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author Robert McIntyre <rlm@mit.edu>
date Fri, 28 Oct 2011 00:03:05 -0700
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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">
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6 <head>
7 <title>Building a Classical Mechanics Library in Clojure</title>
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82 <body>
83
84 <div id="content">
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102 </div>
103
104 <h1 class="title">Building a Classical Mechanics Library in Clojure</h1>
105
106
107
108
109
110
111
112 <div id="table-of-contents">
113 <h2>Table of Contents</h2>
114 <div id="text-table-of-contents">
115 <ul>
116 <li><a href="#sec-1">1 Useful data types </a>
117 <ul>
118 <li><a href="#sec-1-1">1.1 Complex numbers </a></li>
119 <li><a href="#sec-1-2">1.2 Power series </a></li>
120 <li><a href="#sec-1-3">1.3 Tuples and tensors </a>
121 <ul>
122 <li><a href="#sec-1-3-1">1.3.1 Tuples are &ldquo;sequences with spin&rdquo; </a></li>
123 <li><a href="#sec-1-3-2">1.3.2 Matrices </a></li>
124 </ul>
125 </li>
126 <li><a href="#sec-1-4">1.4 Generic Operations </a></li>
127 </ul>
128 </li>
129 <li><a href="#sec-2">2 Operators and Differentiation </a>
130 <ul>
131 <li><a href="#sec-2-1">2.1 Operators </a></li>
132 <li><a href="#sec-2-2">2.2 Differential Terms and Sequences </a></li>
133 </ul>
134 </li>
135 </ul>
136 </div>
137 </div>
138
139 <div id="outline-container-1" class="outline-2">
140 <h2 id="sec-1"><span class="section-number-2">1</span> Useful data types </h2>
141 <div class="outline-text-2" id="text-1">
142
143
144
145 </div>
146
147 <div id="outline-container-1-1" class="outline-3">
148 <h3 id="sec-1-1"><span class="section-number-3">1.1</span> Complex numbers </h3>
149 <div class="outline-text-3" id="text-1-1">
150
151
152 </div>
153
154 </div>
155
156 <div id="outline-container-1-2" class="outline-3">
157 <h3 id="sec-1-2"><span class="section-number-3">1.2</span> Power series </h3>
158 <div class="outline-text-3" id="text-1-2">
159
160
161 </div>
162
163 </div>
164
165 <div id="outline-container-1-3" class="outline-3">
166 <h3 id="sec-1-3"><span class="section-number-3">1.3</span> Tuples and tensors </h3>
167 <div class="outline-text-3" id="text-1-3">
168
169
170
171 </div>
172
173 <div id="outline-container-1-3-1" class="outline-4">
174 <h4 id="sec-1-3-1"><span class="section-number-4">1.3.1</span> Tuples are &ldquo;sequences with spin&rdquo; </h4>
175 <div class="outline-text-4" id="text-1-3-1">
176
177
178
179
180
181 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)
182
183 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">Let some objects have spin</span>
184
185 (<span style="color: #f0dfaf; font-weight: bold;">defprotocol</span> <span style="color: #f0dfaf;">Spinning</span>
186 (up? [this])
187 (down? [this]))
188
189 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">spin</span>
190 <span style="color: #8fb28f;">"Returns the spin of the Spinning s, either :up or :down"</span>
191 [<span style="color: #dfdfbf; font-weight: bold;">#^Spinning</span> s]
192 (<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>))
193
194
195 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">DEFINITION: A tuple is a sequence with spin</span>
196
197 (<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">Tuple</span>
198 [spin coll]
199
200 clojure.lang.Seqable
201 (<span style="color: #8cd0d3;">seq</span> [this] (<span style="color: #8cd0d3;">seq</span> (.coll this)))
202
203 clojure.lang.Counted
204 (<span style="color: #8cd0d3;">count</span> [this] (<span style="color: #8cd0d3;">count</span> (.coll this)))
205
206 Spinning
