Mercurial > dylan
diff sicm/bk/utils.html @ 2:b4de894a1e2e
initial import
| author | Robert McIntyre <rlm@mit.edu> |
|---|---|
| date | Fri, 28 Oct 2011 00:03:05 -0700 |
| parents | |
| children |
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1.122 +/*]]>*///--> 1.123 +</script> 1.124 +</head> 1.125 +<body> 1.126 + 1.127 +<div id="content"> 1.128 + 1.129 + 1.130 + 1.131 +<div class="header"> 1.132 + <div class="float-right"> 1.133 + <!-- 1.134 + <form> 1.135 + <input type="text"/><input type="submit" value="search the blog »"/> 1.136 + </form> 1.137 + --> 1.138 + </div> 1.139 + 1.140 + <h1>aurellem <em>☉</em></h1> 1.141 + <ul class="nav"> 1.142 + <li><a href="/">read the blog »</a></li> 1.143 + <!-- li><a href="#">learn about us »</a></li--> 1.144 + </ul> 1.145 +</div> 1.146 + 1.147 +<h1 class="title">Building a Classical Mechanics Library in Clojure</h1> 1.148 + 1.149 + 1.150 + 1.151 + 1.152 + 1.153 + 1.154 + 1.155 +<div id="table-of-contents"> 1.156 +<h2>Table of Contents</h2> 1.157 +<div id="text-table-of-contents"> 1.158 +<ul> 1.159 +<li><a href="#sec-1">1 Generic Arithmetic </a></li> 1.160 +<li><a href="#sec-2">2 Useful Data Types </a> 1.161 +<ul> 1.162 +<li><a href="#sec-2-1">2.1 Complex Numbers </a></li> 1.163 +<li><a href="#sec-2-2">2.2 Tuples and Tensors </a> 1.164 +<ul> 1.165 +<li><a href="#sec-2-2-1">2.2.1 Contraction </a></li> 1.166 +<li><a href="#sec-2-2-2">2.2.2 Matrices </a></li> 1.167 +</ul> 1.168 +</li> 1.169 +<li><a href="#sec-2-3">2.3 Power Series </a></li> 1.170 +</ul> 1.171 +</li> 1.172 +<li><a href="#sec-3">3 Basic Utilities </a> 1.173 +<ul> 1.174 +<li><a href="#sec-3-1">3.1 Sequence manipulation </a></li> 1.175 +<li><a href="#sec-3-2">3.2 Ranges, Airity and Function Composition </a></li> 1.176 +</ul> 1.177 +</li> 1.178 +<li><a href="#sec-4">4 Numerical Methods </a></li> 1.179 +<li><a href="#sec-5">5 Differentiation </a></li> 1.180 +<li><a href="#sec-6">6 Symbolic Manipulation </a></li> 1.181 +</ul> 1.182 +</div> 1.183 +</div> 1.184 + 1.185 +<div id="outline-container-1" class="outline-2"> 1.186 +<h2 id="sec-1"><span class="section-number-2">1</span> Generic Arithmetic </h2> 1.187 +<div class="outline-text-2" id="text-1"> 1.188 + 1.189 + 1.190 + 1.191 + 1.192 + 1.193 +<pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">ns</span> sicm.utils) 1.194 +(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) 1.195 + 1.196 + 1.197 + 1.198 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">all-equal?</span> [coll] 1.199 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">empty?</span> (<span style="color: #8cd0d3;">rest</span> coll)) true 1.200 + (<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)) 1.201 + (<span style="color: #f0dfaf; font-weight: bold;">recur</span> (<span style="color: #8cd0d3;">rest</span> coll))))) 1.202 + 1.203 + 1.204 +(<span style="color: #f0dfaf; font-weight: bold;">defprotocol</span> <span style="color: #f0dfaf;">Arithmetic</span> 1.205 + (zero [this]) 1.206 + (one [this]) 1.207 + (negate [this]) 1.208 + (invert [this]) 1.209 + (conjugate [this])) 1.210 + 1.211 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">zero?</span> [x] 1.212 + (<span style="color: #8cd0d3;">=</span> x (zero x))) 1.213 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">one?</span> [x] 1.214 + (<span style="color: #8cd0d3;">=</span> x (one x))) 1.215 + 1.216 + 1.217 + 1.218 +(<span style="color: #8cd0d3;">extend-protocol</span> Arithmetic 1.219 + java.lang.Number 1.220 + (zero [x] 0) 1.221 + (one [x] 1) 1.222 + (negate [x] (<span style="color: #8cd0d3;">-</span> x)) 1.223 + (invert [x] (<span style="color: #8cd0d3;">/</span> x)) 1.224 + ) 1.225 + 1.226 +(<span style="color: #8cd0d3;">extend-protocol</span> Arithmetic 1.227 + clojure.lang.IFn 1.228 + (one [f] identity) 1.229 + (negate [f] (<span style="color: #8cd0d3;">comp</span> negate f)) 1.230 + (invert [f] (<span style="color: #8cd0d3;">comp</span> invert f))) 1.231 + 1.232 + 1.233 +(<span style="color: #8cd0d3;">extend-protocol</span> Arithmetic 1.234 + clojure.lang.Seqable 1.235 + (zero [this] (<span style="color: #8cd0d3;">map</span> zero this)) 1.236 + (one [this] (<span style="color: #8cd0d3;">map</span> one this)) 1.237 + (invert [this] (<span style="color: #8cd0d3;">map</span> invert this)) 1.238 + (negate [this] (<span style="color: #8cd0d3;">map</span> negate this))) 1.239 + 1.240 + 1.241 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">ordered-like</span> 1.242 + <span style="color: #8fb28f;">"Create a comparator using the sorted collection as an</span> 1.243 +<span style="color: #8fb28f;"> example. Elements not in the sorted collection are sorted to the</span> 1.244 +<span style="color: #8fb28f;"> end."</span> 1.245 + [sorted-coll] 1.246 + (<span style="color: #f0dfaf; font-weight: bold;">let</span> [ 1.247 + sorted-coll? (<span style="color: #8cd0d3;">set</span> sorted-coll) 1.248 + ascending-pair? 1.249 + (<span style="color: #8cd0d3;">set</span>(<span style="color: #8cd0d3;">reduce</span> concat 1.250 + (map-indexed 1.251 + (<span style="color: #8cd0d3;">fn</span> [n x] 1.252 + (<span style="color: #8cd0d3;">map</span> #(<span style="color: #8cd0d3;">vector</span> x %) (<span style="color: #8cd0d3;">nthnext</span> sorted-coll n))) 1.253 + sorted-coll)))] 1.254 + (<span style="color: #8cd0d3;">fn</span> [x y] 1.255 + (<span style="color: #f0dfaf; font-weight: bold;">cond</span> 1.256 + (<span style="color: #8cd0d3;">=</span> x y) 0 1.257 + (ascending-pair? [x y]) -1 1.258 + (ascending-pair? [y x]) 1 1.259 + (sorted-coll? x) -1 1.260 + (sorted-coll? y) 1)))) 1.261 + 1.262 + 1.263 + 1.264 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">type-precedence</span> 1.265 + (ordered-like [incanter.Matrix])) 1.266 + 1.267 +(<span style="color: #f0dfaf; font-weight: bold;">defmulti</span> <span style="color: #f0dfaf;">add</span> 1.268 + (<span style="color: #8cd0d3;">fn</span> [x y] 1.269 + (<span style="color: #8cd0d3;">sort</span> type-precedence [(<span style="color: #8cd0d3;">type</span> x)(<span style="color: #8cd0d3;">type</span> y)]))) 1.270 + 1.271 +(<span style="color: #f0dfaf; font-weight: bold;">defmulti</span> <span style="color: #f0dfaf;">multiply</span> 1.272 + (<span style="color: #8cd0d3;">fn</span> [x y] 1.273 + (<span style="color: #8cd0d3;">sort</span> type-precedence [(<span style="color: #8cd0d3;">type</span> x) (<span style="color: #8cd0d3;">type</span> y)]))) 1.274 + 1.275 +(<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)) 1.276 +(<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)) 1.277 + 1.278 +(<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">multiply</span> [incanter.Matrix java.lang.Integer] [x y] 1.279 + (<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]) 1.280 + matrix (<span style="color: #8cd0d3;">first</span> args) 1.281 + scalar (<span style="color: #8cd0d3;">second</span> args)] 1.282 + (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)))) 1.283 + 1.284 +</pre> 1.285 + 1.286 + 1.287 + 1.288 + 1.289 + 1.290 +</div> 1.291 + 1.292 +</div> 1.293 + 1.294 +<div id="outline-container-2" class="outline-2"> 1.295 +<h2 id="sec-2"><span class="section-number-2">2</span> Useful Data Types </h2> 1.296 +<div class="outline-text-2" id="text-2"> 1.297 + 1.298 + 1.299 + 1.300 +</div> 1.301 + 1.302 +<div id="outline-container-2-1" class="outline-3"> 1.303 +<h3 id="sec-2-1"><span class="section-number-3">2.1</span> Complex Numbers </h3> 1.304 +<div class="outline-text-3" id="text-2-1"> 1.305 + 1.306 + 1.307 + 1.308 + 1.309 +<pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) 1.310 + 1.311 +(<span style="color: #f0dfaf; font-weight: bold;">defprotocol</span> <span style="color: #f0dfaf;">Complex</span> 1.312 + (real-part [z]) 1.313 + (imaginary-part [z]) 1.314 + (magnitude-squared [z]) 1.315 + (angle [z]) 1.316 + (conjugate [z]) 1.317 + (norm [z])) 1.318 + 1.319 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">complex-rectangular</span> 1.320 + <span style="color: #8fb28f;">"Define a complex number with the given real and imaginary</span> 1.321 +<span style="color: #8fb28f;"> components."