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<pre><a name="line-1"></a><span class='hs-comment'>{-# LANGUAGE CPP #-}</span>
<a name="line-2"></a><span class='hs-comment'>{-# LANGUAGE TypeOperators #-}</span>
<a name="line-3"></a><span class='hs-comment'>{-# LANGUAGE DefaultSignatures #-}</span>
<a name="line-4"></a><span class='hs-cpp'>#if __GLASGOW_HASKELL__ >= 702 && __GLASGOW_HASKELL__ < 710</span>
<a name="line-5"></a><span class='hs-comment'>{-# LANGUAGE Trustworthy #-}</span>
<a name="line-6"></a><span class='hs-cpp'>#endif</span>
<a name="line-7"></a><span class='hs-comment'>---------------------------------------------------------------------------</span>
<a name="line-8"></a><span class='hs-comment'>-- |</span>
<a name="line-9"></a><span class='hs-comment'>-- Copyright : (C) 2012-2015 Edward Kmett</span>
<a name="line-10"></a><span class='hs-comment'>-- License : BSD-style (see the file LICENSE)</span>
<a name="line-11"></a><span class='hs-comment'>--</span>
<a name="line-12"></a><span class='hs-comment'>-- Maintainer : Edward Kmett <ekmett@gmail.com></span>
<a name="line-13"></a><span class='hs-comment'>-- Stability : experimental</span>
<a name="line-14"></a><span class='hs-comment'>-- Portability : non-portable</span>
<a name="line-15"></a><span class='hs-comment'>--</span>
<a name="line-16"></a><span class='hs-comment'>-- Simple matrix operation for low-dimensional primitives.</span>
<a name="line-17"></a><span class='hs-comment'>---------------------------------------------------------------------------</span>
<a name="line-18"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>Linear</span><span class='hs-varop'>.</span><span class='hs-conid'>Trace</span>
<a name="line-19"></a> <span class='hs-layout'>(</span> <span class='hs-conid'>Trace</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span>
<a name="line-20"></a> <span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-21"></a>
<a name="line-22"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>Monad</span>
<a name="line-23"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Linear</span><span class='hs-varop'>.</span><span class='hs-conid'>V0</span>
<a name="line-24"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Linear</span><span class='hs-varop'>.</span><span class='hs-conid'>V1</span>
<a name="line-25"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Linear</span><span class='hs-varop'>.</span><span class='hs-conid'>V2</span>
<a name="line-26"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Linear</span><span class='hs-varop'>.</span><span class='hs-conid'>V3</span>
<a name="line-27"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Linear</span><span class='hs-varop'>.</span><span class='hs-conid'>V4</span>
<a name="line-28"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Linear</span><span class='hs-varop'>.</span><span class='hs-conid'>Plucker</span>
<a name="line-29"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Linear</span><span class='hs-varop'>.</span><span class='hs-conid'>Quaternion</span>
<a name="line-30"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Linear</span><span class='hs-varop'>.</span><span class='hs-conid'>V</span>
<a name="line-31"></a><span class='hs-cpp'>#if defined(__GLASGOW_HASKELL__) && __GLASGOW_HASKELL__ > 704</span>
<a name="line-32"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Complex</span>
<a name="line-33"></a><span class='hs-cpp'>#endif</span>
<a name="line-34"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Distributive</span>
<a name="line-35"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Foldable</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>Foldable</span>
<a name="line-36"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Functor</span><span class='hs-varop'>.</span><span class='hs-conid'>Bind</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>Bind</span>
<a name="line-37"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Functor</span><span class='hs-varop'>.</span><span class='hs-conid'>Compose</span>
<a name="line-38"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Functor</span><span class='hs-varop'>.</span><span class='hs-conid'>Product</span>
<a name="line-39"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Hashable</span>
<a name="line-40"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>HashMap</span><span class='hs-varop'>.</span><span class='hs-conid'>Lazy</span>
<a name="line-41"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>IntMap</span>
<a name="line-42"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Map</span>
<a name="line-43"></a>
<a name="line-44"></a><span class='hs-comment'>-- $setup</span>
<a name="line-45"></a><span class='hs-comment'>-- >>> import Data.Complex</span>
<a name="line-46"></a><span class='hs-comment'>-- >>> import Data.IntMap</span>
<a name="line-47"></a><span class='hs-comment'>-- >>> import Debug.SimpleReflect.Vars</span>
<a name="line-48"></a>
<a name="line-49"></a><a name="Trace"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>Functor</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trace</span> <span class='hs-varid'>m</span> <span class='hs-keyword'>where</span>
<a name="line-50"></a> <span class='hs-comment'>-- | Compute the trace of a matrix</span>
<a name="line-51"></a> <span class='hs-comment'>--</span>
<a name="line-52"></a> <span class='hs-comment'>-- >>> trace (V2 (V2 a b) (V2 c d))</span>
<a name="line-53"></a> <span class='hs-comment'>-- a + d</span>
<a name="line-54"></a> <span class='hs-varid'>trace</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>m</span> <span class='hs-layout'>(</span><span class='hs-varid'>m</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span>
