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| author | Robert McIntyre <rlm@mit.edu> |
|---|---|
| date | Fri, 28 Oct 2011 04:56:48 -0700 |
| parents | b4de894a1e2e |
| children |
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1 <?xml version="1.0" encoding="utf-8"?>2 <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"3 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">4 <html xmlns="http://www.w3.org/1999/xhtml"5 lang="en" xml:lang="en">6 <head>7 <title>Building a Classical Mechanics Library in Clojure</title>8 <meta http-equiv="Content-Type" content="text/html;charset=utf-8"/>9 <meta name="generator" content="Org-mode"/>10 <meta name="generated" content="2011-08-09 18:41:37 EDT"/>11 <meta name="author" content="Robert McIntyre & Dylan Holmes"/>12 <meta name="description" content=""/>13 <meta name="keywords" content=""/>14 <style type="text/css">15 <!--/*--><![CDATA[/*><!--*/16 html { font-family: Times, serif; font-size: 12pt; }17 .title { text-align: center; }18 .todo { color: red; }19 .done { color: green; }20 .tag { background-color: #add8e6; font-weight:normal }21 .target { }22 .timestamp { color: #bebebe; }23 .timestamp-kwd { color: #5f9ea0; }24 .right {margin-left:auto; margin-right:0px; text-align:right;}25 .left {margin-left:0px; 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}55 /*]]>*/-->56 </style>57 <link rel="stylesheet" type="text/css" href="../css/aurellem.css" />58 <script type="text/javascript">59 <!--/*--><![CDATA[/*><!--*/60 function CodeHighlightOn(elem, id)61 {62 var target = document.getElementById(id);63 if(null != target) {64 elem.cacheClassElem = elem.className;65 elem.cacheClassTarget = target.className;66 target.className = "code-highlighted";67 elem.className = "code-highlighted";68 }69 }70 function CodeHighlightOff(elem, id)71 {72 var target = document.getElementById(id);73 if(elem.cacheClassElem)74 elem.className = elem.cacheClassElem;75 if(elem.cacheClassTarget)76 target.className = elem.cacheClassTarget;77 }78 /*]]>*///-->79 </script>80 <script type="text/javascript" src="../MathJax/MathJax.js">81 <!--/*--><![CDATA[/*><!--*/82 MathJax.Hub.Config({83 // Only one of the two following lines, depending on user settings84 // First allows browser-native MathML display, second forces HTML/CSS85 config: ["MMLorHTML.js"], jax: ["input/TeX"],86 // jax: ["input/TeX", "output/HTML-CSS"],87 extensions: ["tex2jax.js","TeX/AMSmath.js","TeX/AMSsymbols.js",88 "TeX/noUndefined.js"],89 tex2jax: {90 inlineMath: [ ["\\(","\\)"] ],91 displayMath: [ ['$$','$$'], ["\\[","\\]"], ["\\begin{displaymath}","\\end{displaymath}"] ],92 skipTags: ["script","noscript","style","textarea","pre","code"],93 ignoreClass: "tex2jax_ignore",94 processEscapes: false,95 processEnvironments: true,96 preview: "TeX"97 },98 showProcessingMessages: true,99 displayAlign: "left",100 displayIndent: "2em",102 "HTML-CSS": {103 scale: 100,104 availableFonts: ["STIX","TeX"],105 preferredFont: "TeX",106 webFont: "TeX",107 imageFont: "TeX",108 showMathMenu: true,109 },110 MMLorHTML: {111 prefer: {112 MSIE: "MML",113 Firefox: "MML",114 Opera: "HTML",115 other: "HTML"116 }117 }118 });119 /*]]>*///-->120 </script>121 </head>122 <body>124 <div id="content">128 <div class="header">129 <div class="float-right">130 <!--131 <form>132 <input type="text"/><input type="submit" value="search the blog »"/>133 </form>134 -->135 </div>137 <h1>aurellem <em>☉</em></h1>138 <ul class="nav">139 <li><a href="/">read the blog »</a></li>140 <!-- li><a href="#">learn about us »</a></li-->141 </ul>142 </div>144 <h1 class="title">Building a Classical Mechanics Library in Clojure</h1>152 <div id="table-of-contents">153 <h2>Table of Contents</h2>154 <div id="text-table-of-contents">155 <ul>156 <li><a href="#sec-1">1 Generic Arithmetic </a></li>157 <li><a href="#sec-2">2 Useful Data Types </a>158 <ul>159 <li><a href="#sec-2-1">2.1 Complex Numbers </a></li>160 <li><a href="#sec-2-2">2.2 Tuples and Tensors </a>161 <ul>162 <li><a href="#sec-2-2-1">2.2.1 Contraction </a></li>163 <li><a href="#sec-2-2-2">2.2.2 Matrices </a></li>164 </ul>165 </li>166 <li><a href="#sec-2-3">2.3 Power Series </a></li>167 </ul>168 </li>169 <li><a href="#sec-3">3 Basic Utilities </a>170 <ul>171 <li><a href="#sec-3-1">3.1 Sequence manipulation </a></li>172 <li><a href="#sec-3-2">3.2 Ranges, Airity and Function Composition </a></li>173 </ul>174 </li>175 <li><a href="#sec-4">4 Numerical Methods </a></li>176 <li><a href="#sec-5">5 Differentiation </a></li>177 <li><a href="#sec-6">6 Symbolic Manipulation </a></li>178 </ul>179 </div>180 </div>182 <div id="outline-container-1" class="outline-2">183 <h2 id="sec-1"><span class="section-number-2">1</span> Generic Arithmetic </h2>184 <div class="outline-text-2" id="text-1">190 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">ns</span> sicm.utils)191 (<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)195 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">all-equal?</span> [coll]196 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">empty?</span> (<span style="color: #8cd0d3;">rest</span> coll)) true197 (<span style="color: #f0dfaf; font-weight: bold;">and</span> (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">first</span> coll) (<span style="color: #8cd0d3;">second</span> coll))198 (<span style="color: #f0dfaf; font-weight: bold;">recur</span> (<span style="color: #8cd0d3;">rest</span> coll)))))201 (<span style="color: #f0dfaf; font-weight: bold;">defprotocol</span> <span style="color: #f0dfaf;">Arithmetic</span>202 (zero [this])203 (one [this])204 (negate [this])205 (invert [this])206 (conjugate [this]))208 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">zero?</span> [x]209 (<span style="color: #8cd0d3;">=</span> x (zero x)))210 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">one?</span> [x]211 (<span style="color: #8cd0d3;">=</span> x (one x)))215 (<span style="color: #8cd0d3;">extend-protocol</span> Arithmetic216 java.lang.Number217 (zero [x] 0)218 (one [x] 1)219 (negate [x] (<span style="color: #8cd0d3;">-</span> x))220 (invert [x] (<span style="color: #8cd0d3;">/</span> x))221 )223 (<span style="color: #8cd0d3;">extend-protocol</span> Arithmetic224 clojure.lang.IFn225 (one [f] identity)226 (negate [f] (<span style="color: #8cd0d3;">comp</span> negate f))227 (invert [f] (<span style="color: #8cd0d3;">comp</span> invert f)))230 (<span style="color: #8cd0d3;">extend-protocol</span> Arithmetic231 clojure.lang.Seqable232 (zero [this] (<span style="color: #8cd0d3;">map</span> zero this))233 (one [this] (<span style="color: #8cd0d3;">map</span> one this))234 (invert [this] (<span style="color: #8cd0d3;">map</span> invert this))235 (negate [this] (<span style="color: #8cd0d3;">map</span> negate this)))238 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">ordered-like</span>239 <span style="color: #8fb28f;">"Create a comparator using the sorted collection as an</span>240 <span style="color: #8fb28f;"> example. Elements not in the sorted collection are sorted to the</span>241 <span style="color: #8fb28f;"> end."</span>242 [sorted-coll]243 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [244 sorted-coll? (<span style="color: #8cd0d3;">set</span> sorted-coll)245 ascending-pair?246 (<span style="color: #8cd0d3;">set</span>(<span style="color: #8cd0d3;">reduce</span> concat247 (map-indexed248 (<span style="color: #8cd0d3;">fn</span> [n x]249 (<span style="color: #8cd0d3;">map</span> #(<span style="color: #8cd0d3;">vector</span> x %) (<span style="color: #8cd0d3;">nthnext</span> sorted-coll n)))250 sorted-coll)))]251 (<span style="color: #8cd0d3;">fn</span> [x y]252 (<span style="color: #f0dfaf; font-weight: bold;">cond</span>253 (<span style="color: #8cd0d3;">=</span> x y) 0254 (ascending-pair? [x y]) -1255 (ascending-pair? [y x]) 1256 (sorted-coll? x) -1257 (sorted-coll? y) 1))))261 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">type-precedence</span>262 (ordered-like [incanter.Matrix]))264 (<span style="color: #f0dfaf; font-weight: bold;">defmulti</span> <span style="color: #f0dfaf;">add</span>265 (<span style="color: #8cd0d3;">fn</span> [x y]266 (<span style="color: #8cd0d3;">sort</span> type-precedence [(<span style="color: #8cd0d3;">type</span> x)(<span style="color: #8cd0d3;">type</span> y)])))268 (<span style="color: #f0dfaf; font-weight: bold;">defmulti</span> <span style="color: #f0dfaf;">multiply</span>269 (<span style="color: #8cd0d3;">fn</span> [x y]270 (<span style="color: #8cd0d3;">sort</span> type-precedence [(<span style="color: #8cd0d3;">type</span> x) (<span style="color: #8cd0d3;">type</span> y)])))272 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">add</span> [java.lang.Number