207 (up? [this] (<span style="color: #8cd0d3;">=</span> <span style="color: #8cd0d3;">::up</span> (.spin this)))
208 (down? [this] (<span style="color: #8cd0d3;">=</span> <span style="color: #8cd0d3;">::down</span> (.spin this))))
209
210 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">print-method</span> Tuple
211 [o w]
212 (<span style="color: #8cd0d3;">print-simple</span>
213 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (up? o)
214 (<span style="color: #8cd0d3;">str</span> <span style="color: #cc9393;">"u"</span> (.coll o))
215 (<span style="color: #8cd0d3;">str</span> <span style="color: #cc9393;">"d"</span> (<span style="color: #8cd0d3;">vec</span>(.coll o))))
216 w))
217
218
219 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">CONSTRUCTORS</span>
220
221 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">up</span>
222 <span style="color: #8fb28f;">"Create a new up-tuple containing the contents of coll."</span>
223 [coll]
224 (Tuple. <span style="color: #8cd0d3;">::up</span> coll))
225
226 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">down</span>
227 <span style="color: #8fb28f;">"Create a new down-tuple containing the contents of coll."</span>
228 [coll]
229 (Tuple. <span style="color: #8cd0d3;">::down</span> coll))
230 </pre>
231
232
233
234
235
236
237 </div>
238
239 </div>
240
241 <div id="outline-container-1-3-2" class="outline-4">
242 <h4 id="sec-1-3-2"><span class="section-number-4">1.3.2</span> Matrices </h4>
243 <div class="outline-text-4" id="text-1-3-2">
244
245
246
247
248 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)
249 (<span style="color: #8cd0d3;">require</span> 'incanter.core) <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">use incanter's fast matrices</span>
250
251
252 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">all-equal?</span> [coll]
253 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">empty?</span> (<span style="color: #8cd0d3;">rest</span> coll)) true
254 (<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))
255 (<span style="color: #f0dfaf; font-weight: bold;">recur</span> (<span style="color: #8cd0d3;">rest</span> coll)))))
256
257
258 (<span style="color: #f0dfaf; font-weight: bold;">defprotocol</span> <span style="color: #f0dfaf;">Matrix</span>
259 (rows [matrix])
260 (cols [matrix])
261 (diagonal [matrix])
262 (trace [matrix])
263 (determinant [matrix])
264 (transpose [matrix])
265 (conjugate [matrix])
266 )
267
268 (<span style="color: #8cd0d3;">extend-protocol</span> Matrix incanter.Matrix
269 (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))))
270 (cols [rs] (<span style="color: #8cd0d3;">map</span> up (<span style="color: #8cd0d3;">apply</span> map vector rs)))
271 (diagonal [matrix] (incanter.core/diag matrix) )
272 (determinant [matrix] (incanter.core/det matrix))
273 (trace [matrix] (incanter.core/trace matrix))
274 (transpose [matrix] (incanter.core/trans matrix)))
275
276 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">count-rows</span> [matrix]
277 ((<span style="color: #8cd0d3;">comp</span> count rows) matrix))
278
279 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">count-cols</span> [matrix]
280 ((<span style="color: #8cd0d3;">comp</span> count cols) matrix))
281
282 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">square?</span> [matrix]
283 (<span style="color: #8cd0d3;">=</span> (count-rows matrix) (count-cols matrix)))
284
285 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">identity-matrix</span>
286 <span style="color: #8fb28f;">"Define a square matrix of size n-by-n with 1s along the diagonal and</span>
287 <span style="color: #8fb28f;"> 0s everywhere else."</span>
288 [n]
289 (incanter.core/identity-matrix n))