</span> 1.322 + [re im] 1.323 + (<span style="color: #8cd0d3;">reify</span> Complex 1.324 + (real-part [z] re) 1.325 + (imaginary-part [z] im) 1.326 + (magnitude-squared [z] (<span style="color: #8cd0d3;">+</span> (<span style="color: #8cd0d3;">*</span> re re) (<span style="color: #8cd0d3;">*</span> im im))) 1.327 + (angle [z] (java.lang.Math/atan2 im re)) 1.328 + (conjugate [z] (complex-rectangular re (<span style="color: #8cd0d3;">-</span> im))) 1.329 + 1.330 + Arithmetic 1.331 + (zero [z] (complex-rectangular 0 0)) 1.332 + (one [z] (complex-rectangular 1 0)) 1.333 + (negate [z] (complex-rectangular (<span style="color: #8cd0d3;">-</span> re) (<span style="color: #8cd0d3;">-</span> im))) 1.334 + (invert [z] (complex-rectangular 1.335 + (<span style="color: #8cd0d3;">/</span> re (magnitude-squared z)) 1.336 + (<span style="color: #8cd0d3;">/</span> (<span style="color: #8cd0d3;">-</span> im) (magnitude-squared z)))) 1.337 + 1.338 + Object 1.339 + (toString [_] 1.340 + (<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) 1.341 + (<span style="color: #8cd0d3;">str</span> 1.342 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not</span>(zero? re)) 1.343 + re) 1.344 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> ((<span style="color: #8cd0d3;">comp</span> not zero?) im) 1.345 + (<span style="color: #8cd0d3;">str</span> 1.346 + (<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>) 1.347 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> ((<span style="color: #8cd0d3;">comp</span> not one?) (java.lang.Math/abs im)) 1.348 + (java.lang.Math/abs im)) 1.349 + <span style="color: #cc9393;">"i"</span>))))))) 1.350 + 1.351 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">complex-polar</span> 1.352 + <span style="color: #8fb28f;">"Define a complex number with the given magnitude and angle."</span> 1.353 + [mag ang] 1.354 + (<span style="color: #8cd0d3;">reify</span> Complex 1.355 + (magnitude-squared [z] (<span style="color: #8cd0d3;">*</span> mag mag)) 1.356 + (angle [z] angle) 1.357 + (real-part [z] (<span style="color: #8cd0d3;">*</span> mag (java.lang.Math/cos ang))) 1.358 + (imaginary-part [z] (<span style="color: #8cd0d3;">*</span> mag (java.lang.Math/sin ang))) 1.359 + (conjugate [z] (complex-polar mag (<span style="color: #8cd0d3;">-</span> ang))) 1.360 + 1.361 + Arithmetic 1.362 + (zero [z] (complex-polar 0 0)) 1.363 + (one [z] (complex-polar 1 0)) 1.364 + (negate [z] (complex-polar (<span style="color: #8cd0d3;">-</span> mag) ang)) 1.365 + (invert [z] (complex-polar (<span style="color: #8cd0d3;">/</span> mag) (<span style="color: #8cd0d3;">-</span> ang))) 1.366 + 1.367 + Object 1.368 + (toString [_] (<span style="color: #8cd0d3;">str</span> mag <span style="color: #cc9393;">" * e^(i"</span> ang<span style="color: #cc9393;">")"</span>)) 1.369 + )) 1.370 + 1.371 + 1.372 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">Numbers are complex quantities</span> 1.373 + 1.374 +(<span style="color: #8cd0d3;">extend-protocol</span> Complex 1.375 + java.lang.Number 1.376 + (real-part [x] x) 1.377 + (imaginary-part [x] 0) 1.378 + (magnitude [x] x) 1.379 + (angle [x] 0) 1.380 + (conjugate [x] x)) 1.381 + 1.382 + 1.383 +</pre> 1.384 + 1.385 + 1.386 + 1.387 + 1.388 + 1.389 + 1.390 +</div> 1.391 + 1.392 +</div> 1.393 + 1.394 +<div id="outline-container-2-2" class="outline-3"> 1.395 +<h3 id="sec-2-2"><span class="section-number-3">2.2</span> Tuples and Tensors </h3> 1.396 +<div class="outline-text-3" id="text-2-2"> 1.397 + 1.398 + 1.399 +<p> 1.400 +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) 1.401 +</p> 1.402 + 1.403 + 1.404 +<pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) 1.405 + 1.406 +(<span style="color: #f0dfaf; font-weight: bold;">defprotocol</span> <span style="color: #f0dfaf;">Spinning</span> 1.407 + (up? [this]) 1.408 + (down? [this])) 1.409 + 1.410 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">spin</span> 1.411 + <span style="color: #8fb28f;">"Returns the spin of the Spinning s, either :up or :down"</span> 1.412 + [<span style="color: #dfdfbf; font-weight: bold;">#^Spinning</span> s] 1.413 + (<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>)) 1.414 + 1.415 + 1.416 +(<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">Tuple</span> 1.417 + [spin coll] 1.418 + clojure.lang.Seqable 1.419 + (<span style="color: #8cd0d3;">seq</span> [this] (<span style="color: #8cd0d3;">seq</span> (.coll this))) 1.420 + clojure.lang.Counted 1.421 + (<span style="color: #8cd0d3;">count</span> [this] (<span style="color: #8cd0d3;">count</span> (.coll this)))) 1.422 + 1.423 +(<span style="color: #8cd0d3;">extend-type</span> Tuple 1.424 + Spinning 1.425 + (up? [this] (<span style="color: #8cd0d3;">=</span> <span style="color: #8cd0d3;">::up</span> (.spin this))) 1.426 + (down? [this] (<span style="color: #8cd0d3;">=</span> <span style="color: #8cd0d3;">::down</span> (.spin this)))) 1.427 + 1.428 +(<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">print-method</span> Tuple 1.429 + [o w] 1.430 + (<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)) 1.431 + 1.432 + 1.433 + 1.434 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">up</span> 1.435 + <span style="color: #8fb28f;">"Create a new up-tuple containing the contents of coll."</span> 1.436 + [coll] 1.437 + (Tuple. <span style="color: #8cd0d3;">::up</span> coll)) 1.438 + 1.439 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">down</span> 1.440 + <span style="color: #8fb28f;">"Create a new down-tuple containing the contents of coll."</span> 1.441 + [coll] 1.442 + (Tuple. <span style="color: #8cd0d3;">::down</span> coll)) 1.443 + 1.444 + 1.445 +</pre> 1.446 + 1.447 + 1.448 + 1.449 + 1.450 + 1.451 +</div> 1.452 + 1.453 +<div id="outline-container-2-2-1" class="outline-4"> 1.454 +<h4 id="sec-2-2-1"><span class="section-number-4">2.2.1</span> Contraction </h4> 1.455 +<div class="outline-text-4" id="text-2-2-1"> 1.456 + 1.457 +<p>Contraction is a binary operation that you can apply to compatible 1.458 + tuples. Tuples are compatible for contraction if they have the same 1.459 + length and opposite spins, and if the corresponding items in each 1.460 + tuple are both numbers or both compatible tuples. 1.461 +</p> 1.462 + 1.463 + 1.464 + 1.465 +<pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) 1.466 + 1.467 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">numbers?</span> 1.468 + <span style="color: #8fb28f;">"Returns true if all arguments are numbers, else false."</span> 1.469 + [& xs] 1.470 + (<span style="color: #8cd0d3;">every?</span> number? xs)) 1.471 + 1.472 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">contractible?</span> 1.473 + <span style="color: #8fb28f;">"Returns true if the tuples a and b are compatible for contraction,</span> 1.474 +<span style="color: #8fb28f;"> else false. Tuples are compatible if they have the same number of</span> 1.475 +<span style="color: #8fb28f;"> components, they have opposite spins, and their elements are</span> 1.476 +<span style="color: #8fb28f;"> pairwise-compatible."</span> 1.477 + [a b] 1.478 + (<span style="color: #f0dfaf; font-weight: bold;">and</span> 1.479 + (<span style="color: #8cd0d3;">isa?</span> (<span style="color: #8cd0d3;">type</span> a) Tuple) 1.480 + (<span style="color: #8cd0d3;">isa?</span> (<span style="color: #8cd0d3;">type</span> b) Tuple) 1.481 + (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">count</span> a) (<span style="color: #8cd0d3;">count</span> b)) 1.482 + (<span style="color: #8cd0d3;">not=</span> (spin a) (spin b)) 1.483 + 1.484 + (<span style="color: #8cd0d3;">not-any?</span> false? 1.485 + (<span style="color: #8cd0d3;">map</span> #(<span style="color: #f0dfaf; font-weight: bold;">or</span> 1.486 + (numbers? %1 %2) 1.487 + (contractible? %1 %2)) 1.488 + a b)))) 1.489 + 1.490 + 1.491 + 1.492 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">contract</span> 1.493 + <span style="color: #8fb28f;">"Contracts two tuples, returning the sum of the</span> 1.494 +<span style="color: #8fb28f;"> products of the corresponding items. Contraction is recursive on</span> 1.495 +<span style="color: #8fb28f;"> nested tuples."</span> 1.496 + [a b] 1.497 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not</span> (contractible? a b)) 1.498 + (<span style="color: #f0dfaf; font-weight: bold;">throw</span> 1.499 + (Exception. <span style="color: #cc9393;">"Not compatible for contraction."