Mercurial > dylan
comparison sicm/bk/utils.html @ 2:b4de894a1e2e
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| author | Robert McIntyre <rlm@mit.edu> |
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| date | Fri, 28 Oct 2011 00:03:05 -0700 |
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| 143 | |
| 144 <h1 class="title">Building a Classical Mechanics Library in Clojure</h1> | |
| 145 | |
| 146 | |
| 147 | |
| 148 | |
| 149 | |
| 150 | |
| 151 | |
| 152 <div id="table-of-contents"> | |
| 153 <h2>Table of Contents</h2> | |
| 154 <div id="text-table-of-contents"> | |
| 155 <ul> | |
| 156 <li><a href="#sec-1">1 Generic Arithmetic </a></li> | |
| 157 <li><a href="#sec-2">2 Useful Data Types </a> | |
| 158 <ul> | |
| 159 <li><a href="#sec-2-1">2.1 Complex Numbers </a></li> | |
| 160 <li><a href="#sec-2-2">2.2 Tuples and Tensors </a> | |
| 161 <ul> | |
| 162 <li><a href="#sec-2-2-1">2.2.1 Contraction </a></li> | |
| 163 <li><a href="#sec-2-2-2">2.2.2 Matrices </a></li> | |
| 164 </ul> | |
| 165 </li> | |
| 166 <li><a href="#sec-2-3">2.3 Power Series </a></li> | |
| 167 </ul> | |
| 168 </li> | |
| 169 <li><a href="#sec-3">3 Basic Utilities </a> | |
| 170 <ul> | |
| 171 <li><a href="#sec-3-1">3.1 Sequence manipulation </a></li> | |
| 172 <li><a href="#sec-3-2">3.2 Ranges, Airity and Function Composition </a></li> | |
| 173 </ul> | |
| 174 </li> | |
| 175 <li><a href="#sec-4">4 Numerical Methods </a></li> | |
| 176 <li><a href="#sec-5">5 Differentiation </a></li> | |
| 177 <li><a href="#sec-6">6 Symbolic Manipulation </a></li> | |
| 178 </ul> | |
| 179 </div> | |
| 180 </div> | |
| 181 | |
| 182 <div id="outline-container-1" class="outline-2"> | |
| 183 <h2 id="sec-1"><span class="section-number-2">1</span> Generic Arithmetic </h2> | |
| 184 <div class="outline-text-2" id="text-1"> | |
| 185 | |
| 186 | |
| 187 | |
| 188 | |
| 189 | |
| 190 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">ns</span> sicm.utils) | |
| 191 (<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) | |
| 192 | |
| 193 | |
| 194 | |
| 195 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">all-equal?</span> [coll] | |
| 196 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">empty?</span> (<span style="color: #8cd0d3;">rest</span> coll)) true | |
| 197 (<span style="color: #f0dfaf; font-weight: bold;">and</span> (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">first</span> coll) (<span style="color: #8cd0d3;">second</span> coll)) | |
| 198 (<span style="color: #f0dfaf; font-weight: bold;">recur</span> (<span style="color: #8cd0d3;">rest</span> coll))))) | |
| 199 | |
| 200 | |
| 201 (<span style="color: #f0dfaf; font-weight: bold;">defprotocol</span> <span style="color: #f0dfaf;">Arithmetic</span> | |
| 202 (zero [this]) | |
| 203 (one [this]) | |
| 204 (negate [this]) | |
| 205 (invert [this]) | |
| 206 (conjugate [this])) | |
| 207 | |
| 208 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">zero?</span> [x] | |
| 209 (<span style="color: #8cd0d3;">=</span> x (zero x))) | |
| 210 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">one?</span> [x] | |
| 211 (<span style="color: #8cd0d3;">=</span> x (one x))) | |
| 212 | |
| 213 | |
| 214 | |
| 215 (<span style="color: #8cd0d3;">extend-protocol</span> Arithmetic | |
| 216 java.lang.Number | |
| 217 (zero [x] 0) | |
| 218 (one [x] 1) | |
| 219 (negate [x] (<span style="color: #8cd0d3;">-</span> x)) | |
| 220 (invert [x] (<span style="color: #8cd0d3;">/</span> x)) | |
| 221 ) | |
| 222 | |
| 223 (<span style="color: #8cd0d3;">extend-protocol</span> Arithmetic | |
| 224 clojure.lang.IFn | |
| 225 (one [f] identity) | |
| 226 (negate [f] (<span style="color: #8cd0d3;">comp</span> negate f)) | |
| 227 (invert [f] (<span style="color: #8cd0d3;">comp</span> invert f))) | |
| 228 | |
| 229 | |
| 230 (<span style="color: #8cd0d3;">extend-protocol</span> Arithmetic | |
| 231 clojure.lang.Seqable | |
| 232 (zero [this] (<span style="color: #8cd0d3;">map</span> zero this)) | |
| 233 (one [this] (<span style="color: #8cd0d3;">map</span> one this)) | |
| 234 (invert [this] (<span style="color: #8cd0d3;">map</span> invert this)) | |
| 235 (negate [this] (<span style="color: #8cd0d3;">map</span> negate this))) | |
| 236 | |
| 237 | |
| 238 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">ordered-like</span> | |
| 239 <span style="color: #8fb28f;">"Create a comparator using the sorted collection as an</span> | |
| 240 <span style="color: #8fb28f;"> example. Elements not in the sorted collection are sorted to the</span> | |
| 241 <span style="color: #8fb28f;"> end."</span> | |
| 242 [sorted-coll] | |
| 243 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [ | |
| 244 sorted-coll? (<span style="color: #8cd0d3;">set</span> sorted-coll) | |
| 245 ascending-pair? | |
| 246 (<span style="color: #8cd0d3;">set</span>(<span style="color: #8cd0d3;">reduce</span> concat | |
| 247 (map-indexed | |
| 248 (<span style="color: #8cd0d3;">fn</span> [n x] | |
| 249 (<span style="color: #8cd0d3;">map</span> #(<span style="color: #8cd0d3;">vector</span> x %) (<span style="color: #8cd0d3;">nthnext</span> sorted-coll n))) | |
| 250 sorted-coll)))] | |
| 251 (<span style="color: #8cd0d3;">fn</span> [x y] | |
| 252 (<span style="color: #f0dfaf; font-weight: bold;">cond</span> | |
| 253 (<span style="color: #8cd0d3;">=</span> x y) 0 | |
| 254 (ascending-pair? [x y]) -1 | |
| 255 (ascending-pair? [y x]) 1 | |
| 256 (sorted-coll? x) -1 | |
| 257 (sorted-coll? y) 1)))) | |
| 258 | |
| 259 | |
| 260 | |
| 261 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">type-precedence</span> | |
| 262 (ordered-like [incanter.Matrix])) | |
| 263 | |
| 264 (<span style="color: #f0dfaf; font-weight: bold;">defmulti</span> <span style="color: #f0dfaf;">add</span> | |
| 265 (<span style="color: #8cd0d3;">fn</span> [x y] | |
| 266 (<span style="color: #8cd0d3;">sort</span> type-precedence [(<span style="color: #8cd0d3;">type</span> x)(<span style="color: #8cd0d3;">type</span> y)]))) | |
| 267 | |
| 268 (<span style="color: #f0dfaf; font-weight: bold;">defmulti</span> <span style="color: #f0dfaf;">multiply</span> | |
| 269 (<span style="color: #8cd0d3;">fn</span> [x y] | |
| 270 (<span style="color: #8cd0d3;">sort</span> type-precedence [(<span style="color: #8cd0d3;">type</span> x) (<span style="color: #8cd0d3;">type</span> y)]))) | |
| 271 | |
| 272 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">add</span> [java.lang.Number java.lang.Number] [x y] (<span style="color: #8cd0d3;">+</span> x y)) | |
| 273 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">multiply</span> [java.lang.Number java.lang.Number] [x y] (<span style="color: #8cd0d3;">*</span> x y)) | |
| 274 | |
| 275 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">multiply</span> [incanter.Matrix java.lang.Integer] [x y] | |
| 276 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [args (<span style="color: #8cd0d3;">sort</span> #(type-precedence (<span style="color: #8cd0d3;">type</span> %1)(<span style="color: #8cd0d3;">type</span> %2)) [x y]) | |
| 277 matrix (<span style="color: #8cd0d3;">first</span> args) | |
| 278 scalar (<span style="color: #8cd0d3;">second</span> args)] | |
| 279 (incanter.core/matrix (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">partial</span> map (<span style="color: #8cd0d3;">partial</span> multiply scalar)) matrix)))) | |
| 280 | |
| 281 </pre> | |
| 282 | |
| 283 | |
| 284 | |
| 285 | |
| 286 | |
| 287 </div> | |
| 288 | |
| 289 </div> | |
| 290 | |
| 291 <div id="outline-container-2" class="outline-2"> | |
| 292 <h2 id="sec-2"><span class="section-number-2">2</span> Useful Data Types </h2> | |
| 293 <div class="outline-text-2" id="text-2"> | |
| 294 | |
| 295 | |
| 296 | |
| 297 </div> | |
| 298 | |
| 299 <div id="outline-container-2-1" class="outline-3"> | |
| 300 <h3 id="sec-2-1"><span class="section-number-3">2.1</span> Complex Numbers </h3> | |
| 301 <div class="outline-text-3" id="text-2-1"> | |
| 302 | |
| 303 | |
| 304 | |
| 305 | |
| 306 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) | |
| 307 | |
| 308 (<span style="color: #f0dfaf; font-weight: bold;">defprotocol</span> <span style="color: #f0dfaf;">Complex</span> | |
| 309 (real-part [z]) | |