290
291
292 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">matrix-by-rows</span>
293 <span style="color: #8fb28f;">"Define a matrix by giving its rows."</span>
294 [&amp; rows]
295 (<span style="color: #f0dfaf; font-weight: bold;">if</span>
296 (<span style="color: #8cd0d3;">not</span> (all-equal? (<span style="color: #8cd0d3;">map</span> count rows)))
297 (<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>))
298 (incanter.core/matrix (<span style="color: #8cd0d3;">vec</span> rows))))
299
300 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">matrix-by-cols</span>
301 <span style="color: #8fb28f;">"Define a matrix by giving its columns"</span>
302 [&amp; cols]
303 (<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)))
304 (<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>))
305 (incanter.core/matrix (<span style="color: #8cd0d3;">vec</span> (<span style="color: #8cd0d3;">apply</span> map vector cols)))))
306
307 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">identity-matrix</span>
308 <span style="color: #8fb28f;">"Define a square matrix of size n-by-n with 1s along the diagonal and</span>
309 <span style="color: #8fb28f;"> 0s everywhere else."</span>
310 [n]
311 (incanter.core/identity-matrix n))
312
313 </pre>
314
315
316
317
318 </div>
319 </div>
320
321 </div>
322
323 <div id="outline-container-1-4" class="outline-3">
324 <h3 id="sec-1-4"><span class="section-number-3">1.4</span> Generic Operations </h3>
325 <div class="outline-text-3" id="text-1-4">
326
327
328
329
330 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)
331 (<span style="color: #8cd0d3;">use</span> 'clojure.contrib.generic.arithmetic
332 'clojure.contrib.generic.collection
333 'clojure.contrib.generic.functor
334 'clojure.contrib.generic.math-functions)
335
336 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">numbers?</span>
337 <span style="color: #8fb28f;">"Returns true if all arguments are numbers, else false."</span>
338 [&amp; xs]
339 (<span style="color: #8cd0d3;">every?</span> number? xs))
340
341 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">tuple-surgery</span>
342 <span style="color: #8fb28f;">"Applies the function f to the items of tuple and the additional</span>
343 <span style="color: #8fb28f;"> arguments, if any. Returns a Tuple of the same type as tuple."</span>
344 [tuple f &amp; xs]
345 ((<span style="color: #f0dfaf; font-weight: bold;">if</span> (up? tuple) up down)
346 (<span style="color: #8cd0d3;">apply</span> f (<span style="color: #8cd0d3;">seq</span> tuple) xs)))
347
348
349
350 <span style="color: #708070;">;;; </span><span style="color: #7f9f7f;">CONTRACTION collapses two compatible tuples into a number.</span>
351
352 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">contractible?</span>
353 <span style="color: #8fb28f;">"Returns true if the tuples a and b are compatible for contraction,</span>
354 <span style="color: #8fb28f;"> else false. Tuples are compatible if they have the same number of</span>
355 <span style="color: #8fb28f;"> components, they have opposite spins, and their elements are</span>
356 <span style="color: #8fb28f;"> pairwise-compatible."</span>
357 [a b]
358 (<span style="color: #f0dfaf; font-weight: bold;">and</span>
359 (<span style="color: #8cd0d3;">isa?</span> (<span style="color: #8cd0d3;">type</span> a) Tuple)
360 (<span style="color: #8cd0d3;">isa?</span> (<span style="color: #8cd0d3;">type</span> b) Tuple)
361 (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">count</span> a) (<span style="color: #8cd0d3;">count</span> b))
362 (<span style="color: #8cd0d3;">not=</span> (spin a) (spin b))
363
364 (<span style="color: #8cd0d3;">not-any?</span> false?