java.lang.Number] [x y] (<span style="color: #8cd0d3;">+</span> x y))273 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">multiply</span> [java.lang.Number java.lang.Number] [x y] (<span style="color: #8cd0d3;">*</span> x y))275 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">multiply</span> [incanter.Matrix java.lang.Integer] [x y]276 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [args (<span style="color: #8cd0d3;">sort</span> #(type-precedence (<span style="color: #8cd0d3;">type</span> %1)(<span style="color: #8cd0d3;">type</span> %2)) [x y])277 matrix (<span style="color: #8cd0d3;">first</span> args)278 scalar (<span style="color: #8cd0d3;">second</span> args)]279 (incanter.core/matrix (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">partial</span> map (<span style="color: #8cd0d3;">partial</span> multiply scalar)) matrix))))281 </pre>287 </div>289 </div>291 <div id="outline-container-2" class="outline-2">292 <h2 id="sec-2"><span class="section-number-2">2</span> Useful Data Types </h2>293 <div class="outline-text-2" id="text-2">297 </div>299 <div id="outline-container-2-1" class="outline-3">300 <h3 id="sec-2-1"><span class="section-number-3">2.1</span> Complex Numbers </h3>301 <div class="outline-text-3" id="text-2-1">306 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)308 (<span style="color: #f0dfaf; font-weight: bold;">defprotocol</span> <span style="color: #f0dfaf;">Complex</span>309 (real-part [z])310 (imaginary-part [z])311 (magnitude-squared [z])312 (angle [z])313 (conjugate [z])314 (norm [z]))316 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">complex-rectangular</span>317 <span style="color: #8fb28f;">"Define a complex number with the given real and imaginary</span>318 <span style="color: #8fb28f;"> components."</span>319 [re im]320 (<span style="color: #8cd0d3;">reify</span> Complex321 (real-part [z] re)322 (imaginary-part [z] im)323 (magnitude-squared [z] (<span style="color: #8cd0d3;">+</span> (<span style="color: #8cd0d3;">*</span> re re) (<span style="color: #8cd0d3;">*</span> im im)))324 (angle [z] (java.lang.Math/atan2 im re))325 (conjugate [z] (complex-rectangular re (<span style="color: #8cd0d3;">-</span> im)))327 Arithmetic328 (zero [z] (complex-rectangular 0 0))329 (one [z] (complex-rectangular 1 0))330 (negate [z] (complex-rectangular (<span style="color: #8cd0d3;">-</span> re) (<span style="color: #8cd0d3;">-</span> im)))331 (invert [z] (complex-rectangular332 (<span style="color: #8cd0d3;">/</span> re (magnitude-squared z))333 (<span style="color: #8cd0d3;">/</span> (<span style="color: #8cd0d3;">-</span> im) (magnitude-squared z))))335 Object336 (toString [_]337 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #f0dfaf; font-weight: bold;">and</span> (zero? re) (zero? im)) (<span style="color: #8cd0d3;">str</span> 0)338 (<span style="color: #8cd0d3;">str</span>339 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not</span>(zero? re))340 re)341 (<span style="color: #f0dfaf; font-weight: bold;">if</span> ((<span style="color: #8cd0d3;">comp</span> not zero?) im)342 (<span style="color: #8cd0d3;">str</span>343 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">neg?</span> im) <span style="color: #cc9393;">"-"</span> <span style="color: #cc9393;">"+"</span>)344 (<span style="color: #f0dfaf; font-weight: bold;">if</span> ((<span style="color: #8cd0d3;">comp</span> not one?) (java.lang.Math/abs im))345 (java.lang.Math/abs im))346 <span style="color: #cc9393;">"i"</span>)))))))348 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">complex-polar</span>349 <span style="color: #8fb28f;">"Define a complex number with the given magnitude and angle."</span>350 [mag ang]351 (<span style="color: #8cd0d3;">reify</span> Complex352 (magnitude-squared [z] (<span style="color: #8cd0d3;">*</span> mag mag))353 (angle [z] angle)354 (real-part [z] (<span style="color: #8cd0d3;">*</span> mag (java.lang.Math/cos ang)))355 (imaginary-part [z] (<span style="color: #8cd0d3;">*</span> mag (java.lang.Math/sin ang)))356 (conjugate [z] (complex-polar mag (<span style="color: #8cd0d3;">-</span> ang)))358 Arithmetic359 (zero [z] (complex-polar 0 0))360 (one [z] (complex-polar 1 0))361 (negate [z] (complex-polar (<span style="color: #8cd0d3;">-</span> mag) ang))362 (invert [z] (complex-polar (<span style="color: #8cd0d3;">/</span> mag) (<span style="color: #8cd0d3;">-</span> ang)))364 Object365 (toString [_] (<span style="color: #8cd0d3;">str</span> mag <span style="color: #cc9393;">" * e^(i"</span> ang<span style="color: #cc9393;">")"</span>))366 ))369 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">Numbers are complex quantities</span>371 (<span style="color: #8cd0d3;">extend-protocol</span> Complex372 java.lang.Number373 (real-part [x] x)374 (imaginary-part [x] 0)375 (magnitude [x] x)376 (angle [x] 0)377 (conjugate [x] x))380 </pre>387 </div>389 </div>391 <div id="outline-container-2-2" class="outline-3">392 <h3 id="sec-2-2"><span class="section-number-3">2.2</span> Tuples and Tensors </h3>393 <div class="outline-text-3" id="text-2-2">396 <p>397 A tuple is a vector which is spinable—it can be either <i>spin up</i> or <i>spin down</i>. (Covariant, contravariant; dual vectors)398 </p>401 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)403 (<span style="color: #f0dfaf; font-weight: bold;">defprotocol</span> <span style="color: #f0dfaf;">Spinning</span>404 (up? [this])405 (down? [this]))407 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">spin</span>408 <span style="color: #8fb28f;">"Returns the spin of the Spinning s, either :up or :down"</span>409 [<span style="color: #dfdfbf; font-weight: bold;">#^Spinning</span> s]410 (<span style="color: #f0dfaf; font-weight: bold;">cond</span> (up? s) <span style="color: #8cd0d3;">:up</span> (down? s) <span style="color: #8cd0d3;">:down</span>))413 (<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">Tuple</span>414 [spin coll]415 clojure.lang.Seqable416 (<span style="color: #8cd0d3;">seq</span> [this] (<span style="color: #8cd0d3;">seq</span> (.coll this)))417 clojure.lang.Counted418 (<span style="color: #8cd0d3;">count</span> [this] (<span style="color: #8cd0d3;">count</span> (.coll this))))420 (<span style="color: #8cd0d3;">extend-type</span> Tuple421 Spinning422 (up? [this] (<span style="color: #8cd0d3;">=</span> <span style="color: #8cd0d3;">::up</span> (.spin this)))423 (down? [this] (<span style="color: #8cd0d3;">=</span> <span style="color: #8cd0d3;">::down</span> (.spin this))))425 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">print-method</span> Tuple426 [o w]427 (<span style="color: #8cd0d3;">print-simple</span> (<span style="color: #8cd0d3;">str</span> (<span style="color: #f0dfaf; font-weight: bold;">if</span> (up? o) 'u 'd) (.coll o)) w))431 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">up</span>432 <span style="color: #8fb28f;">"Create a new up-tuple containing the contents of coll."</span>433 [coll]434 (Tuple. <span style="color: #8cd0d3;">::up</span> coll))436 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">down</span>437 <span style="color: #8fb28f;">"Create a new down-tuple containing the contents of coll."</span>438 [coll]439 (Tuple. <span style="color: #8cd0d3;">::down</span> coll))442 </pre>448 </div>450 <div id="outline-container-2-2-1" class="outline-4">451 <h4 id="sec-2-2-1"><span class="section-number-4">2.2.1</span> Contraction </h4>452 <div class="outline-text-4" id="text-2-2-1">454 <p>Contraction is a binary operation that you can apply to compatible455 tuples. Tuples are compatible for contraction if they have the same456 length and opposite spins, and if the corresponding items in each457 tuple are both numbers or both compatible tuples.458 </p>462 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)464 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">numbers?</span>465 <span style="color: #8fb28f;">"Returns true if all arguments are numbers, else false."</span>466 [& xs]467 (<span style="color: #8cd0d3;">every?</span> number? xs))469 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">contractible?</span>470 <span style="color: #8fb28f;">"Returns true if the tuples a and b are compatible for contraction,</span>471 <span style="color: #8fb28f;"> else false. Tuples are compatible if they have the same number of</span>472 <span style="color: #8fb28f;"> components, they have opposite spins, and their elements are</span>473 <span style="color: #8fb28f;"> pairwise-compatible."</span>474 [a b]475 (<span style="color: #f0dfaf; font-weight: bold;">and</span>476 (<span style="color: #8cd0d3;">isa?</span> (<span style="color: #8cd0d3;">type</span> a) Tuple)477 (<span style="color: #8cd0d3;">isa?