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Add some commentary to Gauss-Seidel function

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sheaf 2024-04-24 01:30:24 +02:00
parent a59f1695fb
commit 32ce7c38bb
2 changed files with 8 additions and 2 deletions
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2
brush-strokes/src/lib/Math/Interval.hs
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@ -115,7 +115,7 @@ bisect x@( 𝕀 x_inf x_sup )
where x_mid = midpoint x where x_mid = midpoint x
infixl 6 ⊖ infixl 6 ⊖
(⊖) :: ( Ring a, Ord a ) => 𝕀 a -> 𝕀 a -> 𝕀 a (⊖) :: ( AbelianGroup a, Ord a ) => 𝕀 a -> 𝕀 a -> 𝕀 a
(⊖) a@( 𝕀 lo1 hi1 ) b@( 𝕀 lo2 hi2 ) (⊖) a@( 𝕀 lo1 hi1 ) b@( 𝕀 lo2 hi2 )
| width a >= width b | width a >= width b
= 𝕀 ( lo1 - lo2 ) ( hi1 - hi2 ) = 𝕀 ( lo1 - lo2 ) ( hi1 - hi2 )

8
brush-strokes/src/lib/Math/Root/Isolation/GaussSeidel.hs
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@ -202,7 +202,13 @@ gaussSeidelStep
-> [ ( T ( 𝕀ℝ n ), Bool ) ] -> [ ( T ( 𝕀ℝ n ), Bool ) ]
gaussSeidelStep as b ( T x0 ) = coerce $ gaussSeidelStep as b ( T x0 ) = coerce $
forEachCoord @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 -- For each i, we have an equation: sum_j a_ij * x_j = b_i
--
-- Re-arrange this with x_i on the left to get an iteration:
-- x_i' = ( b_i - sum { j /= i } a_ij * x_j ) / a_ii
--
-- We perform each iteration in turn (for i = 1, ..., n),
-- **using the latest updated value of each x_j** in each iteration.
let s = b `index` i - sum [ ( as ! j ) `index` i * x `index` j let s = b `index` i - sum [ ( as ! j ) `index` i * x `index` j
| j <- toList ( universe @n ), j /= i ] | j <- toList ( universe @n ), j /= i ]
x_i = x `index` i x_i = x `index` i