diff --git a/LICENSE b/LICENSE
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--- /dev/null
+++ b/LICENSE
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+ GNU GENERAL PUBLIC LICENSE
+ Version 3, 29 June 2007
+
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+Program, unless a warranty or assumption of liability accompanies a
+copy of the Program in return for a fee.
+
+ END OF TERMS AND CONDITIONS
+
+ How to Apply These Terms to Your New Programs
+
+ If you develop a new program, and you want it to be of the greatest
+possible use to the public, the best way to achieve this is to make it
+free software which everyone can redistribute and change under these terms.
+
+ To do so, attach the following notices to the program. It is safest
+to attach them to the start of each source file to most effectively
+state the exclusion of warranty; and each file should have at least
+the "copyright" line and a pointer to where the full notice is found.
+
+
+ Copyright (C)
+
+ This program is free software: you can redistribute it and/or modify
+ it under the terms of the GNU General Public License as published by
+ the Free Software Foundation, either version 3 of the License, or
+ (at your option) any later version.
+
+ This program is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ GNU General Public License for more details.
+
+ You should have received a copy of the GNU General Public License
+ along with this program. If not, see .
+
+Also add information on how to contact you by electronic and paper mail.
+
+ If the program does terminal interaction, make it output a short
+notice like this when it starts in an interactive mode:
+
+ Copyright (C)
+ This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
+ This is free software, and you are welcome to redistribute it
+ under certain conditions; type `show c' for details.
+
+The hypothetical commands `show w' and `show c' should show the appropriate
+parts of the General Public License. Of course, your program's commands
+might be different; for a GUI interface, you would use an "about box".
+
+ You should also get your employer (if you work as a programmer) or school,
+if any, to sign a "copyright disclaimer" for the program, if necessary.
+For more information on this, and how to apply and follow the GNU GPL, see
+.
+
+ The GNU General Public License does not permit incorporating your program
+into proprietary programs. If your program is a subroutine library, you
+may consider it more useful to permit linking proprietary applications with
+the library. If this is what you want to do, use the GNU Lesser General
+Public License instead of this License. But first, please read
+.
diff --git a/Math/LinearMap/Asserted.hs b/Math/LinearMap/Asserted.hs
new file mode 100644
--- /dev/null
+++ b/Math/LinearMap/Asserted.hs
@@ -0,0 +1,147 @@
+-- |
+-- Module : Math.LinearMap.Asserted
+-- Copyright : (c) Justus Sagemüller 2016
+-- License : GPL v3
+--
+-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
+-- Stability : experimental
+-- Portability : portable
+--
+{-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE ConstraintKinds #-}
+{-# LANGUAGE UndecidableInstances #-}
+{-# LANGUAGE FunctionalDependencies #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE Rank2Types #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE PatternSynonyms #-}
+{-# LANGUAGE TupleSections #-}
+{-# LANGUAGE UnicodeSyntax #-}
+{-# LANGUAGE CPP #-}
+{-# LANGUAGE TupleSections #-}
+{-# LANGUAGE StandaloneDeriving #-}
+
+module Math.LinearMap.Asserted where
+
+import Data.VectorSpace
+import Data.Basis
+
+import Prelude ()
+import qualified Prelude as Hask
+
+import Control.Category.Constrained.Prelude
+import Control.Arrow.Constrained
+import Data.Traversable.Constrained
+
+import Data.Coerce
+import Data.Type.Coercion
+
+import Data.VectorSpace.Free
+import qualified Linear.Matrix as Mat
+import qualified Linear.Vector as Mat
+import Math.VectorSpace.ZeroDimensional
+
+
+
+
+-- | A linear map, represented simply as a Haskell function tagged with
+-- the type of scalar with respect to which it is linear. Many (sparse)
+-- linear mappings can actually be calculated much more efficiently
+-- if you don't represent them with any kind of matrix, but
+-- just as a function (which is after all, mathematically speaking,
+-- what a linear map foremostly is).
+--
+-- However, if you sum up many 'LinearFunction's – which you can
+-- simply do with the 'VectorSpace' instance – they will become ever
+-- slower to calculate, because the summand-functions are actually computed
+-- individually and only the results summed. That's where
+-- 'Math.LinearMap.Category.LinearMap' is generally preferrable.
+-- You can always convert between these equivalent categories using 'arr'.
+newtype LinearFunction s v w = LinearFunction { getLinearFunction :: v -> w }
+
+
+
+
+linearFunction :: VectorSpace w => (v->w) -> LinearFunction (Scalar v) v w
+linearFunction = LinearFunction
+
+scaleWith :: (VectorSpace v, Scalar v ~ s) => s -> LinearFunction s v v
+scaleWith μ = LinearFunction (μ*^)
+
+scaleV :: (VectorSpace v, Scalar v ~ s) => v -> LinearFunction s s v
+scaleV v = LinearFunction (*^v)
+
+const0 :: AdditiveGroup w => LinearFunction s v w
+const0 = LinearFunction (const zeroV)
+
+lNegateV :: AdditiveGroup w => LinearFunction s w w
+lNegateV = LinearFunction negateV
+
+addV :: AdditiveGroup w => LinearFunction s (w,w) w
+addV = LinearFunction $ uncurry (^+^)
+
+instance AdditiveGroup w => AdditiveGroup (LinearFunction s v w) where
+ zeroV = const0
+ LinearFunction f ^+^ LinearFunction g = LinearFunction $ \x -> f x ^+^ g x
+ LinearFunction f ^-^ LinearFunction g = LinearFunction $ \x -> f x ^-^ g x
+ negateV (LinearFunction f) = LinearFunction $ negateV . f
+instance VectorSpace w => VectorSpace (LinearFunction s v w) where
+ type Scalar (LinearFunction s v w) = Scalar w
+ μ *^ LinearFunction f = LinearFunction $ (μ*^) . f
+
+instance Functor (LinearFunction s v) Coercion Coercion where
+ fmap Coercion = Coercion
+
+fmapScale :: ( VectorSpace w, Scalar w ~ s, VectorSpace s, Scalar s ~ s
+ , Functor f (LinearFunction s) (LinearFunction s)
+ , Object (LinearFunction s) s
+ , Object (LinearFunction s) w
+ , Object (LinearFunction s) (f s)
+ , Object (LinearFunction s) (f w)
+ , EnhancedCat (->) (LinearFunction s)
+ , VectorSpace (f w), Scalar (f w) ~ s
+ , VectorSpace (f s), Scalar (f s) ~ s )
+ => f s -> LinearFunction s w (f w)
+fmapScale v = LinearFunction $ \w -> fmap (scaleV w) $ v
+
+lCoFst :: (AdditiveGroup w) => LinearFunction s v (v,w)
+lCoFst = LinearFunction (,zeroV)
+lCoSnd :: (AdditiveGroup v) => LinearFunction s w (v,w)
+lCoSnd = LinearFunction (zeroV,)
+
+
+
+-- | Infix synonym of 'LinearFunction', without explicit mention of the scalar type.
+type v-+>w = LinearFunction (Scalar w) v w
+
+-- | A bilinear function is a linear function mapping to a linear function,
+-- or equivalently a 2-argument function that's linear in each argument
+-- independently.
+-- Note that this can /not/ be uncurried to a linear function with a tuple
+-- argument (this would not be linear but quadratic).
+type Bilinear v w y = LinearFunction (Scalar v) v (LinearFunction (Scalar v) w y)
+
+bilinearFunction :: (v -> w -> y) -> Bilinear v w y
+bilinearFunction f = LinearFunction $ LinearFunction . f
+
+flipBilin :: Bilinear v w y -> Bilinear w v y
+flipBilin (LinearFunction f) = LinearFunction
+ $ \w -> LinearFunction $ f >>> \(LinearFunction g) -> g w
+
+scale :: VectorSpace v => Bilinear (Scalar v) v v
+scale = LinearFunction $ \μ -> LinearFunction (μ*^)
+
+-- | @elacs ≡ 'flipBilin' 'scale'@.
+elacs :: VectorSpace v => Bilinear v (Scalar v) v
+elacs = LinearFunction $ \v -> LinearFunction (*^v)
+
+inner :: InnerSpace v => Bilinear v v (Scalar v)
+inner = LinearFunction $ \v -> LinearFunction (v<.>)
+
+biConst0 :: AdditiveGroup v => Bilinear a b v
+biConst0 = LinearFunction $ const const0
+
+lApply :: Bilinear (v-+>w) v w
+lApply = bilinearFunction $ \(LinearFunction f) v -> f v
diff --git a/Math/LinearMap/Category.hs b/Math/LinearMap/Category.hs
new file mode 100644
--- /dev/null
+++ b/Math/LinearMap/Category.hs
@@ -0,0 +1,590 @@
+-- |
+-- Module : Math.LinearMap.Category
+-- Copyright : (c) Justus Sagemüller 2016
+-- License : GPL v3
+--
+-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
+-- Stability : experimental
+-- Portability : portable
+--
+
+
+{-# LANGUAGE CPP #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE StandaloneDeriving #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE UndecidableInstances #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE UnicodeSyntax #-}
+{-# LANGUAGE TupleSections #-}
+{-# LANGUAGE ConstraintKinds #-}
+
+module Math.LinearMap.Category (
+ -- * Linear maps
+ -- $linmapIntro
+
+ -- ** Function implementation
+ LinearFunction (..), (-+>)(), Bilinear
+ -- ** Tensor implementation
+ , LinearMap (..), (+>)()
+ , (⊕), (>+<)
+ , adjoint
+ -- ** Dual vectors
+ -- $dualVectorIntro
+ , (<.>^)
+ -- * Tensor spaces
+ , Tensor (..), (⊗)(), (⊗)
+ -- * Norms
+ -- $metricIntro
+ , Norm(..), Seminorm
+ , spanNorm
+ , euclideanNorm
+ , (|$|)
+ , normSq
+ , (<$|)
+ , scaleNorm
+ , normSpanningSystem
+ , normSpanningSystem'
+ -- ** Variances
+ , Variance, spanVariance, dualNorm
+ , dependence
+ -- ** Utility
+ , densifyNorm
+ -- * Solving linear equations
+ , (\$), pseudoInverse, roughDet
+ -- * Eigenvalue problems
+ , eigen
+ , constructEigenSystem
+ , roughEigenSystem
+ , finishEigenSystem
+ , Eigenvector(..)
+ -- * The classes of suitable vector spaces
+ , LSpace
+ , TensorSpace (..)
+ , LinearSpace (..)
+ -- ** Orthonormal systems
+ , SemiInner (..), cartesianDualBasisCandidates
+ -- ** Finite baseis
+ , FiniteDimensional (..)
+ -- * Utility
+ -- ** Linear primitives
+ , addV, scale, inner, flipBilin, bilinearFunction
+ -- ** Hilbert space operations
+ , DualSpace, riesz, coRiesz, showsPrecAsRiesz, (.<)
+ -- ** Constraint synonyms
+ , HilbertSpace, SimpleSpace
+ , Num', Num'', Num'''
+ , Fractional', Fractional''
+ , RealFrac', RealFloat'
+ -- ** Misc
+ , relaxNorm, transformNorm, transformVariance
+ , findNormalLength, normalLength
+ , summandSpaceNorms, sumSubspaceNorms, sharedNormSpanningSystem
+ ) where
+
+import Math.LinearMap.Category.Class
+import Math.LinearMap.Category.Instances
+import Math.LinearMap.Asserted
+import Math.VectorSpace.Docile
+
+import Data.Tree (Tree(..), Forest)
+import Data.List (sortBy, foldl')
+import qualified Data.Set as Set
+import Data.Set (Set)
+import Data.Ord (comparing)
+import Data.List (maximumBy)
+import Data.Foldable (toList)
+import Data.Semigroup
+
+import Data.VectorSpace
+import Data.Basis
+
+import Prelude ()
+import qualified Prelude as Hask
+
+import Control.Category.Constrained.Prelude hiding ((^))
+import Control.Arrow.Constrained
+
+import Linear ( V0(V0), V1(V1), V2(V2), V3(V3), V4(V4)
+ , _x, _y, _z, _w )
+import Data.VectorSpace.Free
+import Math.VectorSpace.ZeroDimensional
+import qualified Linear.Matrix as Mat
+import qualified Linear.Vector as Mat
+import Control.Lens ((^.))
+
+import Numeric.IEEE
+
+-- $linmapIntro
+-- This library deals with linear functions, i.e. functions @f :: v -> w@
+-- that fulfill
+--
+-- @
+-- f $ μ 'Data.VectorSpace.^*' u 'Data.AdditiveGroup.^+^' v ≡ μ ^* f u ^+^ f v ∀ u,v :: v; μ :: 'Scalar' v
+-- @
+--
+-- Such functions form a cartesian monoidal category (in maths called
+-- ).
+-- This is implemented by 'Control.Arrow.Constrained.PreArrow', which is the
+-- preferred interface for dealing with these mappings. The basic
+-- “matrix operations” are then:
+--
+-- * Identity matrix: 'Control.Category.Constrained.id'
+-- * Matrix addition: 'Data.AdditiveGroup.^+^' (linear maps form an ordinary vector space)
+-- * Matrix-matrix multiplication: 'Control.Category.Constrained.<<<'
+-- (or '>>>' or 'Control.Category.Constrained..')
+-- * Matrix-vector multiplication: 'Control.Arrow.Constrained.$'
+-- * Vertical matrix concatenation: 'Control.Arrow.Constrained.&&&'
+-- * Horizontal matrix concatenation: '⊕' (aka '>+<')
+--
+-- But linear mappings need not necessarily be implemented as matrices:
+
+
+-- $dualVectorIntro
+-- A @'DualVector' v@ is a linear functional or
+-- on the vector space @v@,
+-- i.e. it is a linear function from the vector space into its scalar field.
+-- However, these functions form themselves a vector space, known as the dual space.
+-- In particular, the dual space of any 'InnerSpace' is isomorphic to the
+-- space itself.
+--
+-- (More precisely: the continuous dual space of a
+-- is isomorphic to
+-- that Hilbert space itself; see the 'riesz' isomorphism.)
+--
+-- As a matter of fact, in many applications, no distinction is made between a
+-- space and its dual. Indeed, we have for the basic 'LinearSpace' instances
+-- @'DualVector' v ~ v@, and '<.>^' is simply defined as a scalar product.
+-- In this case, a general 'LinearMap' is just a tensor product / matrix.
+--
+-- However, scalar products are often not as natural as they are made to look:
+--
+-- * A scalar product is only preserved under orthogonal transformations.
+-- It is not preserved under scalings, and certainly not under general linear
+-- transformations. This is very important in applications such as relativity
+-- theory (here, people talk about /covariant/ vs /contravariant/ tensors),
+-- but also relevant for more mundane
+-- like /sphere surfaces/:
+-- on such a surface, the natural symmetry transformations do generally
+-- not preserve a scalar product you might define.
+--
+-- * There may be more than one meaningful scalar product. For instance,
+-- the of weakly
+-- differentiable functions also permits the
+-- scalar product
+-- – each has different and useful properties.
+--
+-- Neither of this is a problem if we keep the dual space a separate type.
+-- Effectively, this enables the type system to prevent you from writing code that
+-- does not behave natural (i.e. that depends on a concrete choice of basis / scalar
+-- product).
+--
+-- For cases when you do have some given notion of orientation/scale in a vector space
+-- and need it for an algorithm, you can always provide a 'Norm', which is essentially
+-- a reified scalar product.
+--
+-- Note that @DualVector (DualVector v) ~ v@ in any 'LSpace': the /double-dual/
+-- space is /naturally/ isomorphic to the original space, by way of
+--
+-- @
+-- v '<.>^' dv ≡ dv '<.>^' v
+-- @
+
+
+
+
+
+-- | For real matrices, this boils down to 'transpose'.