365 (<span style="color: #8cd0d3;">map</span> #(<span style="color: #f0dfaf; font-weight: bold;">or</span>
366 (numbers? %1 %2)
367 (contractible? %1 %2))
368 a b))))
369
370 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">contract</span>
371 <span style="color: #8fb28f;">"Contracts two tuples, returning the sum of the</span>
372 <span style="color: #8fb28f;"> products of the corresponding items. Contraction is recursive on</span>
373 <span style="color: #8fb28f;"> nested tuples."</span>
374 [a b]
375 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not</span> (contractible? a b))
376 (<span style="color: #f0dfaf; font-weight: bold;">throw</span>
377 (Exception. <span style="color: #cc9393;">"Not compatible for contraction."</span>))
378 (<span style="color: #8cd0d3;">reduce</span> +
379 (<span style="color: #8cd0d3;">map</span>
380 (<span style="color: #8cd0d3;">fn</span> [x y]
381 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (numbers? x y)
382 (<span style="color: #8cd0d3;">*</span> x y)
383 (contract x y)))
384 a b))))
385
386
387
388
389
390 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">conj</span> Tuple
391 [tuple &amp; xs]
392 (tuple-surgery tuple #(<span style="color: #8cd0d3;">apply</span> conj % xs)))
393
394 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">fmap</span> Tuple
395 [f tuple]
396 (tuple-surgery tuple (<span style="color: #8cd0d3;">partial</span> map f)))
397
398
399
400 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO: define Scalar, and add it to the hierarchy above Number and Complex</span>
401
402
403 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [Tuple Tuple] <span style="color: #708070;">; </span><span style="color: #7f9f7f;">tuple*tuple</span>
404 [a b]
405 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (contractible? a b)
406 (contract a b)
407 (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">partial</span> * a) b)))
408
409
410 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [java.lang.Number Tuple] <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">scalar * tuple</span>
411 [a x] (fmap (<span style="color: #8cd0d3;">partial</span> * a) x))
412
413 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [Tuple java.lang.Number]
414 [x a] (<span style="color: #8cd0d3;">*</span> a x))
415
416 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [java.lang.Number incanter.Matrix] <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">scalar * matrix</span>
417 [x M] (incanter.core/mult x M))
418
419 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [incanter.Matrix java.lang.Number]
420 [M x] (<span style="color: #8cd0d3;">*</span> x M))
421
422 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [incanter.Matrix incanter.Matrix] <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">matrix * matrix</span>
423 [M1 M2]
424 (incanter.core/mmult M1 M2))
425
426 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [incanter.Matrix Tuple] <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">matrix * tuple</span>
427 [M v]
428 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #f0dfaf; font-weight: bold;">and</span> (<span style="color: #8cd0d3;">apply</span> numbers? v) (up? v))
429 (<span style="color: #8cd0d3;">*</span> M (matrix-by-cols v))
430 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"Currently, you can only multiply a matrix by a tuple of *numbers*"</span>))
431 ))
432
433 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [Tuple incanter.Matrix] <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">tuple * Matrix</span>
434 [v M]
435 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #f0dfaf; font-weight: bold;">and</span> (<span style="color: #8cd0d3;">apply</span> numbers? v) (down? v))
436 (<span style="color: #8cd0d3;">*</span> (matrix-by-rows v) M)
437 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"Currently, you can only multiply a matrix by a tuple of *numbers*"</span>))
438 ))
439
440
441 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">exp</span> incanter.Matrix
442 [M]
443 (incanter.core/exp M))
444
445 </pre>
446
447
448
449
450 </div>
451 </div>
452
453 </div>
454
455 <div id="outline-container-2" class="outline-2">
456 <h2 id="sec-2"><span class="section-number-2">2</span> Operators and Differentiation </h2>
457 <div class="outline-text-2" id="text-2">
458
459
460
461
462 </div>
463
464 <div id="outline-container-2-1" class="outline-3">
465 <h3 id="sec-2-1"><span class="section-number-3">2.1</span> Operators </h3>
466 <div class="outline-text-3" id="text-2-1">
467
468
469 </div>
470
471 </div>
472
473 <div id="outline-container-2-2" class="outline-3">
474 <h3 id="sec-2-2"><span class="section-number-3">2.2</span> Differential Terms and Sequences </h3>
475 <div class="outline-text-3" id="text-2-2">
476
477
478
479
480 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils)
481 (<span style="color: #8cd0d3;">use</span> 'clojure.contrib.seq
482 'clojure.contrib.generic.arithmetic
