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WIP: add Walter's LP approach to interval Newton
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@ -9,6 +9,9 @@ build-type: Simple
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description:
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Computing brush strokes using Bézier curves.
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extra-source-files:
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cbits/**/*.{cpp, hpp}
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flag use-fma
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description: Use fused-muliply add instructions to implement interval arithmetic.
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default: True
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@ -143,8 +146,11 @@ library
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, Math.Root.Isolation.Bisection
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, Math.Root.Isolation.Core
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, Math.Root.Isolation.Degree
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, Math.Root.Isolation.GaussSeidel
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, Math.Root.Isolation.Narrowing
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, Math.Root.Isolation.Newton
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, Math.Root.Isolation.Newton.GaussSeidel
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, Math.Root.Isolation.Newton.LP
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, Math.Root.Isolation.Utils
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, Debug.Utils
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other-modules:
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@ -186,6 +192,22 @@ library
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, transformers
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>= 0.5.6.2 && < 0.7
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-- Extra C++ code for 2D linear systems of interval equations
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include-dirs:
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cbits
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cxx-sources:
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cbits/lp_2d.cpp
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cxx-options:
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-std=c++20
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-mavx2 -mfma
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-frounding-math -fno-math-errno -fno-signed-zeros
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-fno-trapping-math -fno-signaling-nans
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-Wno-unused-result -Wsign-compare -Wno-switch
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-march=native -mtune=native
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-O3 -DNDEBUG
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build-depends:
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system-cxx-std-lib
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--executable convert-metafont
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--
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-- import:
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2035
brush-strokes/cbits/lp_2d.cpp
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2035
brush-strokes/cbits/lp_2d.cpp
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File diff suppressed because it is too large
Load diff
1265
brush-strokes/cbits/lp_2d.hpp
Normal file
1265
brush-strokes/cbits/lp_2d.hpp
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File diff suppressed because it is too large
Load diff
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@ -28,10 +28,12 @@ module Math.Root.Isolation
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, BisectionOptions(..), BisectionCoordPicker
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, defaultBisectionOptions
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-- * The interval Newton method with Gauss–Seidel step
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, GaussSeidel
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-- * The interval Newton method
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, NewtonOptions(..)
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, defaultNewtonOptions
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-- ** Options for the Gauss–Seidel step
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, GaussSeidelOptions(..), Preconditioner(..)
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, defaultGaussSeidelOptions
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, GaussSeidelUpdateMethod(..)
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-- * Box-consistency methods
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@ -79,8 +81,9 @@ import Math.Monomial
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import Math.Root.Isolation.Bisection
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import Math.Root.Isolation.Core
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import Math.Root.Isolation.GaussSeidel
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import Math.Root.Isolation.Narrowing
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import Math.Root.Isolation.Newton
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import Math.Root.Isolation.Newton.GaussSeidel
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--------------------------------------------------------------------------------
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@ -134,16 +137,16 @@ defaultRootIsolationAlgorithms minWidth ε_eq history box =
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-- Currently: we try an interval Gauss–Seidel.
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-- (box(1)- and box(2)-consistency don't seem to help when using
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-- the complete interval union Gauss–Seidel step)
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_ -> Right $ AlgoWithOptions @GaussSeidel _gsOptions
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_ -> Right $ AlgoWithOptions @Newton _newtonOptions
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NE.:| []
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where
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verySmall = and $ ( \ cd -> width cd <= minWidth ) <$> coordinates box
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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 * 100 ) ε_eq
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_box2Options = ( defaultBox2Options @n @d minWidth ε_eq ) { box2LambdaMin = 0.001 }
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_bisOptions = defaultBisectionOptions @n @d minWidth ε_eq box
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_newtonOptions = NewtonLP -- defaultNewtonOptions @n @d history
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_box1Options = defaultBox1Options @n @d ( minWidth * 100 ) ε_eq
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_box2Options = ( defaultBox2Options @n @d minWidth ε_eq ) { box2LambdaMin = 0.001 }
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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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@ -78,7 +78,8 @@ type Box n = 𝕀ℝ n
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-- NB: we require n <= d (no support for under-constrained systems).
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--
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-- NB: in practice, this constraint should specialise away.
