32
brush-strokes/src/lib/Math/Root/Isolation/Narrowing.hs
View file

@ -295,9 +295,9 @@ leftNarrow ε_eq ε_bis ff' = left_narrow
else
let x = 𝕀 ( sup x_left ) x_sup
( _f_x, f'_x ) = ff' x
x's = do δ <- f_x_left `extendedDivide` f'_x
x's = do δ <- f_x_left ⊘ f'_x
let x_new = x_left - δ
map fst $ x_new `intersect` x
map fst $ x_new ∩ x
in
if | null x's
-> []
@ -328,9 +328,9 @@ rightNarrow ε_eq ε_bis ff' = right_narrow
else
let x = 𝕀 x_inf ( inf x_right )
( _f_x, f'_x ) = ff' x
x's = do δ <- f_x_right `extendedDivide` f'_x
x's = do δ <- f_x_right ⊘ f'_x
let x_new = x_right - δ
map fst $ x_new `intersect` x
map fst $ x_new ∩ x
in
if | null x's
-> []
@ -387,7 +387,7 @@ leftShave ε_eq ε_bis
go γ x_sup x_left =
let ( f_x_left, _f'_x_left ) = ff' x_left
in
if 0 `inside` f_x_left
if 0 ∈ f_x_left
-- Box-consistency achieved; finish.
then [ 𝕀 ( inf x_left ) x_sup ]
else
@ -399,7 +399,7 @@ leftShave ε_eq ε_bis
-- NB: this always uses the initial width (to "avoid asymptotic behaviour" according to the paper)
( f_guess, f'_guess ) = ff' guess
x_minus_guess = 𝕀 ( min x_sup ( succFP $ sup guess ) ) x_sup
in if not ( 0 `inside` f_guess )
in if not ( 0 ∈ f_guess )
then
-- We successfully shaved "guess" off; go round again after removing it.
-- TODO: here we could go back to the top with a new "w" maybe?
@ -410,9 +410,9 @@ leftShave ε_eq ε_bis
-- "guesses'" refining where the function can be zero.
let guesses' :: [ ( 𝕀 Double ) ]
guesses' = do
δ <- f_x_left `extendedDivide` f'_guess
δ <- f_x_left ⊘ f'_guess
let guess' = singleton ( inf guess ) - δ
map fst $ guess' `intersect` guess
map fst $ guess' ∩ guess
w_guess = width guess
w_guesses'
| null guesses'
@ -453,30 +453,30 @@ sbc ε ff' = go
| otherwise
= let x_mid = 0.5 * ( x_l + x_r )
( left_done, left_todo )
| 0 `inside` ( fst $ ff' ( 𝕀 x_l x_lp ) )
| 0 ∈ ( fst $ ff' ( 𝕀 x_l x_lp ) )
= ( True, [ ] )
| not $ 0 `inside` ( fst $ ff' i_l )
| not $ 0 ∈ ( fst $ ff' i_l )
= ( False, [ ] )
| otherwise
= let
xls = do
let l = 𝕀 x_lp x_lp
δ <- fst ( ff' l ) `extendedDivide` snd ( ff' i_l )
map fst $ ( l - δ ) `intersect` i_l
δ <- fst ( ff' l ) ⊘ snd ( ff' i_l )
map fst $ ( l - δ ) ∩ i_l
in ( False, xls )
where x_lp = min ( succFP x_l ) x_mid
i_l = 𝕀 x_lp x_mid
( right_done, right_todo )
| 0 `inside` ( fst $ ff' ( 𝕀 x_rm x_r ) )
| 0 ∈ ( fst $ ff' ( 𝕀 x_rm x_r ) )
= ( True, [ ] )
| not $ 0 `inside` ( fst $ ff' i_r )
| not $ 0 ∈ ( fst $ ff' i_r )
= ( False, [ ] )
| otherwise
= let
xrs = do
let r = 𝕀 x_rm x_rm
δ <- fst ( ff' r ) `extendedDivide` snd ( ff' i_r )
map fst $ ( r - δ ) `intersect` i_r
δ <- fst ( ff' r ) ⊘ snd ( ff' i_r )
map fst $ ( r - δ ) ∩ i_r
in ( False, xrs )
where x_rm = max ( prevFP x_r ) x_mid
i_r = 𝕀 x_mid x_rm