| 310 (imaginary-part [z]) | |
| 311 (magnitude-squared [z]) | |
| 312 (angle [z]) | |
| 313 (conjugate [z]) | |
| 314 (norm [z])) | |
| 315 | |
| 316 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">complex-rectangular</span> | |
| 317 <span style="color: #8fb28f;">"Define a complex number with the given real and imaginary</span> | |
| 318 <span style="color: #8fb28f;"> components."</span> | |
| 319 [re im] | |
| 320 (<span style="color: #8cd0d3;">reify</span> Complex | |
| 321 (real-part [z] re) | |
| 322 (imaginary-part [z] im) | |
| 323 (magnitude-squared [z] (<span style="color: #8cd0d3;">+</span> (<span style="color: #8cd0d3;">*</span> re re) (<span style="color: #8cd0d3;">*</span> im im))) | |
| 324 (angle [z] (java.lang.Math/atan2 im re)) | |
| 325 (conjugate [z] (complex-rectangular re (<span style="color: #8cd0d3;">-</span> im))) | |
| 326 | |
| 327 Arithmetic | |
| 328 (zero [z] (complex-rectangular 0 0)) | |
| 329 (one [z] (complex-rectangular 1 0)) | |
| 330 (negate [z] (complex-rectangular (<span style="color: #8cd0d3;">-</span> re) (<span style="color: #8cd0d3;">-</span> im))) | |
| 331 (invert [z] (complex-rectangular | |
| 332 (<span style="color: #8cd0d3;">/</span> re (magnitude-squared z)) | |
| 333 (<span style="color: #8cd0d3;">/</span> (<span style="color: #8cd0d3;">-</span> im) (magnitude-squared z)))) | |
| 334 | |
| 335 Object | |
| 336 (toString [_] | |
| 337 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #f0dfaf; font-weight: bold;">and</span> (zero? re) (zero? im)) (<span style="color: #8cd0d3;">str</span> 0) | |
| 338 (<span style="color: #8cd0d3;">str</span> | |
| 339 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not</span>(zero? re)) | |
| 340 re) | |
| 341 (<span style="color: #f0dfaf; font-weight: bold;">if</span> ((<span style="color: #8cd0d3;">comp</span> not zero?) im) | |
| 342 (<span style="color: #8cd0d3;">str</span> | |
| 343 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">neg?</span> im) <span style="color: #cc9393;">"-"</span> <span style="color: #cc9393;">"+"</span>) | |
| 344 (<span style="color: #f0dfaf; font-weight: bold;">if</span> ((<span style="color: #8cd0d3;">comp</span> not one?) (java.lang.Math/abs im)) | |
| 345 (java.lang.Math/abs im)) | |
| 346 <span style="color: #cc9393;">"i"</span>))))))) | |
| 347 | |
| 348 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">complex-polar</span> | |
| 349 <span style="color: #8fb28f;">"Define a complex number with the given magnitude and angle."</span> | |
| 350 [mag ang] | |
| 351 (<span style="color: #8cd0d3;">reify</span> Complex | |
| 352 (magnitude-squared [z] (<span style="color: #8cd0d3;">*</span> mag mag)) | |
| 353 (angle [z] angle) | |
| 354 (real-part [z] (<span style="color: #8cd0d3;">*</span> mag (java.lang.Math/cos ang))) | |
| 355 (imaginary-part [z] (<span style="color: #8cd0d3;">*</span> mag (java.lang.Math/sin ang))) | |
| 356 (conjugate [z] (complex-polar mag (<span style="color: #8cd0d3;">-</span> ang))) | |
| 357 | |
| 358 Arithmetic | |
| 359 (zero [z] (complex-polar 0 0)) | |
| 360 (one [z] (complex-polar 1 0)) | |
| 361 (negate [z] (complex-polar (<span style="color: #8cd0d3;">-</span> mag) ang)) | |
| 362 (invert [z] (complex-polar (<span style="color: #8cd0d3;">/</span> mag) (<span style="color: #8cd0d3;">-</span> ang))) | |
| 363 | |
| 364 Object | |
| 365 (toString [_] (<span style="color: #8cd0d3;">str</span> mag <span style="color: #cc9393;">" * e^(i"</span> ang<span style="color: #cc9393;">")"</span>)) | |
| 366 )) | |
| 367 | |
| 368 | |
| 369 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">Numbers are complex quantities</span> | |
| 370 | |
| 371 (<span style="color: #8cd0d3;">extend-protocol</span> Complex | |
| 372 java.lang.Number | |
| 373 (real-part [x] x) | |
| 374 (imaginary-part [x] 0) | |
| 375 (magnitude [x] x) | |
| 376 (angle [x] 0) | |
| 377 (conjugate [x] x)) | |
| 378 | |
| 379 | |
| 380 </pre> | |
| 381 | |
| 382 | |
| 383 | |
| 384 | |
| 385 | |
| 386 | |
| 387 </div> | |
| 388 | |
| 389 </div> | |
| 390 | |
| 391 <div id="outline-container-2-2" class="outline-3"> | |
| 392 <h3 id="sec-2-2"><span class="section-number-3">2.2</span> Tuples and Tensors </h3> | |
| 393 <div class="outline-text-3" id="text-2-2"> | |
| 394 | |
| 395 | |
| 396 <p> | |
| 397 A tuple is a vector which is spinable—it can be either <i>spin up</i> or <i>spin down</i>. (Covariant, contravariant; dual vectors) | |
| 398 </p> | |
| 399 | |
| 400 | |
| 401 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) | |
| 402 | |
| 403 (<span style="color: #f0dfaf; font-weight: bold;">defprotocol</span> <span style="color: #f0dfaf;">Spinning</span> | |
| 404 (up? [this]) | |
| 405 (down? [this])) | |
| 406 | |
| 407 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">spin</span> | |
| 408 <span style="color: #8fb28f;">"Returns the spin of the Spinning s, either :up or :down"</span> | |
| 409 [<span style="color: #dfdfbf; font-weight: bold;">#^Spinning</span> s] | |
| 410 (<span style="color: #f0dfaf; font-weight: bold;">cond</span> (up? s) <span style="color: #8cd0d3;">:up</span> (down? s) <span style="color: #8cd0d3;">:down</span>)) | |
| 411 | |
| 412 | |
| 413 (<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">Tuple</span> | |
| 414 [spin coll] | |
| 415 clojure.lang.Seqable | |
| 416 (<span style="color: #8cd0d3;">seq</span> [this] (<span style="color: #8cd0d3;">seq</span> (.coll this))) | |
| 417 clojure.lang.Counted | |
| 418 (<span style="color: #8cd0d3;">count</span> [this] (<span style="color: #8cd0d3;">count</span> (.coll this)))) | |
| 419 | |
| 420 (<span style="color: #8cd0d3;">extend-type</span> Tuple | |
| 421 Spinning | |
| 422 (up? [this] (<span style="color: #8cd0d3;">=</span> <span style="color: #8cd0d3;">::up</span> (.spin this))) | |
| 423 (down? [this] (<span style="color: #8cd0d3;">=</span> <span style="color: #8cd0d3;">::down</span> (.spin this)))) | |
| 424 | |
| 425 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">print-method</span> Tuple | |
| 426 [o w] | |
| 427 (<span style="color: #8cd0d3;">print-simple</span> (<span style="color: #8cd0d3;">str</span> (<span style="color: #f0dfaf; font-weight: bold;">if</span> (up? o) 'u 'd) (.coll o)) w)) | |
| 428 | |
| 429 | |
| 430 | |
| 431 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">up</span> | |
| 432 <span style="color: #8fb28f;">"Create a new up-tuple containing the contents of coll."</span> | |
| 433 [coll] | |
| 434 (Tuple. <span style="color: #8cd0d3;">::up</span> coll)) | |
| 435 | |
| 436 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">down</span> | |
| 437 <span style="color: #8fb28f;">"Create a new down-tuple containing the contents of coll."</span> | |
| 438 [coll] | |
| 439 (Tuple. <span style="color: #8cd0d3;">::down</span> coll)) | |
| 440 | |
| 441 | |
| 442 </pre> | |
| 443 | |
| 444 | |
| 445 | |
| 446 | |
| 447 | |
| 448 </div> | |
| 449 | |
| 450 <div id="outline-container-2-2-1" class="outline-4"> | |
| 451 <h4 id="sec-2-2-1"><span class="section-number-4">2.2.1</span> Contraction </h4> | |
| 452 <div class="outline-text-4" id="text-2-2-1"> | |
| 453 | |
| 454 <p>Contraction is a binary operation that you can apply to compatible | |
| 455 tuples. Tuples are compatible for contraction if they have the same | |
| 456 length and opposite spins, and if the corresponding items in each | |
| 457 tuple are both numbers or both compatible tuples. | |
| 458 </p> | |
| 459 | |
| 460 | |
| 461 | |
| 462 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) | |
| 463 | |
| 464 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">numbers?</span> | |
| 465 <span style="color: #8fb28f;">"Returns true if all arguments are numbers, else false."</span> | |
| 466 [& xs] | |
| 467 (<span style="color: #8cd0d3;">every?</span> number? xs)) | |
| 468 | |
| 469 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">contractible?</span> | |
| 470 <span style="color: #8fb28f;">"Returns true if the tuples a and b are compatible for contraction,</span> | |
| 471 <span style="color: #8fb28f;"> else false. Tuples are compatible if they have the same number of</span> | |
| 472 <span style="color: #8fb28f;"> components, they have opposite spins, and their elements are</span> | |