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type BoxCt n d =
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type BoxCt n d = ( n ~ 2, d ~ 3 )
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{-
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( KnownNat n, KnownNat d
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, 1 <= n, 1 <= d, n <= d
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@ -95,7 +96,7 @@ type BoxCt n d =
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, Module Double ( T ( ℝ d ) )
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, Representable Double ( ℝ d )
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)
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-}
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-- | Boxes we are done with and will not continue processing.
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data DoneBoxes n =
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DoneBoxes
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171
brush-strokes/src/lib/Math/Root/Isolation/Newton.hs
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171
brush-strokes/src/lib/Math/Root/Isolation/Newton.hs
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@ -0,0 +1,171 @@
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{-# LANGUAGE ScopedTypeVariables #-}
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{-# LANGUAGE UndecidableInstances #-}
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module Math.Root.Isolation.Newton
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( -- * The interval Newton method,
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-- with Gauss–Seidel step or explicit linear programming
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Newton
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, intervalNewton
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-- ** Configuration options
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, NewtonOptions(..)
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, defaultNewtonOptions
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)
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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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( Bifunctor(bimap) )
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import Data.Kind
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( Type )
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import Data.List
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( partition )
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import GHC.TypeNats
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( Nat, KnownNat, type (<=) )
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-- transformers
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import Control.Monad.Trans.Writer.CPS
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( Writer, tell )
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-- MetaBrush
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import Math.Algebra.Dual
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( D )
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import Math.Interval
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import Math.Linear
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import Math.Module
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( Module(..) )
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import Math.Monomial
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( MonomialBasis(..), linearMonomial, zeroMonomial )
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import Math.Root.Isolation.Core
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import Math.Root.Isolation.Newton.GaussSeidel
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import Math.Root.Isolation.Newton.LP
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import Math.Root.Isolation.Utils
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--------------------------------------------------------------------------------
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-- Interval Newton
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-- | The interval Newton method; see 'intervalNewton'.
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data Newton
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instance BoxCt n d => RootIsolationAlgorithm Newton n d where
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type instance StepDescription Newton = ()
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type instance RootIsolationAlgorithmOptions Newton n d = NewtonOptions n d
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rootIsolationAlgorithm opts _thisRoundHist _prevRoundsHist eqs box = do
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res <- intervalNewton @n @d opts eqs box
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return ( (), res )
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{-# INLINEABLE rootIsolationAlgorithm #-}
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{-# SPECIALISE rootIsolationAlgorithm
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:: RootIsolationAlgorithmOptions Newton 2 3
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-> [ ( RootIsolationStep, Box 2 ) ]
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-> BoxHistory 2
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-> ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) )
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-> Box 2
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-> Writer ( DoneBoxes 2 ) ( StepDescription Newton, [ Box 2 ] ) #-}
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-- NB: including this to be safe. The specialiser seems to sometimes
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-- be able to generate this specialisation on its own, and sometimes not.
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-- | Options for the interval Newton method.
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type NewtonOptions :: Nat -> Nat -> Type
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data NewtonOptions n d where
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-- | Use the Gauss–Seidel method to solve linear systems.
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NewtonGaussSeidel
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:: GaussSeidelOptions n d -> NewtonOptions n d
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-- | Use linear programming to solve linear systems (2 dimensions only).
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NewtonLP
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:: NewtonOptions 2 d
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-- | Default options for the interval Newton method.
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defaultNewtonOptions
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:: forall n d
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. ( KnownNat n, KnownNat d
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, 1 <= n, 1 <= d, n <= d
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, Representable Double ( ℝ n )
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, Representable Double ( ℝ d )
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)
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=> BoxHistory n
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-> NewtonOptions n d
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defaultNewtonOptions history =
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NewtonGaussSeidel $ defaultGaussSeidelOptions history
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{-# INLINEABLE defaultNewtonOptions #-}
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-- | Interval Newton method with Gauss–Seidel step.