483 'clojure.contrib.generic.collection
484 'clojure.contrib.generic.math-functions)
485
486 <span style="color: #708070;">;;</span><span style="color: #7f9f7f;">&#8706;</span>
487
488 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">DEFINITION : Differential Term</span>
489
490 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">A quantity with infinitesimal components, e.g. x, dxdy, 4dydz. The</span>
491 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">coefficient of the quantity is returned by the 'coefficient' method,</span>
492 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">while the sequence of differential parameters is returned by the</span>
493 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">method 'partials'.</span>
494
495 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">Instead of using (potentially ambiguous) letters to denote</span>
496 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">differential parameters (dx,dy,dz), we use integers. So, dxdz becomes [0 2].</span>
497
498 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">The coefficient can be any arithmetic object; the</span>
499 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">partials must be a nonrepeating sorted sequence of nonnegative</span>
500 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">integers.</span>
501
502 (<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">DifferentialTerm</span> [coefficient partials])
503
504 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">differential-term</span>
505 <span style="color: #8fb28f;">"Make a differential term given pairs of coefficients and partials."</span>
506 [coefficient partials]
507 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #f0dfaf; font-weight: bold;">and</span> (<span style="color: #8cd0d3;">coll?</span> partials) (<span style="color: #8cd0d3;">every?</span> #(<span style="color: #f0dfaf; font-weight: bold;">and</span> (<span style="color: #8cd0d3;">integer?</span> %) (<span style="color: #8cd0d3;">not</span>(<span style="color: #8cd0d3;">neg?</span> %))) partials))
508 (DifferentialTerm. coefficient (<span style="color: #8cd0d3;">set</span> partials))
509 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (java.lang.IllegalArgumentException. <span style="color: #cc9393;">"Partials must be a collection of integers."</span>))))
510
511
512 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">DEFINITION : Differential Sequence</span>
513 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">A differential sequence is a sequence of differential terms, all with different partials.</span>
514 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">Internally, it is a map from the partials of each term to their coefficients.</span>
515
516 (<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">DifferentialSeq</span>
517 [terms]
518 <span style="color: #708070;">;;</span><span style="color: #7f9f7f;">clojure.lang.IPersistentMap</span>
519 clojure.lang.Associative
520 (<span style="color: #8cd0d3;">assoc</span> [this key val]
521 (DifferentialSeq.
522 (<span style="color: #8cd0d3;">cons</span> (differential-term val key) terms)))
523 (<span style="color: #8cd0d3;">cons</span> [this x]
524 (DifferentialSeq. (<span style="color: #8cd0d3;">cons</span> x terms)))
525 (containsKey [this key]
526 (<span style="color: #8cd0d3;">not</span>(<span style="color: #8cd0d3;">nil?</span> (find-first #(<span style="color: #8cd0d3;">=</span> (.partials %) key) terms))))
527 (<span style="color: #8cd0d3;">count</span> [this] (<span style="color: #8cd0d3;">count</span> (.terms this)))
528 (<span style="color: #8cd0d3;">empty</span> [this] (DifferentialSeq. []))
529 (entryAt [this key]
530 ((<span style="color: #8cd0d3;">juxt</span> #(.partials %) #(.coefficient %))
531 (find-first #(<span style="color: #8cd0d3;">=</span> (.partials %) key) terms)))
532 (<span style="color: #8cd0d3;">seq</span> [this] (<span style="color: #8cd0d3;">seq</span> (.terms this))))
533
534
535 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">fmap</span> DifferentialSeq
536 [f dseq]
537 (DifferentialSeq. (fmap f (.terms dseq))))
538
539
540 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">BUILDING DIFFERENTIAL OBJECTS</span>
541
542 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">differential-seq</span>
543 <span style="color: #8fb28f;">"Define a differential sequence by specifying coefficient/partials</span>
544 <span style="color: #8fb28f;">pairs, which are used to create differential terms to populate the</span>
545 <span style="color: #8fb28f;">sequence."</span>
546 ([coefficient partials]
547 (DifferentialSeq. {(<span style="color: #8cd0d3;">set</span> partials) coefficient}))
548 ([coefficient partials &amp; cps]
549 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">odd?</span> (<span style="color: #8cd0d3;">count</span> cps))
550 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"differential-seq requires an even number of terms."</span>))
551 (DifferentialSeq.
