linearmap-category (empty) → 0.1.0.0

raw patch · 9 files changed

+3053/−0 lines, 9 filesdep +basedep +constrained-categoriesdep +containerssetup-changed

Dependencies added: base, constrained-categories, containers, free-vector-spaces, ieee754, lens, linear, semigroups, vector, vector-space

Files

• LICENSE +674/−0
• Math/LinearMap/Asserted.hs +147/−0
• Math/LinearMap/Category.hs +590/−0
• Math/LinearMap/Category/Class.hs +660/−0
• Math/LinearMap/Category/Instances.hs +228/−0
• Math/VectorSpace/Docile.hs +637/−0
• Math/VectorSpace/ZeroDimensional.hs +58/−0
• Setup.hs +2/−0
• linearmap-category.cabal +57/−0

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@@ -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

@@ -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 +-- <https://en.wikipedia.org/wiki/Category_of_modules#Example:_the_category_of_vector_spaces VectK>).+-- 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+-- <https://en.wikipedia.org/wiki/Linear_form linear form> 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+-- <https://en.wikipedia.org/wiki/Hilbert_space Hilbert space> 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+--   <http://hackage.haskell.org/package/manifolds manifolds> 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 <https://en.wikipedia.org/wiki/Sobolev_space Sobolev space> of weakly+--   differentiable functions also permits the+--   <https://en.wikipedia.org/wiki/Square-integrable_function 𝐿²> 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 <https://en.wikipedia.org/wiki/Norm_(mathematics) norm> 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)

@@ -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+-- <https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notationa bra-ket notation>+-- 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+

@@ -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+  (.-.) = (^-^)+  (.+^) = (^+^)++  +

@@ -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+--   <https://en.wikipedia.org/wiki/Galerkin_method Galerkin method> 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<n   = Node (i, dv) (go (k+1) [(i',(zeroAt j av',m)) | (i',(av',m))<-scs])+                                : go k scs+        where (j,_) = maximumBy (comparing snd) $ zip jfus av+              dv = dvs !! j+       go _ _ = []+       +       jfus = [0 .. n-1]+       n = length dvs+       +       zeroAt :: Int -> [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<j   = zip' vs ((j,v'):ds)+        | i==j  = (v,v') : zip' vs ds+       zip' _ _ = []+       candidates = sortBy (comparing fst) . findBest+                             $ dualBasisCandidates vsIxed+        where findBest [] = []+              findBest (Node iv' bv' : _) = iv' : findBest bv'+       vsIxed = zip [0..] vs++instance SemiInner ℝ where+  dualBasisCandidates = fmap ((`Node`[]) . second recip)+                . sortBy (comparing $ negate . abs . snd)+                . filter ((/=0) . snd)++instance (Fractional'' s, Ord s, SemiInner s) => 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 <https://en.wikipedia.org/wiki/Riesz_representation_theorem Riesz representation theorem>+--   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)+                                     +  

@@ -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

@@ -0,0 +1,2 @@+import Distribution.Simple+main = defaultMain

@@ -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