</span>)) 1.500 + (<span style="color: #8cd0d3;">reduce</span> + 1.501 + (<span style="color: #8cd0d3;">map</span> 1.502 + (<span style="color: #8cd0d3;">fn</span> [x y] 1.503 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (numbers? x y) 1.504 + (<span style="color: #8cd0d3;">*</span> x y) 1.505 + (contract x y))) 1.506 + a b)))) 1.507 + 1.508 +</pre> 1.509 + 1.510 + 1.511 + 1.512 + 1.513 +</div> 1.514 + 1.515 +</div> 1.516 + 1.517 +<div id="outline-container-2-2-2" class="outline-4"> 1.518 +<h4 id="sec-2-2-2"><span class="section-number-4">2.2.2</span> Matrices </h4> 1.519 +<div class="outline-text-4" id="text-2-2-2"> 1.520 + 1.521 + 1.522 + 1.523 + 1.524 +<pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) 1.525 +(<span style="color: #8cd0d3;">require</span> 'incanter.core) <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">use incanter's fast matrices</span> 1.526 + 1.527 +(<span style="color: #f0dfaf; font-weight: bold;">defprotocol</span> <span style="color: #f0dfaf;">Matrix</span> 1.528 + (rows [matrix]) 1.529 + (cols [matrix]) 1.530 + (diagonal [matrix]) 1.531 + (trace [matrix]) 1.532 + (determinant [matrix]) 1.533 + (transpose [matrix]) 1.534 + (conjugate [matrix]) 1.535 +) 1.536 + 1.537 +(<span style="color: #8cd0d3;">extend-protocol</span> Matrix 1.538 + incanter.Matrix 1.539 + (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)))) 1.540 + (cols [rs] (<span style="color: #8cd0d3;">map</span> up (<span style="color: #8cd0d3;">apply</span> map vector rs))) 1.541 + (diagonal [matrix] (incanter.core/diag matrix) ) 1.542 + (determinant [matrix] (incanter.core/det matrix)) 1.543 + (trace [matrix] (incanter.core/trace matrix)) 1.544 + (transpose [matrix] (incanter.core/trans matrix)) 1.545 + ) 1.546 + 1.547 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">count-rows</span> [matrix] 1.548 + ((<span style="color: #8cd0d3;">comp</span> count rows) matrix)) 1.549 + 1.550 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">count-cols</span> [matrix] 1.551 + ((<span style="color: #8cd0d3;">comp</span> count cols) matrix)) 1.552 + 1.553 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">square?</span> [matrix] 1.554 + (<span style="color: #8cd0d3;">=</span> (count-rows matrix) (count-cols matrix))) 1.555 + 1.556 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">identity-matrix</span> 1.557 + <span style="color: #8fb28f;">"Define a square matrix of size n-by-n with 1s along the diagonal and</span> 1.558 +<span style="color: #8fb28f;"> 0s everywhere else."</span> 1.559 + [n] 1.560 + (incanter.core/identity-matrix n)) 1.561 + 1.562 + 1.563 + 1.564 + 1.565 + 1.566 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">matrix-by-rows</span> 1.567 + <span style="color: #8fb28f;">"Define a matrix by giving its rows."</span> 1.568 + [& rows] 1.569 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> 1.570 + (<span style="color: #8cd0d3;">not</span> (all-equal? (<span style="color: #8cd0d3;">map</span> count rows))) 1.571 + (<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>)) 1.572 + (incanter.core/matrix (<span style="color: #8cd0d3;">vec</span> rows)))) 1.573 + 1.574 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">matrix-by-cols</span> 1.575 + <span style="color: #8fb28f;">"Define a matrix by giving its columns"</span> 1.576 + [& cols] 1.577 + (<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))) 1.578 + (<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>)) 1.579 + (incanter.core/matrix (<span style="color: #8cd0d3;">vec</span> (<span style="color: #8cd0d3;">apply</span> map vector cols))))) 1.580 + 1.581 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">identity-matrix</span> 1.582 + <span style="color: #8fb28f;">"Define a square matrix of size n-by-n with 1s along the diagonal and</span> 1.583 +<span style="color: #8fb28f;"> 0s everywhere else."</span> 1.584 + [n] 1.585 + (incanter.core/identity-matrix n)) 1.586 + 1.587 + 1.588 + 1.589 +(<span style="color: #8cd0d3;">extend-protocol</span> Arithmetic 1.590 + incanter.Matrix 1.591 + (one [matrix] 1.592 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (square? matrix) 1.593 + (identity-matrix (count-rows matrix)) 1.594 + (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"Non-square matrices have no multiplicative unit."</span>)))) 1.595 + (zero [matrix] 1.596 + (<span style="color: #8cd0d3;">apply</span> matrix-by-rows (<span style="color: #8cd0d3;">map</span> zero (rows matrix)))) 1.597 + (negate [matrix] 1.598 + (<span style="color: #8cd0d3;">apply</span> matrix-by-rows (<span style="color: #8cd0d3;">map</span> negate (rows matrix)))) 1.599 + (invert [matrix] 1.600 + (incanter.core/solve matrix))) 1.601 + 1.602 + 1.603 + 1.604 +(<span style="color: #f0dfaf; font-weight: bold;">defmulti</span> <span style="color: #f0dfaf;">coerce-to-matrix</span> 1.605 + <span style="color: #8fb28f;">"Converts x into a matrix, if possible."</span> 1.606 + type) 1.607 + 1.608 +(<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">coerce-to-matrix</span> incanter.Matrix [x] x) 1.609 +(<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">coerce-to-matrix</span> Tuple [x] 1.610 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">apply</span> numbers? (<span style="color: #8cd0d3;">seq</span> x)) 1.611 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (up? x) 1.612 + (matrix-by-cols (<span style="color: #8cd0d3;">seq</span> x)) 1.613 + (matrix-by-rows (<span style="color: #8cd0d3;">seq</span> x))) 1.614 + (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"Non-numerical tuple cannot be converted into a matrix."</span>)))) 1.615 + 1.616 + 1.617 + 1.618 + 1.619 + 1.620 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(defn matrix-by-cols</span> 1.621 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">"Define a matrix by giving its columns."</span> 1.622 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">[& cols]</span> 1.623 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(cond</span> 1.624 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(not (all-equal? (map count cols)))</span> 1.625 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(throw (Exception. "All columns in a matrix must have the same number of elements."))</span> 1.626 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">:else</span> 1.627 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(reify Matrix</span> 1.628 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(cols [this] (map up cols))</span> 1.629 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(rows [this] (map down (apply map vector cols)))</span> 1.630 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(diagonal [this] (map-indexed (fn [i col] (nth col i) cols)))</span> 1.631 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(trace [this]</span> 1.632 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(if (not= (count-cols this) (count-rows this))</span> 1.633 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(throw (Exception.</span> 1.634 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">"Cannot take the trace of a non-square matrix."))</span> 1.635 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(reduce + (diagonal this))))</span> 1.636 + 1.637 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(determinant [this]</span> 1.638 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(if (not= (count-cols this) (count-rows this))</span> 1.639 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(throw (Exception.</span> 1.640 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">"Cannot take the determinant of a non-square matrix."))</span> 1.641 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(reduce * (map-indexed (fn [i col] (nth col i)) cols))))</span> 1.642 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">)))</span> 1.643 + 1.644 +(<span style="color: #8cd0d3;">extend-protocol</span> Matrix Tuple 1.645 + (rows [this] (<span style="color: #f0dfaf; font-weight: bold;">if</span> (down? this) 1.646 + (<span style="color: #8cd0d3;">list</span> this) 1.647 + (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">comp</span> up vector) this))) 1.648 + 1.649 + (cols [this] (<span style="color: #f0dfaf; font-weight: bold;">if</span> (up? this) 1.650 + (<span style="color: #8cd0d3;">list</span> this) 1.651 + (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">comp</span> down vector) this)) 1.652 + )) 1.653 + 1.654 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">matrix-multiply</span> 1.655 + <span style="color: #8fb28f;">"Returns the matrix resulting from the matrix multiplication of the given arguments."</span> 1.656 + ([A] (coerce-to-matrix A)) 1.657 + ([A B] (incanter.core/mmult (coerce-to-matrix A) (coerce-to-matrix B))) 1.658 + ([M1 M2 & Ms] (<span style="color: #8cd0d3;">reduce</span> matrix-multiply (matrix-multiply M1 M2) Ms))) 1.659 + 1.660 +</pre> 1.661 + 1.662 + 1.663 + 1.664 + 1.665 + 1.666 +</div> 1.667 +</div> 1.668 + 1.669 +</div> 1.670 + 1.671 +<div id="outline-container-2-3" class="outline-3"> 1.672 +<h3 id="sec-2-3"><span class="section-number-3">2.3</span> Power Series </h3> 1.673 +<div class="outline-text-3" id="text-2-3"> 1.674 + 1.675 + 1.676 + 1.677 + 1.678 +<pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) 1.679 +(<span style="color: #8cd0d3;">use</span> 'clojure.contrib.def) 1.680 + 1.681 + 1.682 + 1.683 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">series-fn</span> 1.684 + <span style="color: #8fb28f;">"The function corresponding to the given power series."