+-- For free complex spaces it also incurs complex conjugation.
+--
+-- The signature can also be understood as
+--
+-- @
+-- adjoint :: (v +> w) -> (DualVector w +> DualVector v)
+-- @
+--
+-- Or
+--
+-- @
+-- adjoint :: (DualVector v +> DualVector w) -> (w +> v)
+-- @
+--
+-- But /not/ @(v+>w) -> (w+>v)@, in general (though in a Hilbert space, this too is
+-- equivalent, via 'riesz' isomorphism).
+adjoint :: (LSpace v, LSpace w, Scalar v ~ Scalar w)
+ => (v +> DualVector w) -+> (w +> DualVector v)
+adjoint = arr fromTensor . transposeTensor . arr asTensor
+
+
+
+
+-- $metricIntro
+-- A norm is a way to quantify the magnitude/length of different vectors,
+-- even if they point in different directions.
+--
+-- In an 'InnerSpace', a norm is always given by the scalar product,
+-- but there are spaces without a canonical scalar product (or situations
+-- in which this scalar product does not give the metric you want). Hence,
+-- we let the functions like 'constructEigenSystem', which depend on a norm
+-- for orthonormalisation, accept a 'Norm' as an extra argument instead of
+-- requiring 'InnerSpace'.
+
+-- | A seminorm defined by
+--
+-- @
+-- ‖v‖ = √(∑ᵢ ⟨dᵢ|v⟩²)
+-- @
+--
+-- for some dual vectors @dᵢ@. If given a complete basis of the dual space,
+-- this generates a proper 'Norm'.
+--
+-- If the @dᵢ@ are a complete orthonormal system, you get the 'euclideanNorm'
+-- (in an inefficient form).
+spanNorm :: LSpace v => [DualVector v] -> Seminorm v
+spanNorm dvs = Norm . LinearFunction $ \v -> sumV [dv ^* (dv<.>^v) | dv <- dvs]
+
+spanVariance :: LSpace v => [v] -> Variance v
+spanVariance = spanNorm
+
+-- | Modify a norm in such a way that the given vectors lie within its unit ball.
+-- (Not /optimally/ – the unit ball may be bigger than necessary.)
+relaxNorm :: SimpleSpace v => Norm v -> [v] -> Norm v
+relaxNorm me = \vs -> dualNorm . spanVariance $ vs' ++ vs
+ where vs' = normSpanningSystem' me
+
+-- | Scale the result of a norm with the absolute of the given number.
+--
+-- @
+-- scaleNorm μ n |$| v = abs μ * (n|$|v)
+-- @
+--
+-- Equivalently, this scales the norm's unit ball by the reciprocal of that factor.
+scaleNorm :: LSpace v => Scalar v -> Norm v -> Norm v
+scaleNorm μ (Norm n) = Norm $ μ^2 *^ n
+
+-- | A positive (semi)definite symmetric bilinear form. This gives rise
+-- to a thus:
+--
+-- @
+-- 'Norm' n '|$|' v = √(n v '<.>^' v)
+-- @
+--
+-- Strictly speaking, this type is neither strong enough nor general enough to
+-- deserve the name 'Norm': it includes proper 'Seminorm's (i.e. @m|$|v ≡ 0@ does
+-- not guarantee @v == zeroV@), but not actual norms such as the ℓ₁-norm on ℝⁿ
+-- (Taxcab norm) or the supremum norm.
+-- However, 𝐿₂-like norms are the only ones that can really be formulated without
+-- any basis reference; and guaranteeing positive definiteness through the type
+-- system is scarcely practical.
+newtype Norm v = Norm {
+ applyNorm :: v -+> DualVector v
+ }
+
+-- | A “norm” that may explicitly be degenerate, with @m|$|v ⩵ 0@ for some @v ≠ zeroV@.
+type Seminorm v = Norm v
+
+-- | @(m<>n|$|v)^2 ⩵ (m|$|v)^2 + (n|$|v)^2@
+instance LSpace v => Semigroup (Norm v) where
+ Norm m <> Norm n = Norm $ m^+^n
+-- | @mempty|$|v ≡ 0@
+instance LSpace v => Monoid (Seminorm v) where
+ mempty = Norm zeroV
+ mappend = (<>)
+
+-- | A multidimensional variance of points @v@ with some distribution can be
+-- considered a norm on the dual space, quantifying for a dual vector @dv@ the
+-- expectation value of @(dv<.>^v)^2@.
+type Variance v = Norm (DualVector v)
+
+-- | The canonical standard norm (2-norm) on inner-product / Hilbert spaces.
+euclideanNorm :: HilbertSpace v => Norm v
+euclideanNorm = Norm id
+
+-- | The norm induced from the (arbitrary) choice of basis in a finite space.
+-- Only use this in contexts where you merely need /some/ norm, but don't
+-- care if it might be biased in some unnatural way.
+adhocNorm :: FiniteDimensional v => Norm v
+adhocNorm = Norm uncanonicallyToDual
+
+-- | A proper norm induces a norm on the dual space – the “reciprocal norm”.
+-- (The orthonormal systems of the norm and its dual are mutually conjugate.)
+-- The dual norm of a seminorm is undefined.
+dualNorm :: SimpleSpace v => Norm v -> Variance v
+dualNorm (Norm m) = Norm . arr . pseudoInverse $ arr m
+
+transformNorm :: (LSpace v, LSpace w, Scalar v~Scalar w) => (v+>w) -> Norm w -> Norm v
+transformNorm f (Norm m) = Norm . arr $ (adjoint $ f) . (fmap m $ f)
+
+transformVariance :: (LSpace v, LSpace w, Scalar v~Scalar w)
+ => (v+>w) -> Variance v -> Variance w
+transformVariance f (Norm m) = Norm . arr $ f . (fmap m $ adjoint $ f)
+
+infixl 6 ^%
+(^%) :: (LSpace v, Floating (Scalar v)) => v -> Norm v -> v
+v ^% Norm m = v ^/ sqrt ((m$v)<.>^v)
+
+-- | The unique positive number whose norm is 1 (if the norm is not constant zero).
+findNormalLength :: RealFrac' s => Norm s -> Maybe s
+findNormalLength (Norm m) = case m $ 1 of
+ o | o > 0 -> Just . sqrt $ recip o
+ | otherwise -> Nothing
+
+-- | Unsafe version of 'findNormalLength', only works reliable if the norm
+-- is actually positive definite.
+normalLength :: RealFrac' s => Norm s -> s
+normalLength (Norm m) = case m $ 1 of
+ o | o >= 0 -> sqrt $ recip o
+ | o < 0 -> error "Norm fails to be positive semidefinite."
+ | otherwise -> error "Norm yields NaN."
+
+infixr 0 <$|, |$|
+-- | “Partially apply” a norm, yielding a dual vector
+-- (i.e. a linear form that accepts the second argument of the scalar product).
+--
+-- @
+-- ('euclideanNorm' '<$|' v) '<.>^' w ≡ v '<.>' w
+-- @
+(<$|) :: LSpace v => Norm v -> v -> DualVector v
+Norm m <$| v = m $ v
+
+-- | The squared norm. More efficient than '|$|' because that needs to take
+-- the square root.
+normSq :: LSpace v => Seminorm v -> v -> Scalar v
+normSq (Norm m) v = (m$v)<.>^v
+
+-- | Use a 'Norm' to measure the length / norm of a vector.
+--
+-- @
+-- 'euclideanNorm' |$| v ≡ √(v '<.>' v)
+-- @
+(|$|) :: (LSpace v, Floating (Scalar v)) => Seminorm v -> v -> Scalar v
+(|$|) m = sqrt . normSq m
+
+-- | 'spanNorm' / 'spanVariance' are inefficient if the number of vectors
+-- is similar to the dimension of the space, or even larger than it.
+-- Use this function to optimise the underlying operator to a dense
+-- matrix representation.
+densifyNorm :: LSpace v => Norm v -> Norm v
+densifyNorm (Norm m) = Norm . arr $ sampleLinearFunction $ m
+
+data OrthonormalSystem v = OrthonormalSystem {
+ orthonormalityNorm :: Norm v
+ , orthonormalVectors :: [v]
+ }
+
+orthonormaliseFussily :: (LSpace v, RealFloat (Scalar v))
+ => Scalar v -> Norm v -> [v] -> [v]
+orthonormaliseFussily fuss me = go []
+ where go _ [] = []
+ go ws (v₀:vs)
+ | mvd > fuss = let μ = 1/sqrt mvd
+ v = vd^*μ
+ in v : go ((v,dvd^*μ):ws) vs
+ | otherwise = go ws vs
+ where vd = orthogonalComplementProj' ws $ v₀
+ dvd = applyNorm me $ vd
+ mvd = dvd<.>^vd
+
+orthogonalComplementProj' :: LSpace v => [(v, DualVector v)] -> (v-+>v)
+orthogonalComplementProj' ws = LinearFunction $ \v₀
+ -> foldl' (\v (w,dw) -> v ^-^ w^*(dw<.>^v)) v₀ ws
+
+orthogonalComplementProj :: LSpace v => Norm v -> [v] -> (v-+>v)
+orthogonalComplementProj (Norm m)
+ = orthogonalComplementProj' . map (id &&& (m$))
+
+
+
+data Eigenvector v = Eigenvector {
+ ev_Eigenvalue :: Scalar v -- ^ The estimated eigenvalue @λ@.
+ , ev_Eigenvector :: v -- ^ Normalised vector @v@ that gets mapped to a multiple, namely:
+ , ev_FunctionApplied :: v -- ^ @f $ v ≡ λ *^ v @.
+ , ev_Deviation :: v -- ^ Deviation of these two supposedly equivalent expressions.
+ , ev_Badness :: Scalar v -- ^ Squared norm of the deviation, normalised by the eigenvalue.
+ }
+deriving instance (Show v, Show (Scalar v)) => Show (Eigenvector v)
+
+-- | Lazily compute the eigenbasis of a linear map. The algorithm is essentially
+-- a hybrid of Lanczos/Arnoldi style Krylov-spanning and QR-diagonalisation,
+-- which we don't do separately but /interleave/ at each step.
+--
+-- The size of the eigen-subbasis increases with each step until the space's
+-- dimension is reached. (But the algorithm can also be used for
+-- infinite-dimensional spaces.)
+constructEigenSystem :: (LSpace v, RealFloat (Scalar v))
+ => Norm v -- ^ The notion of orthonormality.
+ -> Scalar v -- ^ Error bound for deviations from eigen-ness.
+ -> (v-+>v) -- ^ Operator to calculate the eigensystem of.
+ -- Must be Hermitian WRT the scalar product
+ -- defined by the given metric.
+ -> [v] -- ^ Starting vector(s) for the power method.
+ -> [[Eigenvector v]] -- ^ Infinite sequence of ever more accurate approximations
+ -- to the eigensystem of the operator.
+constructEigenSystem me@(Norm m) ε₀ f = iterate (
+ sortBy (comparing $
+ negate . abs . ev_Eigenvalue)
+ . map asEV
+ . orthonormaliseFussily (1/4) (Norm m)
+ . newSys)
+ . map (asEV . (^%me))
+ where newSys [] = []
+ newSys (Eigenvector λ v fv dv ε : evs)
+ | ε>ε₀ = case newSys evs of
+ [] -> [fv^/λ, dv^*(sqrt $ λ^2/ε)]
+ vn:vns -> fv^/λ : vn : dv^*(sqrt $ λ^2/ε) : vns
+ | ε>=0 = v : newSys evs
+ | otherwise = newSys evs
+ asEV v = Eigenvector λ v fv dv ε
+ where λ = v'<.>^fv
+ ε = normSq me dv / (λ^2 + ε₀)
+ fv = f $ v
+ dv = v^*λ ^-^ fv
+ v' = m $ v
+
+
+finishEigenSystem :: (LSpace v, RealFloat (Scalar v))
+ => Norm v -> [Eigenvector v] -> [Eigenvector v]
+finishEigenSystem me = go . sortBy (comparing $ negate . ev_Eigenvalue)
+ where go [] = []
+ go [v] = [v]
+ go vs@[Eigenvector λ₀ v₀ fv₀ _dv₀ _ε₀, Eigenvector λ₁ v₁ fv₁ _dv₁ _ε₁]
+ | λ₀>λ₁ = [ asEV v₀' fv₀', asEV v₁' fv₁' ]
+ | otherwise = vs
+ where
+ v₀' = v₀^*μ₀₀ ^+^ v₁^*μ₀₁
+ fv₀' = fv₀^*μ₀₀ ^+^ fv₁^*μ₀₁
+
+ v₁' = v₀^*μ₁₀ ^+^ v₁^*μ₁₁
+ fv₁' = fv₀^*μ₁₀ ^+^ fv₁^*μ₁₁
+
+ fShift₁v₀ = fv₀ ^-^ λ₁*^v₀
+
+ (μ₀₀,μ₀₁) = normalized ( λ₀ - λ₁
+ , (me <$| fShift₁v₀)<.>^v₁ )
+ (μ₁₀,μ₁₁) = (-μ₀₁, μ₀₀)
+
+ go evs = lo'' ++ upper'
+ where l = length evs
+ lChunk = l`quot`3
+ (loEvs, (midEvs, hiEvs)) = second (splitAt $ l - 2*lChunk)
+ $ splitAt lChunk evs
+ (lo',hi') = splitAt lChunk . go $ loEvs++hiEvs
+ (lo'',mid') = splitAt lChunk . go $ lo'++midEvs
+ upper' = go $ mid'++hi'
+
+ asEV v fv = Eigenvector λ v fv dv ε
+ where λ = (me<$|v)<.>^fv
+ dv = v^*λ ^-^ fv
+ ε = normSq me dv / λ^2
+
+
+-- | Find a system of vectors that approximate the eigensytem, in the sense that:
+-- each true eigenvalue is represented by an approximate one, and that is closer
+-- to the true value than all the other approximate EVs.
+--
+-- This function does not make any guarantees as to how well a single eigenvalue
+-- is approximated, though.
+roughEigenSystem :: (FiniteDimensional v, IEEE (Scalar v))
+ => Norm v
+ -> (v+>v)
+ -> [Eigenvector v]
+roughEigenSystem me f = go fBas 0 [[]]
+ where go [] _ (_:evs:_) = evs
+ go [] _ (evs:_) = evs
+ go (v:vs) oldDim (evs:evss)
+ | normSq me vPerp > fpε = case evss of
+ evs':_ | length evs' > oldDim
+ -> go (v:vs) (length evs) evss
+ _ -> let evss' = constructEigenSystem me fpε (arr f)
+ $ map ev_Eigenvector (head $ evss++[evs]) ++ [vPerp]
+ in go vs (length evs) evss'
+ | otherwise = go vs oldDim (evs:evss)
+ where vPerp = orthogonalComplementProj me (ev_Eigenvector<$>evs) $ v
+ fBas = (^%me) <$> snd (decomposeLinMap id) []
+ fpε = epsilon * 8
+
+-- | Simple automatic finding of the eigenvalues and -vectors
+-- of a Hermitian operator, in reasonable approximation.
+--
+-- This works by spanning a QR-stabilised Krylov basis with 'constructEigenSystem'
+-- until it is complete ('roughEigenSystem'), and then properly decoupling the
+-- system with 'finishEigenSystem' (based on two iterations of shifted Givens rotations).
+--
+-- This function is a tradeoff in performance vs. accuracy. Use 'constructEigenSystem'
+-- and 'finishEigenSystem' directly for more quickly computing a (perhaps incomplete)
+-- approximation, or for more precise results.