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intervalNewton
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:: forall n d
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. BoxCt n d
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=> NewtonOptions n d
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-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
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-- ^ equations
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-> 𝕀ℝ n
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-- ^ box
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-> Writer ( DoneBoxes n ) [ 𝕀ℝ n ]
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intervalNewton opts eqs x = case opts of
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NewtonGaussSeidel
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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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let x_mid = singleton $ boxMidpoint x
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f :: 𝕀ℝ n -> 𝕀ℝ n
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f = \ x_0 -> pickEqs $ eqs x_0 `monIndex` zeroMonomial
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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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-- 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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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 ( \ x' -> unT $ x' ^+^ T x_mid ) )
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$ gaussSeidelUpdate gsUpdate a b ( T x ^-^ T x_mid )
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( done, todo ) = bimap ( map fst ) ( map fst )
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$ partition snd gsGuesses
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in -- 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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do tell $ noDoneBoxes { doneSolBoxes = done }
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return todo
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NewtonLP ->
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-- TODO: reduce duplication with the above.
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let x_mid = singleton $ boxMidpoint x
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f :: 𝕀ℝ 2 -> 𝕀ℝ d
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f = \ x_0 -> eqs x_0 `monIndex` zeroMonomial
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f'_x :: Vec 2 ( 𝕀ℝ d )
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f'_x = fmap ( \ i -> eqs x `monIndex` linearMonomial i ) ( universe @2 )
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minus_f_x_mid = unT $ -1 *^ T ( boxMidpoint $ f x_mid )
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( a, b ) = ( f'_x, singleton minus_f_x_mid )
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lpGuesses = map ( first ( \ x' -> unT $ x' ^+^ T x_mid ) )
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$ solveIntervalLinearEquations a b ( T x ^-^ T x_mid )
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( done, todo ) = bimap ( map fst ) ( map fst )
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$ partition snd lpGuesses
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in do tell $ noDoneBoxes { doneSolBoxes = done }
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return todo
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{-# INLINEABLE intervalNewton #-}
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{-
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mbDeg = topologicalDegree 0.005 f x
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det = case f'_x of
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Vec [ c1, c2 ] ->
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let a_11 = c1 `index` Fin 1
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a_12 = c2 `index` Fin 1
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a_21 = c1 `index` Fin 2
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a_22 = c2 `index` Fin 2
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in a_11 * a_22 - a_12 * a_21
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_ -> error "TODO: just testing n=2 here"
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if | not $ 0 ∈ det
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, mbDeg == Just 0
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-> return []
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-- If the Jacobian is invertible over the box, then the topological
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-- degree tells us exactly how many solutions there are in the box.
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-}
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@ -1,33 +1,26 @@
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{-# LANGUAGE ScopedTypeVariables #-}
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{-# LANGUAGE UndecidableInstances #-}
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{-# LANGUAGE ScopedTypeVariables #-}
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module Math.Root.Isolation.GaussSeidel
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( -- * The interval Newton method with Gauss–Seidel step
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GaussSeidel
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, intervalGaussSeidel
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-- ** Configuration options
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, GaussSeidelOptions(..), Preconditioner(..)
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-- | The Gauss–Seidel method for solving systems of
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-- interval linear equations
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module Math.Root.Isolation.Newton.GaussSeidel
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( -- * Gauss–Seidel step
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gaussSeidelUpdate
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-- ** Options for the Gauss–Seidel method
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, GaussSeidelOptions(..)
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, Preconditioner(..), GaussSeidelUpdateMethod(..)
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, defaultGaussSeidelOptions
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, precondition
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)
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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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( Bifunctor(bimap) )
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import Data.Coerce
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( coerce )
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import Data.Kind
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( Type )
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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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( toList )
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import Data.Proxy
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( Proxy(..) )
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import Data.Type.Ord
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|
@ -41,44 +34,16 @@ import qualified Eigen.Matrix as Eigen
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, generate, inverse, unsafeCoeff
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)
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-- transformers
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import Control.Monad.Trans.Writer.CPS
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( Writer, tell )
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-- MetaBrush
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import Math.Algebra.Dual
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( D )
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import Math.Interval
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import Math.Linear
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import Math.Module
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( Module(..) )
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import Math.Monomial
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( MonomialBasis(..), linearMonomial, zeroMonomial )
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import Math.Root.Isolation.Core
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import Math.Root.Isolation.Utils
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--------------------------------------------------------------------------------
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-- Gauss–Seidel
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-- | The interval Newton method with a Gauss–Seidel step; see 'intervalGaussSeidel'.