</span> 1.685 + [series] 1.686 + (<span style="color: #8cd0d3;">fn</span> [x] 1.687 + (<span style="color: #8cd0d3;">reduce</span> + 1.688 + (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)) )) 1.689 + (<span style="color: #8cd0d3;">range</span> 20))))) 1.690 + 1.691 +(<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">PowerSeries</span> 1.692 + [coll] 1.693 + clojure.lang.Seqable 1.694 + (<span style="color: #8cd0d3;">seq</span> [this] (<span style="color: #8cd0d3;">seq</span> (.coll this))) 1.695 + 1.696 + clojure.lang.Indexed 1.697 + (<span style="color: #8cd0d3;">nth</span> [this n] (<span style="color: #8cd0d3;">nth</span> (.coll this) n 0)) 1.698 + (<span style="color: #8cd0d3;">nth</span> [this n not-found] (<span style="color: #8cd0d3;">nth</span> (.coll this) n not-found)) 1.699 + 1.700 + <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">clojure.lang.IFn</span> 1.701 + <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(call [this] (throw(Exception.)))</span> 1.702 + <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(invoke [this & args] args</span> 1.703 + <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(let [f </span> 1.704 + <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">)</span> 1.705 + <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(run [this] (throw(Exception.)))</span> 1.706 + ) 1.707 + 1.708 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">power-series</span> 1.709 + <span style="color: #8fb28f;">"Returns a power series with the items of the coll as its</span> 1.710 +<span style="color: #8fb28f;"> coefficients. Trailing zeros are added to the end of coll."</span> 1.711 + [coeffs] 1.712 + (PowerSeries. coeffs)) 1.713 + 1.714 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">power-series-indexed</span> 1.715 + <span style="color: #8fb28f;">"Returns a power series consisting of the result of mapping f to the non-negative integers."</span> 1.716 + [f] 1.717 + (PowerSeries. (<span style="color: #8cd0d3;">map</span> f (<span style="color: #8cd0d3;">range</span>)))) 1.718 + 1.719 + 1.720 +(<span style="color: #f0dfaf; font-weight: bold;">defn-memo</span> <span style="color: #f0dfaf;">nth-partial-sum</span> 1.721 + ([series n] 1.722 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (zero? n) (<span style="color: #8cd0d3;">first</span> series) 1.723 + (<span style="color: #8cd0d3;">+</span> (<span style="color: #8cd0d3;">nth</span> series n) 1.724 + (nth-partial-sum series (<span style="color: #8cd0d3;">dec</span> n)))))) 1.725 + 1.726 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">partial-sums</span> [series] 1.727 + (<span style="color: #8cd0d3;">lazy-seq</span> (<span style="color: #8cd0d3;">map</span> nth-partial-sum (<span style="color: #8cd0d3;">range</span>)))) 1.728 + 1.729 + 1.730 + 1.731 + 1.732 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">cos-series</span> 1.733 + (power-series-indexed 1.734 + (<span style="color: #8cd0d3;">fn</span>[n] 1.735 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">odd?</span> n) 0 1.736 + (<span style="color: #8cd0d3;">/</span> 1.737 + (<span style="color: #8cd0d3;">reduce</span> * 1.738 + (<span style="color: #8cd0d3;">reduce</span> * (<span style="color: #8cd0d3;">repeat</span> (<span style="color: #8cd0d3;">/</span> n 2) -1)) 1.739 + (<span style="color: #8cd0d3;">range</span> 1 (<span style="color: #8cd0d3;">inc</span> n))) 1.740 + ))))) 1.741 + 1.742 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">sin-series</span> 1.743 + (power-series-indexed 1.744 + (<span style="color: #8cd0d3;">fn</span>[n] 1.745 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">even?</span> n) 0 1.746 + (<span style="color: #8cd0d3;">/</span> 1.747 + (<span style="color: #8cd0d3;">reduce</span> * 1.748 + (<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)) 1.749 + (<span style="color: #8cd0d3;">range</span> 1 (<span style="color: #8cd0d3;">inc</span> n))) 1.750 + ))))) 1.751 + 1.752 +</pre> 1.753 + 1.754 + 1.755 + 1.756 + 1.757 + 1.758 +</div> 1.759 +</div> 1.760 + 1.761 +</div> 1.762 + 1.763 +<div id="outline-container-3" class="outline-2"> 1.764 +<h2 id="sec-3"><span class="section-number-2">3</span> Basic Utilities </h2> 1.765 +<div class="outline-text-2" id="text-3"> 1.766 + 1.767 + 1.768 + 1.769 +</div> 1.770 + 1.771 +<div id="outline-container-3-1" class="outline-3"> 1.772 +<h3 id="sec-3-1"><span class="section-number-3">3.1</span> Sequence manipulation </h3> 1.773 +<div class="outline-text-3" id="text-3-1"> 1.774 + 1.775 + 1.776 + 1.777 + 1.778 + 1.779 +<pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">ns</span> sicm.utils) 1.780 + 1.781 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">do-up</span> 1.782 + <span style="color: #8fb28f;">"Apply f to each number from low to high, presumably for</span> 1.783 +<span style="color: #8fb28f;"> side-effects."</span> 1.784 + [f low high] 1.785 + (<span style="color: #f0dfaf; font-weight: bold;">doseq</span> [i (<span style="color: #8cd0d3;">range</span> low high)] (f i))) 1.786 + 1.787 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">do-down</span> 1.788 + <span style="color: #8fb28f;">"Apply f to each number from high to low, presumably for</span> 1.789 +<span style="color: #8fb28f;"> side-effects."</span> 1.790 + [f high low] 1.791 + (<span style="color: #f0dfaf; font-weight: bold;">doseq</span> [i (<span style="color: #8cd0d3;">range</span> high low -1)] (f i))) 1.792 + 1.793 + 1.794 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">all-equal?</span> [coll] 1.795 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">empty?</span> (<span style="color: #8cd0d3;">rest</span> coll)) true 1.796 + (<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)) 1.797 + (<span style="color: #f0dfaf; font-weight: bold;">recur</span> (<span style="color: #8cd0d3;">rest</span> coll)))))) 1.798 + 1.799 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">multiplier</span> 1.800 + <span style="color: #8fb28f;">"Returns a function that 'multiplies' the members of a collection,</span> 1.801 +<span style="color: #8fb28f;">returning unit if called on an empty collection."</span> 1.802 + [multiply unit] 1.803 + (<span style="color: #8cd0d3;">fn</span> [coll] ((<span style="color: #8cd0d3;">partial</span> reduce multiply unit) coll))) 1.804 + 1.805 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">divider</span> 1.806 + <span style="color: #8fb28f;">"Returns a function that 'divides' the first element of a collection</span> 1.807 +<span style="color: #8fb28f;">by the 'product' of the rest of the collection."</span> 1.808 + [divide multiply invert unit] 1.809 + (<span style="color: #8cd0d3;">fn</span> [coll] 1.810 + (<span style="color: #8cd0d3;">apply</span> 1.811 + (<span style="color: #8cd0d3;">fn</span> 1.812 + ([] unit) 1.813 + ([x] (invert x)) 1.814 + ([x y] (divide x y)) 1.815 + ([x y & zs] (divide x (<span style="color: #8cd0d3;">reduce</span> multiply y zs)))) 1.816 + coll))) 1.817 + 1.818 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">left-circular-shift</span> 1.819 + <span style="color: #8fb28f;">"Remove the first element of coll, adding it to the end of coll."</span> 1.820 + [coll] 1.821 + (<span style="color: #8cd0d3;">concat</span> (<span style="color: #8cd0d3;">rest</span> coll) (<span style="color: #8cd0d3;">take</span> 1 coll))) 1.822 + 1.823 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">right-circular-shift</span> 1.824 + <span style="color: #8fb28f;">"Remove the last element of coll, adding it to the front of coll."</span> 1.825 + [coll] 1.826 + (<span style="color: #8cd0d3;">cons</span> (<span style="color: #8cd0d3;">last</span> coll) (<span style="color: #8cd0d3;">butlast</span> coll))) 1.827 +</pre> 1.828 + 1.829 + 1.830 + 1.831 + 1.832 + 1.833 + 1.834 + 1.835 +</div> 1.836 + 1.837 +</div> 1.838 + 1.839 +<div id="outline-container-3-2" class="outline-3"> 1.840 +<h3 id="sec-3-2"><span class="section-number-3">3.2</span> Ranges, Airity and Function Composition </h3> 1.841 +<div class="outline-text-3" id="text-3-2"> 1.842 + 1.843 + 1.844 + 1.845 + 1.846 +<pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) 1.847 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">infinity</span> Double/POSITIVE_INFINITY) 1.848 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">infinite?</span> [x] (Double/isInfinite x)) 1.849 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">finite?</span> (<span style="color: #8cd0d3;">comp</span> not infinite?)) 