+eigen :: (FiniteDimensional v, HilbertSpace v, IEEE (Scalar v))
+ => (v+>v) -> [(Scalar v, v)]
+eigen f = map (ev_Eigenvalue &&& ev_Eigenvector)
+ $ iterate (finishEigenSystem euclideanNorm) (roughEigenSystem euclideanNorm f) !! 2
+
+
+-- | Approximation of the determinant.
+roughDet :: (FiniteDimensional v, IEEE (Scalar v)) => (v+>v) -> Scalar v
+roughDet = roughEigenSystem adhocNorm >>> map ev_Eigenvalue >>> product
+
+
+orthonormalityError :: LSpace v => Norm v -> [v] -> Scalar v
+orthonormalityError me vs = normSq me $ orthogonalComplementProj me vs $ sumV vs
+
+
+normSpanningSystem :: SimpleSpace v
+ => Norm v -> [DualVector v]
+normSpanningSystem = dualBasis . normSpanningSystem'
+
+normSpanningSystem' :: (FiniteDimensional v, IEEE (Scalar v))
+ => Norm v -> [v]
+normSpanningSystem' me = orthonormaliseFussily 0 me $ enumerateSubBasis entireBasis
+
+
+-- | For any two norms, one can find a system of co-vectors that, with suitable
+-- coefficients, spans /either/ of them: if @shSys = sharedNormSpanningSystem n₀ n₁@,
+-- then
+--
+-- @
+-- n₀ = 'spanNorm' $ fst<$>shSys
+-- @
+--
+-- and
+--
+-- @
+-- n₁ = 'spanNorm' [dv^*η | (dv,η)<-shSys]
+-- @
+sharedNormSpanningSystem :: SimpleSpace v
+ => Norm v -> Norm v -> [(DualVector v, Scalar v)]
+sharedNormSpanningSystem (Norm n) (Norm m)
+ = sep =<< roughEigenSystem (Norm n) (pseudoInverse (arr n) . arr m)
+ where sep (Eigenvector λ _ λv _ _)
+ | λ>0 = [(n$v, sqrt λ)]
+ | otherwise = []
+ where v = λv ^/ λ
+
+
+-- | Interpret a variance as a covariance between two subspaces, and
+-- normalise it by the variance on @u@. The result is effectively
+-- the linear regression coefficient of a simple regression of the vectors
+-- spanning the variance.
+dependence :: (SimpleSpace u, SimpleSpace v, Scalar u~Scalar v)
+ => Variance (u,v) -> (u+>v)
+dependence (Norm m) = fmap ( snd . m . (id&&&zeroV) )
+ $ pseudoInverse (arr $ fst . m . (id&&&zeroV))
+
+
+summandSpaceNorms :: (SimpleSpace u, SimpleSpace v, Scalar u ~ Scalar v)
+ => Norm (u,v) -> (Norm u, Norm v)
+summandSpaceNorms nuv = ( densifyNorm $ spanNorm (fst<$>spanSys)
+ , densifyNorm $ spanNorm (snd<$>spanSys) )
+ where spanSys = normSpanningSystem nuv
+
+sumSubspaceNorms :: (LSpace u, LSpace v, Scalar u~Scalar v)
+ => Norm u -> Norm v -> Norm (u,v)
+sumSubspaceNorms (Norm nu) (Norm nv) = Norm $ nu *** nv
+
+
+
+
+
+instance (SimpleSpace v, Show (DualVector v)) => Show (Norm v) where
+ showsPrec p n = showParen (p>9) $ ("spanNorm "++) . shows (normSpanningSystem n)
diff --git a/Math/LinearMap/Category/Class.hs b/Math/LinearMap/Category/Class.hs
new file mode 100644
--- /dev/null
+++ b/Math/LinearMap/Category/Class.hs
@@ -0,0 +1,660 @@
+-- |
+-- Module : Math.LinearMap.Category.Class
+-- Copyright : (c) Justus Sagemüller 2016
+-- License : GPL v3
+--
+-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
+-- Stability : experimental
+-- Portability : portable
+--
+{-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE ConstraintKinds #-}
+{-# LANGUAGE UndecidableInstances #-}
+{-# LANGUAGE FunctionalDependencies #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE Rank2Types #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE PatternSynonyms #-}
+{-# LANGUAGE ViewPatterns #-}
+{-# LANGUAGE UnicodeSyntax #-}
+{-# LANGUAGE TupleSections #-}
+{-# LANGUAGE StandaloneDeriving #-}
+
+module Math.LinearMap.Category.Class where
+
+import Data.VectorSpace
+
+import Prelude ()
+import qualified Prelude as Hask
+
+import Control.Category.Constrained.Prelude
+import Control.Arrow.Constrained
+
+import Data.Coerce
+import Data.Type.Coercion
+
+import Math.LinearMap.Asserted
+import Math.VectorSpace.ZeroDimensional
+
+type Num' s = (Num s, VectorSpace s, Scalar s ~ s)
+type Num'' s = (Num' s, LinearSpace s)
+type Num''' s = (Num s, InnerSpace s, Scalar s ~ s, LSpace' s, DualVector s ~ s)
+
+class (VectorSpace v) => TensorSpace v where
+ -- | The internal representation of a 'Tensor' product.
+ --
+ -- For euclidean spaces, this is generally constructed by replacing each @s@
+ -- scalar field in the @v@ vector with an entire @w@ vector. I.e., you have
+ -- then a “nested vector” or, if @v@ is a @DualVector@ / “row vector”, a matrix.
+ type TensorProduct v w :: *
+ zeroTensor :: (LSpace w, Scalar w ~ Scalar v)
+ => v ⊗ w
+ toFlatTensor :: v -+> (v ⊗ Scalar v)
+ fromFlatTensor :: (v ⊗ Scalar v) -+> v
+ addTensors :: (LSpace w, Scalar w ~ Scalar v)
+ => (v ⊗ w) -> (v ⊗ w) -> v ⊗ w
+ subtractTensors :: (LSpace v, LSpace w, Num''' (Scalar v), Scalar w ~ Scalar v)
+ => (v ⊗ w) -> (v ⊗ w) -> v ⊗ w
+ subtractTensors m n = addTensors m (negateTensor $ n)
+ scaleTensor :: (LSpace w, Scalar w ~ Scalar v)
+ => Bilinear (Scalar v) (v ⊗ w) (v ⊗ w)
+ negateTensor :: (LSpace w, Scalar w ~ Scalar v)
+ => (v ⊗ w) -+> (v ⊗ w)
+ tensorProduct :: (LSpace w, Scalar w ~ Scalar v)
+ => Bilinear v w (v ⊗ w)
+ transposeTensor :: (LSpace w, Scalar w ~ Scalar v)
+ => (v ⊗ w) -+> (w ⊗ v)
+ fmapTensor :: (LSpace w, LSpace x, Scalar w ~ Scalar v, Scalar x ~ Scalar v)
+ => Bilinear (w -+> x) (v⊗w) (v⊗x)
+ fzipTensorWith :: ( LSpace u, LSpace w, LSpace x
+ , Scalar u ~ Scalar v, Scalar w ~ Scalar v, Scalar x ~ Scalar v )
+ => Bilinear ((w,x) -+> u) (v⊗w, v⊗x) (v⊗u)
+ coerceFmapTensorProduct :: Hask.Functor p
+ => p v -> Coercion a b -> Coercion (TensorProduct v a) (TensorProduct v b)
+
+infixl 7 ⊗
+
+-- | Infix version of 'tensorProduct'.
+(⊗) :: (LSpace v, LSpace w, Scalar w ~ Scalar v)
+ => v -> w -> v ⊗ w
+v⊗w = (tensorProduct $ v) $ w
+
+-- | The class of vector spaces @v@ for which @'LinearMap' s v w@ is well-implemented.
+class ( TensorSpace v, TensorSpace (DualVector v)
+ , Num' (Scalar v), Scalar (DualVector v) ~ Scalar v )
+ => LinearSpace v where
+ -- | Suitable representation of a linear map from the space @v@ to its field.
+ --
+ -- For the usual euclidean spaces, you can just define @'DualVector' v = v@.
+ -- (In this case, a dual vector will be just a “row vector” if you consider
+ -- @v@-vectors as “column vectors”. 'LinearMap' will then effectively have
+ -- a matrix layout.)
+ type DualVector v :: *
+
+ linearId :: v +> v
+
+ idTensor :: LSpace v => v ⊗ DualVector v
+ idTensor = transposeTensor $ asTensor $ linearId
+
+ sampleLinearFunction :: (LSpace v, LSpace w, Scalar v ~ Scalar w)
+ => (v-+>w) -+> (v+>w)
+ sampleLinearFunction = LinearFunction $ \f -> fmap f $ id
+
+ toLinearForm :: LSpace v => DualVector v -+> (v+>Scalar v)
+ toLinearForm = toFlatTensor >>> arr fromTensor
+
+ fromLinearForm :: LSpace v => (v+>Scalar v) -+> DualVector v
+ fromLinearForm = arr asTensor >>> fromFlatTensor
+
+ coerceDoubleDual :: Coercion v (DualVector (DualVector v))
+
+ blockVectSpan :: (LSpace w, Scalar w ~ Scalar v)
+ => w -+> (v⊗(v+>w))
+ blockVectSpan' :: (LSpace v, LSpace w, Num''' (Scalar v), Scalar v ~ Scalar w)
+ => w -+> (v+>(v⊗w))
+ blockVectSpan' = LinearFunction $ \w -> fmap (flipBilin tensorProduct $ w) $ id
+
+ trace :: LSpace v => (v+>v) -+> Scalar v
+ trace = flipBilin contractLinearMapAgainst $ id
+
+ contractTensorMap :: (LSpace w, Scalar w ~ Scalar v)
+ => (v+>(v⊗w)) -+> w
+ contractMapTensor :: (LSpace w, Scalar w ~ Scalar v)
+ => (v⊗(v+>w)) -+> w
+ contractFnTensor :: (LSpace v, LSpace w, Scalar w ~ Scalar v)
+ => (v⊗(v-+>w)) -+> w
+ contractFnTensor = fmap sampleLinearFunction >>> contractMapTensor
+ contractTensorFn :: (LSpace v, LSpace w, Scalar w ~ Scalar v)
+ => (v-+>(v⊗w)) -+> w
+ contractTensorFn = sampleLinearFunction >>> contractTensorMap
+ contractTensorWith :: (LSpace v, LSpace w, Scalar w ~ Scalar v)
+ => Bilinear (v⊗w) (DualVector w) v
+ contractTensorWith = flipBilin $ LinearFunction
+ (\dw -> fromFlatTensor . fmap (flipBilin applyDualVector$dw))
+ contractLinearMapAgainst :: (LSpace w, Scalar w ~ Scalar v)
+ => Bilinear (v+>w) (w-+>v) (Scalar v)
+
+ applyDualVector :: LSpace v
+ => Bilinear (DualVector v) v (Scalar v)
+
+ applyLinear :: (LSpace w, Scalar w ~ Scalar v)
+ => Bilinear (v+>w) v w
+ composeLinear :: ( LSpace w, LSpace x
+ , Scalar w ~ Scalar v, Scalar x ~ Scalar v )
+ => Bilinear (w+>x) (v+>w) (v+>x)
+
+
+instance Num''' s => TensorSpace (ZeroDim s) where
+ type TensorProduct (ZeroDim s) v = ZeroDim s
+ zeroTensor = Tensor Origin
+ toFlatTensor = LinearFunction $ \Origin -> Tensor Origin
+ fromFlatTensor = LinearFunction $ \(Tensor Origin) -> Origin
+ negateTensor = const0
+ scaleTensor = biConst0
+ addTensors (Tensor Origin) (Tensor Origin) = Tensor Origin
+ subtractTensors (Tensor Origin) (Tensor Origin) = Tensor Origin
+ tensorProduct = biConst0
+ transposeTensor = const0
+ fmapTensor = biConst0
+ fzipTensorWith = biConst0
+ coerceFmapTensorProduct _ Coercion = Coercion
+instance Num''' s => LinearSpace (ZeroDim s) where
+ type DualVector (ZeroDim s) = ZeroDim s
+ linearId = LinearMap Origin
+ idTensor = Tensor Origin
+ fromLinearForm = const0
+ coerceDoubleDual = Coercion
+ contractTensorMap = const0
+ contractMapTensor = const0
+ contractTensorWith = biConst0
+ contractLinearMapAgainst = biConst0
+ blockVectSpan = const0
+ applyDualVector = biConst0
+ applyLinear = biConst0
+ composeLinear = biConst0
+
+
+-- | The tensor product between one space's dual space and another space is the
+-- space spanned by vector–dual-vector pairs, in
+--
+-- written as
+--
+-- @
+-- m = ∑ |w⟩⟨v|
+-- @
+--
+-- Any linear mapping can be written as such a (possibly infinite) sum. The
+-- 'TensorProduct' data structure only stores the linear independent parts
+-- though; for simple finite-dimensional spaces this means e.g. @'LinearMap' ℝ ℝ³ ℝ³@
+-- effectively boils down to an ordinary matrix type, namely an array of
+-- column-vectors @|w⟩@.
+--
+-- (The @⟨v|@ dual-vectors are then simply assumed to come from the canonical basis.)
+--
+-- For bigger spaces, the tensor product may be implemented in a more efficient
+-- sparse structure; this can be defined in the 'TensorSpace' instance.
+newtype LinearMap s v w = LinearMap {getLinearMap :: TensorProduct (DualVector v) w}
+
+-- | Tensor products are most interesting because they can be used to implement
+-- linear mappings, but they also form a useful vector space on their own right.
+newtype Tensor s v w = Tensor {getTensorProduct :: TensorProduct v w}
+
+asTensor :: Coercion (LinearMap s v w) (Tensor s (DualVector v) w)
+asTensor = Coercion
+fromTensor :: Coercion (Tensor s (DualVector v) w) (LinearMap s v w)
+fromTensor = Coercion
+
+asLinearMap :: ∀ s v w . (LSpace v, Scalar v ~ s)
+ => Coercion (Tensor s v w) (LinearMap s (DualVector v) w)
+asLinearMap = Coercion
+fromLinearMap :: ∀ s v w . (LSpace v, Scalar v ~ s)
+ => Coercion (LinearMap s (DualVector v) w) (Tensor s v w)
+fromLinearMap = Coercion
+
+-- | Infix synonym for 'LinearMap', without explicit mention of the scalar type.
+type v +> w = LinearMap (Scalar v) v w
+
+-- | Infix synonym for 'Tensor', without explicit mention of the scalar type.
+type v ⊗ w = Tensor (Scalar v) v w
+
+type LSpace' v = ( LinearSpace v, LinearSpace (Scalar v)
+ , LinearSpace (DualVector v), DualVector (DualVector v) ~ v )
+
+-- | The workhorse of this package: most functions here work on vector
+-- spaces that fulfill the @'LSpace' v@ constraint. In summary, this is:
+--
+-- * A 'VectorSpace' whose 'Scalar' is a 'Num'''' (i.e. a number type that
+-- has itself all the vector-space instances).
+-- * You have an implementation for @'TensorProduct' v w@, for any other space @w@.
+-- * You have a 'DualVector' space that fulfills @'DualVector' ('DualVector' v) ~ v@.
+--
+-- To make a new space of yours an 'LSpace', you must define instances of
+-- 'TensorSpace' and 'LinearSpace'.