</span> (<span style="color: #8cd0d3;">type</span> b) Tuple)478 (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">count</span> a) (<span style="color: #8cd0d3;">count</span> b))479 (<span style="color: #8cd0d3;">not=</span> (spin a) (spin b))481 (<span style="color: #8cd0d3;">not-any?</span> false?482 (<span style="color: #8cd0d3;">map</span> #(<span style="color: #f0dfaf; font-weight: bold;">or</span>483 (numbers? %1 %2)484 (contractible? %1 %2))485 a b))))489 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">contract</span>490 <span style="color: #8fb28f;">"Contracts two tuples, returning the sum of the</span>491 <span style="color: #8fb28f;"> products of the corresponding items. Contraction is recursive on</span>492 <span style="color: #8fb28f;"> nested tuples."</span>493 [a b]494 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not</span> (contractible? a b))495 (<span style="color: #f0dfaf; font-weight: bold;">throw</span>496 (Exception. <span style="color: #cc9393;">"Not compatible for contraction."</span>))497 (<span style="color: #8cd0d3;">reduce</span> +498 (<span style="color: #8cd0d3;">map</span>499 (<span style="color: #8cd0d3;">fn</span> [x y]500 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (numbers? x y)501 (<span style="color: #8cd0d3;">*</span> x y)502 (contract x y)))503 a b))))505 </pre>510 </div>512 </div>514 <div id="outline-container-2-2-2" class="outline-4">515 <h4 id="sec-2-2-2"><span class="section-number-4">2.2.2</span> Matrices </h4>516 <div class="outline-text-4" id="text-2-2-2">521 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)522 (<span style="color: #8cd0d3;">require</span> 'incanter.core) <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">use incanter's fast matrices</span>524 (<span style="color: #f0dfaf; font-weight: bold;">defprotocol</span> <span style="color: #f0dfaf;">Matrix</span>525 (rows [matrix])526 (cols [matrix])527 (diagonal [matrix])528 (trace [matrix])529 (determinant [matrix])530 (transpose [matrix])531 (conjugate [matrix])532 )534 (<span style="color: #8cd0d3;">extend-protocol</span> Matrix535 incanter.Matrix536 (rows [rs] (<span style="color: #8cd0d3;">map</span> down (<span style="color: #8cd0d3;">apply</span> map vector (<span style="color: #8cd0d3;">apply</span> map vector rs))))537 (cols [rs] (<span style="color: #8cd0d3;">map</span> up (<span style="color: #8cd0d3;">apply</span> map vector rs)))538 (diagonal [matrix] (incanter.core/diag matrix) )539 (determinant [matrix] (incanter.core/det matrix))540 (trace [matrix] (incanter.core/trace matrix))541 (transpose [matrix] (incanter.core/trans matrix))542 )544 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">count-rows</span> [matrix]545 ((<span style="color: #8cd0d3;">comp</span> count rows) matrix))547 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">count-cols</span> [matrix]548 ((<span style="color: #8cd0d3;">comp</span> count cols) matrix))550 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">square?</span> [matrix]551 (<span style="color: #8cd0d3;">=</span> (count-rows matrix) (count-cols matrix)))553 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">identity-matrix</span>554 <span style="color: #8fb28f;">"Define a square matrix of size n-by-n with 1s along the diagonal and</span>555 <span style="color: #8fb28f;"> 0s everywhere else."</span>556 [n]557 (incanter.core/identity-matrix n))563 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">matrix-by-rows</span>564 <span style="color: #8fb28f;">"Define a matrix by giving its rows."</span>565 [& rows]566 (<span style="color: #f0dfaf; font-weight: bold;">if</span>567 (<span style="color: #8cd0d3;">not</span> (all-equal? (<span style="color: #8cd0d3;">map</span> count rows)))568 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"All rows in a matrix must have the same number of elements."</span>))569 (incanter.core/matrix (<span style="color: #8cd0d3;">vec</span> rows))))571 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">matrix-by-cols</span>572 <span style="color: #8fb28f;">"Define a matrix by giving its columns"</span>573 [& cols]574 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not</span> (all-equal? (<span style="color: #8cd0d3;">map</span> count cols)))575 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"All columns in a matrix must have the same number of elements."</span>))576 (incanter.core/matrix (<span style="color: #8cd0d3;">vec</span> (<span style="color: #8cd0d3;">apply</span> map vector cols)))))578 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">identity-matrix</span>579 <span style="color: #8fb28f;">"Define a square matrix of size n-by-n with 1s along the diagonal and</span>580 <span style="color: #8fb28f;"> 0s everywhere else."</span>581 [n]582 (incanter.core/identity-matrix n))586 (<span style="color: #8cd0d3;">extend-protocol</span> Arithmetic587 incanter.Matrix588 (one [matrix]589 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (square? matrix)590 (identity-matrix (count-rows matrix))591 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"Non-square matrices have no multiplicative unit."</span>))))592 (zero [matrix]593 (<span style="color: #8cd0d3;">apply</span> matrix-by-rows (<span style="color: #8cd0d3;">map</span> zero (rows matrix))))594 (negate [matrix]595 (<span style="color: #8cd0d3;">apply</span> matrix-by-rows (<span style="color: #8cd0d3;">map</span> negate (rows matrix))))596 (invert [matrix]597 (incanter.core/solve matrix)))601 (<span style="color: #f0dfaf; font-weight: bold;">defmulti</span> <span style="color: #f0dfaf;">coerce-to-matrix</span>602 <span style="color: #8fb28f;">"Converts x into a matrix, if possible."</span>603 type)605 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">coerce-to-matrix</span> incanter.Matrix [x] x)606 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">coerce-to-matrix</span> Tuple [x]607 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">apply</span> numbers? (<span style="color: #8cd0d3;">seq</span> x))608 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (up? x)609 (matrix-by-cols (<span style="color: #8cd0d3;">seq</span> x))610 (matrix-by-rows (<span style="color: #8cd0d3;">seq</span> x)))611 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"Non-numerical tuple cannot be converted into a matrix."</span>))))617 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(defn matrix-by-cols</span>618 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">"Define a matrix by giving its columns."</span>619 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">[& cols]</span>620 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(cond</span>621 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(not (all-equal? (map count cols)))</span>622 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(throw (Exception. "All columns in a matrix must have the same number of elements."))</span>623 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">:else</span>624 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(reify Matrix</span>625 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(cols [this] (map up cols))</span>626 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(rows [this] (map down (apply map vector cols)))</span>627 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(diagonal [this] (map-indexed (fn [i col] (nth col i) cols)))</span>628 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(trace [this]</span>629 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(if (not= (count-cols this) (count-rows this))</span>630 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(throw (Exception.</span>631 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">"Cannot take the trace of a non-square matrix."))</span>632 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(reduce + (diagonal this))))</span>634 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(determinant [this]</span>635 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(if (not= (count-cols this) (count-rows this))</span>636 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(throw (Exception.</span>637 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">"Cannot take the determinant of a non-square matrix."))