1.850 + 1.851 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">arity-min</span> 1.852 + <span style="color: #8fb28f;">"Returns the smallest number of arguments f can take."</span> 1.853 + [f] 1.854 + (<span style="color: #8cd0d3;">apply</span> 1.855 + min 1.856 + (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">comp</span> alength #(.getParameterTypes %)) 1.857 + (<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 %)) 1.858 + (.getDeclaredMethods (<span style="color: #8cd0d3;">class</span> f)))))) 1.859 + 1.860 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">arity-max</span> 1.861 + <span style="color: #8fb28f;">"Returns the largest number of arguments f can take, possibly</span> 1.862 +<span style="color: #8fb28f;"> Infinity."</span> 1.863 + [f] 1.864 + (<span style="color: #f0dfaf; font-weight: bold;">let</span> [methods (.getDeclaredMethods (<span style="color: #8cd0d3;">class</span> f))] 1.865 + (<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)) 1.866 + (<span style="color: #8cd0d3;">apply</span> max 1.867 + (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">comp</span> alength #(.getParameterTypes %)) 1.868 + (<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))) 1.869 + infinity))) 1.870 + 1.871 + 1.872 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">^</span>{<span style="color: #8cd0d3;">:arglists</span> '([f]) 1.873 + <span style="color: #8cd0d3;">:doc</span> <span style="color: #cc9393;">"Returns a two-element list containing the minimum and</span> 1.874 +<span style="color: #cc9393;"> maximum number of args that f can take."</span>} 1.875 + arity-interval 1.876 + (<span style="color: #8cd0d3;">juxt</span> arity-min arity-max)) 1.877 + 1.878 + 1.879 + 1.880 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">--- intervals</span> 1.881 + 1.882 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">intersect</span> 1.883 + <span style="color: #8fb28f;">"Returns the interval of overlap between interval-1 and interval-2"</span> 1.884 + [interval-1 interval-2] 1.885 + (<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)) [] 1.886 + (<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)) 1.887 + right (<span style="color: #8cd0d3;">min</span> (<span style="color: #8cd0d3;">second</span> interval-1) (<span style="color: #8cd0d3;">second</span> interval-2))] 1.888 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">></span> left right) [] 1.889 + [left right])))) 1.890 + 1.891 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">same-endpoints?</span> 1.892 + <span style="color: #8fb28f;">"Returns true if the left endpoint is the same as the right</span> 1.893 +<span style="color: #8fb28f;"> endpoint."</span> 1.894 + [interval] 1.895 + (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">first</span> interval) (<span style="color: #8cd0d3;">second</span> interval))) 1.896 + 1.897 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">naturals?</span> 1.898 + <span style="color: #8fb28f;">"Returns true if the left endpoint is 0 and the right endpoint is</span> 1.899 +<span style="color: #8fb28f;">infinite."</span> 1.900 + [interval] 1.901 + (<span style="color: #f0dfaf; font-weight: bold;">and</span> (zero? (<span style="color: #8cd0d3;">first</span> interval)) 1.902 + (infinite? (<span style="color: #8cd0d3;">second</span> interval)))) 1.903 + 1.904 + 1.905 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">fan-in</span> 1.906 + <span style="color: #8fb28f;">"Returns a function that pipes its input to each of the gs, then</span> 1.907 +<span style="color: #8fb28f;"> applies f to the list of results. Consequently, f must be able to</span> 1.908 +<span style="color: #8fb28f;"> take a number of arguments equal to the number of gs."</span> 1.909 + [f & gs] 1.910 + (<span style="color: #8cd0d3;">fn</span> [& args] 1.911 + (<span style="color: #8cd0d3;">apply</span> f (<span style="color: #8cd0d3;">apply</span> (<span style="color: #8cd0d3;">apply</span> juxt gs) args)))) 1.912 + 1.913 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">fan-in</span> 1.914 + <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> 1.915 + [f & gs] 1.916 + (<span style="color: #8cd0d3;">comp</span> (<span style="color: #8cd0d3;">partial</span> apply f) (<span style="color: #8cd0d3;">apply</span> juxt gs))) 1.917 + 1.918 + 1.919 + 1.920 +(<span style="color: #f0dfaf; font-weight: bold;">defmacro</span> <span style="color: #f0dfaf;">airty-blah-sad</span> [f n more?] 1.921 + (<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))) 1.922 + optional (<span style="color: #8cd0d3;">gensym</span> <span style="color: #cc9393;">"xs"</span>)] 1.923 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> more? 1.924 + `(<span style="color: #8cd0d3;">fn</span> <span style="color: #f0dfaf;">~</span>(<span style="color: #8cd0d3;">conj</span> syms '& optional) 1.925 + (<span style="color: #8cd0d3;">apply</span> ~f ~@syms ~optional)) 1.926 + `(<span style="color: #8cd0d3;">fn</span> <span style="color: #f0dfaf;">~syms</span> (~f ~@syms))))) 1.927 + 1.928 +(<span style="color: #f0dfaf; font-weight: bold;">defmacro</span> <span style="color: #f0dfaf;">airt-whaa*</span> [f n more?] 1.929 + `(airty-blah-sad ~f ~n ~more?)) 1.930 + 1.931 + 1.932 + 1.933 + 1.934 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">fan-in*</span> 1.935 + <span style="color: #8fb28f;">"Returns a function that pipes its input to each of the gs, then</span> 1.936 +<span style="color: #8fb28f;"> applies f to the list of results. Unlike fan-in, fan-in* strictly</span> 1.937 +<span style="color: #8fb28f;"> enforces arity: it will fail if the gs do not have compatible</span> 1.938 +<span style="color: #8fb28f;"> arities."</span> 1.939 + [f & gs] 1.940 + (<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)) 1.941 + left (<span style="color: #8cd0d3;">first</span> arity-in) 1.942 + right (<span style="color: #8cd0d3;">second</span> arity-in) 1.943 + composite (fan-in f gs) 1.944 + ] 1.945 + (<span style="color: #f0dfaf; font-weight: bold;">cond</span> 1.946 + (<span style="color: #8cd0d3;">empty?</span> arity-in) 1.947 + (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"Cannot compose functions with incompatible arities."</span>)) 1.948 + 1.949 + (<span style="color: #8cd0d3;">not</span> 1.950 + (<span style="color: #f0dfaf; font-weight: bold;">or</span> (<span style="color: #8cd0d3;">=</span> left right) 1.951 + (<span style="color: #f0dfaf; font-weight: bold;">and</span> (finite? left) 1.952 + (<span style="color: #8cd0d3;">=</span> right infinity)))) 1.953 + 1.954 + (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. 1.955 + <span style="color: #cc9393;">"Compose can only handle arities of the form [n n] or [n infinity]"</span>)) 1.956 + <span style="color: #8cd0d3;">:else</span> 1.957 + (airty-blah-sad composite left (<span style="color: #8cd0d3;">=</span> right infinity))))) 1.958 + 1.959 + 1.960 + 1.961 +(<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> 1.962 + ([] identity) 1.963 + ([f] f) 1.964 + ([f & fs] 1.965 + (<span style="color: #f0dfaf; font-weight: bold;">let</span> [fns (<span style="color: #8cd0d3;">cons</span> f fs)] 1.966 + (compose-bin (<span style="color: #8cd0d3;">reduce</span> fan-in (<span style="color: #8cd0d3;">butlast</span> fs)) (<span style="color: #8cd0d3;">last</span> fs)))) 1.967 +) 1.968 + 1.969 + 1.970 + 1.971 + 1.972 + 1.973 + 1.974 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">iterated</span> 1.975 + ([f n id] (<span style="color: #8cd0d3;">reduce</span> comp id (<span style="color: #8cd0d3;">repeat</span> n f))) 1.976 + ([f n] (<span style="color: #8cd0d3;">reduce</span> comp identity (<span style="color: #8cd0d3;">repeat</span> n f)))) 1.977 + 1.978 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">iterate-until-stable</span> 1.979 + <span style="color: #8fb28f;">"Repeatedly applies f to x, returning the first result that is close</span> 1.980 +<span style="color: #8fb28f;">enough to its predecessor."</span> 1.981 + [f close-enough? x] 1.982 + (<span style="color: #8cd0d3;">second</span> (swank.util/find-first 1.983 + (<span style="color: #8cd0d3;">partial</span> apply close-enough?) 1.984 + (<span style="color: #8cd0d3;">partition</span> 2 1 (<span style="color: #8cd0d3;">iterate</span> f x))))) 1.985 + 1.986 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">lexical<</span> [x y] 1.987 + (<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)))) 1.988 + 1.989 + 1.990 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">do-up</span> 1.991 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">do-down</span> 1.992 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">make-pairwise-test</span> comparator) 1.993 +<span style="color: #708070;">;;</span><span style="color: #7f9f7f;">all-equal?