+type LSpace v = (LSpace' v, Num''' (Scalar v))
+
+instance (LSpace v, LSpace w, Scalar v~s, Scalar w~s)
+ => AdditiveGroup (LinearMap s v w) where
+ zeroV = fromTensor $ zeroTensor
+ m^+^n = fromTensor $ (asTensor$m) ^+^ (asTensor$n)
+ m^-^n = fromTensor $ (asTensor$m) ^-^ (asTensor$n)
+ negateV = (fromTensor$) . negateV . (asTensor$)
+instance (LSpace v, LSpace w, Scalar v~s, Scalar w~s)
+ => VectorSpace (LinearMap s v w) where
+ type Scalar (LinearMap s v w) = s
+ μ*^v = arr fromTensor . (scaleTensor$μ) . arr asTensor $ v
+
+instance (LSpace v, LSpace w, Scalar v~s, Scalar w~s)
+ => AdditiveGroup (Tensor s v w) where
+ zeroV = zeroTensor
+ (^+^) = addTensors
+ (^-^) = subtractTensors
+ negateV = arr negateTensor
+instance (LSpace v, LSpace w, Scalar v~s, Scalar w~s)
+ => VectorSpace (Tensor s v w) where
+ type Scalar (Tensor s v w) = s
+ μ*^t = (scaleTensor $ μ) $ t
+
+infixr 6 ⊕, >+<, <⊕
+
+(<⊕) :: (u⊗w) -> (v⊗w) -> (u,v)⊗w
+m <⊕ n = Tensor $ (m, n)
+
+-- | The dual operation to the tuple constructor, or rather to the
+-- '&&&' fanout operation: evaluate two (linear) functions in parallel
+-- and sum up the results.
+-- The typical use is to concatenate “row vectors” in a matrix definition.
+(⊕) :: (u+>w) -> (v+>w) -> (u,v)+>w
+LinearMap m ⊕ LinearMap n = LinearMap $ (Tensor m, Tensor n)
+
+-- | ASCII version of '⊕'
+(>+<) :: (u+>w) -> (v+>w) -> (u,v)+>w
+(>+<) = (⊕)
+
+
+instance Category (LinearMap s) where
+ type Object (LinearMap s) v = (LSpace v, Scalar v ~ s)
+ id = linearId
+ (.) = arr . arr composeLinear
+instance Num''' s => Cartesian (LinearMap s) where
+ type UnitObject (LinearMap s) = ZeroDim s
+ swap = (fmap (const0&&&id) $ id) ⊕ (fmap (id&&&const0) $ id)
+ attachUnit = fmap (id&&&const0) $ id
+ detachUnit = fst
+ regroup = sampleLinearFunction $ LinearFunction regroup
+ regroup' = sampleLinearFunction $ LinearFunction regroup'
+instance Num''' s => Morphism (LinearMap s) where
+ f *** g = (fmap (id&&&const0) $ f) ⊕ (fmap (const0&&&id) $ g)
+instance Num''' s => PreArrow (LinearMap s) where
+ f &&& g = fromTensor $ (fzipTensorWith$id) $ (asTensor $ f, asTensor $ g)
+ terminal = zeroV
+ fst = sampleLinearFunction $ fst
+ snd = sampleLinearFunction $ snd
+instance Num''' s => EnhancedCat (->) (LinearMap s) where
+ arr m = arr $ applyLinear $ m
+instance Num''' s => EnhancedCat (LinearFunction s) (LinearMap s) where
+ arr m = applyLinear $ m
+instance Num''' s => EnhancedCat (LinearMap s) (LinearFunction s) where
+ arr m = sampleLinearFunction $ m
+
+
+
+
+
+
+instance ∀ u v . ( Num''' (Scalar v), LSpace u, LSpace v, Scalar u ~ Scalar v )
+ => TensorSpace (u,v) where
+ type TensorProduct (u,v) w = (u⊗w, v⊗w)
+ zeroTensor = zeroTensor <⊕ zeroTensor
+ scaleTensor = scaleTensor&&&scaleTensor >>> LinearFunction (
+ uncurry (***) >>> pretendLike Tensor )
+ negateTensor = pretendLike Tensor $ negateTensor *** negateTensor
+ addTensors (Tensor (fu, fv)) (Tensor (fu', fv')) = (fu ^+^ fu') <⊕ (fv ^+^ fv')
+ subtractTensors (Tensor (fu, fv)) (Tensor (fu', fv'))
+ = (fu ^-^ fu') <⊕ (fv ^-^ fv')
+ toFlatTensor = follow Tensor <<< toFlatTensor *** toFlatTensor
+ fromFlatTensor = flout Tensor >>> fromFlatTensor *** fromFlatTensor
+ tensorProduct = LinearFunction $ \(u,v) ->
+ (tensorProduct$u) &&& (tensorProduct$v) >>> follow Tensor
+ transposeTensor = flout Tensor >>> transposeTensor *** transposeTensor >>> fzip
+ fmapTensor = LinearFunction $
+ \f -> pretendLike Tensor $ (fmapTensor$f) *** (fmapTensor$f)
+ fzipTensorWith = bilinearFunction
+ $ \f (Tensor (uw, vw), Tensor (ux, vx))
+ -> Tensor ( (fzipTensorWith$f)$(uw,ux)
+ , (fzipTensorWith$f)$(vw,vx) )
+ coerceFmapTensorProduct p cab = case
+ ( coerceFmapTensorProduct (fst<$>p) cab
+ , coerceFmapTensorProduct (snd<$>p) cab ) of
+ (Coercion, Coercion) -> Coercion
+instance ∀ u v . ( LinearSpace u, LinearSpace (DualVector u), DualVector (DualVector u) ~ u
+ , LinearSpace v, LinearSpace (DualVector v), DualVector (DualVector v) ~ v
+ , Scalar u ~ Scalar v, Num''' (Scalar u) )
+ => LinearSpace (u,v) where
+ type DualVector (u,v) = (DualVector u, DualVector v)
+ linearId = (fmap (id&&&const0) $ id) ⊕ (fmap (const0&&&id) $ id)
+ -- idTensor = fmapTensor linearCoFst idTensor <⊕ fmapTensor linearCoSnd idTensor
+ sampleLinearFunction = LinearFunction $ \f -> (sampleLinearFunction $ f . lCoFst)
+ ⊕ (sampleLinearFunction $ f . lCoSnd)
+ coerceDoubleDual = Coercion
+ blockVectSpan = (blockVectSpan >>> fmap lfstBlock) &&& (blockVectSpan >>> fmap lsndBlock)
+ >>> follow Tensor
+ contractTensorMap = flout LinearMap
+ >>> contractTensorMap . fmap (fst . flout Tensor) . arr fromTensor
+ ***contractTensorMap . fmap (snd . flout Tensor) . arr fromTensor
+ >>> addV
+ contractMapTensor = flout Tensor
+ >>> contractMapTensor . fmap (arr fromTensor . fst . flout LinearMap)
+ ***contractMapTensor . fmap (arr fromTensor . snd . flout LinearMap)
+ >>> addV
+ contractTensorWith = LinearFunction $ \(Tensor (fu, fv))
+ -> (contractTensorWith$fu) &&& (contractTensorWith$fv)
+ contractLinearMapAgainst = flout LinearMap >>> bilinearFunction
+ (\(mu,mv) f -> ((contractLinearMapAgainst$fromTensor$mu)$(fst.f))
+ + ((contractLinearMapAgainst$fromTensor$mv)$(snd.f)) )
+ applyDualVector = LinearFunction $ \(du,dv)
+ -> (applyDualVector$du) *** (applyDualVector$dv) >>> addV
+ applyLinear = LinearFunction $ \(LinearMap (fu, fv)) ->
+ (applyLinear $ (asLinearMap $ fu)) *** (applyLinear $ (asLinearMap $ fv))
+ >>> addV
+ composeLinear = bilinearFunction $ \f (LinearMap (fu, fv))
+ -> f . (asLinearMap $ fu) ⊕ f . (asLinearMap $ fv)
+
+lfstBlock :: ( LSpace u, LSpace v, LSpace w
+ , Scalar u ~ Scalar v, Scalar v ~ Scalar w )
+ => (u+>w) -+> ((u,v)+>w)
+lfstBlock = LinearFunction (⊕zeroV)
+lsndBlock :: ( LSpace u, LSpace v, LSpace w
+ , Scalar u ~ Scalar v, Scalar v ~ Scalar w )
+ => (v+>w) -+> ((u,v)+>w)
+lsndBlock = LinearFunction (zeroV⊕)
+
+
+-- | @(u+>(v⊗w)) -> (u+>v)⊗w@
+deferLinearMap :: Coercion (LinearMap s u (Tensor s v w)) (Tensor s (LinearMap s u v) w)
+deferLinearMap = Coercion
+
+-- | @(u+>v)⊗w -> u+>(v⊗w)@
+hasteLinearMap :: Coercion (Tensor s (LinearMap s u v) w) (LinearMap s u (Tensor s v w))
+hasteLinearMap = Coercion
+
+
+lassocTensor :: Coercion (Tensor s u (Tensor s v w)) (Tensor s (Tensor s u v) w)
+lassocTensor = Coercion
+rassocTensor :: Coercion (Tensor s (Tensor s u v) w) (Tensor s u (Tensor s v w))
+rassocTensor = Coercion
+
+instance ∀ s u v . ( Num''' s, LSpace u, LSpace v, Scalar u ~ s, Scalar v ~ s )
+ => TensorSpace (LinearMap s u v) where
+ type TensorProduct (LinearMap s u v) w = TensorProduct (DualVector u) (Tensor s v w)
+ zeroTensor = deferLinearMap $ zeroV
+ toFlatTensor = arr deferLinearMap . fmap toFlatTensor
+ fromFlatTensor = fmap fromFlatTensor . arr hasteLinearMap
+ addTensors t₁ t₂ = deferLinearMap $ (hasteLinearMap$t₁) ^+^ (hasteLinearMap$t₂)
+ subtractTensors t₁ t₂ = deferLinearMap $ (hasteLinearMap$t₁) ^-^ (hasteLinearMap$t₂)
+ scaleTensor = LinearFunction $ \μ -> arr deferLinearMap . scaleWith μ . arr hasteLinearMap
+ negateTensor = arr deferLinearMap . lNegateV . arr hasteLinearMap
+ transposeTensor -- t :: (u +> v) ⊗ w
+ = arr hasteLinearMap -- u +> (v ⊗ w)
+ >>> fmap transposeTensor -- u +> (w ⊗ v)
+ >>> arr asTensor -- u' ⊗ (w ⊗ v)
+ >>> transposeTensor -- (w ⊗ v) ⊗ u'
+ >>> arr rassocTensor -- w ⊗ (v ⊗ u')
+ >>> fmap transposeTensor -- w ⊗ (u' ⊗ v)
+ >>> arr (fmap fromTensor) -- w ⊗ (u +> v)
+ tensorProduct = LinearFunction $ \t -> arr deferLinearMap
+ . (flipBilin composeLinear $ t) . blockVectSpan'
+ fmapTensor = LinearFunction $ \f
+ -> arr deferLinearMap <<< fmap (fmap f) <<< arr hasteLinearMap
+ fzipTensorWith = LinearFunction $ \f
+ -> arr deferLinearMap <<< fzipWith (fzipWith f)
+ <<< arr hasteLinearMap *** arr hasteLinearMap
+ coerceFmapTensorProduct = cftlp
+ where cftlp :: ∀ a b p . p (LinearMap s u v) -> Coercion a b
+ -> Coercion (TensorProduct (DualVector u) (Tensor s v a))
+ (TensorProduct (DualVector u) (Tensor s v b))
+ cftlp _ c = coerceFmapTensorProduct ([]::[DualVector u])
+ (fmap c :: Coercion (v⊗a) (v⊗b))
+
+-- | @((u+>v)+>w) -> v+>(u⊗w)@
+coCurryLinearMap :: Coercion (LinearMap s (LinearMap s u v) w) (LinearMap s v (Tensor s u w))
+coCurryLinearMap = Coercion
+
+-- | @(u+>(v⊗w)) -> (v+>u)+>w@
+coUncurryLinearMap :: Coercion (LinearMap s u (Tensor s v w)) (LinearMap s (LinearMap s v u) w)
+coUncurryLinearMap = Coercion
+
+curryLinearMap :: (Num''' s, LSpace u, Scalar u ~ s)
+ => Coercion (LinearMap s (Tensor s u v) w) (LinearMap s u (LinearMap s v w))
+curryLinearMap = fmap fromTensor . fromTensor . rassocTensor . asTensor
+
+uncurryLinearMap :: (Num''' s, LSpace u, Scalar u ~ s)
+ => Coercion (LinearMap s u (LinearMap s v w)) (LinearMap s (Tensor s u v) w)
+uncurryLinearMap = fromTensor . lassocTensor . asTensor . fmap asTensor
+
+uncurryLinearFn :: ( Num''' s, LSpace u, LSpace v, LSpace w
+ , Scalar u ~ s, Scalar v ~ s, Scalar w ~ s )
+ => LinearFunction s u (LinearMap s v w) -+> LinearFunction s (Tensor s u v) w
+uncurryLinearFn = bilinearFunction
+ $ \f t -> contractMapTensor . fmap f . transposeTensor $ t
+
+instance ∀ s u v . (Num''' s, LSpace u, LSpace v, Scalar u ~ s, Scalar v ~ s)
+ => LinearSpace (LinearMap s u v) where
+ type DualVector (LinearMap s u v) = LinearMap s v u
+ linearId = coUncurryLinearMap $ fmap blockVectSpan $ id
+ coerceDoubleDual = Coercion
+ blockVectSpan = arr deferLinearMap
+ . fmap (arr (fmap coUncurryLinearMap) . blockVectSpan)
+ . blockVectSpan'
+ applyLinear = bilinearFunction $ \f g -> contractTensorMap $ (coCurryLinearMap$f) . g
+ applyDualVector = contractLinearMapAgainst >>> LinearFunction (. applyLinear)
+ composeLinear = bilinearFunction $ \f g
+ -> coUncurryLinearMap $ fmap (fmap $ applyLinear $ f) $ (coCurryLinearMap$g)
+ contractTensorMap = contractTensorMap . fmap (contractMapTensor . arr (fmap hasteLinearMap))
+ . arr coCurryLinearMap
+ contractMapTensor = contractTensorMap . fmap (contractMapTensor . arr (fmap coCurryLinearMap))
+ . arr hasteLinearMap
+ contractTensorWith = arr hasteLinearMap >>> bilinearFunction (\l dw
+ -> fmap (flipBilin contractTensorWith $ dw) $ l )