</span>638 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(reduce * (map-indexed (fn [i col] (nth col i)) cols))))</span>639 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">)))</span>641 (<span style="color: #8cd0d3;">extend-protocol</span> Matrix Tuple642 (rows [this] (<span style="color: #f0dfaf; font-weight: bold;">if</span> (down? this)643 (<span style="color: #8cd0d3;">list</span> this)644 (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">comp</span> up vector) this)))646 (cols [this] (<span style="color: #f0dfaf; font-weight: bold;">if</span> (up? this)647 (<span style="color: #8cd0d3;">list</span> this)648 (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">comp</span> down vector) this))649 ))651 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">matrix-multiply</span>652 <span style="color: #8fb28f;">"Returns the matrix resulting from the matrix multiplication of the given arguments."</span>653 ([A] (coerce-to-matrix A))654 ([A B] (incanter.core/mmult (coerce-to-matrix A) (coerce-to-matrix B)))655 ([M1 M2 & Ms] (<span style="color: #8cd0d3;">reduce</span> matrix-multiply (matrix-multiply M1 M2) Ms)))657 </pre>663 </div>664 </div>666 </div>668 <div id="outline-container-2-3" class="outline-3">669 <h3 id="sec-2-3"><span class="section-number-3">2.3</span> Power Series </h3>670 <div class="outline-text-3" id="text-2-3">675 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)676 (<span style="color: #8cd0d3;">use</span> 'clojure.contrib.def)680 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">series-fn</span>681 <span style="color: #8fb28f;">"The function corresponding to the given power series."</span>682 [series]683 (<span style="color: #8cd0d3;">fn</span> [x]684 (<span style="color: #8cd0d3;">reduce</span> +685 (map-indexed (<span style="color: #8cd0d3;">fn</span>[n x] (<span style="color: #8cd0d3;">*</span> (<span style="color: #8cd0d3;">float</span> (<span style="color: #8cd0d3;">nth</span> series n)) (<span style="color: #8cd0d3;">float</span>(java.lang.Math/pow (<span style="color: #8cd0d3;">float</span> x) n)) ))686 (<span style="color: #8cd0d3;">range</span> 20)))))688 (<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">PowerSeries</span>689 [coll]690 clojure.lang.Seqable691 (<span style="color: #8cd0d3;">seq</span> [this] (<span style="color: #8cd0d3;">seq</span> (.coll this)))693 clojure.lang.Indexed694 (<span style="color: #8cd0d3;">nth</span> [this n] (<span style="color: #8cd0d3;">nth</span> (.coll this) n 0))695 (<span style="color: #8cd0d3;">nth</span> [this n not-found] (<span style="color: #8cd0d3;">nth</span> (.coll this) n not-found))697 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">clojure.lang.IFn</span>698 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(call [this] (throw(Exception.)))</span>699 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(invoke [this & args] args</span>700 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(let [f </span>701 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">)</span>702 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(run [this] (throw(Exception.)))</span>703 )705 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">power-series</span>706 <span style="color: #8fb28f;">"Returns a power series with the items of the coll as its</span>707 <span style="color: #8fb28f;"> coefficients. Trailing zeros are added to the end of coll."</span>708 [coeffs]709 (PowerSeries. coeffs))711 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">power-series-indexed</span>712 <span style="color: #8fb28f;">"Returns a power series consisting of the result of mapping f to the non-negative integers."</span>713 [f]714 (PowerSeries. (<span style="color: #8cd0d3;">map</span> f (<span style="color: #8cd0d3;">range</span>))))717 (<span style="color: #f0dfaf; font-weight: bold;">defn-memo</span> <span style="color: #f0dfaf;">nth-partial-sum</span>718 ([series n]719 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (zero? n) (<span style="color: #8cd0d3;">first</span> series)720 (<span style="color: #8cd0d3;">+</span> (<span style="color: #8cd0d3;">nth</span> series n)721 (nth-partial-sum series (<span style="color: #8cd0d3;">dec</span> n))))))723 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">partial-sums</span> [series]724 (<span style="color: #8cd0d3;">lazy-seq</span> (<span style="color: #8cd0d3;">map</span> nth-partial-sum (<span style="color: #8cd0d3;">range</span>))))729 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">cos-series</span>730 (power-series-indexed731 (<span style="color: #8cd0d3;">fn</span>[n]732 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">odd?</span> n) 0733 (<span style="color: #8cd0d3;">/</span>734 (<span style="color: #8cd0d3;">reduce</span> *735 (<span style="color: #8cd0d3;">reduce</span> * (<span style="color: #8cd0d3;">repeat</span> (<span style="color: #8cd0d3;">/</span> n 2) -1))736 (<span style="color: #8cd0d3;">range</span> 1 (<span style="color: #8cd0d3;">inc</span> n)))737 )))))739 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">sin-series</span>740 (power-series-indexed741 (<span style="color: #8cd0d3;">fn</span>[n]742 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">even?</span> n) 0743 (<span style="color: #8cd0d3;">/</span>744 (<span style="color: #8cd0d3;">reduce</span> *745 (<span style="color: #8cd0d3;">reduce</span> * (<span style="color: #8cd0d3;">repeat</span> (<span style="color: #8cd0d3;">/</span> (<span style="color: #8cd0d3;">dec</span> n) 2) -1))746 (<span style="color: #8cd0d3;">range</span> 1 (<span style="color: #8cd0d3;">inc</span> n)))747 )))))749 </pre>755 </div>756 </div>758 </div>760 <div id="outline-container-3" class="outline-2">761 <h2 id="sec-3"><span class="section-number-2">3</span> Basic Utilities </h2>762 <div class="outline-text-2" id="text-3">766 </div>768 <div id="outline-container-3-1" class="outline-3">769 <h3 id="sec-3-1"><span class="section-number-3">3.1</span> Sequence manipulation </h3>770 <div class="outline-text-3" id="text-3-1">776 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">ns</span> sicm.utils)778 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">do-up</span>779 <span style="color: #8fb28f;">"Apply f to each number from low to high, presumably for</span>780 <span style="color: #8fb28f;"> side-effects."</span>781 [f low high]782 (<span style="color: #f0dfaf; font-weight: bold;">doseq</span> [i (<span style="color: #8cd0d3;">range</span> low high)] (f i)))784 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">do-down</span>785 <span style="color: #8fb28f;">"Apply f to each number from high to low, presumably for</span>786 <span style="color: #8fb28f;"> side-effects."</span>787 [f high low]788 (<span style="color: #f0dfaf; font-weight: bold;">doseq</span> [i (<span style="color: #8cd0d3;">range</span> high low -1)] (f i)))791 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">all-equal?</span> [coll]792 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">empty?</span> (<span style="color: #8cd0d3;">rest</span> coll)) true793 (<span style="color: #f0dfaf; font-weight: bold;">and</span> (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">first</span> coll) (<span style="color: #8cd0d3;">second</span> coll))794 (<span style="color: #f0dfaf; font-weight: bold;">recur</span> (<span style="color: #8cd0d3;">rest</span> coll))))))796 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">multiplier</span>797 <span style="color: #8fb28f;">"Returns a function that 'multiplies' the members of a collection,</span>798 <span style="color: #8fb28f;">returning unit if called on an empty collection."</span>799 [multiply unit]800 (<span style="color: #8cd0d3;">fn</span> [coll] ((<span style="color: #8cd0d3;">partial</span> reduce multiply unit) coll)))802 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">divider</span>803 <span style="color: #8fb28f;">"Returns a function that 'divides' the first element of a collection</span>804 <span style="color: #8fb28f;">by the 'product' of the rest of the collection."</span>805 [divide multiply invert unit]806 (<span style="color: #8cd0d3;">fn</span> [coll]807 (<span style="color: #8cd0d3;">apply</span>808 (<span style="color: #8cd0d3;">fn</span>809 ([] unit)810 ([x] (invert x))811 ([x y] (divide x y))812 ([x y & zs] (divide x (<span style="color: #8cd0d3;">reduce</span> multiply y zs))))813 coll)))815 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">left-circular-shift</span>816 <span style="color: #8fb28f;">"Remove the first element of coll, adding it to the end of coll."</span>817 [coll]818 (<span style="color: #8cd0d3;">concat</span> (<span style="color: #8cd0d3;">rest</span> coll) (<span style="color: #8cd0d3;">take</span> 1 coll)))820 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">right-circular-shift</span>821 <span style="color: #8fb28f;">"Remove the last element of coll, adding it to the front of coll."