| 473 <span style="color: #8fb28f;"> pairwise-compatible."</span> | |
| 474 [a b] | |
| 475 (<span style="color: #f0dfaf; font-weight: bold;">and</span> | |
| 476 (<span style="color: #8cd0d3;">isa?</span> (<span style="color: #8cd0d3;">type</span> a) Tuple) | |
| 477 (<span style="color: #8cd0d3;">isa?</span> (<span style="color: #8cd0d3;">type</span> b) Tuple) | |
| 478 (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">count</span> a) (<span style="color: #8cd0d3;">count</span> b)) | |
| 479 (<span style="color: #8cd0d3;">not=</span> (spin a) (spin b)) | |
| 480 | |
| 481 (<span style="color: #8cd0d3;">not-any?</span> false? | |
| 482 (<span style="color: #8cd0d3;">map</span> #(<span style="color: #f0dfaf; font-weight: bold;">or</span> | |
| 483 (numbers? %1 %2) | |
| 484 (contractible? %1 %2)) | |
| 485 a b)))) | |
| 486 | |
| 487 | |
| 488 | |
| 489 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">contract</span> | |
| 490 <span style="color: #8fb28f;">"Contracts two tuples, returning the sum of the</span> | |
| 491 <span style="color: #8fb28f;"> products of the corresponding items. Contraction is recursive on</span> | |
| 492 <span style="color: #8fb28f;"> nested tuples."</span> | |
| 493 [a b] | |
| 494 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not</span> (contractible? a b)) | |
| 495 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> | |
| 496 (Exception. <span style="color: #cc9393;">"Not compatible for contraction."</span>)) | |
| 497 (<span style="color: #8cd0d3;">reduce</span> + | |
| 498 (<span style="color: #8cd0d3;">map</span> | |
| 499 (<span style="color: #8cd0d3;">fn</span> [x y] | |
| 500 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (numbers? x y) | |
| 501 (<span style="color: #8cd0d3;">*</span> x y) | |
| 502 (contract x y))) | |
| 503 a b)))) | |
| 504 | |
| 505 </pre> | |
| 506 | |
| 507 | |
| 508 | |
| 509 | |
| 510 </div> | |
| 511 | |
| 512 </div> | |
| 513 | |
| 514 <div id="outline-container-2-2-2" class="outline-4"> | |
| 515 <h4 id="sec-2-2-2"><span class="section-number-4">2.2.2</span> Matrices </h4> | |
| 516 <div class="outline-text-4" id="text-2-2-2"> | |
| 517 | |
| 518 | |
| 519 | |
| 520 | |
| 521 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) | |
| 522 (<span style="color: #8cd0d3;">require</span> 'incanter.core) <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">use incanter's fast matrices</span> | |
| 523 | |
| 524 (<span style="color: #f0dfaf; font-weight: bold;">defprotocol</span> <span style="color: #f0dfaf;">Matrix</span> | |
| 525 (rows [matrix]) | |
| 526 (cols [matrix]) | |
| 527 (diagonal [matrix]) | |
| 528 (trace [matrix]) | |
| 529 (determinant [matrix]) | |
| 530 (transpose [matrix]) | |
| 531 (conjugate [matrix]) | |
| 532 ) | |
| 533 | |
| 534 (<span style="color: #8cd0d3;">extend-protocol</span> Matrix | |
| 535 incanter.Matrix | |
| 536 (rows [rs] (<span style="color: #8cd0d3;">map</span> down (<span style="color: #8cd0d3;">apply</span> map vector (<span style="color: #8cd0d3;">apply</span> map vector rs)))) | |
| 537 (cols [rs] (<span style="color: #8cd0d3;">map</span> up (<span style="color: #8cd0d3;">apply</span> map vector rs))) | |
| 538 (diagonal [matrix] (incanter.core/diag matrix) ) | |
| 539 (determinant [matrix] (incanter.core/det matrix)) | |
| 540 (trace [matrix] (incanter.core/trace matrix)) | |
| 541 (transpose [matrix] (incanter.core/trans matrix)) | |
| 542 ) | |
| 543 | |
| 544 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">count-rows</span> [matrix] | |
| 545 ((<span style="color: #8cd0d3;">comp</span> count rows) matrix)) | |
| 546 | |
| 547 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">count-cols</span> [matrix] | |
| 548 ((<span style="color: #8cd0d3;">comp</span> count cols) matrix)) | |
| 549 | |
| 550 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">square?</span> [matrix] | |
| 551 (<span style="color: #8cd0d3;">=</span> (count-rows matrix) (count-cols matrix))) | |
| 552 | |
| 553 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">identity-matrix</span> | |
| 554 <span style="color: #8fb28f;">"Define a square matrix of size n-by-n with 1s along the diagonal and</span> | |
| 555 <span style="color: #8fb28f;"> 0s everywhere else."</span> | |
| 556 [n] | |
| 557 (incanter.core/identity-matrix n)) | |
| 558 | |
| 559 | |
| 560 | |
| 561 | |
| 562 | |
| 563 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">matrix-by-rows</span> | |
| 564 <span style="color: #8fb28f;">"Define a matrix by giving its rows."</span> | |
| 565 [& rows] | |
| 566 (<span style="color: #f0dfaf; font-weight: bold;">if</span> | |
| 567 (<span style="color: #8cd0d3;">not</span> (all-equal? (<span style="color: #8cd0d3;">map</span> count rows))) | |
| 568 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"All rows in a matrix must have the same number of elements."</span>)) | |
| 569 (incanter.core/matrix (<span style="color: #8cd0d3;">vec</span> rows)))) | |
| 570 | |
| 571 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">matrix-by-cols</span> | |
| 572 <span style="color: #8fb28f;">"Define a matrix by giving its columns"</span> | |
| 573 [& cols] | |
| 574 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not</span> (all-equal? (<span style="color: #8cd0d3;">map</span> count cols))) | |
| 575 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"All columns in a matrix must have the same number of elements."</span>)) | |
| 576 (incanter.core/matrix (<span style="color: #8cd0d3;">vec</span> (<span style="color: #8cd0d3;">apply</span> map vector cols))))) | |
| 577 | |
| 578 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">identity-matrix</span> | |
| 579 <span style="color: #8fb28f;">"Define a square matrix of size n-by-n with 1s along the diagonal and</span> | |
| 580 <span style="color: #8fb28f;"> 0s everywhere else."</span> | |
| 581 [n] | |
| 582 (incanter.core/identity-matrix n)) | |
| 583 | |
| 584 | |
| 585 | |
| 586 (<span style="color: #8cd0d3;">extend-protocol</span> Arithmetic | |
| 587 incanter.Matrix | |
| 588 (one [matrix] | |
| 589 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (square? matrix) | |
| 590 (identity-matrix (count-rows matrix)) | |
| 591 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"Non-square matrices have no multiplicative unit."</span>)))) | |
| 592 (zero [matrix] | |
| 593 (<span style="color: #8cd0d3;">apply</span> matrix-by-rows (<span style="color: #8cd0d3;">map</span> zero (rows matrix)))) | |
| 594 (negate [matrix] | |
| 595 (<span style="color: #8cd0d3;">apply</span> matrix-by-rows (<span style="color: #8cd0d3;">map</span> negate (rows matrix)))) | |
| 596 (invert [matrix] | |
| 597 (incanter.core/solve matrix))) | |
| 598 | |
| 599 | |
| 600 | |
| 601 (<span style="color: #f0dfaf; font-weight: bold;">defmulti</span> <span style="color: #f0dfaf;">coerce-to-matrix</span> | |
| 602 <span style="color: #8fb28f;">"Converts x into a matrix, if possible."</span> | |
| 603 type) | |
| 604 | |
| 605 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">coerce-to-matrix</span> incanter.Matrix [x] x) | |
| 606 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">coerce-to-matrix</span> Tuple [x] | |
| 607 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">apply</span> numbers? (<span style="color: #8cd0d3;">seq</span> x)) | |
| 608 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (up? x) | |
| 609 (matrix-by-cols (<span style="color: #8cd0d3;">seq</span> x)) | |
| 610 (matrix-by-rows (<span style="color: #8cd0d3;">seq</span> x))) | |
| 611 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"Non-numerical tuple cannot be converted into a matrix."</span>)))) | |
| 612 | |
| 613 | |
| 614 | |
| 615 | |
| 616 | |
| 617 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(defn matrix-by-cols</span> | |
| 618 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">"Define a matrix by giving its columns."</span> | |
| 619 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">[& cols]</span> | |
| 620 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(cond</span> | |
| 621 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(not (all-equal? (map count cols)))</span> | |