+ contractLinearMapAgainst = arr coCurryLinearMap >>> bilinearFunction (\l f
+ -> (contractLinearMapAgainst . fmap transposeTensor $ l)
+ . uncurryLinearFn $f )
+
+instance ∀ s u v . (Num''' s, LSpace u, LSpace v, Scalar u ~ s, Scalar v ~ s)
+ => TensorSpace (Tensor s u v) where
+ type TensorProduct (Tensor s u v) w = TensorProduct u (Tensor s v w)
+ zeroTensor = lassocTensor $ zeroTensor
+ toFlatTensor = arr lassocTensor . fmap toFlatTensor
+ fromFlatTensor = fmap fromFlatTensor . arr rassocTensor
+ addTensors t₁ t₂ = lassocTensor $ (rassocTensor$t₁) ^+^ (rassocTensor$t₂)
+ subtractTensors t₁ t₂ = lassocTensor $ (rassocTensor$t₁) ^-^ (rassocTensor$t₂)
+ scaleTensor = LinearFunction $ \μ -> arr lassocTensor . scaleWith μ . arr rassocTensor
+ negateTensor = arr lassocTensor . lNegateV . arr rassocTensor
+ tensorProduct = flipBilin $ LinearFunction $ \w
+ -> arr lassocTensor . fmap (flipBilin tensorProduct $ w)
+ transposeTensor = fmap transposeTensor . arr rassocTensor
+ . transposeTensor . fmap transposeTensor . arr rassocTensor
+ fmapTensor = LinearFunction $ \f
+ -> arr lassocTensor <<< fmap (fmap f) <<< arr rassocTensor
+ fzipTensorWith = LinearFunction $ \f
+ -> arr lassocTensor <<< fzipWith (fzipWith f)
+ <<< arr rassocTensor *** arr rassocTensor
+ coerceFmapTensorProduct = cftlp
+ where cftlp :: ∀ a b p . p (Tensor s u v) -> Coercion a b
+ -> Coercion (TensorProduct u (Tensor s v a))
+ (TensorProduct u (Tensor s v b))
+ cftlp _ c = coerceFmapTensorProduct ([]::[u])
+ (fmap c :: Coercion (v⊗a) (v⊗b))
+instance ∀ s u v . (Num''' s, LSpace u, LSpace v, Scalar u ~ s, Scalar v ~ s)
+ => LinearSpace (Tensor s u v) where
+ type DualVector (Tensor s u v) = Tensor s (DualVector u) (DualVector v)
+ linearId = uncurryLinearMap $ fmap (fmap transposeTensor . blockVectSpan') $ id
+ coerceDoubleDual = Coercion
+ blockVectSpan = arr lassocTensor . arr (fmap $ fmap uncurryLinearMap)
+ . fmap (transposeTensor . arr deferLinearMap) . blockVectSpan
+ . arr deferLinearMap . fmap transposeTensor . blockVectSpan'
+ applyLinear = LinearFunction $ \f -> contractMapTensor
+ . fmap (applyLinear$curryLinearMap$f) . transposeTensor
+ applyDualVector = bilinearFunction $ \f t
+ -> (contractLinearMapAgainst $ (fromTensor$f))
+ . contractTensorWith $ t
+ composeLinear = bilinearFunction $ \f g
+ -> uncurryLinearMap $ fmap (fmap $ applyLinear $ f) $ (curryLinearMap$g)
+ contractTensorMap = contractTensorMap
+ . fmap (transposeTensor . contractTensorMap
+ . fmap (arr rassocTensor . transposeTensor . arr rassocTensor))
+ . arr curryLinearMap
+ contractMapTensor = contractTensorMap . fmap transposeTensor . contractMapTensor
+ . fmap (arr (curryLinearMap . hasteLinearMap) . transposeTensor)
+ . arr rassocTensor
+ contractTensorWith = arr rassocTensor >>> bilinearFunction (\l dw
+ -> fmap (flipBilin contractTensorWith $ dw) $ l )
+ contractLinearMapAgainst = arr curryLinearMap >>> bilinearFunction (\l f
+ -> (contractLinearMapAgainst $ l)
+ $ contractTensorMap . fmap (transposeTensor . f) )
+
+
+
+type DualSpace v = v+>Scalar v
+
+type Fractional' s = (Fractional s, Eq s, VectorSpace s, Scalar s ~ s)
+type Fractional'' s = (Fractional' s, LSpace s)
+
+
+
+instance (Num''' s, LSpace v, Scalar v ~ s)
+ => Functor (Tensor s v) (LinearFunction s) (LinearFunction s) where
+ fmap f = fmapTensor $ f
+instance (Num''' s, LSpace v, Scalar v ~ s)
+ => Monoidal (Tensor s v) (LinearFunction s) (LinearFunction s) where
+ pureUnit = const0
+ fzipWith f = fzipTensorWith $ f
+
+instance (Num''' s, LSpace v, Scalar v ~ s)
+ => Functor (LinearMap s v) (LinearFunction s) (LinearFunction s) where
+ fmap f = arr fromTensor . fmap f . arr asTensor
+instance (Num''' s, LSpace v, Scalar v ~ s)
+ => Monoidal (LinearMap s v) (LinearFunction s) (LinearFunction s) where
+ pureUnit = const0
+ fzipWith f = arr asTensor *** arr asTensor >>> fzipWith f >>> arr fromTensor
+
+instance (Num''' s, TensorSpace v, Scalar v ~ s)
+ => Functor (Tensor s v) Coercion Coercion where
+ fmap = crcFmap
+ where crcFmap :: ∀ s v a b . (TensorSpace v, Scalar v ~ s)
+ => Coercion a b -> Coercion (Tensor s v a) (Tensor s v b)
+ crcFmap f = case coerceFmapTensorProduct ([]::[v]) f of
+ Coercion -> Coercion
+
+instance (LSpace v, Num''' s, Scalar v ~ s)
+ => Functor (LinearMap s v) Coercion Coercion where
+ fmap = crcFmap
+ where crcFmap :: ∀ s v a b . (LSpace v, Num''' s, Scalar v ~ s)
+ => Coercion a b -> Coercion (LinearMap s v a) (LinearMap s v b)
+ crcFmap f = case coerceFmapTensorProduct ([]::[DualVector v]) f of
+ Coercion -> Coercion
+
+instance Category (LinearFunction s) where
+ type Object (LinearFunction s) v = (LSpace v, Scalar v ~ s)
+ id = LinearFunction id
+ LinearFunction f . LinearFunction g = LinearFunction $ f.g
+instance Num''' s => Cartesian (LinearFunction s) where
+ type UnitObject (LinearFunction s) = ZeroDim s
+ swap = LinearFunction swap
+ attachUnit = LinearFunction (, Origin)
+ detachUnit = LinearFunction fst
+ regroup = LinearFunction regroup
+ regroup' = LinearFunction regroup'
+instance Num''' s => Morphism (LinearFunction s) where
+ LinearFunction f***LinearFunction g = LinearFunction $ f***g
+instance Num''' s => PreArrow (LinearFunction s) where
+ LinearFunction f&&&LinearFunction g = LinearFunction $ f&&&g
+ fst = LinearFunction fst; snd = LinearFunction snd
+ terminal = const0
+instance EnhancedCat (->) (LinearFunction s) where
+ arr = getLinearFunction
+instance EnhancedCat (LinearFunction s) Coercion where
+ arr = LinearFunction . coerceWith
+
+instance (LSpace w, Scalar w ~ s)
+ => Functor (LinearFunction s w) (LinearFunction s) (LinearFunction s) where
+ fmap f = LinearFunction (f.)
+
+
+deferLinearFn :: Coercion (LinearFunction s u (Tensor s v w))
+ (Tensor s (LinearFunction s u v) w)
+deferLinearFn = Coercion
+
+hasteLinearFn :: Coercion (Tensor s (LinearFunction s u v) w)
+ (LinearFunction s u (Tensor s v w))
+hasteLinearFn = Coercion
+
+
+instance (LSpace u, LSpace v, Scalar u ~ s, Scalar v ~ s)
+ => TensorSpace (LinearFunction s u v) where
+ type TensorProduct (LinearFunction s u v) w = LinearFunction s u (Tensor s v w)
+ zeroTensor = deferLinearFn $ const0
+ toFlatTensor = arr deferLinearFn . fmap toFlatTensor
+ fromFlatTensor = fmap fromFlatTensor . arr hasteLinearFn
+ addTensors t s = deferLinearFn $ (hasteLinearFn$t)^+^(hasteLinearFn$s)
+ subtractTensors t s = deferLinearFn $ (hasteLinearFn$t)^-^(hasteLinearFn$s)
+ scaleTensor = LinearFunction $ \μ -> arr deferLinearFn . scaleWith μ . arr hasteLinearFn
+ negateTensor = arr deferLinearFn . lNegateV . arr hasteLinearFn
+ tensorProduct = flipBilin $ LinearFunction $
+ \w -> arr deferLinearFn . fmap (flipBilin tensorProduct $ w)
+ transposeTensor = arr hasteLinearFn >>> LinearFunction tp
+ where tp f = fmap (LinearFunction $ \dw -> (flipBilin contractTensorWith$dw) . f)
+ $ idTensor
+ fmapTensor = bilinearFunction $ \f g
+ -> deferLinearFn $ fmap f . (hasteLinearFn$g)
+ fzipTensorWith = bilinearFunction $ \f (g,h)
+ -> deferLinearFn $ fzipWith f
+ <<< (hasteLinearFn$g)&&&(hasteLinearFn$h)
+ coerceFmapTensorProduct = cftpLf
+ where cftpLf :: ∀ s u v a b p . TensorSpace v
+ => p (LinearFunction s u v) -> Coercion a b
+ -> Coercion (LinearFunction s u (Tensor s v a))
+ (LinearFunction s u (Tensor s v b))
+ cftpLf p c = case coerceFmapTensorProduct ([]::[v]) c of
+ Coercion -> Coercion
+
+coCurryLinearFn :: Coercion (LinearMap s (LinearFunction s u v) w)
+ (LinearFunction s v (Tensor s u w))
+coCurryLinearFn = Coercion
+
+coUncurryLinearFn :: Coercion (LinearFunction s u (Tensor s v w))
+ (LinearMap s (LinearFunction s v u) w)
+coUncurryLinearFn = Coercion
+
+instance (LSpace u, LSpace v, Scalar u ~ s, Scalar v ~ s)
+ => LinearSpace (LinearFunction s u v) where
+ type DualVector (LinearFunction s u v) = LinearFunction s v u
+ linearId = coUncurryLinearFn $ LinearFunction $
+ \v -> fmap (fmap (scaleV v) . applyDualVector) $ idTensor
+ coerceDoubleDual = Coercion
+ blockVectSpan = arr deferLinearFn . bilinearFunction (\w u
+ -> fmap ( arr coUncurryLinearFn
+ . fmap (flipBilin tensorProduct$w) . applyLinear )
+ $ (blockVectSpan$u) )
+ contractTensorMap = arr coCurryLinearFn
+ >>> arr (fmap (fmap hasteLinearFn))
+ >>> sampleLinearFunction
+ >>> fmap contractFnTensor
+ >>> contractTensorMap
+ contractMapTensor = arr hasteLinearFn
+ >>> arr (fmap (fmap coCurryLinearFn))
+ >>> sampleLinearFunction
+ >>> fmap contractFnTensor
+ >>> contractTensorMap
+ contractLinearMapAgainst = arr coCurryLinearFn
+ >>> bilinearFunction (\v2uw w2uv
+ -> trace . fmap (contractTensorFn . fmap v2uw)
+ . sampleLinearFunction $ w2uv )
+ applyDualVector = sampleLinearFunction >>> contractLinearMapAgainst
+ applyLinear = arr coCurryLinearFn >>> LinearFunction (\f
+ -> contractTensorFn . fmap f)
+ composeLinear = LinearFunction $ \f
+ -> arr coCurryLinearFn >>> fmap (fmap $ applyLinear $ f)
+ >>> arr coUncurryLinearFn
+
diff --git a/Math/LinearMap/Category/Instances.hs b/Math/LinearMap/Category/Instances.hs
new file mode 100644
--- /dev/null
+++ b/Math/LinearMap/Category/Instances.hs
@@ -0,0 +1,228 @@
+-- |
+-- Module : Math.LinearMap.Category.Instances
+-- Copyright : (c) Justus Sagemüller 2016
+-- License : GPL v3
+--
+-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
+-- Stability : experimental
+-- Portability : portable
+--
+{-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE ConstraintKinds #-}
+{-# LANGUAGE UndecidableInstances #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE UnicodeSyntax #-}
+{-# LANGUAGE CPP #-}
+{-# LANGUAGE TupleSections #-}
+
+module Math.LinearMap.Category.Instances where
+
+import Math.LinearMap.Category.Class
+
+import Data.VectorSpace
+import Data.Basis
+
+import Prelude ()
+import qualified Prelude as Hask
+
+import Control.Category.Constrained.Prelude
+import Control.Arrow.Constrained
+
+import Data.Coerce
+import Data.Type.Coercion
+
+import Data.Foldable (foldl')
+
+import Data.VectorSpace.Free
+import qualified Linear.Matrix as Mat
+import qualified Linear.Vector as Mat
+import qualified Linear.Metric as Mat
+import Linear ( V0(V0), V1(V1), V2(V2), V3(V3), V4(V4)
+ , _x, _y, _z, _w )
+import Control.Lens ((^.))