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data GaussSeidel
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instance BoxCt n d => RootIsolationAlgorithm GaussSeidel n d where
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type instance StepDescription GaussSeidel = ()
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type instance RootIsolationAlgorithmOptions GaussSeidel n d = GaussSeidelOptions n d
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rootIsolationAlgorithm opts _thisRoundHist _prevRoundsHist eqs box = do
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res <- intervalGaussSeidel @n @d opts eqs box
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return ( (), res )
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{-# INLINEABLE rootIsolationAlgorithm #-}
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{-# SPECIALISE rootIsolationAlgorithm
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:: RootIsolationAlgorithmOptions GaussSeidel 2 3
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-> [ ( RootIsolationStep, Box 2 ) ]
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-> BoxHistory 2
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-> ( 𝕀ℝ 2 -> D 1 ( 𝕀ℝ 2 ) ( 𝕀ℝ 3 ) )
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-> Box 2
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-> Writer ( DoneBoxes 2 ) ( StepDescription GaussSeidel, [ Box 2 ] ) #-}
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-- NB: including this to be safe. The specialiser seems to sometimes
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-- be able to generate this specialisation on its own, and sometimes not.
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-- | Options for the interval Gauss–Seidel method.
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-- | Options for the Gauss–Seidel method.
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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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|
@ -98,7 +63,7 @@ data GaussSeidelUpdateMethod
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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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-- | Default options for the Gauss–Seidel step.
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defaultGaussSeidelOptions
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:: forall n d
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. ( KnownNat n, KnownNat d
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|
@ -133,72 +98,6 @@ data Preconditioner
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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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intervalGaussSeidel
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:: forall n d
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. BoxCt n d
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=> GaussSeidelOptions n d
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-> ( 𝕀ℝ n -> D 1 ( 𝕀ℝ n ) ( 𝕀ℝ d ) )
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-- ^ equations
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-> 𝕀ℝ n
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-- ^ box
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-> Writer ( DoneBoxes n ) [ 𝕀ℝ n ]
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intervalGaussSeidel
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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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x
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| let x_mid = singleton $ boxMidpoint x
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f :: 𝕀ℝ n -> 𝕀ℝ n
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f = \ x_0 -> pickEqs $ eqs x_0 `monIndex` zeroMonomial
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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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= 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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|
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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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f'_x ( singleton minus_f_x_mid )
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|
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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 ( \ x' -> unT $ x' ^+^ T x_mid ) )
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$ gaussSeidelUpdate gsUpdate a b ( T x ^-^ T x_mid )
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( done, todo ) = bimap ( map fst ) ( map fst )
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$ partition snd gsGuesses
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in -- 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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do tell $ noDoneBoxes { doneSolBoxes = done }
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return todo
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{-# INLINEABLE intervalGaussSeidel #-}
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{-
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mbDeg = topologicalDegree 0.005 f x
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det = case f'_x of
|
||||
Vec [ c1, c2 ] ->
|
||||
let a_11 = c1 `index` Fin 1
|
||||
a_12 = c2 `index` Fin 1
|
||||
a_21 = c1 `index` Fin 2
|
||||
a_22 = c2 `index` Fin 2
|
||||
in a_11 * a_22 - a_12 * a_21
|
||||
_ -> error "TODO: just testing n=2 here"
|
||||
|
||||
if | not $ 0 ∈ det
|
||||
, mbDeg == Just 0
|
||||
-> return []
|
||||
-- If the Jacobian is invertible over the box, then the topological
|
||||
-- degree tells us exactly how many solutions there are in the box.
|
||||
-}
|
||||
|
||||
-- | A partial or complete Gauss–Seidel step for the equation \( A X = B \),
|
||||
-- refining the initial guess box for \( X \) into up to \( 2^n \) (disjoint) new boxes.
|
||||
gaussSeidelUpdate
|
||||
|
@ -291,22 +190,6 @@ gaussSeidelStep_Complete as b ( T x0 ) = coerce $ do
|
|||
return ( x', and subs )
|
||||
{-# INLINEABLE gaussSeidelStep_Complete #-}
|
||||
|
||||
fromComponents
|
||||
:: forall n
|
||||
. ( Representable Double ( ℝ n ), n ~ RepDim ( ℝ n ) )
|
||||
=> ( Fin n -> [ ( 𝕀 Double, Bool ) ] ) -> [ ( 𝕀ℝ n, Vec n Bool ) ]
|
||||
fromComponents f = do
|
||||
( xs, bs ) <- unzip <$> traverse f ( universe @n )
|
||||
return $ ( tabulate $ \ i -> xs ! i, bs )
|
||||
-- TODO: this could be more efficient.