| 622 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(throw (Exception. "All columns in a matrix must have the same number of elements."))</span> | |
| 623 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">:else</span> | |
| 624 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(reify Matrix</span> | |
| 625 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(cols [this] (map up cols))</span> | |
| 626 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(rows [this] (map down (apply map vector cols)))</span> | |
| 627 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(diagonal [this] (map-indexed (fn [i col] (nth col i) cols)))</span> | |
| 628 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(trace [this]</span> | |
| 629 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(if (not= (count-cols this) (count-rows this))</span> | |
| 630 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(throw (Exception.</span> | |
| 631 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">"Cannot take the trace of a non-square matrix."))</span> | |
| 632 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(reduce + (diagonal this))))</span> | |
| 633 | |
| 634 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(determinant [this]</span> | |
| 635 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(if (not= (count-cols this) (count-rows this))</span> | |
| 636 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(throw (Exception.</span> | |
| 637 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">"Cannot take the determinant of a non-square matrix."))</span> | |
| 638 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(reduce * (map-indexed (fn [i col] (nth col i)) cols))))</span> | |
| 639 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">)))</span> | |
| 640 | |
| 641 (<span style="color: #8cd0d3;">extend-protocol</span> Matrix Tuple | |
| 642 (rows [this] (<span style="color: #f0dfaf; font-weight: bold;">if</span> (down? this) | |
| 643 (<span style="color: #8cd0d3;">list</span> this) | |
| 644 (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">comp</span> up vector) this))) | |
| 645 | |
| 646 (cols [this] (<span style="color: #f0dfaf; font-weight: bold;">if</span> (up? this) | |
| 647 (<span style="color: #8cd0d3;">list</span> this) | |
| 648 (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">comp</span> down vector) this)) | |
| 649 )) | |
| 650 | |
| 651 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">matrix-multiply</span> | |
| 652 <span style="color: #8fb28f;">"Returns the matrix resulting from the matrix multiplication of the given arguments."</span> | |
| 653 ([A] (coerce-to-matrix A)) | |
| 654 ([A B] (incanter.core/mmult (coerce-to-matrix A) (coerce-to-matrix B))) | |
| 655 ([M1 M2 & Ms] (<span style="color: #8cd0d3;">reduce</span> matrix-multiply (matrix-multiply M1 M2) Ms))) | |
| 656 | |
| 657 </pre> | |
| 658 | |
| 659 | |
| 660 | |
| 661 | |
| 662 | |
| 663 </div> | |
| 664 </div> | |
| 665 | |
| 666 </div> | |
| 667 | |
| 668 <div id="outline-container-2-3" class="outline-3"> | |
| 669 <h3 id="sec-2-3"><span class="section-number-3">2.3</span> Power Series </h3> | |
| 670 <div class="outline-text-3" id="text-2-3"> | |
| 671 | |
| 672 | |
| 673 | |
| 674 | |
| 675 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) | |
| 676 (<span style="color: #8cd0d3;">use</span> 'clojure.contrib.def) | |
| 677 | |
| 678 | |
| 679 | |
| 680 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">series-fn</span> | |
| 681 <span style="color: #8fb28f;">"The function corresponding to the given power series."</span> | |
| 682 [series] | |
| 683 (<span style="color: #8cd0d3;">fn</span> [x] | |
| 684 (<span style="color: #8cd0d3;">reduce</span> + | |
| 685 (map-indexed (<span style="color: #8cd0d3;">fn</span>[n x] (<span style="color: #8cd0d3;">*</span> (<span style="color: #8cd0d3;">float</span> (<span style="color: #8cd0d3;">nth</span> series n)) (<span style="color: #8cd0d3;">float</span>(java.lang.Math/pow (<span style="color: #8cd0d3;">float</span> x) n)) )) | |
| 686 (<span style="color: #8cd0d3;">range</span> 20))))) | |
| 687 | |
| 688 (<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">PowerSeries</span> | |
| 689 [coll] | |
| 690 clojure.lang.Seqable | |
| 691 (<span style="color: #8cd0d3;">seq</span> [this] (<span style="color: #8cd0d3;">seq</span> (.coll this))) | |
| 692 | |
| 693 clojure.lang.Indexed | |
| 694 (<span style="color: #8cd0d3;">nth</span> [this n] (<span style="color: #8cd0d3;">nth</span> (.coll this) n 0)) | |
| 695 (<span style="color: #8cd0d3;">nth</span> [this n not-found] (<span style="color: #8cd0d3;">nth</span> (.coll this) n not-found)) | |
| 696 | |
| 697 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">clojure.lang.IFn</span> | |
| 698 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(call [this] (throw(Exception.)))</span> | |
| 699 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(invoke [this & args] args</span> | |
| 700 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(let [f </span> | |
| 701 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">)</span> | |
| 702 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(run [this] (throw(Exception.)))</span> | |
| 703 ) | |
| 704 | |
| 705 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">power-series</span> | |
| 706 <span style="color: #8fb28f;">"Returns a power series with the items of the coll as its</span> | |
| 707 <span style="color: #8fb28f;"> coefficients. Trailing zeros are added to the end of coll."</span> | |
| 708 [coeffs] | |
| 709 (PowerSeries. coeffs)) | |
| 710 | |
| 711 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">power-series-indexed</span> | |
| 712 <span style="color: #8fb28f;">"Returns a power series consisting of the result of mapping f to the non-negative integers."</span> | |
| 713 [f] | |
| 714 (PowerSeries. (<span style="color: #8cd0d3;">map</span> f (<span style="color: #8cd0d3;">range</span>)))) | |
| 715 | |
| 716 | |
| 717 (<span style="color: #f0dfaf; font-weight: bold;">defn-memo</span> <span style="color: #f0dfaf;">nth-partial-sum</span> | |
| 718 ([series n] | |
| 719 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (zero? n) (<span style="color: #8cd0d3;">first</span> series) | |
| 720 (<span style="color: #8cd0d3;">+</span> (<span style="color: #8cd0d3;">nth</span> series n) | |
| 721 (nth-partial-sum series (<span style="color: #8cd0d3;">dec</span> n)))))) | |
| 722 | |
| 723 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">partial-sums</span> [series] | |
| 724 (<span style="color: #8cd0d3;">lazy-seq</span> (<span style="color: #8cd0d3;">map</span> nth-partial-sum (<span style="color: #8cd0d3;">range</span>)))) | |
| 725 | |
| 726 | |
| 727 | |
| 728 | |
| 729 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">cos-series</span> | |
| 730 (power-series-indexed | |
| 731 (<span style="color: #8cd0d3;">fn</span>[n] | |
| 732 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">odd?</span> n) 0 | |
| 733 (<span style="color: #8cd0d3;">/</span> | |
| 734 (<span style="color: #8cd0d3;">reduce</span> * | |
| 735 (<span style="color: #8cd0d3;">reduce</span> * (<span style="color: #8cd0d3;">repeat</span> (<span style="color: #8cd0d3;">/</span> n 2) -1)) | |
| 736 (<span style="color: #8cd0d3;">range</span> 1 (<span style="color: #8cd0d3;">inc</span> n))) | |
| 737 ))))) | |
| 738 | |
| 739 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">sin-series</span> | |
| 740 (power-series-indexed | |
| 741 (<span style="color: #8cd0d3;">fn</span>[n] | |
| 742 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">even?</span> n) 0 | |
| 743 (<span style="color: #8cd0d3;">/</span> | |
| 744 (<span style="color: #8cd0d3;">reduce</span> * | |
| 745 (<span style="color: #8cd0d3;">reduce</span> * (<span style="color: #8cd0d3;">repeat</span> (<span style="color: #8cd0d3;">/</span> (<span style="color: #8cd0d3;">dec</span> n) 2) -1)) | |
| 746 (<span style="color: #8cd0d3;">range</span> 1 (<span style="color: #8cd0d3;">inc</span> n))) | |
| 747 ))))) | |
| 748 | |
| 749 </pre> | |
| 750 | |
| 751 | |
| 752 | |
| 753 | |
| 754 | |
| 755 </div> | |
| 756 </div> | |
| 757 | |
| 758 </div> | |