+
+import Math.LinearMap.Asserted
+import Math.VectorSpace.ZeroDimensional
+
+
+type ℝ = Double
+
+instance TensorSpace ℝ where
+ type TensorProduct ℝ w = w
+ zeroTensor = Tensor zeroV
+ scaleTensor = LinearFunction (pretendLike Tensor) . scale
+ addTensors (Tensor v) (Tensor w) = Tensor $ v ^+^ w
+ subtractTensors (Tensor v) (Tensor w) = Tensor $ v ^-^ w
+ negateTensor = pretendLike Tensor lNegateV
+ toFlatTensor = follow Tensor
+ fromFlatTensor = flout Tensor
+ tensorProduct = LinearFunction $ \μ -> follow Tensor . scaleWith μ
+ transposeTensor = toFlatTensor . flout Tensor
+ fmapTensor = LinearFunction $ pretendLike Tensor
+ fzipTensorWith = LinearFunction
+ $ \f -> follow Tensor <<< f <<< flout Tensor *** flout Tensor
+ coerceFmapTensorProduct _ Coercion = Coercion
+instance LinearSpace ℝ where
+ type DualVector ℝ = ℝ
+ linearId = LinearMap 1
+ idTensor = Tensor 1
+ fromLinearForm = flout LinearMap
+ coerceDoubleDual = Coercion
+ contractTensorMap = flout Tensor . flout LinearMap
+ contractMapTensor = flout LinearMap . flout Tensor
+ contractTensorWith = flout Tensor >>> applyDualVector
+ contractLinearMapAgainst = flout LinearMap >>> flipBilin lApply
+ blockVectSpan = follow Tensor . follow LinearMap
+ applyDualVector = scale
+ applyLinear = elacs . flout LinearMap
+ composeLinear = LinearFunction $ \f -> follow LinearMap . arr f . flout LinearMap
+
+#define FreeLinearSpace(V, LV, tp, bspan, tenspl, dspan, contraction, contraaction) \
+instance Num''' s => TensorSpace (V s) where { \
+ type TensorProduct (V s) w = V w; \
+ zeroTensor = Tensor $ pure zeroV; \
+ addTensors (Tensor m) (Tensor n) = Tensor $ liftA2 (^+^) m n; \
+ subtractTensors (Tensor m) (Tensor n) = Tensor $ liftA2 (^-^) m n; \
+ negateTensor = LinearFunction $ Tensor . fmap negateV . getTensorProduct; \
+ scaleTensor = bilinearFunction \
+ $ \μ -> Tensor . fmap (μ*^) . getTensorProduct; \
+ toFlatTensor = follow Tensor; \
+ fromFlatTensor = flout Tensor; \
+ tensorProduct = bilinearFunction $ \w v -> Tensor $ fmap (*^v) w; \
+ transposeTensor = LinearFunction (tp); \
+ fmapTensor = bilinearFunction $ \
+ \(LinearFunction f) -> pretendLike Tensor $ fmap f; \
+ fzipTensorWith = bilinearFunction $ \
+ \(LinearFunction f) (Tensor vw, Tensor vx) \
+ -> Tensor $ liftA2 (curry f) vw vx; \
+ coerceFmapTensorProduct _ Coercion = Coercion }; \
+instance Num''' s => LinearSpace (V s) where { \
+ type DualVector (V s) = V s; \
+ linearId = LV Mat.identity; \
+ idTensor = Tensor Mat.identity; \
+ coerceDoubleDual = Coercion; \
+ fromLinearForm = flout LinearMap; \
+ blockVectSpan = LinearFunction $ Tensor . (bspan); \
+ contractTensorMap = LinearFunction $ (contraction) . coerce . getLinearMap; \
+ contractMapTensor = LinearFunction $ (contraction) . coerce . getTensorProduct; \
+ contractTensorWith = bilinearFunction $ \
+ \(Tensor wv) dw -> fmap (arr $ applyDualVector $ dw) wv; \
+ contractLinearMapAgainst = bilinearFunction $ getLinearMap >>> (contraaction); \
+ applyDualVector = bilinearFunction Mat.dot; \
+ applyLinear = bilinearFunction $ \(LV m) \
+ -> foldl' (^+^) zeroV . liftA2 (^*) m; \
+ composeLinear = bilinearFunction $ \
+ \f (LinearMap g) -> LinearMap $ fmap (f$) g }
+FreeLinearSpace( V0
+ , LinearMap
+ , \(Tensor V0) -> zeroV
+ , \_ -> V0
+ , \_ -> LinearMap V0
+ , LinearMap V0
+ , \V0 -> zeroV
+ , \V0 _ -> 0 )
+FreeLinearSpace( V1
+ , LinearMap
+ , \(Tensor (V1 w₀)) -> w₀⊗V1 1
+ , \w -> V1 (LinearMap $ V1 w)
+ , \w -> LinearMap $ V1 (Tensor $ V1 w)
+ , LinearMap . V1 . blockVectSpan $ V1 1
+ , \(V1 (V1 w)) -> w
+ , \(V1 x) f -> (f$x)^._x )
+FreeLinearSpace( V2
+ , LinearMap
+ , \(Tensor (V2 w₀ w₁)) -> w₀⊗V2 1 0
+ ^+^ w₁⊗V2 0 1
+ , \w -> V2 (LinearMap $ V2 w zeroV)
+ (LinearMap $ V2 zeroV w)
+ , \w -> LinearMap $ V2 (Tensor $ V2 w zeroV)
+ (Tensor $ V2 zeroV w)
+ , LinearMap $ V2 (blockVectSpan $ V2 1 0)
+ (blockVectSpan $ V2 0 1)
+ , \(V2 (V2 w₀ _)
+ (V2 _ w₁)) -> w₀^+^w₁
+ , \(V2 x y) f -> (f$x)^._x + (f$y)^._y )
+FreeLinearSpace( V3
+ , LinearMap
+ , \(Tensor (V3 w₀ w₁ w₂)) -> w₀⊗V3 1 0 0
+ ^+^ w₁⊗V3 0 1 0
+ ^+^ w₂⊗V3 0 0 1
+ , \w -> V3 (LinearMap $ V3 w zeroV zeroV)
+ (LinearMap $ V3 zeroV w zeroV)
+ (LinearMap $ V3 zeroV zeroV w)
+ , \w -> LinearMap $ V3 (Tensor $ V3 w zeroV zeroV)
+ (Tensor $ V3 zeroV w zeroV)
+ (Tensor $ V3 zeroV zeroV w)
+ , LinearMap $ V3 (blockVectSpan $ V3 1 0 0)
+ (blockVectSpan $ V3 0 1 0)
+ (blockVectSpan $ V3 0 0 1)
+ , \(V3 (V3 w₀ _ _)
+ (V3 _ w₁ _)
+ (V3 _ _ w₂)) -> w₀^+^w₁^+^w₂
+ , \(V3 x y z) f -> (f$x)^._x + (f$y)^._y + (f$z)^._z )
+FreeLinearSpace( V4
+ , LinearMap
+ , \(Tensor (V4 w₀ w₁ w₂ w₃)) -> w₀⊗V4 1 0 0 0
+ ^+^ w₁⊗V4 0 1 0 0
+ ^+^ w₂⊗V4 0 0 1 0
+ ^+^ w₃⊗V4 0 0 0 1
+ , \w -> V4 (LinearMap $ V4 w zeroV zeroV zeroV)
+ (LinearMap $ V4 zeroV w zeroV zeroV)
+ (LinearMap $ V4 zeroV zeroV w zeroV)
+ (LinearMap $ V4 zeroV zeroV zeroV w)
+ , \w -> LinearMap $ V4 (Tensor $ V4 w zeroV zeroV zeroV)
+ (Tensor $ V4 zeroV w zeroV zeroV)
+ (Tensor $ V4 zeroV zeroV w zeroV)
+ (Tensor $ V4 zeroV zeroV zeroV w)
+ , LinearMap $ V4 (blockVectSpan $ V4 1 0 0 0)
+ (blockVectSpan $ V4 0 1 0 0)
+ (blockVectSpan $ V4 0 0 1 0)
+ (blockVectSpan $ V4 0 0 0 1)
+ , \(V4 (V4 w₀ _ _ _)
+ (V4 _ w₁ _ _)
+ (V4 _ _ w₂ _)
+ (V4 _ _ _ w₃)) -> w₀^+^w₁^+^w₂^+^w₃
+ , \(V4 x y z w) f -> (f$x)^._x + (f$y)^._y + (f$z)^._z + (f$w)^._w )
+
+
+
+instance (Num''' n, TensorProduct (DualVector n) n ~ n) => Num (LinearMap n n n) where
+ LinearMap n + LinearMap m = LinearMap $ n + m
+ LinearMap n - LinearMap m = LinearMap $ n - m
+ LinearMap n * LinearMap m = LinearMap $ n * m
+ abs (LinearMap n) = LinearMap $ abs n
+ signum (LinearMap n) = LinearMap $ signum n
+ fromInteger = LinearMap . fromInteger
+
+instance (Fractional'' n, TensorProduct (DualVector n) n ~ n)
+ => Fractional (LinearMap n n n) where
+ LinearMap n / LinearMap m = LinearMap $ n / m
+ recip (LinearMap n) = LinearMap $ recip n
+ fromRational = LinearMap . fromRational
+
+
+
+
+
+
+instance (LSpace u, LSpace v, s~Scalar u, s~Scalar v)
+ => AffineSpace (Tensor s u v) where
+ type Diff (Tensor s u v) = Tensor s u v
+ (.-.) = (^-^)
+ (.+^) = (^+^)
+instance (LSpace u, LSpace v, s~Scalar u, s~Scalar v)
+ => AffineSpace (LinearMap s u v) where
+ type Diff (LinearMap s u v) = LinearMap s u v
+ (.-.) = (^-^)
+ (.+^) = (^+^)
+instance (LSpace u, LSpace v, s~Scalar u, s~Scalar v)
+ => AffineSpace (LinearFunction s u v) where
+ type Diff (LinearFunction s u v) = LinearFunction s u v
+ (.-.) = (^-^)
+ (.+^) = (^+^)
+
+
+
diff --git a/Math/VectorSpace/Docile.hs b/Math/VectorSpace/Docile.hs
new file mode 100644
--- /dev/null
+++ b/Math/VectorSpace/Docile.hs
@@ -0,0 +1,637 @@
+-- |
+-- Module : Math.VectorSpace.Docile
+-- Copyright : (c) Justus Sagemüller 2016
+-- License : GPL v3
+--
+-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
+-- Stability : experimental
+-- Portability : portable
+--
+
+
+{-# LANGUAGE CPP #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE StandaloneDeriving #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE UndecidableInstances #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE UnicodeSyntax #-}
+{-# LANGUAGE TupleSections #-}
+{-# LANGUAGE LambdaCase #-}
+{-# LANGUAGE ConstraintKinds #-}
+
+module Math.VectorSpace.Docile where
+
+import Math.LinearMap.Category.Class
+import Math.LinearMap.Category.Instances
+import Math.LinearMap.Asserted
+
+import Data.Tree (Tree(..), Forest)
+import Data.List (sortBy, foldl')
+import qualified Data.Set as Set
+import Data.Set (Set)
+import Data.Ord (comparing)
+import Data.List (maximumBy, unfoldr)
+import Data.Foldable (toList)
+import Data.Semigroup
+
+import Data.VectorSpace
+import Data.Basis
+
+import Prelude ()
+import qualified Prelude as Hask
+
+import Control.Category.Constrained.Prelude hiding ((^))
+import Control.Arrow.Constrained
+
+import Linear ( V0(V0), V1(V1), V2(V2), V3(V3), V4(V4)
+ , _x, _y, _z, _w )
+import qualified Data.Vector.Unboxed as UArr
+import Data.VectorSpace.Free
+import Math.VectorSpace.ZeroDimensional
+import qualified Linear.Matrix as Mat
+import qualified Linear.Vector as Mat
+import Control.Lens ((^.))
+import Data.Coerce
+
+import Numeric.IEEE
+
+
+
+
+-- | 'SemiInner' is the class of vector spaces with finite subspaces in which
+-- you can define a basis that can be used to project from the whole space
+-- into the subspace. The usual application is for using a kind of
+-- to
+-- give an approximate solution (see '\$') to a linear equation in a possibly
+-- infinite-dimensional space.
+--
+-- Of course, this also works for spaces which are already finite-dimensional themselves.
+class LSpace v => SemiInner v where
+ -- | Lazily enumerate choices of a basis of functionals that can be made dual
+ -- to the given vectors, in order of preference (which roughly means, large in
+ -- the normal direction.) I.e., if the vector @𝑣@ is assigned early to the
+ -- dual vector @𝑣'@, then @(𝑣' $ 𝑣)@ should be large and all the other products
+ -- comparably small.
+ --
+ -- The purpose is that we should be able to make this basis orthonormal
+ -- with a ~Gaussian-elimination approach, in a way that stays numerically
+ -- stable. This is otherwise known as the /choice of a pivot element/.
+ --
+ -- For simple finite-dimensional array-vectors, you can easily define this
+ -- method using 'cartesianDualBasisCandidates'.
+ dualBasisCandidates :: [(Int,v)] -> Forest (Int, DualVector v)
+
+cartesianDualBasisCandidates
+ :: [DualVector v] -- ^ Set of canonical basis functionals.
+ -> (v -> [ℝ]) -- ^ Decompose a vector in /absolute value/ components.
+ -- the list indices should correspond to those in
+ -- the functional list.
+ -> ([(Int,v)] -> Forest (Int, DualVector v))
+ -- ^ Suitable definition of 'dualBasisCandidates'.
+cartesianDualBasisCandidates dvs abss vcas = go 0 sorted
+ where sorted = sortBy (comparing $ negate . snd . snd)
+ [ (i, (av, maximum av)) | (i,v)<-vcas, let av = abss v ]
+ go k ((i,(av,_)):scs)
+ | k [ℝ] -> [ℝ]
+ zeroAt _ [] = []
+ zeroAt 0 (_:l) = (-1/0):l
+ zeroAt j (e:l) = e : zeroAt (j-1) l
+
+instance (Fractional'' s, SemiInner s) => SemiInner (ZeroDim s) where
+ dualBasisCandidates _ = []
+instance (Fractional'' s, SemiInner s) => SemiInner (V0 s) where
+ dualBasisCandidates _ = []
+
+(<.>^) :: LSpace v => DualVector v -> v -> Scalar v
+f<.>^v = (applyDualVector$f)$v
+
+orthonormaliseDuals :: (SemiInner v, LSpace v, Fractional'' (Scalar v))
+ => [(v, DualVector v)] -> [(v,DualVector v)]
+orthonormaliseDuals [] = []
+orthonormaliseDuals ((v,v'₀):ws)
+ = (v,v') : [(w, w' ^-^ (w'<.>^v)*^v') | (w,w')<-wssys]
+ where wssys = orthonormaliseDuals ws
+ v'₁ = foldl' (\v'i (w,w') -> v'i ^-^ (v'i<.>^w)*^w') v'₀ wssys
+ v' = v'₁ ^/ (v'₁<.>^v)
+
+dualBasis :: (SemiInner v, LSpace v, Fractional'' (Scalar v)) => [v] -> [DualVector v]
+dualBasis vs = snd <$> orthonormaliseDuals (zip' vsIxed candidates)
+ where zip' ((i,v):vs) ((j,v'):ds)
+ | i SemiInner (V1 s) where
+ dualBasisCandidates = fmap ((`Node`[]) . second recip)
+ . sortBy (comparing $ negate . abs . snd)
+ . filter ((/=0) . snd)
+
+#define FreeSemiInner(V, sabs) \
+instance SemiInner (V) where { \
+ dualBasisCandidates \
+ = cartesianDualBasisCandidates Mat.basis (fmap sabs . toList) }
+FreeSemiInner(V2 ℝ, abs)
+FreeSemiInner(V3 ℝ, abs)
+FreeSemiInner(V4 ℝ, abs)
+
+instance ∀ u v . ( SemiInner u, SemiInner v, Scalar u ~ Scalar v ) => SemiInner (u,v) where
+ dualBasisCandidates = fmap (\(i,(u,v))->((i,u),(i,v))) >>> unzip
+ >>> dualBasisCandidates *** dualBasisCandidates
+ >>> combineBaseis False mempty
+ where combineBaseis :: Bool -> Set Int
+ -> ( Forest (Int, DualVector u)
+ , Forest (Int, DualVector v) )
+ -> Forest (Int, (DualVector u, DualVector v))
+ combineBaseis _ _ ([], []) = []
+ combineBaseis False forbidden (Node (i,du) bu' : abu, bv)
+ | i`Set.member`forbidden = combineBaseis False forbidden (abu, bv)
+ | otherwise
+ = Node (i, (du, zeroV))
+ (combineBaseis True (Set.insert i forbidden) (bu', bv))
+ : combineBaseis False forbidden (abu, bv)
+ combineBaseis True forbidden (bu, Node (i,dv) bv' : abv)
+ | i`Set.member`forbidden = combineBaseis True forbidden (bu, abv)
+ | otherwise
+ = Node (i, (zeroV, dv))
+ (combineBaseis False (Set.insert i forbidden) (bu, bv'))
+ : combineBaseis True forbidden (bu, abv)
+ combineBaseis _ forbidden (bu, []) = combineBaseis False forbidden (bu,[])
+ combineBaseis _ forbidden ([], bv) = combineBaseis True forbidden ([],bv)
+
+
+instance ∀ s u v . ( LSpace u, FiniteDimensional (DualVector u), SemiInner (DualVector u)
+ , SemiInner v, FiniteDimensional v
+ , Scalar u ~ s, Scalar v ~ s, RealFrac' s )
+ => SemiInner (Tensor s u v) where
+ dualBasisCandidates = map (fmap (second $ arr transposeTensor . arr asTensor))
+ . dualBasisCandidates
+ . map (second $ arr asLinearMap)
+
+instance ∀ s u v . ( SemiInner u, FiniteDimensional u, Scalar u ~ s
+ , SemiInner v, FiniteDimensional v, Scalar v ~ s, RealFrac' s )
+ => SemiInner (LinearMap s u v) where
+ dualBasisCandidates = sequenceForest
+ . map (second pseudoInverse) -- this is not efficient
+ where sequenceForest [] = []
+ sequenceForest (x:xs) = [Node x $ sequenceForest xs]
+
+(^/^) :: (InnerSpace v, Eq (Scalar v), Fractional (Scalar v)) => v -> v -> Scalar v
+v^/^w = case (v<.>w) of
+ 0 -> 0
+ vw -> vw / (w<.>w)
+
+type DList x = [x]->[x]
+
+class (LSpace v, LSpace (Scalar v)) => FiniteDimensional v where
+ -- | Whereas 'Basis'-values refer to a single basis vector, a single
+ -- 'SubBasis' value represents a collection of such basis vectors,
+ -- which can be used to associate a vector with a list of coefficients.
+ --
+ -- For spaces with a canonical finite basis, 'SubBasis' does not actually
+ -- need to contain any information, it can simply have the full finite
+ -- basis as its only value. Even for large sparse spaces, it should only
+ -- have a very coarse structure that can be shared by many vectors.
