<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html> <head> <!-- Generated by HsColour, http://www.cs.york.ac.uk/fp/darcs/hscolour/ --> <title>src/ForSyDe/Shallow/PolyArith.hs</title> <link type='text/css' rel='stylesheet' href='hscolour.css' /> </head> <body> <pre><a name="line-1"></a><span class='hs-comment'>-----------------------------------------------------------------------------</span> <a name="line-2"></a><span class='hs-comment'>-- |</span> <a name="line-3"></a><span class='hs-comment'>-- Module : ForSyDe.Shallow.PolyArith</span> <a name="line-4"></a><span class='hs-comment'>-- Copyright : (c) SAM Group, KTH/ICT/ECS 2007-2008</span> <a name="line-5"></a><span class='hs-comment'>-- License : BSD-style (see the file LICENSE)</span> <a name="line-6"></a><span class='hs-comment'>-- </span> <a name="line-7"></a><span class='hs-comment'>-- Maintainer : forsyde-dev@ict.kth.se</span> <a name="line-8"></a><span class='hs-comment'>-- Stability : experimental</span> <a name="line-9"></a><span class='hs-comment'>-- Portability : portable</span> <a name="line-10"></a><span class='hs-comment'>--</span> <a name="line-11"></a><span class='hs-comment'>-- This is the polynomial arithematic library. The arithematic operations include </span> <a name="line-12"></a><span class='hs-comment'>-- addition, multiplication, division and power. However, the computation time is </span> <a name="line-13"></a><span class='hs-comment'>-- not optimized for multiplication and is O(n2), which could be considered to be </span> <a name="line-14"></a><span class='hs-comment'>-- optimized by FFT algorithms later on.</span> <a name="line-15"></a><span class='hs-comment'>-----------------------------------------------------------------------------</span> <a name="line-16"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>ForSyDe</span><span class='hs-varop'>.</span><span class='hs-conid'>Shallow</span><span class='hs-varop'>.</span><span class='hs-conid'>PolyArith</span><span class='hs-layout'>(</span> <a name="line-17"></a> <span class='hs-comment'>-- *Polynomial data type</span> <a name="line-18"></a> <span class='hs-conid'>Poly</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <a name="line-19"></a> <span class='hs-comment'>-- *Addition, DmMultiplication, division and power operations</span> <a name="line-20"></a> <span class='hs-varid'>addPoly</span><span class='hs-layout'>,</span> <span class='hs-varid'>mulPoly</span><span class='hs-layout'>,</span> <span class='hs-varid'>divPoly</span><span class='hs-layout'>,</span> <span class='hs-varid'>powerPoly</span><span class='hs-layout'>,</span> <a name="line-21"></a> <span class='hs-comment'>-- *Some helper functions</span> <a name="line-22"></a> <span class='hs-varid'>getCoef</span><span class='hs-layout'>,</span> <span class='hs-varid'>scalePoly</span><span class='hs-layout'>,</span> <span class='hs-varid'>addPolyCoef</span><span class='hs-layout'>,</span> <span class='hs-varid'>subPolyCoef</span><span class='hs-layout'>,</span> <span class='hs-varid'>scalePolyCoef</span> <a name="line-23"></a> <span class='hs-layout'>)</span> <a name="line-24"></a> <span class='hs-keyword'>where</span> <a name="line-25"></a> <a name="line-26"></a><a name="Poly"></a><span class='hs-comment'>-- |Polynomial data type.</span> <a name="line-27"></a><a name="Poly"></a><span class='hs-keyword'>data</span> <span class='hs-conid'>Num</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=></span> <span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Poly</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <a name="line-28"></a> <span class='hs-keyglyph'>|</span> <span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyword'>deriving</span> <span class='hs-layout'>(</span><span class='hs-conid'>Eq</span><span class='hs-layout'>)</span> <a name="line-29"></a> <a name="line-30"></a> <a name="line-31"></a><a name="mulPoly"></a><span class='hs-comment'>-- |Multiplication operation of polynomials.</span> <a name="line-32"></a><span class='hs-definition'>mulPoly</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-conid'>Poly</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span> <a name="line-33"></a><span class='hs-definition'>mulPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Poly</span> <span class='hs-conid'>[]</span> <a name="line-34"></a><span class='hs-definition'>mulPoly</span> <span class='hs-keyword'>_</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Poly</span> <span class='hs-conid'>[]</span> <a name="line-35"></a><span class='hs-comment'>-- Here is the O(n^2) version of polynomial multiplication</span> <a name="line-36"></a><span class='hs-definition'>mulPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Poly</span> <span class='hs-varop'>$</span> <span class='hs-varid'>foldr</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>y</span> <span class='hs-varid'>zs</span> <span class='hs-keyglyph'>-></span> <a name="line-37"></a> <span class='hs-keyword'>let</span> <span class='hs-layout'>(</span><span class='hs-varid'>v</span><span class='hs-conop'>:</span><span class='hs-varid'>vs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>scalePolyCoef</span> <span class='hs-varid'>y</span> <span class='hs-varid'>xs</span> <span class='hs-keyword'>in</span> <span class='hs-varid'>v</span> <span class='hs-conop'>:</span><span