| 759 | |
| 760 <div id="outline-container-3" class="outline-2"> | |
| 761 <h2 id="sec-3"><span class="section-number-2">3</span> Basic Utilities </h2> | |
| 762 <div class="outline-text-2" id="text-3"> | |
| 763 | |
| 764 | |
| 765 | |
| 766 </div> | |
| 767 | |
| 768 <div id="outline-container-3-1" class="outline-3"> | |
| 769 <h3 id="sec-3-1"><span class="section-number-3">3.1</span> Sequence manipulation </h3> | |
| 770 <div class="outline-text-3" id="text-3-1"> | |
| 771 | |
| 772 | |
| 773 | |
| 774 | |
| 775 | |
| 776 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">ns</span> sicm.utils) | |
| 777 | |
| 778 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">do-up</span> | |
| 779 <span style="color: #8fb28f;">"Apply f to each number from low to high, presumably for</span> | |
| 780 <span style="color: #8fb28f;"> side-effects."</span> | |
| 781 [f low high] | |
| 782 (<span style="color: #f0dfaf; font-weight: bold;">doseq</span> [i (<span style="color: #8cd0d3;">range</span> low high)] (f i))) | |
| 783 | |
| 784 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">do-down</span> | |
| 785 <span style="color: #8fb28f;">"Apply f to each number from high to low, presumably for</span> | |
| 786 <span style="color: #8fb28f;"> side-effects."</span> | |
| 787 [f high low] | |
| 788 (<span style="color: #f0dfaf; font-weight: bold;">doseq</span> [i (<span style="color: #8cd0d3;">range</span> high low -1)] (f i))) | |
| 789 | |
| 790 | |
| 791 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">all-equal?</span> [coll] | |
| 792 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">empty?</span> (<span style="color: #8cd0d3;">rest</span> coll)) true | |
| 793 (<span style="color: #f0dfaf; font-weight: bold;">and</span> (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">first</span> coll) (<span style="color: #8cd0d3;">second</span> coll)) | |
| 794 (<span style="color: #f0dfaf; font-weight: bold;">recur</span> (<span style="color: #8cd0d3;">rest</span> coll)))))) | |
| 795 | |
| 796 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">multiplier</span> | |
| 797 <span style="color: #8fb28f;">"Returns a function that 'multiplies' the members of a collection,</span> | |
| 798 <span style="color: #8fb28f;">returning unit if called on an empty collection."</span> | |
| 799 [multiply unit] | |
| 800 (<span style="color: #8cd0d3;">fn</span> [coll] ((<span style="color: #8cd0d3;">partial</span> reduce multiply unit) coll))) | |
| 801 | |
| 802 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">divider</span> | |
| 803 <span style="color: #8fb28f;">"Returns a function that 'divides' the first element of a collection</span> | |
| 804 <span style="color: #8fb28f;">by the 'product' of the rest of the collection."</span> | |
| 805 [divide multiply invert unit] | |
| 806 (<span style="color: #8cd0d3;">fn</span> [coll] | |
| 807 (<span style="color: #8cd0d3;">apply</span> | |
| 808 (<span style="color: #8cd0d3;">fn</span> | |
| 809 ([] unit) | |
| 810 ([x] (invert x)) | |
| 811 ([x y] (divide x y)) | |
| 812 ([x y & zs] (divide x (<span style="color: #8cd0d3;">reduce</span> multiply y zs)))) | |
| 813 coll))) | |
| 814 | |
| 815 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">left-circular-shift</span> | |
| 816 <span style="color: #8fb28f;">"Remove the first element of coll, adding it to the end of coll."</span> | |
| 817 [coll] | |
| 818 (<span style="color: #8cd0d3;">concat</span> (<span style="color: #8cd0d3;">rest</span> coll) (<span style="color: #8cd0d3;">take</span> 1 coll))) | |
| 819 | |
| 820 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">right-circular-shift</span> | |
| 821 <span style="color: #8fb28f;">"Remove the last element of coll, adding it to the front of coll."</span> | |
| 822 [coll] | |
| 823 (<span style="color: #8cd0d3;">cons</span> (<span style="color: #8cd0d3;">last</span> coll) (<span style="color: #8cd0d3;">butlast</span> coll))) | |
| 824 </pre> | |
| 825 | |
| 826 | |
| 827 | |
| 828 | |
| 829 | |
| 830 | |
| 831 | |
| 832 </div> | |
| 833 | |
| 834 </div> | |
| 835 | |
| 836 <div id="outline-container-3-2" class="outline-3"> | |
| 837 <h3 id="sec-3-2"><span class="section-number-3">3.2</span> Ranges, Airity and Function Composition </h3> | |
| 838 <div class="outline-text-3" id="text-3-2"> | |
| 839 | |
| 840 | |
| 841 | |
| 842 | |
| 843 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) | |
| 844 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">infinity</span> Double/POSITIVE_INFINITY) | |
| 845 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">infinite?</span> [x] (Double/isInfinite x)) | |
| 846 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">finite?</span> (<span style="color: #8cd0d3;">comp</span> not infinite?)) | |
| 847 | |
| 848 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">arity-min</span> | |
| 849 <span style="color: #8fb28f;">"Returns the smallest number of arguments f can take."</span> | |
| 850 [f] | |
| 851 (<span style="color: #8cd0d3;">apply</span> | |
| 852 min | |
| 853 (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">comp</span> alength #(.getParameterTypes %)) | |
| 854 (<span style="color: #8cd0d3;">filter</span> (<span style="color: #8cd0d3;">comp</span> (<span style="color: #8cd0d3;">partial</span> = <span style="color: #cc9393;">"invoke"</span>) #(.getName %)) | |
| 855 (.getDeclaredMethods (<span style="color: #8cd0d3;">class</span> f)))))) | |
| 856 | |
| 857 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">arity-max</span> | |
| 858 <span style="color: #8fb28f;">"Returns the largest number of arguments f can take, possibly</span> | |
| 859 <span style="color: #8fb28f;"> Infinity."</span> | |
| 860 [f] | |
| 861 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [methods (.getDeclaredMethods (<span style="color: #8cd0d3;">class</span> f))] | |
| 862 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">not-any?</span> (<span style="color: #8cd0d3;">partial</span> = <span style="color: #cc9393;">"doInvoke"</span>) (<span style="color: #8cd0d3;">map</span> #(.getName %) methods)) | |
| 863 (<span style="color: #8cd0d3;">apply</span> max | |
| 864 (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">comp</span> alength #(.getParameterTypes %)) | |
| 865 (<span style="color: #8cd0d3;">filter</span> (<span style="color: #8cd0d3;">comp</span> (<span style="color: #8cd0d3;">partial</span> = <span style="color: #cc9393;">"invoke"</span>) #(.getName %)) methods))) | |
| 866 infinity))) | |
| 867 | |
| 868 | |
| 869 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">^</span>{<span style="color: #8cd0d3;">:arglists</span> '([f]) | |
| 870 <span style="color: #8cd0d3;">:doc</span> <span style="color: #cc9393;">"Returns a two-element list containing the minimum and</span> | |
| 871 <span style="color: #cc9393;"> maximum number of args that f can take."</span>} | |
| 872 arity-interval | |
| 873 (<span style="color: #8cd0d3;">juxt</span> arity-min arity-max)) | |
| 874 | |
| 875 | |
| 876 | |
| 877 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">--- intervals</span> | |
| 878 | |
| 879 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">intersect</span> | |
| 880 <span style="color: #8fb28f;">"Returns the interval of overlap between interval-1 and interval-2"</span> | |
| 881 [interval-1 interval-2] | |
| 882 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #f0dfaf; font-weight: bold;">or</span> (<span style="color: #8cd0d3;">empty?</span> interval-1) (<span style="color: #8cd0d3;">empty?</span> interval-2)) [] | |
| 883 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [left (<span style="color: #8cd0d3;">max</span> (<span style="color: #8cd0d3;">first</span> interval-1) (<span style="color: #8cd0d3;">first</span> interval-2)) | |
| 884 right (<span style="color: #8cd0d3;">min</span> (<span style="color: #8cd0d3;">second</span> interval-1) (<span style="color: #8cd0d3;">second</span> interval-2))] | |
| 885 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">></span> left right) [] | |
| 886 [left right])))) | |
| 887 | |
| 888 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">same-endpoints?</span> | |