</span> 1.994 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">accumulation</span> multiplier) 1.995 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">inverse-accumulation</span> divider) 1.996 +<span style="color: #708070;">;;</span><span style="color: #7f9f7f;">left-circular-shift</span> 1.997 +<span style="color: #708070;">;;</span><span style="color: #7f9f7f;">right-circular-shift</span> 1.998 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">exactly-n?</span> same-endpoints?) 1.999 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">any-number?</span> naturals?) 1.1000 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO compose</span> 1.1001 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO compose-n</span> 1.1002 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">identity</span> 1.1003 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">compose-2</span> fan-in) 1.1004 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">compose-bin</span> fan-in*) 1.1005 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">any?</span> (<span style="color: #8cd0d3;">constantly</span> true)) 1.1006 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">none?</span> (<span style="color: #8cd0d3;">constantly</span> false)) 1.1007 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">constant</span> constantly) 1.1008 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">joint-arity</span> intersect) 1.1009 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">a-reduce</span> reduce) 1.1010 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">filter</span> 1.1011 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">make-map</span> (<span style="color: #8cd0d3;">partial</span> partial map) ) 1.1012 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">bracket</span> juxt) 1.1013 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO apply-to-all</span> 1.1014 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO nary-combine</span> 1.1015 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO binary-combine</span> 1.1016 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO unary-combine</span> 1.1017 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">iterated</span> 1.1018 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">iterate-until-stable</span> 1.1019 +(<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)) 1.1020 +(<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)))) 1.1021 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">alphaless</span> lexical<) 1.1022 + 1.1023 +</pre> 1.1024 + 1.1025 + 1.1026 + 1.1027 + 1.1028 + 1.1029 + 1.1030 + 1.1031 + 1.1032 + 1.1033 + 1.1034 + 1.1035 + 1.1036 +</div> 1.1037 +</div> 1.1038 + 1.1039 +</div> 1.1040 + 1.1041 +<div id="outline-container-4" class="outline-2"> 1.1042 +<h2 id="sec-4"><span class="section-number-2">4</span> Numerical Methods </h2> 1.1043 +<div class="outline-text-2" id="text-4"> 1.1044 + 1.1045 + 1.1046 + 1.1047 + 1.1048 +<pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) 1.1049 +(<span style="color: #f0dfaf; font-weight: bold;">import</span> java.lang.Math) 1.1050 +(<span style="color: #8cd0d3;">use</span> 'clojure.contrib.def) 1.1051 + 1.1052 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">---- USEFUL CONSTANTS</span> 1.1053 + 1.1054 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">machine-epsilon</span> 1.1055 + <span style="color: #8fb28f;">"Smallest value representable on your machine, as determined by</span> 1.1056 +<span style="color: #8fb28f;">successively dividing a number in half until consecutive results are</span> 1.1057 +<span style="color: #8fb28f;">indistinguishable."</span> 1.1058 + [] 1.1059 + (<span style="color: #8cd0d3;">ffirst</span> 1.1060 + (<span style="color: #8cd0d3;">drop-while</span> 1.1061 + (<span style="color: #8cd0d3;">comp</span> not zero? second) 1.1062 + (<span style="color: #8cd0d3;">partition</span> 2 1 1.1063 + (<span style="color: #8cd0d3;">iterate</span> (<span style="color: #8cd0d3;">partial</span> * 0.5) 1))))) 1.1064 + 1.1065 + 1.1066 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">pi</span> (Math/PI)) 1.1067 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">two-pi</span> (<span style="color: #8cd0d3;">*</span> 2 pi)) 1.1068 + 1.1069 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">eulers-gamma</span> 0.5772156649015328606065) 1.1070 + 1.1071 +(<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)) 1.1072 + 1.1073 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">ln2</span> (Math/log 2)) 1.1074 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">ln10</span> (Math/log 10)) 1.1075 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">exp10</span> #(Math/pow 10 %)) 1.1076 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">exp2</span> #(Math/pow 2 %)) 1.1077 + 1.1078 + 1.1079 +<span style="color: #708070;">;;</span> 1.1080 + 1.1081 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">---- ANGLES AND TRIGONOMETRY</span> 1.1082 + 1.1083 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">angle-restrictor</span> 1.1084 + <span style="color: #8fb28f;">"Returns a function that ensures that angles lie in the specified interval of length two-pi."</span> 1.1085 + [max-angle] 1.1086 + (<span style="color: #f0dfaf; font-weight: bold;">let</span> [min-angle (<span style="color: #8cd0d3;">-</span> max-angle two-pi)] 1.1087 + (<span style="color: #8cd0d3;">fn</span> [x] 1.1088 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #f0dfaf; font-weight: bold;">and</span> 1.1089 + (<span style="color: #8cd0d3;"><=</span> min-angle x) 1.1090 + (<span style="color: #8cd0d3;"><</span> x max-angle)) 1.1091 + x 1.1092 + (<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))))] 1.1093 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;"><</span> corrected-x max-angle) 1.1094 + corrected-x 1.1095 + (<span style="color: #8cd0d3;">-</span> corrected-x two-pi))))))) 1.1096 + 1.1097 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">angle-restrict-pi</span> 1.1098 + <span style="color: #8fb28f;">"Coerces angles to lie in the interval from -pi to pi."</span> 1.1099 + [angle] 1.1100 + ((angle-restrictor pi) angle)) 1.1101 + 1.1102 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">angle-restrict-two-pi</span> 1.1103 + <span style="color: #8fb28f;">"Coerces angles to lie in the interval from zero to two-pi"</span> 1.1104 + [angle] 1.1105 + ((angle-restrictor two-pi) angle)) 1.1106 + 1.1107 + 1.1108 + 1.1109 + 1.1110 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">invert</span> [x] (<span style="color: #8cd0d3;">/</span> x)) 1.1111 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">negate</span> [x] (<span style="color: #8cd0d3;">-</span> x)) 1.1112 + 1.1113 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">exp</span> [x] (Math/exp x)) 1.1114 + 1.1115 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">sin</span> [x] (Math/sin x)) 1.1116 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">cos</span> [x] (Math/cos x)) 1.1117 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">tan</span> [x] (Math/tan x)) 1.1118 + 1.1119 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">sec</span> (<span style="color: #8cd0d3;">comp</span> invert cos)) 1.1120 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">csc</span> (<span style="color: #8cd0d3;">comp</span> invert sin)) 1.1121 + 1.1122 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">sinh</span> [x] (Math/sinh x)) 1.1123 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">cosh</span> [x] (Math/cosh x)) 1.1124 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">tanh</span> [x] (Math/tanh x)) 1.1125 + 1.1126 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">sech</span> (<span style="color: #8cd0d3;">comp</span> invert cosh)) 1.1127 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">csch</span> (<span style="color: #8cd0d3;">comp</span> invert sinh)) 1.1128 + 1.1129 + 1.1130 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">------------</span> 1.1131 + 1.1132 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">factorial</span> 1.1133 + <span style="color: #8fb28f;">"Computes the factorial of the nonnegative integer n."</span> 1.1134 + [n] 1.1135 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">neg?</span> n) 1.1136 + (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"Cannot compute the factorial of a negative number."