<a name="line-55"></a><span class='hs-cpp'>#ifndef HLINT</span>
<a name="line-56"></a> <span class='hs-keyword'>default</span> <span class='hs-varid'>trace</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Foldable</span> <span class='hs-varid'>m</span><span class='hs-layout'>,</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>m</span> <span class='hs-layout'>(</span><span class='hs-varid'>m</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>a</span>
<a name="line-57"></a> <span class='hs-varid'>trace</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Foldable</span><span class='hs-varop'>.</span><span class='hs-varid'>sum</span> <span class='hs-varop'>.</span> <span class='hs-varid'>diagonal</span>
<a name="line-58"></a> <span class='hs-comment'>{-# INLINE trace #-}</span>
<a name="line-59"></a><span class='hs-cpp'>#endif</span>
<a name="line-60"></a>
<a name="line-61"></a> <span class='hs-comment'>-- | Compute the diagonal of a matrix</span>
<a name="line-62"></a> <span class='hs-comment'>--</span>
<a name="line-63"></a> <span class='hs-comment'>-- >>> diagonal (V2 (V2 a b) (V2 c d))</span>
<a name="line-64"></a> <span class='hs-comment'>-- V2 a d</span>
<a name="line-65"></a> <span class='hs-varid'>diagonal</span> <span class='hs-keyglyph'>::</span> <span class='hs-varid'>m</span> <span class='hs-layout'>(</span><span class='hs-varid'>m</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>m</span> <span class='hs-varid'>a</span>
<a name="line-66"></a><span class='hs-cpp'>#ifndef HLINT</span>
<a name="line-67"></a> <span class='hs-keyword'>default</span> <span class='hs-varid'>diagonal</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Monad</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=></span> <span class='hs-varid'>m</span> <span class='hs-layout'>(</span><span class='hs-varid'>m</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>m</span> <span class='hs-varid'>a</span>
<a name="line-68"></a> <span class='hs-varid'>diagonal</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Monad</span><span class='hs-varop'>.</span><span class='hs-varid'>join</span>
<a name="line-69"></a> <span class='hs-comment'>{-# INLINE diagonal #-}</span>
<a name="line-70"></a><span class='hs-cpp'>#endif</span>
<a name="line-71"></a>
<a name="line-72"></a><a name="instance%20Trace%20IntMap"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trace</span> <span class='hs-conid'>IntMap</span> <span class='hs-keyword'>where</span>
<a name="line-73"></a> <span class='hs-varid'>diagonal</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Bind</span><span class='hs-varop'>.</span><span class='hs-varid'>join</span>
<a name="line-74"></a> <span class='hs-comment'>{-# INLINE diagonal #-}</span>
<a name="line-75"></a>
<a name="line-76"></a><a name="instance%20Trace%20(Map%20k)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Ord</span> <span class='hs-varid'>k</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trace</span> <span class='hs-layout'>(</span><span class='hs-conid'>Map</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-77"></a> <span class='hs-varid'>diagonal</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Bind</span><span class='hs-varop'>.</span><span class='hs-varid'>join</span>
<a name="line-78"></a> <span class='hs-comment'>{-# INLINE diagonal #-}</span>
<a name="line-79"></a>
<a name="line-80"></a><a name="instance%20Trace%20(HashMap%20k)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span> <span class='hs-varid'>k</span><span class='hs-layout'>,</span> <span class='hs-conid'>Hashable</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trace</span> <span class='hs-layout'>(</span><span class='hs-conid'>HashMap</span> <span class='hs-varid'>k</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-81"></a> <span class='hs-varid'>diagonal</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Bind</span><span class='hs-varop'>.</span><span class='hs-varid'>join</span>
<a name="line-82"></a> <span class='hs-comment'>{-# INLINE diagonal #-}</span>
<a name="line-83"></a>
<a name="line-84"></a><a name="instance%20Trace%20(V%20n)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Dim</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trace</span> <span class='hs-layout'>(</span><span class='hs-conid'>V</span> <span class='hs-varid'>n</span><span class='hs-layout'>)</span>
<a name="line-85"></a><a name="instance%20Trace%20V0"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trace</span> <span class='hs-conid'>V0</span>
<a name="line-86"></a><a name="instance%20Trace%20V1"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trace</span> <span class='hs-conid'>V1</span>
<a name="line-87"></a><a name="instance%20Trace%20V2"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trace</span> <span class='hs-conid'>V2</span>
<a name="line-88"></a><a name="instance%20Trace%20V3"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trace</span> <span class='hs-conid'>V3</span>
<a name="line-89"></a><a name="instance%20Trace%20V4"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trace</span> <span class='hs-conid'>V4</span>
<a name="line-90"></a><a name="instance%20Trace%20Plucker"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trace</span> <span class='hs-conid'>Plucker</span>
<a name="line-91"></a><a name="instance%20Trace%20Quaternion"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trace</span> <span class='hs-conid'>Quaternion</span>
<a name="line-92"></a>
<a name="line-93"></a><span class='hs-cpp'>#if defined(__GLASGOW_HASKELL__) && __GLASGOW_HASKELL__ > 704</span>