+ data SubBasis v :: *
+
+ entireBasis :: SubBasis v
+
+ enumerateSubBasis :: SubBasis v -> [v]
+
+ subbasisDimension :: SubBasis v -> Int
+ subbasisDimension = length . enumerateSubBasis
+
+ -- | Split up a linear map in “column vectors” WRT some suitable basis.
+ decomposeLinMap :: (LSpace w, Scalar w ~ Scalar v) => (v+>w) -> (SubBasis v, DList w)
+
+ -- | Expand in the given basis, if possible. Else yield a superbasis of the given
+ -- one, in which this /is/ possible, and the decomposition therein.
+ decomposeLinMapWithin :: (LSpace w, Scalar w ~ Scalar v)
+ => SubBasis v -> (v+>w) -> Either (SubBasis v, DList w) (DList w)
+
+ -- | Assemble a vector from coefficients in some basis. Return any excess coefficients.
+ recomposeSB :: SubBasis v -> [Scalar v] -> (v, [Scalar v])
+
+ recomposeSBTensor :: (FiniteDimensional w, Scalar w ~ Scalar v)
+ => SubBasis v -> SubBasis w -> [Scalar v] -> (v⊗w, [Scalar v])
+
+ recomposeLinMap :: (LSpace w, Scalar w~Scalar v) => SubBasis v -> [w] -> (v+>w, [w])
+
+ -- | Given a function that interprets a coefficient-container as a vector representation,
+ -- build a linear function mapping to that space.
+ recomposeContraLinMap :: (LinearSpace w, Scalar w ~ Scalar v, Hask.Functor f)
+ => (f (Scalar w) -> w) -> f (DualVector v) -> v+>w
+
+ recomposeContraLinMapTensor
+ :: ( FiniteDimensional u, LinearSpace w
+ , Scalar u ~ Scalar v, Scalar w ~ Scalar v, Hask.Functor f )
+ => (f (Scalar w) -> w) -> f (DualVector v⊗DualVector u) -> (v⊗u)+>w
+
+ -- | The existance of a finite basis gives us an isomorphism between a space
+ -- and its dual space. Note that this isomorphism is not natural (i.e. it
+ -- depends on the actual choice of basis, unlike everything else in this
+ -- library).
+ uncanonicallyFromDual :: DualVector v -+> v
+ uncanonicallyToDual :: v -+> DualVector v
+
+
+
+instance (Num''' s) => FiniteDimensional (ZeroDim s) where
+ data SubBasis (ZeroDim s) = ZeroBasis
+ entireBasis = ZeroBasis
+ enumerateSubBasis ZeroBasis = []
+ subbasisDimension ZeroBasis = 0
+ recomposeSB ZeroBasis l = (Origin, l)
+ recomposeSBTensor ZeroBasis _ l = (Tensor Origin, l)
+ recomposeLinMap ZeroBasis l = (LinearMap Origin, l)
+ decomposeLinMap _ = (ZeroBasis, id)
+ decomposeLinMapWithin ZeroBasis _ = pure id
+ recomposeContraLinMap _ _ = LinearMap Origin
+ recomposeContraLinMapTensor _ _ = LinearMap Origin
+ uncanonicallyFromDual = id
+ uncanonicallyToDual = id
+
+instance (Num''' s, LinearSpace s) => FiniteDimensional (V0 s) where
+ data SubBasis (V0 s) = V0Basis
+ entireBasis = V0Basis
+ enumerateSubBasis V0Basis = []
+ subbasisDimension V0Basis = 0
+ recomposeSB V0Basis l = (V0, l)
+ recomposeSBTensor V0Basis _ l = (Tensor V0, l)
+ recomposeLinMap V0Basis l = (LinearMap V0, l)
+ decomposeLinMap _ = (V0Basis, id)
+ decomposeLinMapWithin V0Basis _ = pure id
+ recomposeContraLinMap _ _ = LinearMap V0
+ recomposeContraLinMapTensor _ _ = LinearMap V0
+ uncanonicallyFromDual = id
+ uncanonicallyToDual = id
+
+instance FiniteDimensional ℝ where
+ data SubBasis ℝ = RealsBasis
+ entireBasis = RealsBasis
+ enumerateSubBasis RealsBasis = [1]
+ subbasisDimension RealsBasis = 1
+ recomposeSB RealsBasis [] = (0, [])
+ recomposeSB RealsBasis (μ:cs) = (μ, cs)
+ recomposeSBTensor RealsBasis bw = first Tensor . recomposeSB bw
+ recomposeLinMap RealsBasis (w:ws) = (LinearMap w, ws)
+ decomposeLinMap (LinearMap v) = (RealsBasis, (v:))
+ decomposeLinMapWithin RealsBasis (LinearMap v) = pure (v:)
+ recomposeContraLinMap fw = LinearMap . fw
+ recomposeContraLinMapTensor fw = arr uncurryLinearMap . LinearMap
+ . recomposeContraLinMap fw . fmap getTensorProduct
+ uncanonicallyFromDual = id
+ uncanonicallyToDual = id
+
+#define FreeFiniteDimensional(V, VB, dimens, take, give) \
+instance (Num''' s, LSpace s) \
+ => FiniteDimensional (V s) where { \
+ data SubBasis (V s) = VB; \
+ entireBasis = VB; \
+ enumerateSubBasis VB = toList $ Mat.identity; \
+ subbasisDimension VB = dimens; \
+ uncanonicallyFromDual = id; \
+ uncanonicallyToDual = id; \
+ recomposeSB _ (take:cs) = (give, cs); \
+ recomposeSB b cs = recomposeSB b $ cs ++ [0]; \
+ recomposeSBTensor VB bw cs = case recomposeMultiple bw dimens cs of \
+ {(take:[], cs') -> (Tensor (give), cs')}; \
+ recomposeLinMap VB (take:ws') = (LinearMap (give), ws'); \
+ decomposeLinMap (LinearMap m) = (VB, (toList m ++)); \
+ decomposeLinMapWithin VB (LinearMap m) = pure (toList m ++); \
+ recomposeContraLinMap fw mv \
+ = LinearMap $ (\v -> fw $ fmap (<.>^v) mv) <$> Mat.identity; \
+ recomposeContraLinMapTensor fw mv = LinearMap $ \
+ (\v -> fromLinearMap $ recomposeContraLinMap fw \
+ $ fmap (\(Tensor q) -> foldl' (^+^) zeroV $ liftA2 (*^) v q) mv) \
+ <$> Mat.identity }
+FreeFiniteDimensional(V1, V1Basis, 1, c₀ , V1 c₀ )
+FreeFiniteDimensional(V2, V2Basis, 2, c₀:c₁ , V2 c₀ c₁ )
+FreeFiniteDimensional(V3, V3Basis, 3, c₀:c₁:c₂ , V3 c₀ c₁ c₂ )
+FreeFiniteDimensional(V4, V4Basis, 4, c₀:c₁:c₂:c₃, V4 c₀ c₁ c₂ c₃)
+
+recomposeMultiple :: FiniteDimensional w
+ => SubBasis w -> Int -> [Scalar w] -> ([w], [Scalar w])
+recomposeMultiple bw n dc
+ | n<1 = ([], dc)
+ | otherwise = case recomposeSB bw dc of
+ (w, dc') -> first (w:) $ recomposeMultiple bw (n-1) dc'
+
+deriving instance Show (SubBasis ℝ)
+
+instance ( FiniteDimensional u, FiniteDimensional v
+ , Scalar u ~ Scalar v )
+ => FiniteDimensional (u,v) where
+ data SubBasis (u,v) = TupleBasis !(SubBasis u) !(SubBasis v)
+ entireBasis = TupleBasis entireBasis entireBasis
+ enumerateSubBasis (TupleBasis bu bv)
+ = ((,zeroV)<$>enumerateSubBasis bu) ++ ((zeroV,)<$>enumerateSubBasis bv)
+ subbasisDimension (TupleBasis bu bv) = subbasisDimension bu + subbasisDimension bv
+ decomposeLinMap (LinearMap (fu, fv))
+ = case (decomposeLinMap (asLinearMap$fu), decomposeLinMap (asLinearMap$fv)) of
+ ((bu, du), (bv, dv)) -> (TupleBasis bu bv, du . dv)
+ decomposeLinMapWithin (TupleBasis bu bv) (LinearMap (fu, fv))
+ = case ( decomposeLinMapWithin bu (asLinearMap$fu)
+ , decomposeLinMapWithin bv (asLinearMap$fv) ) of
+ (Left (bu', du), Left (bv', dv)) -> Left (TupleBasis bu' bv', du . dv)
+ (Left (bu', du), Right dv) -> Left (TupleBasis bu' bv, du . dv)
+ (Right du, Left (bv', dv)) -> Left (TupleBasis bu bv', du . dv)
+ (Right du, Right dv) -> Right $ du . dv
+ recomposeSB (TupleBasis bu bv) coefs = case recomposeSB bu coefs of
+ (u, coefs') -> case recomposeSB bv coefs' of
+ (v, coefs'') -> ((u,v), coefs'')
+ recomposeSBTensor (TupleBasis bu bv) bw cs = case recomposeSBTensor bu bw cs of
+ (tuw, cs') -> case recomposeSBTensor bv bw cs' of
+ (tvw, cs'') -> (Tensor (tuw, tvw), cs'')
+ recomposeLinMap (TupleBasis bu bv) ws = case recomposeLinMap bu ws of
+ (lmu, ws') -> first (lmu⊕) $ recomposeLinMap bv ws'
+ recomposeContraLinMap fw dds
+ = recomposeContraLinMap fw (fst<$>dds)
+ ⊕ recomposeContraLinMap fw (snd<$>dds)
+ recomposeContraLinMapTensor fw dds
+ = uncurryLinearMap
+ $ LinearMap ( fromLinearMap . curryLinearMap
+ $ recomposeContraLinMapTensor fw (fmap (\(Tensor(tu,_))->tu) dds)
+ , fromLinearMap . curryLinearMap
+ $ recomposeContraLinMapTensor fw (fmap (\(Tensor(_,tv))->tv) dds) )
+ uncanonicallyFromDual = uncanonicallyFromDual *** uncanonicallyFromDual
+ uncanonicallyToDual = uncanonicallyToDual *** uncanonicallyToDual
+
+deriving instance (Show (SubBasis u), Show (SubBasis v))
+ => Show (SubBasis (u,v))
+
+
+instance ∀ s u v .
+ ( FiniteDimensional u, FiniteDimensional v
+ , Scalar u~s, Scalar v~s, Fractional' (Scalar v) )
+ => FiniteDimensional (Tensor s u v) where
+ data SubBasis (Tensor s u v) = TensorBasis !(SubBasis u) !(SubBasis v)
+ entireBasis = TensorBasis entireBasis entireBasis
+ enumerateSubBasis (TensorBasis bu bv)
+ = [ u⊗v | u <- enumerateSubBasis bu, v <- enumerateSubBasis bv ]
+ subbasisDimension (TensorBasis bu bv) = subbasisDimension bu * subbasisDimension bv
+ decomposeLinMap muvw = case decomposeLinMap $ curryLinearMap $ muvw of
+ (bu, mvwsg) -> first (TensorBasis bu) . go id $ mvwsg []
+ where (go, _) = tensorLinmapDecompositionhelpers
+ decomposeLinMapWithin (TensorBasis bu bv) muvw
+ = case decomposeLinMapWithin bu $ curryLinearMap $ muvw of
+ Left (bu', mvwsg) -> let (_, (bv', ws)) = goWith bv id (mvwsg []) id
+ in Left (TensorBasis bu' bv', ws)
+ where (_, goWith) = tensorLinmapDecompositionhelpers
+ recomposeSB (TensorBasis bu bv) = recomposeSBTensor bu bv
+ recomposeSBTensor (TensorBasis bu bv) bw
+ = first (arr lassocTensor) . recomposeSBTensor bu (TensorBasis bv bw)
+ recomposeLinMap (TensorBasis bu bv) ws =
+ ( uncurryLinearMap $ fst . recomposeLinMap bu $ unfoldr (pure . recomposeLinMap bv) ws
+ , drop (subbasisDimension bu * subbasisDimension bv) ws )
+ recomposeContraLinMap = recomposeContraLinMapTensor
+ recomposeContraLinMapTensor fw dds
+ = uncurryLinearMap . uncurryLinearMap . fmap (curryLinearMap) . curryLinearMap
+ $ recomposeContraLinMapTensor fw $ fmap (arr rassocTensor) dds
+ uncanonicallyToDual = fmap uncanonicallyToDual
+ >>> transposeTensor >>> fmap uncanonicallyToDual
+ >>> transposeTensor
+ uncanonicallyFromDual = fmap uncanonicallyFromDual
+ >>> transposeTensor >>> fmap uncanonicallyFromDual
+ >>> transposeTensor
+
+tensorLinmapDecompositionhelpers
+ :: ( FiniteDimensional v, LSpace w , Scalar v~s, Scalar w~s )
+ => ( DList w -> [v+>w] -> (SubBasis v, DList w)
+ , SubBasis v -> DList w -> [v+>w] -> DList (v+>w)
+ -> (Bool, (SubBasis v, DList w)) )
+tensorLinmapDecompositionhelpers = (go, goWith)
+ where go _ [] = decomposeLinMap zeroV
+ go prevdc (mvw:mvws) = case decomposeLinMap mvw of
+ (bv, cfs) -> snd (goWith bv prevdc mvws (mvw:))
+ goWith bv prevdc [] prevs = (False, (bv, prevdc))
+ goWith bv prevdc (mvw:mvws) prevs = case decomposeLinMapWithin bv mvw of
+ Right cfs -> goWith bv (prevdc . cfs) mvws (prevs . (mvw:))
+ Left (bv', cfs) -> first (const True)
+ ( goWith bv' (regoWith bv' (prevs[]) . cfs)
+ mvws (prevs . (mvw:)) )
+ regoWith _ [] = id
+ regoWith bv (mvw:mvws) = case decomposeLinMapWithin bv mvw of
+ Right cfs -> cfs . regoWith bv mvws
+ Left _ -> error $
+ "Misbehaved FiniteDimensional instance: `decomposeLinMapWithin` should,\
+ \\nif it cannot decompose in the given basis, do so in a proper\
+ \\nsuperbasis of the given one (so that any vector that could be\
+ \\ndecomposed in the old basis can also be decomposed in the new one)."
+
+
+instance ∀ s u v .