552 (<span style="color: #8cd0d3;">reduce</span>
553 #(<span style="color: #8cd0d3;">assoc</span> %1 (<span style="color: #8cd0d3;">set</span> (<span style="color: #8cd0d3;">second</span> %2)) (<span style="color: #8cd0d3;">first</span> %2))
554 {(<span style="color: #8cd0d3;">set</span> partials) coefficient}
555 (<span style="color: #8cd0d3;">partition</span> 2 cps))))))
556
557
558
559 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">big-part</span>
560 <span style="color: #8fb28f;">"Returns the part of the differential sequence that is finite,</span>
561 <span style="color: #8fb28f;"> i.e. not infinitely small, as a differential sequence. If given a</span>
562 <span style="color: #8fb28f;"> non-differential object, returns that object."</span>
563 [dseq]
564 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not=</span> (<span style="color: #8cd0d3;">type</span> dseq) DifferentialSeq) dseq
565 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [m (.terms dseq)
566 keys (<span style="color: #8cd0d3;">sort-by</span> count (<span style="color: #8cd0d3;">keys</span> m))
567 smallest-var (<span style="color: #8cd0d3;">last</span> (<span style="color: #8cd0d3;">last</span> keys))]
568 (DifferentialSeq.
569 (<span style="color: #8cd0d3;">reduce</span>
570 #(<span style="color: #8cd0d3;">assoc</span> %1 (<span style="color: #8cd0d3;">first</span> %2) (<span style="color: #8cd0d3;">second</span> %2))
571 {}
572 (<span style="color: #8cd0d3;">remove</span> #((<span style="color: #8cd0d3;">first</span> %) smallest-var) m))))))
573
574
575 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">small-part</span>
576 <span style="color: #8fb28f;">"Returns the part of the differential sequence that infinitely</span>
577 <span style="color: #8fb28f;"> small, as a differential sequence. If given a non-differential</span>
578 <span style="color: #8fb28f;"> object, returns that object."</span>
579 [dseq]
580 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not=</span> (<span style="color: #8cd0d3;">type</span> dseq) DifferentialSeq) 0
581 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [m (.terms dseq)
582 keys (<span style="color: #8cd0d3;">sort-by</span> count (<span style="color: #8cd0d3;">keys</span> m))
583 smallest-var (<span style="color: #8cd0d3;">last</span> (<span style="color: #8cd0d3;">last</span> keys))]
584 (DifferentialSeq.
585 (<span style="color: #8cd0d3;">reduce</span>
586 #(<span style="color: #8cd0d3;">assoc</span> %1 (<span style="color: #8cd0d3;">first</span> %2) (<span style="color: #8cd0d3;">second</span> %2)) {}
587 (<span style="color: #8cd0d3;">filter</span> #((<span style="color: #8cd0d3;">first</span> %) smallest-var) m))))))
588
589
590
591 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">cartesian-product</span> [set1 set2]
592 (<span style="color: #8cd0d3;">reduce</span> concat
593 (<span style="color: #f0dfaf; font-weight: bold;">for</span> [x set1]
594 (<span style="color: #f0dfaf; font-weight: bold;">for</span> [y set2]
595 [x y]))))
596
597 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">differential-multiply</span>
598 <span style="color: #8fb28f;">"Multiply two differential sequences. The square of any differential</span>
599 <span style="color: #8fb28f;"> variable is zero since differential variables are infinitesimally</span>
600 <span style="color: #8fb28f;"> small."</span>
601 [dseq1 dseq2]
602 (DifferentialSeq.