<a name="line-94"></a><a name="instance%20Trace%20Complex"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Trace</span> <span class='hs-conid'>Complex</span> <span class='hs-keyword'>where</span>
<a name="line-95"></a> <span class='hs-varid'>trace</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-conop'>:+</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-conop'>:+</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span> <span class='hs-conop'>:+</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>a</span> <span class='hs-varop'>+</span> <span class='hs-varid'>b</span>
<a name="line-96"></a> <span class='hs-comment'>{-# INLINE trace #-}</span>
<a name="line-97"></a> <span class='hs-varid'>diagonal</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-conop'>:+</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-conop'>:+</span> <span class='hs-layout'>(</span><span class='hs-keyword'>_</span> <span class='hs-conop'>:+</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>a</span> <span class='hs-conop'>:+</span> <span class='hs-varid'>b</span>
<a name="line-98"></a> <span class='hs-comment'>{-# INLINE diagonal #-}</span>
<a name="line-99"></a><span class='hs-cpp'>#endif</span>
<a name="line-100"></a>
<a name="line-101"></a><a name="instance%20Trace%20(Product%20f%20g)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Trace</span> <span class='hs-varid'>f</span><span class='hs-layout'>,</span> <span class='hs-conid'>Trace</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trace</span> <span class='hs-layout'>(</span><span class='hs-conid'>Product</span> <span class='hs-varid'>f</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-102"></a> <span class='hs-varid'>trace</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pair</span> <span class='hs-varid'>xx</span> <span class='hs-varid'>yy</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>trace</span> <span class='hs-layout'>(</span><span class='hs-varid'>pfst</span> <span class='hs-varop'><$></span> <span class='hs-varid'>xx</span><span class='hs-layout'>)</span> <span class='hs-varop'>+</span> <span class='hs-varid'>trace</span> <span class='hs-layout'>(</span><span class='hs-varid'>psnd</span> <span class='hs-varop'><$></span> <span class='hs-varid'>yy</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-103"></a> <span class='hs-varid'>pfst</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pair</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span>
<a name="line-104"></a> <span class='hs-varid'>psnd</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pair</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>y</span>
<a name="line-105"></a> <span class='hs-comment'>{-# INLINE trace #-}</span>
<a name="line-106"></a> <span class='hs-varid'>diagonal</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pair</span> <span class='hs-varid'>xx</span> <span class='hs-varid'>yy</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>diagonal</span> <span class='hs-layout'>(</span><span class='hs-varid'>pfst</span> <span class='hs-varop'><$></span> <span class='hs-varid'>xx</span><span class='hs-layout'>)</span> <span class='hs-varop'>`Pair`</span> <span class='hs-varid'>diagonal</span> <span class='hs-layout'>(</span><span class='hs-varid'>psnd</span> <span class='hs-varop'><$></span> <span class='hs-varid'>yy</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-107"></a> <span class='hs-varid'>pfst</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pair</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>x</span>
<a name="line-108"></a> <span class='hs-varid'>psnd</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pair</span> <span class='hs-keyword'>_</span> <span class='hs-varid'>y</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>y</span>
<a name="line-109"></a> <span class='hs-comment'>{-# INLINE diagonal #-}</span>
<a name="line-110"></a>
<a name="line-111"></a><a name="instance%20Trace%20(Compose%20g%20f)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Distributive</span> <span class='hs-varid'>g</span><span class='hs-layout'>,</span> <span class='hs-conid'>Trace</span> <span class='hs-varid'>g</span><span class='hs-layout'>,</span> <span class='hs-conid'>Trace</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Trace</span> <span class='hs-layout'>(</span><span class='hs-conid'>Compose</span> <span class='hs-varid'>g</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-112"></a> <span class='hs-varid'>trace</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>trace</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fmap</span> <span class='hs-layout'>(</span><span class='hs-varid'>fmap</span> <span class='hs-varid'>trace</span> <span class='hs-varop'>.</span> <span class='hs-varid'>distribute</span><span class='hs-layout'>)</span> <span class='hs-varop'>.</span> <span class='hs-varid'>getCompose</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fmap</span> <span class='hs-varid'>getCompose</span>
<a name="line-113"></a> <span class='hs-comment'>{-# INLINE trace #-}</span>
<a name="line-114"></a> <span class='hs-varid'>diagonal</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Compose</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fmap</span> <span class='hs-varid'>diagonal</span> <span class='hs-varop'>.</span> <span class='hs-varid'>diagonal</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fmap</span> <span class='hs-varid'>distribute</span> <span class='hs-varop'>.</span> <span class='hs-varid'>getCompose</span> <span class='hs-varop'>.</span> <span class='hs-varid'>fmap</span> <span class='hs-varid'>getCompose</span>
<a name="line-115"></a> <span class='hs-comment'>{-# INLINE diagonal #-}</span>
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