|
||||
{-# INLINEABLE fromComponents #-}
|
||||
|
||||
-- | The midpoint of a box.
|
||||
boxMidpoint :: Representable Double ( ℝ n ) => 𝕀ℝ n -> ℝ n
|
||||
boxMidpoint box =
|
||||
tabulate $ \ i -> midpoint ( box `index` i )
|
||||
{-# INLINEABLE boxMidpoint #-}
|
||||
|
||||
-- | Pre-condition the system \( AX = B \).
|
||||
precondition
|
||||
:: forall n
|
||||
|
@ -340,16 +223,3 @@ precondition meth as b =
|
|||
)
|
||||
( universe @n )
|
||||
{-# INLINEABLE precondition #-}
|
||||
|
||||
-- | Matrix multiplication \( A v \).
|
||||
matMulVec
|
||||
:: forall n m
|
||||
. ( Representable Double ( ℝ n ), Representable Double ( ℝ m ) )
|
||||
=> Vec m ( ℝ n ) -- ^ columns of the matrix \( A )
|
||||
-> 𝕀ℝ m -- ^ vector \( v \)
|
||||
-> 𝕀ℝ n
|
||||
matMulVec as v = tabulate $ \ r ->
|
||||
sum [ scaleInterval ( a `index` r ) ( index v c )
|
||||
| ( c, a ) <- toList ( (,) <$> universe @m <*> as )
|
||||
]
|
||||
{-# INLINEABLE matMulVec #-}
|
132
brush-strokes/src/lib/Math/Root/Isolation/Newton/LP.hs
Normal file
132
brush-strokes/src/lib/Math/Root/Isolation/Newton/LP.hs
Normal file
|
@ -0,0 +1,132 @@
|
|||
{-# LANGUAGE CApiFFI #-}
|
||||
{-# LANGUAGE ForeignFunctionInterface #-}
|
||||
{-# LANGUAGE ParallelListComp #-}
|
||||
{-# LANGUAGE ScopedTypeVariables #-}
|
||||
|
||||
-- | A linear programming approach for solving systems of
|
||||
-- interval linear equations.
|
||||
module Math.Root.Isolation.Newton.LP
|
||||
( solveIntervalLinearEquations )
|
||||
where
|
||||
|
||||
-- base
|
||||
import Control.Arrow
|
||||
( (&&&) )
|
||||
import Data.Coerce
|
||||
( coerce )
|
||||
import Data.Foldable
|
||||
( toList )
|
||||
import Foreign.C.Types
|
||||
( CDouble(..), CInt(..), CUInt(..) )
|
||||
import Foreign.Marshal
|
||||
( allocaArray, peekArray, pokeArray, with, withArray )
|
||||
import Foreign.Ptr
|
||||
( Ptr, castPtr )
|
||||
import Foreign.Storable
|
||||
( Storable(..) )
|
||||
import GHC.TypeNats
|
||||
( KnownNat, natVal' )
|
||||
import GHC.Exts
|
||||
( proxy# )
|
||||
import System.IO.Unsafe
|
||||
( unsafePerformIO )
|
||||
|
||||
-- MetaBrush
|
||||
import Math.Interval
|
||||
import Math.Linear
|
||||
|
||||
--------------------------------------------------------------------------------
|
||||
-- Linear programming approach to solving systems of linear equations
|
||||
-- (in two variables).
|
||||
|
||||
-- | Solve the system of linear equations \( A X = B \)
|
||||
-- using linear programming.