</span>822 [coll]823 (<span style="color: #8cd0d3;">cons</span> (<span style="color: #8cd0d3;">last</span> coll) (<span style="color: #8cd0d3;">butlast</span> coll)))824 </pre>832 </div>834 </div>836 <div id="outline-container-3-2" class="outline-3">837 <h3 id="sec-3-2"><span class="section-number-3">3.2</span> Ranges, Airity and Function Composition </h3>838 <div class="outline-text-3" id="text-3-2">843 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)844 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">infinity</span> Double/POSITIVE_INFINITY)845 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">infinite?</span> [x] (Double/isInfinite x))846 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">finite?</span> (<span style="color: #8cd0d3;">comp</span> not infinite?))848 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">arity-min</span>849 <span style="color: #8fb28f;">"Returns the smallest number of arguments f can take."</span>850 [f]851 (<span style="color: #8cd0d3;">apply</span>852 min853 (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">comp</span> alength #(.getParameterTypes %))854 (<span style="color: #8cd0d3;">filter</span> (<span style="color: #8cd0d3;">comp</span> (<span style="color: #8cd0d3;">partial</span> = <span style="color: #cc9393;">"invoke"</span>) #(.getName %))855 (.getDeclaredMethods (<span style="color: #8cd0d3;">class</span> f))))))857 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">arity-max</span>858 <span style="color: #8fb28f;">"Returns the largest number of arguments f can take, possibly</span>859 <span style="color: #8fb28f;"> Infinity."</span>860 [f]861 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [methods (.getDeclaredMethods (<span style="color: #8cd0d3;">class</span> f))]862 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not-any?</span> (<span style="color: #8cd0d3;">partial</span> = <span style="color: #cc9393;">"doInvoke"</span>) (<span style="color: #8cd0d3;">map</span> #(.getName %) methods))863 (<span style="color: #8cd0d3;">apply</span> max864 (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">comp</span> alength #(.getParameterTypes %))865 (<span style="color: #8cd0d3;">filter</span> (<span style="color: #8cd0d3;">comp</span> (<span style="color: #8cd0d3;">partial</span> = <span style="color: #cc9393;">"invoke"</span>) #(.getName %)) methods)))866 infinity)))869 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">^</span>{<span style="color: #8cd0d3;">:arglists</span> '([f])870 <span style="color: #8cd0d3;">:doc</span> <span style="color: #cc9393;">"Returns a two-element list containing the minimum and</span>871 <span style="color: #cc9393;"> maximum number of args that f can take."</span>}872 arity-interval873 (<span style="color: #8cd0d3;">juxt</span> arity-min arity-max))877 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">--- intervals</span>879 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">intersect</span>880 <span style="color: #8fb28f;">"Returns the interval of overlap between interval-1 and interval-2"</span>881 [interval-1 interval-2]882 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #f0dfaf; font-weight: bold;">or</span> (<span style="color: #8cd0d3;">empty?</span> interval-1) (<span style="color: #8cd0d3;">empty?</span> interval-2)) []883 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [left (<span style="color: #8cd0d3;">max</span> (<span style="color: #8cd0d3;">first</span> interval-1) (<span style="color: #8cd0d3;">first</span> interval-2))884 right (<span style="color: #8cd0d3;">min</span> (<span style="color: #8cd0d3;">second</span> interval-1) (<span style="color: #8cd0d3;">second</span> interval-2))]885 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">></span> left right) []886 [left right]))))888 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">same-endpoints?</span>889 <span style="color: #8fb28f;">"Returns true if the left endpoint is the same as the right</span>890 <span style="color: #8fb28f;"> endpoint."</span>891 [interval]892 (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">first</span> interval) (<span style="color: #8cd0d3;">second</span> interval)))894 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">naturals?</span>895 <span style="color: #8fb28f;">"Returns true if the left endpoint is 0 and the right endpoint is</span>896 <span style="color: #8fb28f;">infinite."</span>897 [interval]898 (<span style="color: #f0dfaf; font-weight: bold;">and</span> (zero? (<span style="color: #8cd0d3;">first</span> interval))899 (infinite? (<span style="color: #8cd0d3;">second</span> interval))))902 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">fan-in</span>903 <span style="color: #8fb28f;">"Returns a function that pipes its input to each of the gs, then</span>904 <span style="color: #8fb28f;"> applies f to the list of results. Consequently, f must be able to</span>905 <span style="color: #8fb28f;"> take a number of arguments equal to the number of gs."</span>906 [f & gs]907 (<span style="color: #8cd0d3;">fn</span> [& args]908 (<span style="color: #8cd0d3;">apply</span> f (<span style="color: #8cd0d3;">apply</span> (<span style="color: #8cd0d3;">apply</span> juxt gs) args))))910 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">fan-in</span>911 <span style="color: #8fb28f;">"Returns a function that pipes its input to each of the gs, then applies f to the list of results. The resulting function takes any number of arguments, but will fail if given arguments that are incompatible with any of the gs."</span>912 [f & gs]913 (<span style="color: #8cd0d3;">comp</span> (<span style="color: #8cd0d3;">partial</span> apply f) (<span style="color: #8cd0d3;">apply</span> juxt gs)))917 (<span style="color: #f0dfaf; font-weight: bold;">defmacro</span> <span style="color: #f0dfaf;">airty-blah-sad</span> [f n more?]918 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [syms (<span style="color: #8cd0d3;">vec</span> (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">comp</span> gensym (<span style="color: #8cd0d3;">partial</span> str <span style="color: #cc9393;">"x"</span>)) (<span style="color: #8cd0d3;">range</span> n)))919 optional (<span style="color: #8cd0d3;">gensym</span> <span style="color: #cc9393;">"xs"</span>)]920 (<span style="color: #f0dfaf; font-weight: bold;">if</span> more?921 `(<span style="color: #8cd0d3;">fn</span> <span style="color: #f0dfaf;">~</span>(<span style="color: #8cd0d3;">conj</span> syms '& optional)922 (<span style="color: #8cd0d3;">apply</span> ~f ~@syms ~optional))923 `(<span style="color: #8cd0d3;">fn</span> <span style="color: #f0dfaf;">~syms</span> (~f ~@syms)))))925 (<span style="color: #f0dfaf; font-weight: bold;">defmacro</span> <span style="color: #f0dfaf;">airt-whaa*</span> [f n more?]926 `(airty-blah-sad ~f ~n ~more?))931 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">fan-in*</span>932 <span style="color: #8fb28f;">"Returns a function that pipes its input to each of the gs, then</span>933 <span style="color: #8fb28f;"> applies f to the list of results. Unlike fan-in, fan-in* strictly</span>934 <span style="color: #8fb28f;"> enforces arity: it will fail if the gs do not have compatible</span>935 <span style="color: #8fb28f;"> arities."</span>936 [f & gs]937 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [arity-in (<span style="color: #8cd0d3;">reduce</span> intersect (<span style="color: #8cd0d3;">map</span> arity-interval gs))938 left (<span style="color: #8cd0d3;">first</span> arity-in)939 right (<span style="color: #8cd0d3;">second</span> arity-in)940 composite (fan-in f gs)941 ]942 (<span style="color: #f0dfaf; font-weight: bold;">cond</span>943 (<span style="color: #8cd0d3;">empty?</span> arity-in)944 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"Cannot compose functions with incompatible arities."</span>))946 (<span style="color: #8cd0d3;">not</span>947 (<span style="color: #f0dfaf; font-weight: bold;">or</span> (<span style="color: #8cd0d3;">=</span> left right)948 (<span style="color: #f0dfaf; font-weight: bold;">and</span> (finite? left)949 (<span style="color: #8cd0d3;">=</span> right infinity))))951 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception.952 <span style="color: #cc9393;">"Compose can only handle arities of the form [n n] or [n infinity]"</span>))953 <span style="color: #8cd0d3;">:else</span>954 (airty-blah-sad composite left (<span style="color: #8cd0d3;">=</span> right infinity)))))958 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">compose-n</span> <span style="color: #8fb28f;">"Compose any number of functions together."