class='hs-varid'>addPolyCoef</span> <span class='hs-varid'>vs</span> <span class='hs-varid'>zs</span><span class='hs-layout'>)</span> <span class='hs-conid'>[]</span> <span class='hs-varid'>ys</span> <a name="line-38"></a><span class='hs-definition'>mulPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>c</span><span class='hs-layout'>,</span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <a name="line-39"></a> <span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>mulPoly</span> <span class='hs-varid'>a</span> <span class='hs-varid'>c</span><span class='hs-layout'>,</span> <span class='hs-varid'>mulPoly</span> <span class='hs-varid'>b</span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span> <a name="line-40"></a><span class='hs-definition'>mulPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <a name="line-41"></a> <span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>mulPoly</span> <span class='hs-varid'>a</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <a name="line-42"></a><span class='hs-definition'>mulPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <a name="line-43"></a> <span class='hs-varid'>mulPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <a name="line-44"></a> <a name="line-45"></a><a name="divPoly"></a><span class='hs-comment'>-- |Division operation of polynomials.</span> <a name="line-46"></a><span class='hs-definition'>divPoly</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-conid'>Poly</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span> <a name="line-47"></a><span class='hs-definition'>divPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span><span class='hs-layout'>,</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <a name="line-48"></a><span class='hs-definition'>divPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>c</span><span class='hs-layout'>,</span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <a name="line-49"></a> <span class='hs-varid'>mulPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>d</span><span class='hs-layout'>,</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <a name="line-50"></a><span class='hs-definition'>divPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <a name="line-51"></a> <span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-varid'>mulPoly</span> <span class='hs-varid'>b</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <a name="line-52"></a><span class='hs-definition'>divPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <a name="line-53"></a> <span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>mulPoly</span> <span class='hs-varid'>b</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <a name="line-54"></a> <a name="line-55"></a><a name="addPoly"></a><span class='hs-comment'>-- |Addition operations of polynomials.</span> <a name="line-56"></a><span class='hs-definition'>addPoly</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-conid'>Poly</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span> <a name="line-57"></a><span class='hs-definition'>addPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Poly</span> <span class='hs-varop'>$</span> <span class='hs-varid'>addPolyCoef</span> <span class='hs-varid'>a</span> <span class='hs-varid'>b</span> <a name="line-58"></a><span class='hs-definition'>addPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>c</span><span class='hs-layout'>,</span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <a name="line-59"></a> <span class='hs-keyword'>if</span> <span class='hs-varid'>b</span><span class='hs-varop'>==</span><span class='hs-varid'>d</span> <span class='hs-keyword'>then</span> <span class='hs-comment'>-- simplifyPolyPair $</span> <a name="line-60"></a> <span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>addPoly</span> <span class='hs-varid'>a</span> <span class='hs-varid'>c</span><span class='hs-layout'>,</span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span> <a name="line-61"></a> <span class='hs-keyword'>else</span> <span class='hs-comment'>-- simplifyPolyPair $</span> <a name="line-62"></a> <span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>dividedPoly</span><span class='hs-layout'>,</span> <span class='hs-varid'>divisorPoly</span><span class='hs-layout'>)</span> <a name="line-63"></a> <span class='hs-keyword'>where</span> <a name="line-64"></a> <span class='hs-varid'>divisorPoly</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>b</span> <span class='hs-varop'>==</span><span class='hs-varid'>d</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>b</span> <span class='hs-keyword'>else</span> <span class='hs-varid'>mulPoly</span> <span class='hs-varid'>b</span> <span class='hs-varid'>d</span> <a name="line-65"></a> <span class='hs-varid'>dividedPoly</span> <span class='hs-keyglyph'>=</span> <span class='hs-keyword'>if</span> <span class='hs-varid'>b</span> <span class='hs-varop'>==</span> <span class='hs-varid'>d</span> <span class='hs-keyword'>then</span> <span class='hs-varid'>addPoly</span> <span class='hs-varid'>a</span> <span class='hs-varid'>c</span> <a name="line-66"></a> <span class='hs-keyword'>else</span> <span class='hs-varid'>addPoly</span> <span class='hs-layout'>(</span><span class='hs-varid'>mulPoly</span> <span class='hs-varid'>a</span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>mulPoly</span> <span