| 889 <span style="color: #8fb28f;">"Returns true if the left endpoint is the same as the right</span> | |
| 890 <span style="color: #8fb28f;"> endpoint."</span> | |
| 891 [interval] | |
| 892 (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">first</span> interval) (<span style="color: #8cd0d3;">second</span> interval))) | |
| 893 | |
| 894 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">naturals?</span> | |
| 895 <span style="color: #8fb28f;">"Returns true if the left endpoint is 0 and the right endpoint is</span> | |
| 896 <span style="color: #8fb28f;">infinite."</span> | |
| 897 [interval] | |
| 898 (<span style="color: #f0dfaf; font-weight: bold;">and</span> (zero? (<span style="color: #8cd0d3;">first</span> interval)) | |
| 899 (infinite? (<span style="color: #8cd0d3;">second</span> interval)))) | |
| 900 | |
| 901 | |
| 902 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">fan-in</span> | |
| 903 <span style="color: #8fb28f;">"Returns a function that pipes its input to each of the gs, then</span> | |
| 904 <span style="color: #8fb28f;"> applies f to the list of results. Consequently, f must be able to</span> | |
| 905 <span style="color: #8fb28f;"> take a number of arguments equal to the number of gs."</span> | |
| 906 [f & gs] | |
| 907 (<span style="color: #8cd0d3;">fn</span> [& args] | |
| 908 (<span style="color: #8cd0d3;">apply</span> f (<span style="color: #8cd0d3;">apply</span> (<span style="color: #8cd0d3;">apply</span> juxt gs) args)))) | |
| 909 | |
| 910 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">fan-in</span> | |
| 911 <span style="color: #8fb28f;">"Returns a function that pipes its input to each of the gs, then applies f to the list of results. The resulting function takes any number of arguments, but will fail if given arguments that are incompatible with any of the gs."</span> | |
| 912 [f & gs] | |
| 913 (<span style="color: #8cd0d3;">comp</span> (<span style="color: #8cd0d3;">partial</span> apply f) (<span style="color: #8cd0d3;">apply</span> juxt gs))) | |
| 914 | |
| 915 | |
| 916 | |
| 917 (<span style="color: #f0dfaf; font-weight: bold;">defmacro</span> <span style="color: #f0dfaf;">airty-blah-sad</span> [f n more?] | |
| 918 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [syms (<span style="color: #8cd0d3;">vec</span> (<span style="color: #8cd0d3;">map</span> (<span style="color: #8cd0d3;">comp</span> gensym (<span style="color: #8cd0d3;">partial</span> str <span style="color: #cc9393;">"x"</span>)) (<span style="color: #8cd0d3;">range</span> n))) | |
| 919 optional (<span style="color: #8cd0d3;">gensym</span> <span style="color: #cc9393;">"xs"</span>)] | |
| 920 (<span style="color: #f0dfaf; font-weight: bold;">if</span> more? | |
| 921 `(<span style="color: #8cd0d3;">fn</span> <span style="color: #f0dfaf;">~</span>(<span style="color: #8cd0d3;">conj</span> syms '& optional) | |
| 922 (<span style="color: #8cd0d3;">apply</span> ~f ~@syms ~optional)) | |
| 923 `(<span style="color: #8cd0d3;">fn</span> <span style="color: #f0dfaf;">~syms</span> (~f ~@syms))))) | |
| 924 | |
| 925 (<span style="color: #f0dfaf; font-weight: bold;">defmacro</span> <span style="color: #f0dfaf;">airt-whaa*</span> [f n more?] | |
| 926 `(airty-blah-sad ~f ~n ~more?)) | |
| 927 | |
| 928 | |
| 929 | |
| 930 | |
| 931 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">fan-in*</span> | |
| 932 <span style="color: #8fb28f;">"Returns a function that pipes its input to each of the gs, then</span> | |
| 933 <span style="color: #8fb28f;"> applies f to the list of results. Unlike fan-in, fan-in* strictly</span> | |
| 934 <span style="color: #8fb28f;"> enforces arity: it will fail if the gs do not have compatible</span> | |
| 935 <span style="color: #8fb28f;"> arities."</span> | |
| 936 [f & gs] | |
| 937 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [arity-in (<span style="color: #8cd0d3;">reduce</span> intersect (<span style="color: #8cd0d3;">map</span> arity-interval gs)) | |
| 938 left (<span style="color: #8cd0d3;">first</span> arity-in) | |
| 939 right (<span style="color: #8cd0d3;">second</span> arity-in) | |
| 940 composite (fan-in f gs) | |
| 941 ] | |
| 942 (<span style="color: #f0dfaf; font-weight: bold;">cond</span> | |
| 943 (<span style="color: #8cd0d3;">empty?</span> arity-in) | |
| 944 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"Cannot compose functions with incompatible arities."</span>)) | |
| 945 | |
| 946 (<span style="color: #8cd0d3;">not</span> | |
| 947 (<span style="color: #f0dfaf; font-weight: bold;">or</span> (<span style="color: #8cd0d3;">=</span> left right) | |
| 948 (<span style="color: #f0dfaf; font-weight: bold;">and</span> (finite? left) | |
| 949 (<span style="color: #8cd0d3;">=</span> right infinity)))) | |
| 950 | |
| 951 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. | |
| 952 <span style="color: #cc9393;">"Compose can only handle arities of the form [n n] or [n infinity]"</span>)) | |
| 953 <span style="color: #8cd0d3;">:else</span> | |
| 954 (airty-blah-sad composite left (<span style="color: #8cd0d3;">=</span> right infinity))))) | |
| 955 | |
| 956 | |
| 957 | |
| 958 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">compose-n</span> <span style="color: #8fb28f;">"Compose any number of functions together."</span> | |
| 959 ([] identity) | |
| 960 ([f] f) | |
| 961 ([f & fs] | |
| 962 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [fns (<span style="color: #8cd0d3;">cons</span> f fs)] | |
| 963 (compose-bin (<span style="color: #8cd0d3;">reduce</span> fan-in (<span style="color: #8cd0d3;">butlast</span> fs)) (<span style="color: #8cd0d3;">last</span> fs)))) | |
| 964 ) | |
| 965 | |
| 966 | |
| 967 | |
| 968 | |
| 969 | |
| 970 | |
| 971 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">iterated</span> | |
| 972 ([f n id] (<span style="color: #8cd0d3;">reduce</span> comp id (<span style="color: #8cd0d3;">repeat</span> n f))) | |
| 973 ([f n] (<span style="color: #8cd0d3;">reduce</span> comp identity (<span style="color: #8cd0d3;">repeat</span> n f)))) | |
| 974 | |
| 975 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">iterate-until-stable</span> | |
| 976 <span style="color: #8fb28f;">"Repeatedly applies f to x, returning the first result that is close</span> | |
| 977 <span style="color: #8fb28f;">enough to its predecessor."</span> | |
| 978 [f close-enough? x] | |
| 979 (<span style="color: #8cd0d3;">second</span> (swank.util/find-first | |
| 980 (<span style="color: #8cd0d3;">partial</span> apply close-enough?) | |
| 981 (<span style="color: #8cd0d3;">partition</span> 2 1 (<span style="color: #8cd0d3;">iterate</span> f x))))) | |
| 982 | |
| 983 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">lexical<</span> [x y] | |
| 984 (<span style="color: #8cd0d3;">neg?</span> (<span style="color: #8cd0d3;">compare</span> (<span style="color: #8cd0d3;">str</span> x) (<span style="color: #8cd0d3;">str</span> y)))) | |
| 985 | |
| 986 | |
| 987 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">do-up</span> | |
| 988 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">do-down</span> | |
| 989 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">make-pairwise-test</span> comparator) | |
| 990 <span style="color: #708070;">;;</span><span style="color: #7f9f7f;">all-equal?</span> | |
| 991 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">accumulation</span> multiplier) | |
| 992 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">inverse-accumulation</span> divider) | |
| 993 <span style="color: #708070;">;;</span><span style="color: #7f9f7f;">left-circular-shift</span> | |
| 994 <span style="color: #708070;">;;</span><span style="color: #7f9f7f;">right-circular-shift</span> | |
| 995 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">exactly-n?</span> same-endpoints?) | |
| 996 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">any-number?</span> naturals?) | |
| 997 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO compose</span> | |
| 998 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO compose-n</span> | |
| 999 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">identity</span> | |
| 1000 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">compose-2</span> fan-in) | |