</span>959 ([] identity)960 ([f] f)961 ([f & fs]962 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [fns (<span style="color: #8cd0d3;">cons</span> f fs)]963 (compose-bin (<span style="color: #8cd0d3;">reduce</span> fan-in (<span style="color: #8cd0d3;">butlast</span> fs)) (<span style="color: #8cd0d3;">last</span> fs))))964 )971 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">iterated</span>972 ([f n id] (<span style="color: #8cd0d3;">reduce</span> comp id (<span style="color: #8cd0d3;">repeat</span> n f)))973 ([f n] (<span style="color: #8cd0d3;">reduce</span> comp identity (<span style="color: #8cd0d3;">repeat</span> n f))))975 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">iterate-until-stable</span>976 <span style="color: #8fb28f;">"Repeatedly applies f to x, returning the first result that is close</span>977 <span style="color: #8fb28f;">enough to its predecessor."</span>978 [f close-enough? x]979 (<span style="color: #8cd0d3;">second</span> (swank.util/find-first980 (<span style="color: #8cd0d3;">partial</span> apply close-enough?)981 (<span style="color: #8cd0d3;">partition</span> 2 1 (<span style="color: #8cd0d3;">iterate</span> f x)))))983 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">lexical<</span> [x y]984 (<span style="color: #8cd0d3;">neg?</span> (<span style="color: #8cd0d3;">compare</span> (<span style="color: #8cd0d3;">str</span> x) (<span style="color: #8cd0d3;">str</span> y))))987 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">do-up</span>988 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">do-down</span>989 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">make-pairwise-test</span> comparator)990 <span style="color: #708070;">;;</span><span style="color: #7f9f7f;">all-equal?</span>991 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">accumulation</span> multiplier)992 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">inverse-accumulation</span> divider)993 <span style="color: #708070;">;;</span><span style="color: #7f9f7f;">left-circular-shift</span>994 <span style="color: #708070;">;;</span><span style="color: #7f9f7f;">right-circular-shift</span>995 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">exactly-n?</span> same-endpoints?)996 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">any-number?</span> naturals?)997 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO compose</span>998 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO compose-n</span>999 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">identity</span>1000 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">compose-2</span> fan-in)1001 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">compose-bin</span> fan-in*)1002 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">any?</span> (<span style="color: #8cd0d3;">constantly</span> true))1003 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">none?</span> (<span style="color: #8cd0d3;">constantly</span> false))1004 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">constant</span> constantly)1005 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">joint-arity</span> intersect)1006 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">a-reduce</span> reduce)1007 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">filter</span>1008 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">make-map</span> (<span style="color: #8cd0d3;">partial</span> partial map) )1009 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">bracket</span> juxt)1010 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO apply-to-all</span>1011 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO nary-combine</span>1012 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO binary-combine</span>1013 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO unary-combine</span>1014 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">iterated</span>1015 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">iterate-until-stable</span>1016 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">make-function-of-vector</span> (<span style="color: #8cd0d3;">partial</span> partial map))1017 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">make-function-of-arguments</span> (<span style="color: #8cd0d3;">fn</span> [f] (<span style="color: #8cd0d3;">fn</span> [& args] (f args))))1018 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">alphaless</span> lexical<)1020 </pre>1033 </div>1034 </div>1036 </div>1038 <div id="outline-container-4" class="outline-2">1039 <h2 id="sec-4"><span class="section-number-2">4</span> Numerical Methods </h2>1040 <div class="outline-text-2" id="text-4">1045 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)1046 (<span style="color: #f0dfaf; font-weight: bold;">import</span> java.lang.Math)1047 (<span style="color: #8cd0d3;">use</span> 'clojure.contrib.def)1049 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">---- USEFUL CONSTANTS</span>1051 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">machine-epsilon</span>1052 <span style="color: #8fb28f;">"Smallest value representable on your machine, as determined by</span>1053 <span style="color: #8fb28f;">successively dividing a number in half until consecutive results are</span>1054 <span style="color: #8fb28f;">indistinguishable."</span>1055 []1056 (<span style="color: #8cd0d3;">ffirst</span>1057 (<span style="color: #8cd0d3;">drop-while</span>1058 (<span style="color: #8cd0d3;">comp</span> not zero? second)1059 (<span style="color: #8cd0d3;">partition</span> 2 11060 (<span style="color: #8cd0d3;">iterate</span> (<span style="color: #8cd0d3;">partial</span> * 0.5) 1)))))1063 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">pi</span> (Math/PI))1064 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">two-pi</span> (<span style="color: #8cd0d3;">*</span> 2 pi))1066 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">eulers-gamma</span> 0.5772156649015328606065)1068 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">phi</span> (<span style="color: #8cd0d3;">/</span> (<span style="color: #8cd0d3;">inc</span> (Math/sqrt 5)) 2))1070 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">ln2</span> (Math/log 2))1071 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">ln10</span> (Math/log 10))1072 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">exp10</span> #(Math/pow 10 %))1073 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">exp2</span> #(Math/pow 2 %))1076 <span style="color: #708070;">;;</span>1078 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">---- ANGLES AND TRIGONOMETRY</span>1080 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">angle-restrictor</span>1081 <span style="color: #8fb28f;">"Returns a function that ensures that angles lie in the specified interval of length two-pi."</span>1082 [max-angle]1083 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [min-angle (<span style="color: #8cd0d3;">-</span> max-angle two-pi)]1084 (<span style="color: #8cd0d3;">fn</span> [x]1085 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #f0dfaf; font-weight: bold;">and</span>1086 (<span style="color: #8cd0d3;"><=</span> min-angle x)1087 (<span style="color: #8cd0d3;"><</span> x max-angle))1088 x1089 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [corrected-x (<span style="color: #8cd0d3;">-</span> x (<span style="color: #8cd0d3;">*</span> two-pi (Math/floor (<span style="color: #8cd0d3;">/</span> x two-pi))))]1090 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;"><</span> corrected-x max-angle)1091 corrected-x1092 (<span style="color: #8cd0d3;">-</span> corrected-x two-pi)))))))1094 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">angle-restrict-pi</span>1095 <span style="color: #8fb28f;">"Coerces angles to lie in the interval from -pi to pi."</span>1096 [angle]1097 ((angle-restrictor pi) angle))1099 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">angle-restrict-two-pi</span>1100 <span style="color: #8fb28f;">"Coerces angles to lie in the interval from zero to two-pi"</span>1101 [angle]1102 ((angle-restrictor two-pi) angle))1107 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">invert</span> [x] (<span style="color: #8cd0d3;">/</span> x))1108 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">negate</span> [x] (<span style="color: #8cd0d3;">-</span> x))1110 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">exp</span> [x] (Math/exp x))1112 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">sin</span> [x] (Math/sin x))1113 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">cos</span> [x] (Math/cos x))1114 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">tan</span> [x] (Math/tan x))1116 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">sec</span> (<span style="color: #8cd0d3;">comp</span> invert cos))1117 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">csc</span> (<span style="color: #8cd0d3;">comp</span> invert sin))1119 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">sinh</span> [x] (Math/sinh x))1120 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">cosh</span> [x] (Math/cosh x))1121 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">tanh</span> [x] (Math/tanh x))1123 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">sech</span> (<span style="color: #8cd0d3;">comp</span> invert cosh))1124 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">csch</span> (<span style="color: #8cd0d3;">comp</span> invert sinh))1127 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">------------</span>1129 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">factorial</span>1130 <span style="color: #8fb28f;">"Computes the factorial of the nonnegative integer n."