|
||||
solveIntervalLinearEquations
|
||||
:: ( KnownNat d, Representable Double ( ℝ d ), Show ( ℝ d ) )
|
||||
=> Vec 2 ( 𝕀ℝ d ) -- ^ columns of \( A \)
|
||||
-> 𝕀ℝ d -- ^ \( B \)
|
||||
-> T ( 𝕀ℝ 2 ) -- ^ initial box \( X \)
|
||||
-> [ ( T ( 𝕀ℝ 2 ), Bool ) ]
|
||||
solveIntervalLinearEquations a b x =
|
||||
let !sols = unsafePerformIO $ intervalSystem2D_LP a b x
|
||||
in if
|
||||
| any hasNaNs sols
|
||||
-> [ ( x, False ) ]
|
||||
| length sols <= 1
|
||||
-> map ( id &&& isSubBox x ) sols
|
||||
| otherwise
|
||||
-> map ( , False ) sols
|
||||
|
||||
-- Assuming the second box is a subset of the second box, returns whether it
|
||||
-- is in fact a strict subset.
|
||||
isSubBox :: T ( 𝕀ℝ 2 ) -> T ( 𝕀ℝ 2 ) -> Bool
|
||||
isSubBox
|
||||
( T ( 𝕀 ( ℝ2 x1_min y1_min ) ( ℝ2 x1_max y1_max ) ) )
|
||||
( T ( 𝕀 ( ℝ2 x2_min y2_min ) ( ℝ2 x2_max y2_max ) ) )
|
||||
= ( ( x2_min > x1_min && x2_max < x1_max ) || x2_min == x2_max )
|
||||
&& ( ( y2_min > y1_min && y2_max < y1_max ) || y2_min == y2_max )
|
||||
|
||||
hasNaNs :: T (𝕀ℝ 2) -> Bool
|
||||
hasNaNs ( T ( 𝕀 ( ℝ2 x_min y_min ) ( ℝ2 x_max y_max ) ) ) =
|
||||
any ( \ x -> isNaN x || isInfinite x ) [ x_min, y_min, x_max, y_max ]
|
||||
|
||||
intervalSystem2D_LP
|
||||
:: forall d
|
||||
. ( KnownNat d, Representable Double ( ℝ d ) )
|
||||
=> Vec 2 ( 𝕀ℝ d )
|
||||
-> 𝕀ℝ d
|
||||
-> T ( 𝕀ℝ 2 )
|
||||
-> IO [ T ( 𝕀ℝ 2 ) ]
|
||||
intervalSystem2D_LP a b x =
|
||||
allocaArray 4 \ ptrSolutions ->
|
||||
with ( CBox $ unT $ x ) \ ptrBox ->
|
||||
withArray ( mkEquationArray a b ) \ ptrEqs -> do
|
||||
CInt nbSols <-
|
||||
interval_system_2d ptrSolutions ptrBox ptrEqs ( fromIntegral d )
|
||||
if nbSols < 0
|
||||
then
|
||||
error $ unlines
|
||||
[ "interval_system_2d returned with exit code " ++ show nbSols
|
||||
, "This probably means it was given invalid input." ]
|
||||
else
|
||||
coerce <$> peekArray ( fromIntegral nbSols ) ptrSolutions
|
||||
where
|
||||
d = natVal' @d proxy#
|
||||
|
||||
mkEquationArray
|
||||
:: ( KnownNat d, Representable Double ( ℝ d ) )
|
||||
=> Vec 2 ( 𝕀ℝ d ) -> 𝕀ℝ d -> [ CEqn ]
|
||||
mkEquationArray ( Vec [ a_x, a_y ] ) b =
|
||||
[ CEqn a_x_i a_y_i b_i
|
||||
| a_x_i <- toList $ coordinates a_x
|
||||
| a_y_i <- toList $ coordinates a_y
|
||||
| b_i <- toList $ coordinates b
|
||||
]
|
||||
mkEquationArray _ _ = error "impossible"
|
||||
|
||||
foreign import ccall "interval_system_2d"
|
||||
interval_system_2d :: Ptr CBox -> Ptr CBox -> Ptr CEqn -> CUInt -> IO CInt
|
||||
|
||||
data CEqn = CEqn !( 𝕀 Double ) !( 𝕀 Double ) !( 𝕀 Double )
|
||||
instance Storable CEqn where
|
||||
sizeOf _ = 6 * sizeOf @Double undefined
|
||||
alignment _ = 4 * alignment @Double undefined
|
||||
peek ptr = do
|
||||