| 1001 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">compose-bin</span> fan-in*) | |
| 1002 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">any?</span> (<span style="color: #8cd0d3;">constantly</span> true)) | |
| 1003 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">none?</span> (<span style="color: #8cd0d3;">constantly</span> false)) | |
| 1004 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">constant</span> constantly) | |
| 1005 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">joint-arity</span> intersect) | |
| 1006 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">a-reduce</span> reduce) | |
| 1007 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">filter</span> | |
| 1008 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">make-map</span> (<span style="color: #8cd0d3;">partial</span> partial map) ) | |
| 1009 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">bracket</span> juxt) | |
| 1010 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO apply-to-all</span> | |
| 1011 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO nary-combine</span> | |
| 1012 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO binary-combine</span> | |
| 1013 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">TODO unary-combine</span> | |
| 1014 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">iterated</span> | |
| 1015 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">iterate-until-stable</span> | |
| 1016 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">make-function-of-vector</span> (<span style="color: #8cd0d3;">partial</span> partial map)) | |
| 1017 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">make-function-of-arguments</span> (<span style="color: #8cd0d3;">fn</span> [f] (<span style="color: #8cd0d3;">fn</span> [& args] (f args)))) | |
| 1018 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">alphaless</span> lexical<) | |
| 1019 | |
| 1020 </pre> | |
| 1021 | |
| 1022 | |
| 1023 | |
| 1024 | |
| 1025 | |
| 1026 | |
| 1027 | |
| 1028 | |
| 1029 | |
| 1030 | |
| 1031 | |
| 1032 | |
| 1033 </div> | |
| 1034 </div> | |
| 1035 | |
| 1036 </div> | |
| 1037 | |
| 1038 <div id="outline-container-4" class="outline-2"> | |
| 1039 <h2 id="sec-4"><span class="section-number-2">4</span> Numerical Methods </h2> | |
| 1040 <div class="outline-text-2" id="text-4"> | |
| 1041 | |
| 1042 | |
| 1043 | |
| 1044 | |
| 1045 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) | |
| 1046 (<span style="color: #f0dfaf; font-weight: bold;">import</span> java.lang.Math) | |
| 1047 (<span style="color: #8cd0d3;">use</span> 'clojure.contrib.def) | |
| 1048 | |
| 1049 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">---- USEFUL CONSTANTS</span> | |
| 1050 | |
| 1051 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">machine-epsilon</span> | |
| 1052 <span style="color: #8fb28f;">"Smallest value representable on your machine, as determined by</span> | |
| 1053 <span style="color: #8fb28f;">successively dividing a number in half until consecutive results are</span> | |
| 1054 <span style="color: #8fb28f;">indistinguishable."</span> | |
| 1055 [] | |
| 1056 (<span style="color: #8cd0d3;">ffirst</span> | |
| 1057 (<span style="color: #8cd0d3;">drop-while</span> | |
| 1058 (<span style="color: #8cd0d3;">comp</span> not zero? second) | |
| 1059 (<span style="color: #8cd0d3;">partition</span> 2 1 | |
| 1060 (<span style="color: #8cd0d3;">iterate</span> (<span style="color: #8cd0d3;">partial</span> * 0.5) 1))))) | |
| 1061 | |
| 1062 | |
| 1063 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">pi</span> (Math/PI)) | |
| 1064 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">two-pi</span> (<span style="color: #8cd0d3;">*</span> 2 pi)) | |
| 1065 | |
| 1066 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">eulers-gamma</span> 0.5772156649015328606065) | |
| 1067 | |
| 1068 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">phi</span> (<span style="color: #8cd0d3;">/</span> (<span style="color: #8cd0d3;">inc</span> (Math/sqrt 5)) 2)) | |
| 1069 | |
| 1070 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">ln2</span> (Math/log 2)) | |
| 1071 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">ln10</span> (Math/log 10)) | |
| 1072 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">exp10</span> #(Math/pow 10 %)) | |
| 1073 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">exp2</span> #(Math/pow 2 %)) | |
| 1074 | |
| 1075 | |
| 1076 <span style="color: #708070;">;;</span> | |
| 1077 | |
| 1078 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">---- ANGLES AND TRIGONOMETRY</span> | |
| 1079 | |
| 1080 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">angle-restrictor</span> | |
| 1081 <span style="color: #8fb28f;">"Returns a function that ensures that angles lie in the specified interval of length two-pi."</span> | |
| 1082 [max-angle] | |
| 1083 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [min-angle (<span style="color: #8cd0d3;">-</span> max-angle two-pi)] | |
| 1084 (<span style="color: #8cd0d3;">fn</span> [x] | |
| 1085 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #f0dfaf; font-weight: bold;">and</span> | |
| 1086 (<span style="color: #8cd0d3;"><=</span> min-angle x) | |
| 1087 (<span style="color: #8cd0d3;"><</span> x max-angle)) | |
| 1088 x | |
| 1089 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [corrected-x (<span style="color: #8cd0d3;">-</span> x (<span style="color: #8cd0d3;">*</span> two-pi (Math/floor (<span style="color: #8cd0d3;">/</span> x two-pi))))] | |
| 1090 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;"><</span> corrected-x max-angle) | |
| 1091 corrected-x | |
| 1092 (<span style="color: #8cd0d3;">-</span> corrected-x two-pi))))))) | |
| 1093 | |
| 1094 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">angle-restrict-pi</span> | |
| 1095 <span style="color: #8fb28f;">"Coerces angles to lie in the interval from -pi to pi."</span> | |
| 1096 [angle] | |
| 1097 ((angle-restrictor pi) angle)) | |
| 1098 | |
| 1099 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">angle-restrict-two-pi</span> | |
| 1100 <span style="color: #8fb28f;">"Coerces angles to lie in the interval from zero to two-pi"</span> | |
| 1101 [angle] | |
| 1102 ((angle-restrictor two-pi) angle)) | |
| 1103 | |
| 1104 | |
| 1105 | |
| 1106 | |
| 1107 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">invert</span> [x] (<span style="color: #8cd0d3;">/</span> x)) | |
| 1108 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">negate</span> [x] (<span style="color: #8cd0d3;">-</span> x)) | |
| 1109 | |
| 1110 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">exp</span> [x] (Math/exp x)) | |
| 1111 | |
| 1112 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">sin</span> [x] (Math/sin x)) | |
| 1113 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">cos</span> [x] (Math/cos x)) | |
| 1114 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">tan</span> [x] (Math/tan x)) | |
| 1115 | |
| 1116 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">sec</span> (<span style="color: #8cd0d3;">comp</span> invert cos)) | |
| 1117 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">csc</span> (<span style="color: #8cd0d3;">comp</span> invert sin)) | |
| 1118 | |
| 1119 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">sinh</span> [x] (Math/sinh x)) | |
| 1120 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">cosh</span> [x] (Math/cosh x)) | |
| 1121 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">tanh</span> [x] (Math/tanh x)) | |
| 1122 | |
| 1123 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">sech</span> (<span style="color: #8cd0d3;">comp</span> invert cosh)) | |
| 1124 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">csch</span> (<span style="color: #8cd0d3;">comp</span> invert sinh)) | |
| 1125 | |
| 1126 | |
| 1127 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">------------</span> | |
| 1128 | |
| 1129 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">factorial</span> | |
| 1130 <span style="color: #8fb28f;">"Computes the factorial of the nonnegative integer n."</span> | |
| 1131 [n] | |