</span>1131 [n]1132 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">neg?</span> n)1133 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"Cannot compute the factorial of a negative number."</span>))1134 (<span style="color: #8cd0d3;">reduce</span> * 1 (<span style="color: #8cd0d3;">range</span> 1 (<span style="color: #8cd0d3;">inc</span> n)))))1136 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">exact-quotient</span> [n d] (<span style="color: #8cd0d3;">/</span> n d))1138 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">binomial-coefficient</span>1139 <span style="color: #8fb28f;">"Computes the number of different ways to choose m elements from n."</span>1140 [n m]1141 (<span style="color: #8cd0d3;">assert</span> (<span style="color: #8cd0d3;"><=</span> 0 m n))1142 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [difference (<span style="color: #8cd0d3;">-</span> n m)]1143 (exact-quotient1144 (<span style="color: #8cd0d3;">reduce</span> * (<span style="color: #8cd0d3;">range</span> n (<span style="color: #8cd0d3;">max</span> difference m) -1 ))1145 (factorial (<span style="color: #8cd0d3;">min</span> difference m)))))1147 (<span style="color: #f0dfaf; font-weight: bold;">defn-memo</span> <span style="color: #f0dfaf;">stirling-1</span>1148 <span style="color: #cc9393;">"Stirling numbers of the first kind: the number of permutations of n</span>1149 <span style="color: #cc9393;"> elements with exactly m permutation cycles. "</span>1150 [n k]1151 <span style="color: #708070;">;</span><span style="color: #7f9f7f;">(assert (<= 1 k n))</span>1152 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (zero? n)1153 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (zero? k) 1 0)1154 (<span style="color: #8cd0d3;">+</span> (stirling-1 (<span style="color: #8cd0d3;">dec</span> n) (<span style="color: #8cd0d3;">dec</span> k))1155 (<span style="color: #8cd0d3;">*</span> (<span style="color: #8cd0d3;">dec</span> n) (stirling-1 (<span style="color: #8cd0d3;">dec</span> n) k)))))1157 (<span style="color: #f0dfaf; font-weight: bold;">defn-memo</span> <span style="color: #f0dfaf;">stirling-2</span> <span style="color: #708070;">;;</span><span style="color: #7f9f7f;">count-partitions</span>1158 <span style="color: #cc9393;">"Stirling numbers of the second kind: the number of ways to partition a set of n elements into k subsets."</span>1159 [n k]1160 (<span style="color: #f0dfaf; font-weight: bold;">cond</span>1161 (<span style="color: #8cd0d3;">=</span> k 1) 11162 (<span style="color: #8cd0d3;">=</span> k n) 11163 <span style="color: #8cd0d3;">:else</span> (<span style="color: #8cd0d3;">+</span> (stirling-2 (<span style="color: #8cd0d3;">dec</span> n) (<span style="color: #8cd0d3;">dec</span> k))1164 (<span style="color: #8cd0d3;">*</span> k (stirling-2 (<span style="color: #8cd0d3;">dec</span> n) k)))))1166 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">harmonic-number</span> [n]1167 (<span style="color: #8cd0d3;">/</span> (stirling-1 (<span style="color: #8cd0d3;">inc</span> n) 2)1168 (factorial n)))1171 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">sum</span>1172 [f low high]1173 (<span style="color: #8cd0d3;">reduce</span> + (<span style="color: #8cd0d3;">map</span> f (<span style="color: #8cd0d3;">range</span> low (<span style="color: #8cd0d3;">inc</span> high)))))1175 </pre>1191 </div>1193 </div>1195 <div id="outline-container-5" class="outline-2">1196 <h2 id="sec-5"><span class="section-number-2">5</span> Differentiation </h2>1197 <div class="outline-text-2" id="text-5">1200 <p>1201 We compute derivatives by passing special <b>differential objects</b> \([x,1202 dx]\) through functions. Roughly speaking, applying a function \(f\) to a1203 differential object \([x, dx]\) should produce a new differential1204 object \([f(x),\,Df(x)\cdot dx]\).1205 </p>1208 \([x,\,dx]\xrightarrow{\quad f \quad}[f(x),\,Df(x)\cdot dx]\)1209 <p>1210 Notice that you can obtain the derivative of \(f\) from this1211 differential object, as it is the coefficient of the \(dx\) term. Also,1212 as you apply successive functions using this rule, you get the1213 chain-rule answer you expect:1214 </p>1217 \([f(x),\,Df(x)\cdot dx]\xrightarrow{\quad g\quad} [gf(x),\,1218 Dgf(x)\cdot Df(x) \cdot dx ]\)1220 <p>1221 In order to generalize to multiple variables and multiple derivatives,1222 we use a <b>power series of differentials</b>, a sortred infinite sequence which1223 contains all terms like \(dx\cdot dy\), \(dx^2\cdot dy\), etc.1224 </p>1229 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)1231 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">differential-seq</span>1232 <span style="color: #8fb28f;">"Constructs a sequence of differential terms from a numerical</span>1233 <span style="color: #8fb28f;">coefficient and a list of keys for variables. If no coefficient is supplied, uses 1."</span>1234 ([variables] (differential-seq* 1 variables))1235 ([coefficient variables & cvs]1236 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">number?</span> coefficient)1237 (<span style="color: #8cd0d3;">conj</span> (<span style="color: #8cd0d3;">assoc</span> {} (<span style="color: #8cd0d3;">apply</span> sorted-set variables) coefficient)1238 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">empty?</span> cvs)1239 nil1240 (<span style="color: #8cd0d3;">apply</span> differential-seq* cvs)))1241 (<span style="color: #8cd0d3;">apply</span> differential-seq* 1 coefficient 1 variables cvs)1242 )))1246 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">differential-add</span>1247 <span style="color: #8fb28f;">"Add two differential sequences by combining like terms."</span>1248 [dseq1 dseq2]1249 (<span style="color: #8cd0d3;">merge-with</span> + dseq1 dseq2))1251 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">differential-multiply</span>1252 <span style="color: #8fb28f;">"Multiply two differential sequences. The square of any differential variable is zero since differential variables are infinitesimally small."</span>1253 [dseq1 dseq2]1254 (<span style="color: #8cd0d3;">reduce</span>1255 (<span style="color: #8cd0d3;">fn</span> [m [[vars1 coeff1] [vars2 coeff2]]]1256 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">empty?</span> (clojure.set/<span style="color: #dfdfbf; font-weight: bold;">intersection</span> vars1 vars2))1257 (<span style="color: #8cd0d3;">assoc</span> m (clojure.set/<span style="color: #dfdfbf; font-weight: bold;">union</span> vars1 vars2) (<span style="color: #8cd0d3;">*</span> coeff1 coeff2))1258 m))1259 {}1260 (clojure.contrib.combinatorics/cartesian-product1261 dseq11262 dseq2)))1266 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">big-part</span>1267 <span style="color: #8fb28f;">"Returns the 'finite' part of the differential sequence."</span>1268 [dseq]1269 (<span style="color: #f0dfaf; font-weight: bold;">let</span>1270 [keys (<span style="color: #8cd0d3;">sort-by</span> count (<span style="color: #8cd0d3;">keys</span> dseq))1271 pivot-key (<span style="color: #8cd0d3;">first</span> (<span style="color: #8cd0d3;">last</span> keys))]1273 (<span style="color: #8cd0d3;">apply</span> hash-map1274 (<span style="color: #8cd0d3;">reduce</span> concat1275 (<span style="color: #8cd0d3;">filter</span> (<span style="color: #8cd0d3;">comp</span> pivot-key first) dseq)1276 ))))1279 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">small-part</span>1280 <span style="color: #8fb28f;">"Returns the 'infinitesimal' part of the differential sequence."