603 (<span style="color: #8cd0d3;">reduce</span>
604 (<span style="color: #8cd0d3;">fn</span> [m [[vars1 coeff1] [vars2 coeff2]]]
605 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not</span> (<span style="color: #8cd0d3;">empty?</span> (clojure.set/<span style="color: #dfdfbf; font-weight: bold;">intersection</span> vars1 vars2)))
606 m
607 (<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))))
608 {}
609 (cartesian-product (.terms dseq1) (.terms dseq2)))))
610
611
612
613 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [DifferentialSeq DifferentialSeq]
614 [dseq1 dseq2]
615 (differential-multiply dseq1 dseq2))
616
617 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">+</span> [DifferentialSeq DifferentialSeq]
618 [dseq1 dseq2]
619 (DifferentialSeq.
620 (<span style="color: #8cd0d3;">merge-with</span> + (.terms dseq1) (.terms dseq2))))
621
622 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [java.lang.Number DifferentialSeq]
623 [x dseq]
624 (fmap (<span style="color: #8cd0d3;">partial</span> * x) dseq))
625
626 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">*</span> [DifferentialSeq java.lang.Number]
627 [dseq x]
628 (fmap (<span style="color: #8cd0d3;">partial</span> * x) dseq))
629 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">DERIVATIVES</span>
630
631
632
633 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">linear-approximator</span>
634 <span style="color: #8fb28f;">"Returns an operator that linearly approximates the given function."</span>
635 ([f df|dx]
636 (<span style="color: #8cd0d3;">fn</span> [x]
637 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [big-part (big-part x)
638 small-part (small-part x)]
639 (<span style="color: #8cd0d3;">+</span> (fmap f big-part)
640 (<span style="color: #8cd0d3;">*</span> (fmap df|dx big-part) small-part)))))
641 ([f df|dx df|dy]
642 (<span style="color: #8cd0d3;">fn</span> [x y]
643 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [X (big-part x)
644 Y (big-part y)
645 DX (small-part x)
646 DY (small-part y)]
647 (<span style="color: #8cd0d3;">+</span> (f X Y)
648 (<span style="color: #8cd0d3;">+</span> (<span style="color: #8cd0d3;">*</span> DX (df|dx X Y))
649 (<span style="color: #8cd0d3;">*</span> DY (df|dy X Y)))))))
650 )
651
652
653
654
655
656 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">D</span>[f]
657 (<span style="color: #8cd0d3;">fn</span>[x]
658 (f (differential-seq x [] 1 [0]))))
659
660 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">d</span>[f partials]
661 (<span style="color: #8cd0d3;">fn</span> [x]
662 (<span style="color: #8cd0d3;">get</span>
663 (.terms ((D f)x))
664 (<span style="color: #8cd0d3;">set</span> partials)
665 0
666 )))
667
668 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">exp</span> DifferentialSeq [x]
669 ((linear-approximator exp exp) x))
670
671 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">sin</span> DifferentialSeq
672 [x]
673 ((linear-approximator sin cos) x))
674
675 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">cos</span> DifferentialSeq
676 [x]
677 ((linear-approximator cos #(<span style="color: #8cd0d3;">-</span> (sin %))) x))
678
679
680
681 </pre>
682
683
684
685
686
687
688
689
690
691 </div>
692 </div>
693 </div>
694 <div id="postamble">
695 <p class="date">Date: 2011-08-11 04:10:36 EDT</p>
696 <p class="author">Author: Dylan Holmes</p>
697 <p class="creator">Org version 7.6 with Emacs version 23</p>
698 <a href="http://validator.w3.org/check?uri=referer">Validate XHTML 1.0</a>
699 </div>
700 </div>
701 </body>
702 </html>