class='hs-varid'>b</span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <a name="line-67"></a><span class='hs-definition'>addPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>c</span><span class='hs-layout'>,</span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span> <span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <a name="line-68"></a> <span class='hs-varid'>addPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>multiPolyHelper</span><span class='hs-layout'>,</span> <span class='hs-varid'>d</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-varid'>c</span><span class='hs-layout'>,</span><span class='hs-varid'>d</span><span class='hs-layout'>)</span> <span class='hs-layout'>)</span> <a name="line-69"></a> <span class='hs-keyword'>where</span> <a name="line-70"></a> <span class='hs-varid'>multiPolyHelper</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>mulPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-varid'>d</span> <a name="line-71"></a><span class='hs-definition'>addPoly</span> <span class='hs-varid'>abPoly</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-varid'>cPoly</span><span class='hs-keyglyph'>@</span><span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-keyword'>_</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>addPoly</span> <span class='hs-varid'>cPoly</span> <span class='hs-varid'>abPoly</span> <a name="line-72"></a> <a name="line-73"></a><a name="powerPoly"></a><span class='hs-comment'>-- |Power operation of polynomials.</span> <a name="line-74"></a><span class='hs-definition'>powerPoly</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-conid'>Poly</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span> <a name="line-75"></a><span class='hs-definition'>powerPoly</span> <span class='hs-varid'>p</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>powerX'</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-varid'>p</span> <span class='hs-varid'>n</span> <a name="line-76"></a> <span class='hs-keyword'>where</span> <a name="line-77"></a> <span class='hs-varid'>powerX'</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-conid'>Poly</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Int</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span> <a name="line-78"></a> <span class='hs-varid'>powerX'</span> <span class='hs-varid'>p'</span> <span class='hs-keyword'>_</span> <span class='hs-num'>0</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>p'</span> <a name="line-79"></a> <span class='hs-varid'>powerX'</span> <span class='hs-varid'>p'</span> <span class='hs-varid'>p</span> <span class='hs-varid'>n</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>powerX'</span> <span class='hs-layout'>(</span><span class='hs-varid'>mulPoly</span> <span class='hs-varid'>p'</span> <span class='hs-varid'>p</span><span class='hs-layout'>)</span> <span class='hs-varid'>p</span> <span class='hs-layout'>(</span><span class='hs-varid'>n</span><span class='hs-comment'>-</span><span class='hs-num'>1</span><span class='hs-layout'>)</span> <a name="line-80"></a> <a name="line-81"></a><span class='hs-comment'>-- |Some helper functions below.</span> <a name="line-82"></a> <a name="line-83"></a><a name="getCoef"></a><span class='hs-comment'>-- |To get the coefficients of the polynomial.</span> <a name="line-84"></a><span class='hs-definition'>getCoef</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-conid'>Poly</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>,</span><span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <a name="line-85"></a><span class='hs-definition'>getCoef</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>xs</span><span class='hs-layout'>,</span><span class='hs-keyglyph'>[</span><span class='hs-num'>1</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <a name="line-86"></a><span class='hs-definition'>getCoef</span> <span class='hs-layout'>(</span><span class='hs-conid'>PolyPair</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>xs</span><span class='hs-layout'>,</span><span class='hs-conid'>Poly</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-layout'>(</span><span class='hs-varid'>xs</span><span class='hs-layout'>,</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <a name="line-87"></a><span class='hs-definition'>getCoef</span> <span class='hs-keyword'>_</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>error</span> <span class='hs-str'>"getCoef: Nested fractions found"</span> <a name="line-88"></a> <a name="line-89"></a><a name="scalePoly"></a><span class='hs-definition'>scalePoly</span> <span class='hs-keyglyph'>::</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'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-conid'>Poly</span> <span class='hs-varid'>a</span> <a name="line-90"></a><span class='hs-definition'>scalePoly</span> <span class='hs-varid'>s</span> <span class='hs-varid'>p</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>mulPoly</span> <span class='hs-layout'>(</span><span class='hs-conid'>Poly</span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>s</span><span class='hs-keyglyph'>]</span><span class='hs-layout'>)</span> <span class='hs-varid'>p</span> <a name="line-91"></a> <a name="line-92"></a><a name="addPolyCoef"></a><span