+ ( LSpace u, FiniteDimensional (DualVector u), FiniteDimensional v
+ , Scalar u~s, Scalar v~s, Fractional' (Scalar v) )
+ => FiniteDimensional (LinearMap s u v) where
+ data SubBasis (LinearMap s u v) = LinMapBasis !(SubBasis (DualVector u)) !(SubBasis v)
+ entireBasis = case entireBasis of TensorBasis bu bv -> LinMapBasis bu bv
+ enumerateSubBasis (LinMapBasis bu bv)
+ = arr (fmap asLinearMap) . enumerateSubBasis $ TensorBasis bu bv
+ subbasisDimension (LinMapBasis bu bv) = subbasisDimension bu * subbasisDimension bv
+ decomposeLinMap = first (\(TensorBasis bv bu)->LinMapBasis bu bv)
+ . decomposeLinMap . coerce
+ decomposeLinMapWithin (LinMapBasis bu bv) m
+ = case decomposeLinMapWithin (TensorBasis bv bu) (coerce m) of
+ Right ws -> Right ws
+ Left (TensorBasis bv' bu', ws) -> Left (LinMapBasis bu' bv', ws)
+ recomposeSB (LinMapBasis bu bv)
+ = recomposeSB (TensorBasis bu bv) >>> first (arr fromTensor)
+ recomposeSBTensor (LinMapBasis bu bv) bw
+ = recomposeSBTensor (TensorBasis bu bv) bw >>> first coerce
+ recomposeLinMap (LinMapBasis bu bv) ws =
+ ( coUncurryLinearMap . fmap asTensor $ fst . recomposeLinMap bv
+ $ unfoldr (pure . recomposeLinMap bu) ws
+ , drop (subbasisDimension bu * subbasisDimension bv) ws )
+ recomposeContraLinMap fw dds = coUncurryLinearMap . fmap fromLinearMap . curryLinearMap
+ $ recomposeContraLinMapTensor fw $ fmap (arr asTensor) dds
+ recomposeContraLinMapTensor fw dds
+ = uncurryLinearMap . coUncurryLinearMap
+ . fmap (fromLinearMap . curryLinearMap) . curryLinearMap
+ $ recomposeContraLinMapTensor fw $ fmap (arr $ asTensor . hasteLinearMap) dds
+ uncanonicallyToDual = fmap uncanonicallyToDual >>> arr asTensor
+ >>> transposeTensor >>> arr fromTensor >>> fmap uncanonicallyToDual
+ uncanonicallyFromDual = fmap uncanonicallyFromDual >>> arr asTensor
+ >>> transposeTensor >>> arr fromTensor >>> fmap uncanonicallyFromDual
+
+
+infixr 0 \$
+
+-- | Inverse function application, aka solving of a linear system:
+--
+-- @
+-- f '\$' f '$' v ≡ v
+--
+-- f '$' f '\$' u ≡ u
+-- @
+--
+-- If @f@ does not have full rank, the behaviour is undefined (but we expect
+-- it to be reasonably well-behaved or even give a least-squares solution).
+--
+-- If you want to solve for multiple RHS vectors, be sure to partially
+-- apply this operator to the linear map, like
+--
+-- @
+-- map (f '\$') [v₁, v₂, ...]
+-- @
+--
+-- Since most of the work is actually done in triangularising the operator,
+-- this may be much faster than
+--
+-- @
+-- [f '\$' v₁, f '\$' v₂, ...]
+-- @
+(\$) :: ( FiniteDimensional u, FiniteDimensional v, SemiInner v
+ , Scalar u ~ Scalar v, Fractional' (Scalar v) )
+ => (u+>v) -> v -> u
+(\$) m = fst . \v -> recomposeSB mbas [v'<.>^v | v' <- v's]
+ where v's = dualBasis $ mdecomp []
+ (mbas, mdecomp) = decomposeLinMap m
+
+
+pseudoInverse :: ( FiniteDimensional u, FiniteDimensional v, SemiInner v
+ , Scalar u ~ Scalar v, Fractional' (Scalar v) )
+ => (u+>v) -> v+>u
+pseudoInverse m = recomposeContraLinMap (fst . recomposeSB mbas) v's
+ where v's = dualBasis $ mdecomp []
+ (mbas, mdecomp) = decomposeLinMap m
+
+
+-- | The
+-- provides an isomorphism between a Hilbert space and its (continuous) dual space.
+riesz :: (FiniteDimensional v, InnerSpace v) => DualVector v -+> v
+riesz = LinearFunction $ \dv ->
+ let (bas, compos) = decomposeLinMap $ sampleLinearFunction $ applyDualVector $ dv
+ in fst . recomposeSB bas $ compos []
+
+sRiesz :: (FiniteDimensional v, InnerSpace v) => DualSpace v -+> v
+sRiesz = LinearFunction $ \dv ->
+ let (bas, compos) = decomposeLinMap $ dv
+ in fst . recomposeSB bas $ compos []
+
+coRiesz :: (LSpace v, Num''' (Scalar v), InnerSpace v) => v -+> DualVector v
+coRiesz = fromFlatTensor . arr asTensor . sampleLinearFunction . inner
+
+-- | Functions are generally a pain to display, but since linear functionals
+-- in a Hilbert space can be represented by /vectors/ in that space,
+-- this can be used for implementing a 'Show' instance.
+showsPrecAsRiesz :: ( FiniteDimensional v, InnerSpace v, Show v
+ , HasBasis (Scalar v), Basis (Scalar v) ~ () )
+ => Int -> DualSpace v -> ShowS
+showsPrecAsRiesz p dv = showParen (p>0) $ ("().<"++)
+ . showsPrec 7 (sRiesz$dv)
+
+instance Show (LinearMap ℝ (V0 ℝ) ℝ) where showsPrec = showsPrecAsRiesz
+instance Show (LinearMap ℝ (V1 ℝ) ℝ) where showsPrec = showsPrecAsRiesz
+instance Show (LinearMap ℝ (V2 ℝ) ℝ) where showsPrec = showsPrecAsRiesz
+instance Show (LinearMap ℝ (V3 ℝ) ℝ) where showsPrec = showsPrecAsRiesz
+instance Show (LinearMap ℝ (V4 ℝ) ℝ) where showsPrec = showsPrecAsRiesz
+
+
+infixl 7 .<
+
+-- | Outer product of a general @v@-vector and a basis element from @w@.
+-- Note that this operation is in general pretty inefficient; it is
+-- provided mostly to lay out matrix definitions neatly.
+(.<) :: ( FiniteDimensional v, Num''' (Scalar v)
+ , InnerSpace v, LSpace w, HasBasis w, Scalar v ~ Scalar w )
+ => Basis w -> v -> v+>w
+bw .< v = sampleLinearFunction $ LinearFunction $ \v' -> recompose [(bw, v<.>v')]
+
+instance Show (LinearMap s v (V0 s)) where
+ show _ = "zeroV"
+instance (FiniteDimensional v, InnerSpace v, Scalar v ~ ℝ, Show v)
+ => Show (LinearMap ℝ v (V1 ℝ)) where
+ showsPrec p m = showParen (p>6) $ ("ex .< "++)
+ . showsPrec 7 (sRiesz $ fmap (LinearFunction (^._x)) $ m)
+instance (FiniteDimensional v, InnerSpace v, Scalar v ~ ℝ, Show v)
+ => Show (LinearMap ℝ v (V2 ℝ)) where
+ showsPrec p m = showParen (p>6)
+ $ ("ex.<"++) . showsPrec 7 (sRiesz $ fmap (LinearFunction (^._x)) $ m)
+ . (" ^+^ ey.<"++) . showsPrec 7 (sRiesz $ fmap (LinearFunction (^._y)) $ m)
+instance (FiniteDimensional v, InnerSpace v, Scalar v ~ ℝ, Show v)
+ => Show (LinearMap ℝ v (V3 ℝ)) where
+ showsPrec p m = showParen (p>6)
+ $ ("ex.<"++) . showsPrec 7 (sRiesz $ fmap (LinearFunction (^._x)) $ m)
+ . (" ^+^ ey.<"++) . showsPrec 7 (sRiesz $ fmap (LinearFunction (^._y)) $ m)
+ . (" ^+^ ez.<"++) . showsPrec 7 (sRiesz $ fmap (LinearFunction (^._z)) $ m)
+instance (FiniteDimensional v, InnerSpace v, Scalar v ~ ℝ, Show v)
+ => Show (LinearMap ℝ v (V4 ℝ)) where
+ showsPrec p m = showParen (p>6)
+ $ ("ex.<"++) . showsPrec 7 (sRiesz $ fmap (LinearFunction (^._x)) $ m)
+ . (" ^+^ ey.<"++) . showsPrec 7 (sRiesz $ fmap (LinearFunction (^._y)) $ m)
+ . (" ^+^ ez.<"++) . showsPrec 7 (sRiesz $ fmap (LinearFunction (^._z)) $ m)
+ . (" ^+^ ew.<"++) . showsPrec 7 (sRiesz $ fmap (LinearFunction (^._w)) $ m)
+
+
+
+
+
+(^) :: Num a => a -> Int -> a
+(^) = (Hask.^)
+
+
+type HilbertSpace v = (LSpace v, InnerSpace v, DualVector v ~ v)
+
+type RealFrac' s = (IEEE s, HilbertSpace s, Scalar s ~ s)
+type RealFloat' s = (RealFrac' s, Floating s)
+
+type SimpleSpace v = ( FiniteDimensional v, FiniteDimensional (DualVector v)
+ , SemiInner v, SemiInner (DualVector v)
+ , RealFrac' (Scalar v) )
+
+
+instance ∀ s u v .
+ ( FiniteDimensional u, LSpace v, FiniteFreeSpace v
+ , Scalar u~s, Scalar v~s ) => FiniteFreeSpace (LinearMap s u v) where
+ freeDimension _ = subbasisDimension (entireBasis :: SubBasis u)
+ * freeDimension ([]::[v])
+ toFullUnboxVect = decomposeLinMapWithin entireBasis >>> \case
+ Right l -> UArr.concat $ toFullUnboxVect <$> l []
+ unsafeFromFullUnboxVect arrv = fst . recomposeLinMap entireBasis
+ $ [unsafeFromFullUnboxVect $ UArr.slice (dv*j) dv arrv | j <- [0 .. du-1]]
+ where du = subbasisDimension (entireBasis :: SubBasis u)
+ dv = freeDimension ([]::[v])
+
+instance ∀ s u v .
+ ( LSpace u, FiniteDimensional (DualVector u), LSpace v, FiniteFreeSpace v
+ , Scalar u~s, Scalar v~s ) => FiniteFreeSpace (Tensor s u v) where
+ freeDimension _ = subbasisDimension (entireBasis :: SubBasis (DualVector u))
+ * freeDimension ([]::[v])
+ toFullUnboxVect = arr asLinearMap >>> decomposeLinMapWithin entireBasis >>> \case
+ Right l -> UArr.concat $ toFullUnboxVect <$> l []
+ unsafeFromFullUnboxVect arrv = fromLinearMap $ fst . recomposeLinMap entireBasis
+ $ [unsafeFromFullUnboxVect $ UArr.slice (dv*j) dv arrv | j <- [0 .. du-1]]
+ where du = subbasisDimension (entireBasis :: SubBasis (DualVector u))
+ dv = freeDimension ([]::[v])
+
+instance ∀ s u v .
+ ( FiniteDimensional u, LSpace v, FiniteFreeSpace v
+ , Scalar u~s, Scalar v~s ) => FiniteFreeSpace (LinearFunction s u v) where
+ freeDimension _ = subbasisDimension (entireBasis :: SubBasis u)
+ * freeDimension ([]::[v])
+ toFullUnboxVect f = toFullUnboxVect (arr f :: LinearMap s u v)
+ unsafeFromFullUnboxVect arrv = arr (unsafeFromFullUnboxVect arrv :: LinearMap s u v)
+
+
diff --git a/Math/VectorSpace/ZeroDimensional.hs b/Math/VectorSpace/ZeroDimensional.hs
new file mode 100644
--- /dev/null
+++ b/Math/VectorSpace/ZeroDimensional.hs
@@ -0,0 +1,58 @@
+-- |
+-- Module : Math.VectorSpace.ZeroDimensional
+-- Copyright : (c) Justus Sagemüller 2016
+-- License : GPL v3
+--
+-- Maintainer : (@) sagemueller $ geo.uni-koeln.de
+-- Stability : experimental
+-- Portability : portable
+--
+{-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE ConstraintKinds #-}
+{-# LANGUAGE UndecidableInstances #-}
+{-# LANGUAGE FunctionalDependencies #-}
+{-# LANGUAGE TypeOperators #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE Rank2Types #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE PatternSynonyms #-}
+{-# LANGUAGE ViewPatterns #-}
+{-# LANGUAGE UnicodeSyntax #-}
+{-# LANGUAGE CPP #-}
+{-# LANGUAGE TupleSections #-}
+{-# LANGUAGE StandaloneDeriving #-}
+
+module Math.VectorSpace.ZeroDimensional (
+ ZeroDim (..)
+ ) where
+
+import Data.AffineSpace
+import Data.VectorSpace
+import Data.Basis
+import Data.Void
+
+
+
+data ZeroDim s = Origin
+
+instance Monoid (ZeroDim s) where
+ mempty = Origin
+ mappend Origin Origin = Origin
+
+instance AffineSpace (ZeroDim s) where
+ type Diff (ZeroDim s) = ZeroDim s
+ Origin .+^ Origin = Origin
+ Origin .-. Origin = Origin
+instance AdditiveGroup (ZeroDim s) where
+ zeroV = Origin
+ Origin ^+^ Origin = Origin
+ negateV Origin = Origin
+instance VectorSpace (ZeroDim s) where
+ type Scalar (ZeroDim s) = s
+ _ *^ Origin = Origin
+instance HasBasis (ZeroDim s) where
+ type Basis (ZeroDim k) = Void
+ basisValue = absurd
+ decompose Origin = []
+ decompose' Origin = absurd
diff --git a/Setup.hs b/Setup.hs
new file mode 100644
--- /dev/null
+++ b/Setup.hs
@@ -0,0 +1,2 @@
+import Distribution.Simple
+main = defaultMain
diff --git a/linearmap-category.cabal b/linearmap-category.cabal
new file mode 100644
--- /dev/null
+++ b/linearmap-category.cabal
@@ -0,0 +1,57 @@
+-- Initial linearmap-family.cabal generated by cabal init. For further
+-- documentation, see http://haskell.org/cabal/users-guide/
+
+name: linearmap-category
+version: 0.1.0.0
+synopsis: Native, complete, matrix-free linear algebra.
+description: The term /numerical linear algebra/ is often used almost
+ synonymous with /matrix modifications/. However, what's interesting
+ for most applications are really just /points in some vector space/
+ and linear mappings between them, not matrices (which represent
+ points or mappings, but inherently depend on a particular choice
+ of basis / coordinate system).
+ .
+ This library implements the crucial LA operations like solving
+ linear equations and eigenvalue problems, without requiring
+ that the vectors are represented in some particular basis. Apart
+ from conceptual elegance (only operations that are actually
+ geometrically sensible will typecheck – this is far stronger than
+ just confirming that the dimensions match, as some other libraries
+ do), this also opens up good optimisation possibilities: the
+ vectors can be unboxed, use dedicated sparse compression, possibly
+ carry out the computations on accelerated hardware (GPU etc.).
+ The spaces can even be infinite-dimensional (e.g. function spaces).
+ .
+ The linear algebra algorithms in this package only require the
+ vectors to support fundamental operations like addition, scalar
+ products, double-dual-space coercion and tensor products; none of
+ this requires a basis representation.
+homepage: https://github.com/leftaroundabout/linearmap-family
+license: GPL-3
+license-file: LICENSE
+author: Justus Sagemüller
+maintainer: (@) sagemueller $ geo.uni-koeln.de
+-- copyright:
+category: Math
+build-type: Simple
+-- extra-source-files:
+cabal-version: >=1.10
+
+library
+ exposed-modules: Math.LinearMap.Category
+ Math.VectorSpace.ZeroDimensional
+ other-modules: Math.LinearMap.Category.Class
+ Math.LinearMap.Asserted
+ Math.LinearMap.Category.Instances
+ Math.VectorSpace.Docile
+ other-extensions: FlexibleInstances, UndecidableInstances, FunctionalDependencies, TypeOperators, TypeFamilies
+ build-depends: base >=4.8 && <4.9,
+ vector-space >=0.10 && <0.11,
+ constrained-categories >=0.3 && <0.4,
+ containers, vector,
+ free-vector-spaces >= 0.1.1 && < 0.2,
+ linear, lens,
+ semigroups,
+ ieee754 >= 0.7 && < 0.9
+ -- hs-source-dirs:
+ default-language: Haskell2010