[ CDouble a_min, CDouble a_max, CDouble b_min, CDouble b_max, CDouble c_min, CDouble c_max ]
|
||||
<- peekArray 6 ( castPtr ptr :: Ptr CDouble )
|
||||
return $
|
||||
CEqn ( 𝕀 a_min a_max ) ( 𝕀 b_min b_max ) ( 𝕀 c_min c_max )
|
||||
poke ptr ( CEqn ( 𝕀 a_min a_max ) ( 𝕀 b_min b_max ) ( 𝕀 c_min c_max ) )
|
||||
= pokeArray ( castPtr ptr ) [ CDouble a_min, CDouble a_max, CDouble b_min, CDouble b_max, CDouble c_min, CDouble c_max ]
|
||||
|
||||
|
||||
newtype CBox = CBox ( 𝕀ℝ 2 )
|
||||
instance Storable CBox where
|
||||
sizeOf _ = 4 * sizeOf @Double undefined
|
||||
alignment _ = 4 * alignment @Double undefined
|
||||
peek ptr = do
|
||||
[ CDouble x_min, CDouble x_max, CDouble y_min, CDouble y_max ] <- peekArray 4 ( castPtr ptr :: Ptr CDouble )
|
||||
return $
|
||||
CBox ( 𝕀 ( ℝ2 x_min y_min ) ( ℝ2 x_max y_max ) )
|
||||
poke ptr ( CBox ( 𝕀 ( ℝ2 x_min y_min ) ( ℝ2 x_max y_max ) ) ) =
|
||||
pokeArray ( castPtr ptr ) [ CDouble x_min, CDouble x_max, CDouble y_min, CDouble y_max ]
|
48
brush-strokes/src/lib/Math/Root/Isolation/Utils.hs
Normal file
48
brush-strokes/src/lib/Math/Root/Isolation/Utils.hs
Normal file
|
@ -0,0 +1,48 @@
|
|||
{-# LANGUAGE ScopedTypeVariables #-}
|
||||
|
||||
-- | Utilities for root isolation
|
||||
module Math.Root.Isolation.Utils
|
||||
( fromComponents, matMulVec, boxMidpoint )
|
||||
where
|
||||
|
||||
-- base
|
||||
import Prelude hiding ( unzip )
|
||||
import Data.Foldable
|
||||
( toList )
|
||||
import Data.List.NonEmpty
|
||||
( unzip )
|
||||
|
||||
-- MetaBrush
|
||||
import Math.Interval
|
||||
import Math.Linear
|
||||
|
||||
--------------------------------------------------------------------------------
|
||||
|
||||
fromComponents
|
||||
:: forall n
|
||||
. ( Representable Double ( ℝ n ), n ~ RepDim ( ℝ n ) )
|
||||
=> ( Fin n -> [ ( 𝕀 Double, Bool ) ] ) -> [ ( 𝕀ℝ n, Vec n Bool ) ]
|
||||
fromComponents f = do
|
||||
( xs, bs ) <- unzip <$> traverse f ( universe @n )
|
||||
return $ ( tabulate $ \ i -> xs ! i, bs )
|
||||
-- TODO: this could be more efficient.
|
||||
{-# INLINEABLE fromComponents #-}
|
||||
|
||||
-- | The midpoint of a box.
|
||||
boxMidpoint :: Representable Double ( ℝ n ) => 𝕀ℝ n -> ℝ n
|
||||
boxMidpoint box =
|
||||
tabulate $ \ i -> midpoint ( box `index` i )
|
||||
{-# INLINEABLE boxMidpoint #-}
|
||||
|
||||
-- | Matrix multiplication \( A v \).
|
||||
matMulVec
|
||||
:: forall n m
|
||||
. ( Representable Double ( ℝ n ), Representable Double ( ℝ m ) )
|
||||
=> Vec m ( ℝ n ) -- ^ columns of the matrix \( A )
|
||||
-> 𝕀ℝ m -- ^ vector \( v \)
|
||||
-> 𝕀ℝ n
|
||||
matMulVec as v = tabulate $ \ r ->
|
||||
sum [ scaleInterval ( a `index` r ) ( index v c )
|
||||
| ( c, a ) <- toList ( (,) <$> universe @m <*> as )
|
||||
]
|
||||
{-# INLINEABLE matMulVec #-}
|
Loading…
Reference in a new issue