</span>)) 1.1137 + (<span style="color: #8cd0d3;">reduce</span> * 1 (<span style="color: #8cd0d3;">range</span> 1 (<span style="color: #8cd0d3;">inc</span> n))))) 1.1138 + 1.1139 +(<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)) 1.1140 + 1.1141 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">binomial-coefficient</span> 1.1142 + <span style="color: #8fb28f;">"Computes the number of different ways to choose m elements from n."</span> 1.1143 + [n m] 1.1144 + (<span style="color: #8cd0d3;">assert</span> (<span style="color: #8cd0d3;"><=</span> 0 m n)) 1.1145 + (<span style="color: #f0dfaf; font-weight: bold;">let</span> [difference (<span style="color: #8cd0d3;">-</span> n m)] 1.1146 + (exact-quotient 1.1147 + (<span style="color: #8cd0d3;">reduce</span> * (<span style="color: #8cd0d3;">range</span> n (<span style="color: #8cd0d3;">max</span> difference m) -1 )) 1.1148 + (factorial (<span style="color: #8cd0d3;">min</span> difference m))))) 1.1149 + 1.1150 +(<span style="color: #f0dfaf; font-weight: bold;">defn-memo</span> <span style="color: #f0dfaf;">stirling-1</span> 1.1151 + <span style="color: #cc9393;">"Stirling numbers of the first kind: the number of permutations of n</span> 1.1152 +<span style="color: #cc9393;"> elements with exactly m permutation cycles. "</span> 1.1153 + [n k] 1.1154 + <span style="color: #708070;">;</span><span style="color: #7f9f7f;">(assert (<= 1 k n))</span> 1.1155 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (zero? n) 1.1156 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (zero? k) 1 0) 1.1157 + (<span style="color: #8cd0d3;">+</span> (stirling-1 (<span style="color: #8cd0d3;">dec</span> n) (<span style="color: #8cd0d3;">dec</span> k)) 1.1158 + (<span style="color: #8cd0d3;">*</span> (<span style="color: #8cd0d3;">dec</span> n) (stirling-1 (<span style="color: #8cd0d3;">dec</span> n) k))))) 1.1159 + 1.1160 +(<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> 1.1161 + <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> 1.1162 + [n k] 1.1163 + (<span style="color: #f0dfaf; font-weight: bold;">cond</span> 1.1164 + (<span style="color: #8cd0d3;">=</span> k 1) 1 1.1165 + (<span style="color: #8cd0d3;">=</span> k n) 1 1.1166 + <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)) 1.1167 + (<span style="color: #8cd0d3;">*</span> k (stirling-2 (<span style="color: #8cd0d3;">dec</span> n) k))))) 1.1168 + 1.1169 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">harmonic-number</span> [n] 1.1170 + (<span style="color: #8cd0d3;">/</span> (stirling-1 (<span style="color: #8cd0d3;">inc</span> n) 2) 1.1171 + (factorial n))) 1.1172 + 1.1173 + 1.1174 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">sum</span> 1.1175 + [f low high] 1.1176 + (<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))))) 1.1177 + 1.1178 +</pre> 1.1179 + 1.1180 + 1.1181 + 1.1182 + 1.1183 + 1.1184 + 1.1185 + 1.1186 + 1.1187 + 1.1188 + 1.1189 + 1.1190 + 1.1191 + 1.1192 + 1.1193 + 1.1194 +</div> 1.1195 + 1.1196 +</div> 1.1197 + 1.1198 +<div id="outline-container-5" class="outline-2"> 1.1199 +<h2 id="sec-5"><span class="section-number-2">5</span> Differentiation </h2> 1.1200 +<div class="outline-text-2" id="text-5"> 1.1201 + 1.1202 + 1.1203 +<p> 1.1204 +We compute derivatives by passing special <b>differential objects</b> \([x, 1.1205 +dx]\) through functions. Roughly speaking, applying a function \(f\) to a 1.1206 +differential object \([x, dx]\) should produce a new differential 1.1207 +object \([f(x),\,Df(x)\cdot dx]\). 1.1208 +</p> 1.1209 + 1.1210 + 1.1211 +\([x,\,dx]\xrightarrow{\quad f \quad}[f(x),\,Df(x)\cdot dx]\) 1.1212 +<p> 1.1213 +Notice that you can obtain the derivative of \(f\) from this 1.1214 +differential object, as it is the coefficient of the \(dx\) term. Also, 1.1215 +as you apply successive functions using this rule, you get the 1.1216 +chain-rule answer you expect: 1.1217 +</p> 1.1218 + 1.1219 + 1.1220 +\([f(x),\,Df(x)\cdot dx]\xrightarrow{\quad g\quad} [gf(x),\, 1.1221 +Dgf(x)\cdot Df(x) \cdot dx ]\) 1.1222 + 1.1223 +<p> 1.1224 +In order to generalize to multiple variables and multiple derivatives, 1.1225 +we use a <b>power series of differentials</b>, a sortred infinite sequence which 1.1226 +contains all terms like \(dx\cdot dy\), \(dx^2\cdot dy\), etc. 1.1227 +</p> 1.1228 + 1.1229 + 1.1230 + 1.1231 + 1.1232 +<pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) 1.1233 + 1.1234 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">differential-seq</span> 1.1235 + <span style="color: #8fb28f;">"Constructs a sequence of differential terms from a numerical</span> 1.1236 +<span style="color: #8fb28f;">coefficient and a list of keys for variables. If no coefficient is supplied, uses 1."</span> 1.1237 + ([variables] (differential-seq* 1 variables)) 1.1238 + ([coefficient variables & cvs] 1.1239 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">number?</span> coefficient) 1.1240 + (<span style="color: #8cd0d3;">conj</span> (<span style="color: #8cd0d3;">assoc</span> {} (<span style="color: #8cd0d3;">apply</span> sorted-set variables) coefficient) 1.1241 + (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">empty?</span> cvs) 1.1242 + nil 1.1243 + (<span style="color: #8cd0d3;">apply</span> differential-seq* cvs))) 1.1244 + (<span style="color: #8cd0d3;">apply</span> differential-seq* 1 coefficient 1 variables cvs) 1.1245 + ))) 1.1246 + 1.1247 + 1.1248 + 1.1249 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">differential-add</span> 1.1250 + <span style="color: #8fb28f;">"Add two differential sequences by combining like terms."</span> 1.1251 + [dseq1 dseq2] 1.1252 + (<span style="color: #8cd0d3;">merge-with</span> + dseq1 dseq2)) 1.1253 + 1.1254 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">differential-multiply</span> 1.1255 + <span style="color: #8fb28f;">"Multiply two differential sequences. The square of any differential variable is zero since differential variables are infinitesimally small."</span> 1.1256 + [dseq1 dseq2] 1.1257 + (<span style="color: #8cd0d3;">reduce</span> 1.1258 + (<span style="color: #8cd0d3;">fn</span> [m [[vars1 coeff1] [vars2 coeff2]]] 1.1259 + (<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)) 1.1260 + (<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)) 1.1261 + m)) 1.1262 + {} 1.1263 + (clojure.contrib.combinatorics/cartesian-product 1.1264 + dseq1 1.1265 + dseq2))) 1.1266 + 1.1267 + 1.1268 + 1.1269 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">big-part</span> 1.1270 + <span style="color: #8fb28f;">"Returns the 'finite' part of the differential sequence."</span> 1.1271 + [dseq] 1.1272 + (<span style="color: #f0dfaf; font-weight: bold;">let</span> 1.1273 + [keys (<span style="color: #8cd0d3;">sort-by</span> count (<span style="color: #8cd0d3;">keys</span> dseq)) 1.1274 + pivot-key (<span style="color: #8cd0d3;">first</span> (<span style="color: #8cd0d3;">last</span> keys))] 1.1275 + 1.1276 + (<span style="color: #8cd0d3;">apply</span> hash-map 1.1277 + (<span style="color: #8cd0d3;">reduce</span> concat 1.1278 + (<span style="color: #8cd0d3;">filter</span> (<span style="color: #8cd0d3;">comp</span> pivot-key first) dseq) 1.1279 + )))) 1.1280 + 1.1281 + 1.1282 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">small-part</span> 1.1283 + <span style="color: #8fb28f;">"Returns the 'infinitesimal' part of the differential sequence."</span> 1.1284 + [dseq] 1.1285 + (<span style="color: #f0dfaf; font-weight: bold;">let</span> 1.1286 + [keys (<span style="color: #8cd0d3;">sort-by</span> count (<span style="color: #8cd0d3;">keys</span> dseq)) 1.1287 + pivot-key (<span style="color: #8cd0d3;">first</span> (<span style="color: #8cd0d3;">last</span> keys))] 1.1288 + 1.1289 + (<span style="color: #8cd0d3;">apply</span> hash-map 1.1290 + (<span style="color: #8cd0d3;">reduce</span> concat 1.1291 + (<span style="color: #8cd0d3;">remove</span> (<span style="color: #8cd0d3;">comp</span> pivot-key first) dseq) 1.1292 + )))) 1.1293 + 1.1294 + 1.1295 + 1.1296 + 1.1297 + 1.1298 + 1.1299 + 1.1300 + 1.1301 + 1.1302 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">;; A differential term consists of a numerical coefficient and a</span> 1.1303 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">;; sorted </span> 1.1304 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(defrecord DifferentialTerm [coefficient variables])</span> 1.1305 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(defmethod print-method DifferentialTerm</span> 1.1306 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">[o w]</span> 1.1307 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(print-simple</span> 1.1308 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(apply str (.coefficient o)(map (comp (partial str "d") name) (.variables o)))</span> 1.1309 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">w))</span> 1.1310 + 1.1311 + 1.1312 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(defn differential-seq</span> 1.1313 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">"Constructs a sequence of differential terms from a numerical</span> 1.1314 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">coefficient and a list of keywords for variables. If no coefficient is</span> 1.1315 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">supplied, uses 1."