class='hs-definition'>addPolyCoef</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-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <a name="line-93"></a><span class='hs-definition'>addPolyCoef</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>zipWithExt</span> <span class='hs-layout'>(</span><span class='hs-num'>0</span><span class='hs-layout'>,</span><span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varop'>+</span><span class='hs-layout'>)</span> <a name="line-94"></a><a name="subPolyCoef"></a><span class='hs-definition'>subPolyCoef</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>RealFloat</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>=></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <a name="line-95"></a><span class='hs-definition'>subPolyCoef</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>zipWithExt</span> <span class='hs-layout'>(</span><span class='hs-num'>0</span><span class='hs-layout'>,</span><span class='hs-num'>0</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-comment'>-</span><span class='hs-layout'>)</span> <a name="line-96"></a> <a name="line-97"></a><a name="scalePolyCoef"></a><span class='hs-definition'>scalePolyCoef</span> <span class='hs-keyglyph'>::</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'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <a name="line-98"></a><span class='hs-definition'>scalePolyCoef</span> <span class='hs-varid'>s</span> <span class='hs-varid'>p</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>map</span> <span class='hs-layout'>(</span><span class='hs-varid'>s</span><span class='hs-varop'>*</span><span class='hs-layout'>)</span> <span class='hs-varid'>p</span> <a name="line-99"></a> <a name="line-100"></a><a name="zipWithExt"></a><span class='hs-comment'>-- |Extended version of 'zipWith', which will add zeros to the shorter list.</span> <a name="line-101"></a><span class='hs-definition'>zipWithExt</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span><span class='hs-varid'>b</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-></span> <span class='hs-varid'>c</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>a</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>b</span><span class='hs-keyglyph'>]</span> <span class='hs-keyglyph'>-></span> <span class='hs-keyglyph'>[</span><span class='hs-varid'>c</span><span class='hs-keyglyph'>]</span> <a name="line-102"></a><span class='hs-definition'>zipWithExt</span> <span class='hs-keyword'>_</span> <span class='hs-keyword'>_</span> <span class='hs-conid'>[]</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>[]</span> <a name="line-103"></a><span class='hs-definition'>zipWithExt</span> <span class='hs-layout'>(</span><span class='hs-varid'>x0</span><span class='hs-layout'>,</span><span class='hs-varid'>y0</span><span class='hs-layout'>)</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-conid'>[]</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>f</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y0</span> <span class='hs-conop'>:</span> <span class='hs-layout'>(</span><span class='hs-varid'>zipWithExt</span> <span class='hs-layout'>(</span><span class='hs-varid'>x0</span><span class='hs-layout'>,</span><span class='hs-varid'>y0</span><span class='hs-layout'>)</span> <span class='hs-varid'>f</span> <span class='hs-varid'>xs</span> <span class='hs-conid'>[]</span><span class='hs-layout'>)</span> <a name="line-104"></a><span class='hs-definition'>zipWithExt</span> <span class='hs-layout'>(</span><span class='hs-varid'>x0</span><span class='hs-layout'>,</span><span class='hs-varid'>y0</span><span class='hs-layout'>)</span> <span class='hs-varid'>f</span> <span class='hs-conid'>[]</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-conop'>:</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>f</span> <span class='hs-varid'>x0</span> <span class='hs-varid'>y</span> <span class='hs-conop'>:</span> <span class='hs-layout'>(</span><span class='hs-varid'>zipWithExt</span> <span class='hs-layout'>(</span><span class='hs-varid'>x0</span><span class='hs-layout'>,</span><span class='hs-varid'>y0</span><span class='hs-layout'>)</span> <span class='hs-varid'>f</span> <span class='hs-conid'>[]</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <a name="line-105"></a><span class='hs-definition'>zipWithExt</span> <span class='hs-layout'>(</span><span class='hs-varid'>x0</span><span class='hs-layout'>,</span><span class='hs-varid'>y0</span><span class='hs-layout'>)</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>x</span><span class='hs-conop'>:</span><span class='hs-varid'>xs</span><span class='hs-layout'>)</span> <span class='hs-layout'>(</span><span class='hs-varid'>y</span><span class='hs-conop'>:</span><span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>f</span> <span class='hs-varid'>x</span> <span class='hs-varid'>y</span> <span class='hs-conop'>:</span> <span class='hs-layout'>(</span><span class='hs-varid'>zipWithExt</span> <span class='hs-layout'>(</span><span class='hs-varid'>x0</span><span class='hs-layout'>,</span><span class='hs-varid'>y0</span><span class='hs-layout'>)</span> <span class='hs-varid'>f</span> <span class='hs-varid'>xs</span> <span class='hs-varid'>ys</span><span class='hs-layout'>)</span> <a name="line-106"></a> </pre></body> </html>
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