</span>1281 [dseq]1282 (<span style="color: #f0dfaf; font-weight: bold;">let</span>1283 [keys (<span style="color: #8cd0d3;">sort-by</span> count (<span style="color: #8cd0d3;">keys</span> dseq))1284 pivot-key (<span style="color: #8cd0d3;">first</span> (<span style="color: #8cd0d3;">last</span> keys))]1286 (<span style="color: #8cd0d3;">apply</span> hash-map1287 (<span style="color: #8cd0d3;">reduce</span> concat1288 (<span style="color: #8cd0d3;">remove</span> (<span style="color: #8cd0d3;">comp</span> pivot-key first) dseq)1289 ))))1299 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">;; A differential term consists of a numerical coefficient and a</span>1300 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">;; sorted </span>1301 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(defrecord DifferentialTerm [coefficient variables])</span>1302 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(defmethod print-method DifferentialTerm</span>1303 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">[o w]</span>1304 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(print-simple</span>1305 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(apply str (.coefficient o)(map (comp (partial str "d") name) (.variables o)))</span>1306 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">w))</span>1309 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(defn differential-seq</span>1310 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">"Constructs a sequence of differential terms from a numerical</span>1311 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">coefficient and a list of keywords for variables. If no coefficient is</span>1312 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">supplied, uses 1."</span>1313 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">([variables] (differential-seq 1 variables))</span>1314 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">([coefficient variables]</span>1315 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(list</span>1316 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(DifferentialTerm. coefficient (apply sorted-set variables))))</span>1317 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">([coefficient variables & cvs]</span>1318 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(sort-by</span>1319 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">#(vec(.variables %))</span>1320 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(concat (differential-seq coefficient variables) (apply differential-seq cvs)))))</span>1322 </pre>1339 </div>1341 </div>1343 <div id="outline-container-6" class="outline-2">1344 <h2 id="sec-6"><span class="section-number-2">6</span> Symbolic Manipulation </h2>1345 <div class="outline-text-2" id="text-6">1351 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)1353 (<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">Symbolic</span> [type expression]1354 Object1355 (equals [this that]1356 (<span style="color: #f0dfaf; font-weight: bold;">cond</span>1357 (<span style="color: #8cd0d3;">=</span> (.expression this) (.expression that)) true1358 <span style="color: #8cd0d3;">:else</span>1359 (Symbolic.1360 java.lang.Boolean1361 (<span style="color: #8cd0d3;">list</span> '= (.expression this) (.expression that)))1362 )))1367 (<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">AbstractSet</span> [glyph membership-test])1368 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">member?</span> [abstract-set x]1369 ((.membership-test abstract-set) x))1371 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">------------ Some important AbstractSets</span>1374 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">Real</span>1375 (AbstractSet.1376 'R1377 (<span style="color: #8cd0d3;">fn</span>[x](<span style="color: #8cd0d3;">number?</span> x))))1380 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">------------ Create new AbstractSets from existing ones</span>1382 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">abstract-product</span>1383 <span style="color: #8fb28f;">"Gives the cartesian product of abstract sets."</span>1384 ([sets]1385 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">count</span> sets) 1) (<span style="color: #8cd0d3;">first</span> sets)1386 (AbstractSet.1387 (<span style="color: #8cd0d3;">symbol</span>1388 (<span style="color: #8cd0d3;">apply</span> str1389 (<span style="color: #8cd0d3;">interpose</span> 'x (<span style="color: #8cd0d3;">map</span> #(.glyph %) sets))))1390 (<span style="color: #8cd0d3;">fn</span> [x]1391 (<span style="color: #f0dfaf; font-weight: bold;">and</span>1392 (<span style="color: #8cd0d3;">coll?</span> x)1393 (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">count</span> sets) (<span style="color: #8cd0d3;">count</span> x))1394 (<span style="color: #8cd0d3;">reduce</span> #(<span style="color: #f0dfaf; font-weight: bold;">and</span> %1 %2)1395 true1396 (<span style="color: #8cd0d3;">map</span> #(member? %1 %2) sets x)))))))1397 ([abstract-set n]1398 (abstract-product (<span style="color: #8cd0d3;">repeat</span> n abstract-set))))1402 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">abstract-up</span>1403 <span style="color: #8fb28f;">"Returns the abstract set of all up-tuples whose items belong to the</span>1404 <span style="color: #8fb28f;"> corresponding abstract sets in coll."</span>1405 ([coll]1406 (AbstractSet.1407 (<span style="color: #8cd0d3;">symbol</span> (<span style="color: #8cd0d3;">str</span> <span style="color: #cc9393;">"u["</span>1408 (<span style="color: #8cd0d3;">apply</span> str1409 (<span style="color: #8cd0d3;">interpose</span> <span style="color: #cc9393;">" "</span>1410 (<span style="color: #8cd0d3;">map</span> #(.glyph %) coll)))1411 <span style="color: #cc9393;">"]"</span>))1412 (<span style="color: #8cd0d3;">fn</span> [x]1413 (<span style="color: #f0dfaf; font-weight: bold;">and</span>1414 (<span style="color: #8cd0d3;">satisfies?</span> Spinning x)1415 (up? x)1416 (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">count</span> coll) (<span style="color: #8cd0d3;">count</span> x))1417 (<span style="color: #8cd0d3;">reduce</span>1418 #(<span style="color: #f0dfaf; font-weight: bold;">and</span> %1 %2)1419 true1420 (<span style="color: #8cd0d3;">map</span> #(member? %1 %2) coll x))))))1421 ([abstract-set n]1422 (abstract-up (<span style="color: #8cd0d3;">repeat</span> n abstract-set))))1425 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">abstract-down</span>1426 <span style="color: #8fb28f;">"Returns the abstract set of all down-tuples whose items belong to the</span>1427 <span style="color: #8fb28f;"> corresponding abstract sets in coll."</span>1428 ([coll]1429 (AbstractSet.1430 (<span style="color: #8cd0d3;">symbol</span> (<span style="color: #8cd0d3;">str</span> <span style="color: #cc9393;">"d["</span>1431 (<span style="color: #8cd0d3;">apply</span> str1432 (<span style="color: #8cd0d3;">interpose</span> <span style="color: #cc9393;">" "</span>1433 (<span style="color: #8cd0d3;">map</span> #(.glyph %) coll)))1434 <span style="color: #cc9393;">"]"</span>))1435 (<span style="color: #8cd0d3;">fn</span> [x]1436 (<span style="color: #f0dfaf; font-weight: bold;">and</span>1437 (<span style="color: #8cd0d3;">satisfies?</span> Spinning x)1438 (down? x)1439 (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">count</span> coll) (<span style="color: #8cd0d3;">count</span> x))1440 (<span style="color: #8cd0d3;">reduce</span>1441 #(<span style="color: #f0dfaf; font-weight: bold;">and</span> %1 %2)1442 true1443 (<span style="color: #8cd0d3;">map</span> #(member? %1 %2) coll x))))))1444 ([abstract-set n]1445 (abstract-down (<span style="color: #8cd0d3;">repeat</span> n abstract-set))))1451 <span style="color: #708070;">;</span><span style="color: #7f9f7f;">-------ABSTRACT FUNCTIONS</span>1452 (<span style="color: #f0dfaf; font-weight: bold;">defrecord</span> <span style="color: #f0dfaf;">AbstractFn</span>1453 [<span style="color: #dfdfbf; font-weight: bold;">#^AbstractSet</span> domain <span style="color: #dfdfbf; font-weight: bold;">#^AbstractSet</span> codomain])1456 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">print-method</span> AbstractFn1457 [o w]1458 (<span style="color: #8cd0d3;">print-simple</span>1459 (<span style="color: #8cd0d3;">str</span>1460 <span style="color: #cc9393;">"f:"</span>1461 (.glyph (<span style="color: #8cd0d3;">:domain</span> o))1462 <span style="color: #cc9393;">"-->"</span>1463 (.glyph (<span style="color: #8cd0d3;">:codomain</span> o))) w))1464 </pre>1472 </div>1473 </div>1474 <div id="postamble">1475 <p class="date">Date: 2011-08-09 18:41:37 EDT</p>1476 <p class="author">Author: Robert McIntyre & Dylan Holmes</p>1477 <p class="creator">Org version 7.6 with Emacs version 23</p>1478 <a href="http://validator.w3.org/check?uri=referer">Validate XHTML 1.0</a>1479 </div>1480 </div>1481 </body>1482 </html>