</span> 1.1316 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">([variables] (differential-seq 1 variables))</span> 1.1317 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">([coefficient variables]</span> 1.1318 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(list</span> 1.1319 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(DifferentialTerm. coefficient (apply sorted-set variables))))</span> 1.1320 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">([coefficient variables & cvs]</span> 1.1321 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(sort-by</span> 1.1322 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">#(vec(.variables %))</span> 1.1323 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(concat (differential-seq coefficient variables) (apply differential-seq cvs)))))</span> 1.1324 + 1.1325 +</pre> 1.1326 + 1.1327 + 1.1328 + 1.1329 + 1.1330 + 1.1331 + 1.1332 + 1.1333 + 1.1334 + 1.1335 + 1.1336 + 1.1337 + 1.1338 + 1.1339 + 1.1340 + 1.1341 + 1.1342 +</div> 1.1343 + 1.1344 +</div> 1.1345 + 1.1346 +<div id="outline-container-6" class="outline-2"> 1.1347 +<h2 id="sec-6"><span class="section-number-2">6</span> Symbolic Manipulation </h2> 1.1348 +<div class="outline-text-2" id="text-6"> 1.1349 + 1.1350 + 1.1351 + 1.1352 + 1.1353 + 1.1354 +<pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) 1.1355 + 1.1356 +(<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">Symbolic</span> [type expression] 1.1357 + Object 1.1358 + (equals [this that] 1.1359 + (<span style="color: #f0dfaf; font-weight: bold;">cond</span> 1.1360 + (<span style="color: #8cd0d3;">=</span> (.expression this) (.expression that)) true 1.1361 + <span style="color: #8cd0d3;">:else</span> 1.1362 + (Symbolic. 1.1363 + java.lang.Boolean 1.1364 + (<span style="color: #8cd0d3;">list</span> '= (.expression this) (.expression that))) 1.1365 + ))) 1.1366 + 1.1367 + 1.1368 + 1.1369 + 1.1370 +(<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">AbstractSet</span> [glyph membership-test]) 1.1371 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">member?</span> [abstract-set x] 1.1372 + ((.membership-test abstract-set) x)) 1.1373 + 1.1374 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">------------ Some important AbstractSets</span> 1.1375 + 1.1376 + 1.1377 +(<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">Real</span> 1.1378 + (AbstractSet. 1.1379 + 'R 1.1380 + (<span style="color: #8cd0d3;">fn</span>[x](<span style="color: #8cd0d3;">number?</span> x)))) 1.1381 + 1.1382 + 1.1383 +<span style="color: #708070;">;; </span><span style="color: #7f9f7f;">------------ Create new AbstractSets from existing ones</span> 1.1384 + 1.1385 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">abstract-product</span> 1.1386 + <span style="color: #8fb28f;">"Gives the cartesian product of abstract sets."</span> 1.1387 + ([sets] 1.1388 + (<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) 1.1389 + (AbstractSet. 1.1390 + (<span style="color: #8cd0d3;">symbol</span> 1.1391 + (<span style="color: #8cd0d3;">apply</span> str 1.1392 + (<span style="color: #8cd0d3;">interpose</span> 'x (<span style="color: #8cd0d3;">map</span> #(.glyph %) sets)))) 1.1393 + (<span style="color: #8cd0d3;">fn</span> [x] 1.1394 + (<span style="color: #f0dfaf; font-weight: bold;">and</span> 1.1395 + (<span style="color: #8cd0d3;">coll?</span> x) 1.1396 + (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">count</span> sets) (<span style="color: #8cd0d3;">count</span> x)) 1.1397 + (<span style="color: #8cd0d3;">reduce</span> #(<span style="color: #f0dfaf; font-weight: bold;">and</span> %1 %2) 1.1398 + true 1.1399 + (<span style="color: #8cd0d3;">map</span> #(member? %1 %2) sets x))))))) 1.1400 + ([abstract-set n] 1.1401 + (abstract-product (<span style="color: #8cd0d3;">repeat</span> n abstract-set)))) 1.1402 + 1.1403 + 1.1404 + 1.1405 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">abstract-up</span> 1.1406 + <span style="color: #8fb28f;">"Returns the abstract set of all up-tuples whose items belong to the</span> 1.1407 +<span style="color: #8fb28f;"> corresponding abstract sets in coll."</span> 1.1408 + ([coll] 1.1409 + (AbstractSet. 1.1410 + (<span style="color: #8cd0d3;">symbol</span> (<span style="color: #8cd0d3;">str</span> <span style="color: #cc9393;">"u["</span> 1.1411 + (<span style="color: #8cd0d3;">apply</span> str 1.1412 + (<span style="color: #8cd0d3;">interpose</span> <span style="color: #cc9393;">" "</span> 1.1413 + (<span style="color: #8cd0d3;">map</span> #(.glyph %) coll))) 1.1414 + <span style="color: #cc9393;">"]"</span>)) 1.1415 + (<span style="color: #8cd0d3;">fn</span> [x] 1.1416 + (<span style="color: #f0dfaf; font-weight: bold;">and</span> 1.1417 + (<span style="color: #8cd0d3;">satisfies?</span> Spinning x) 1.1418 + (up? x) 1.1419 + (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">count</span> coll) (<span style="color: #8cd0d3;">count</span> x)) 1.1420 + (<span style="color: #8cd0d3;">reduce</span> 1.1421 + #(<span style="color: #f0dfaf; font-weight: bold;">and</span> %1 %2) 1.1422 + true 1.1423 + (<span style="color: #8cd0d3;">map</span> #(member? %1 %2) coll x)))))) 1.1424 + ([abstract-set n] 1.1425 + (abstract-up (<span style="color: #8cd0d3;">repeat</span> n abstract-set)))) 1.1426 + 1.1427 + 1.1428 +(<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">abstract-down</span> 1.1429 + <span style="color: #8fb28f;">"Returns the abstract set of all down-tuples whose items belong to the</span> 1.1430 +<span style="color: #8fb28f;"> corresponding abstract sets in coll."</span> 1.1431 + ([coll] 1.1432 + (AbstractSet. 1.1433 + (<span style="color: #8cd0d3;">symbol</span> (<span style="color: #8cd0d3;">str</span> <span style="color: #cc9393;">"d["</span> 1.1434 + (<span style="color: #8cd0d3;">apply</span> str 1.1435 + (<span style="color: #8cd0d3;">interpose</span> <span style="color: #cc9393;">" "</span> 1.1436 + (<span style="color: #8cd0d3;">map</span> #(.glyph %) coll))) 1.1437 + <span style="color: #cc9393;">"]"</span>)) 1.1438 + (<span style="color: #8cd0d3;">fn</span> [x] 1.1439 + (<span style="color: #f0dfaf; font-weight: bold;">and</span> 1.1440 + (<span style="color: #8cd0d3;">satisfies?</span> Spinning x) 1.1441 + (down? x) 1.1442 + (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">count</span> coll) (<span style="color: #8cd0d3;">count</span> x)) 1.1443 + (<span style="color: #8cd0d3;">reduce</span> 1.1444 + #(<span style="color: #f0dfaf; font-weight: bold;">and</span> %1 %2) 1.1445 + true 1.1446 + (<span style="color: #8cd0d3;">map</span> #(member? %1 %2) coll x)))))) 1.1447 + ([abstract-set n] 1.1448 + (abstract-down (<span style="color: #8cd0d3;">repeat</span> n abstract-set)))) 1.1449 + 1.1450 + 1.1451 + 1.1452 + 1.1453 + 1.1454 + <span style="color: #708070;">;</span><span style="color: #7f9f7f;">-------ABSTRACT FUNCTIONS</span> 1.1455 +(<span style="color: #f0dfaf; font-weight: bold;">defrecord</span> <span style="color: #f0dfaf;">AbstractFn</span> 1.1456 + [<span style="color: #dfdfbf; font-weight: bold;">#^AbstractSet</span> domain <span style="color: #dfdfbf; font-weight: bold;">#^AbstractSet</span> codomain]) 1.1457 + 1.1458 + 1.1459 +(<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">print-method</span> AbstractFn 1.1460 + [o w] 1.1461 + (<span style="color: #8cd0d3;">print-simple</span> 1.1462 + (<span style="color: #8cd0d3;">str</span> 1.1463 + <span style="color: #cc9393;">"f:"</span> 1.1464 + (.glyph (<span style="color: #8cd0d3;">:domain</span> o)) 1.1465 + <span style="color: #cc9393;">"-->"</span> 1.1466 + (.glyph (<span style="color: #8cd0d3;">:codomain</span> o))) w)) 1.1467 +</pre> 1.1468 + 1.1469 + 1.1470 + 1.1471 + 1.1472 + 1.1473 + 1.1474 + 1.1475 +</div> 1.1476 +</div> 1.1477 +<div id="postamble"> 1.1478 +<p class="date">Date: 2011-08-09 18:41:37 EDT</p> 1.1479 +<p class="author">Author: Robert McIntyre & Dylan Holmes</p> 1.1480 +<p class="creator">Org version 7.6 with Emacs version 23</p> 1.1481 +<a href="http://validator.w3.org/check?uri=referer">Validate XHTML 1.0</a> 1.1482 +</div> 1.1483 +</div> 1.1484 +</body> 1.1485 +</html>