| 1132 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">neg?</span> n) | |
| 1133 (<span style="color: #f0dfaf; font-weight: bold;">throw</span> (Exception. <span style="color: #cc9393;">"Cannot compute the factorial of a negative number."</span>)) | |
| 1134 (<span style="color: #8cd0d3;">reduce</span> * 1 (<span style="color: #8cd0d3;">range</span> 1 (<span style="color: #8cd0d3;">inc</span> n))))) | |
| 1135 | |
| 1136 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">exact-quotient</span> [n d] (<span style="color: #8cd0d3;">/</span> n d)) | |
| 1137 | |
| 1138 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">binomial-coefficient</span> | |
| 1139 <span style="color: #8fb28f;">"Computes the number of different ways to choose m elements from n."</span> | |
| 1140 [n m] | |
| 1141 (<span style="color: #8cd0d3;">assert</span> (<span style="color: #8cd0d3;"><=</span> 0 m n)) | |
| 1142 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [difference (<span style="color: #8cd0d3;">-</span> n m)] | |
| 1143 (exact-quotient | |
| 1144 (<span style="color: #8cd0d3;">reduce</span> * (<span style="color: #8cd0d3;">range</span> n (<span style="color: #8cd0d3;">max</span> difference m) -1 )) | |
| 1145 (factorial (<span style="color: #8cd0d3;">min</span> difference m))))) | |
| 1146 | |
| 1147 (<span style="color: #f0dfaf; font-weight: bold;">defn-memo</span> <span style="color: #f0dfaf;">stirling-1</span> | |
| 1148 <span style="color: #cc9393;">"Stirling numbers of the first kind: the number of permutations of n</span> | |
| 1149 <span style="color: #cc9393;"> elements with exactly m permutation cycles. "</span> | |
| 1150 [n k] | |
| 1151 <span style="color: #708070;">;</span><span style="color: #7f9f7f;">(assert (<= 1 k n))</span> | |
| 1152 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (zero? n) | |
| 1153 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (zero? k) 1 0) | |
| 1154 (<span style="color: #8cd0d3;">+</span> (stirling-1 (<span style="color: #8cd0d3;">dec</span> n) (<span style="color: #8cd0d3;">dec</span> k)) | |
| 1155 (<span style="color: #8cd0d3;">*</span> (<span style="color: #8cd0d3;">dec</span> n) (stirling-1 (<span style="color: #8cd0d3;">dec</span> n) k))))) | |
| 1156 | |
| 1157 (<span style="color: #f0dfaf; font-weight: bold;">defn-memo</span> <span style="color: #f0dfaf;">stirling-2</span> <span style="color: #708070;">;;</span><span style="color: #7f9f7f;">count-partitions</span> | |
| 1158 <span style="color: #cc9393;">"Stirling numbers of the second kind: the number of ways to partition a set of n elements into k subsets."</span> | |
| 1159 [n k] | |
| 1160 (<span style="color: #f0dfaf; font-weight: bold;">cond</span> | |
| 1161 (<span style="color: #8cd0d3;">=</span> k 1) 1 | |
| 1162 (<span style="color: #8cd0d3;">=</span> k n) 1 | |
| 1163 <span style="color: #8cd0d3;">:else</span> (<span style="color: #8cd0d3;">+</span> (stirling-2 (<span style="color: #8cd0d3;">dec</span> n) (<span style="color: #8cd0d3;">dec</span> k)) | |
| 1164 (<span style="color: #8cd0d3;">*</span> k (stirling-2 (<span style="color: #8cd0d3;">dec</span> n) k))))) | |
| 1165 | |
| 1166 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">harmonic-number</span> [n] | |
| 1167 (<span style="color: #8cd0d3;">/</span> (stirling-1 (<span style="color: #8cd0d3;">inc</span> n) 2) | |
| 1168 (factorial n))) | |
| 1169 | |
| 1170 | |
| 1171 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">sum</span> | |
| 1172 [f low high] | |
| 1173 (<span style="color: #8cd0d3;">reduce</span> + (<span style="color: #8cd0d3;">map</span> f (<span style="color: #8cd0d3;">range</span> low (<span style="color: #8cd0d3;">inc</span> high))))) | |
| 1174 | |
| 1175 </pre> | |
| 1176 | |
| 1177 | |
| 1178 | |
| 1179 | |
| 1180 | |
| 1181 | |
| 1182 | |
| 1183 | |
| 1184 | |
| 1185 | |
| 1186 | |
| 1187 | |
| 1188 | |
| 1189 | |
| 1190 | |
| 1191 </div> | |
| 1192 | |
| 1193 </div> | |
| 1194 | |
| 1195 <div id="outline-container-5" class="outline-2"> | |
| 1196 <h2 id="sec-5"><span class="section-number-2">5</span> Differentiation </h2> | |
| 1197 <div class="outline-text-2" id="text-5"> | |
| 1198 | |
| 1199 | |
| 1200 <p> | |
| 1201 We compute derivatives by passing special <b>differential objects</b> \([x, | |
| 1202 dx]\) through functions. Roughly speaking, applying a function \(f\) to a | |
| 1203 differential object \([x, dx]\) should produce a new differential | |
| 1204 object \([f(x),\,Df(x)\cdot dx]\). | |
| 1205 </p> | |
| 1206 | |
| 1207 | |
| 1208 \([x,\,dx]\xrightarrow{\quad f \quad}[f(x),\,Df(x)\cdot dx]\) | |
| 1209 <p> | |
| 1210 Notice that you can obtain the derivative of \(f\) from this | |
| 1211 differential object, as it is the coefficient of the \(dx\) term. Also, | |
| 1212 as you apply successive functions using this rule, you get the | |
| 1213 chain-rule answer you expect: | |
| 1214 </p> | |
| 1215 | |
| 1216 | |
| 1217 \([f(x),\,Df(x)\cdot dx]\xrightarrow{\quad g\quad} [gf(x),\, | |
| 1218 Dgf(x)\cdot Df(x) \cdot dx ]\) | |
| 1219 | |
| 1220 <p> | |
| 1221 In order to generalize to multiple variables and multiple derivatives, | |
| 1222 we use a <b>power series of differentials</b>, a sortred infinite sequence which | |
| 1223 contains all terms like \(dx\cdot dy\), \(dx^2\cdot dy\), etc. | |
| 1224 </p> | |
| 1225 | |
| 1226 | |
| 1227 | |
| 1228 | |
| 1229 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) | |
| 1230 | |
| 1231 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">differential-seq</span> | |
| 1232 <span style="color: #8fb28f;">"Constructs a sequence of differential terms from a numerical</span> | |
| 1233 <span style="color: #8fb28f;">coefficient and a list of keys for variables. If no coefficient is supplied, uses 1."</span> | |
| 1234 ([variables] (differential-seq* 1 variables)) | |
| 1235 ([coefficient variables & cvs] | |
| 1236 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">number?</span> coefficient) | |
| 1237 (<span style="color: #8cd0d3;">conj</span> (<span style="color: #8cd0d3;">assoc</span> {} (<span style="color: #8cd0d3;">apply</span> sorted-set variables) coefficient) | |
| 1238 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">empty?</span> cvs) | |
| 1239 nil | |
| 1240 (<span style="color: #8cd0d3;">apply</span> differential-seq* cvs))) | |
| 1241 (<span style="color: #8cd0d3;">apply</span> differential-seq* 1 coefficient 1 variables cvs) | |
| 1242 ))) | |
| 1243 | |
| 1244 | |
| 1245 | |
| 1246 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">differential-add</span> | |
| 1247 <span style="color: #8fb28f;">"Add two differential sequences by combining like terms."</span> | |
| 1248 [dseq1 dseq2] | |
| 1249 (<span style="color: #8cd0d3;">merge-with</span> + dseq1 dseq2)) | |
| 1250 | |
| 1251 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">differential-multiply</span> | |
| 1252 <span style="color: #8fb28f;">"Multiply two differential sequences. The square of any differential variable is zero since differential variables are infinitesimally small."</span> | |
| 1253 [dseq1 dseq2] | |
| 1254 (<span style="color: #8cd0d3;">reduce</span> | |
| 1255 (<span style="color: #8cd0d3;">fn</span> [m [[vars1 coeff1] [vars2 coeff2]]] | |
| 1256 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">empty?</span> (clojure.set/<span style="color: #dfdfbf; font-weight: bold;">intersection</span> vars1 vars2)) | |
| 1257 (<span style="color: #8cd0d3;">assoc</span> m (clojure.set/<span style="color: #dfdfbf; font-weight: bold;">union</span> vars1 vars2) (<span style="color: #8cd0d3;">*</span> coeff1 coeff2)) | |
| 1258 m)) | |
| 1259 {} | |
| 1260 (clojure.contrib.combinatorics/cartesian-product | |
| 1261 dseq1 | |
| 1262 dseq2))) | |
| 1263 | |
| 1264 | |
| 1265 | |
| 1266 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">big-part</span> | |
| 1267 <span style="color: #8fb28f;">"Returns the 'finite' part of the differential sequence."</span> | |
| 1268 [dseq] | |
| 1269 (<span style="color: #f0dfaf; font-weight: bold;">let</span> | |
| 1270 [keys (<span style="color: #8cd0d3;">sort-by</span> count (<span style="color: #8cd0d3;">keys</span> dseq)) | |
| 1271 pivot-key (<span style="color: #8cd0d3;">first</span> (<span style="color: #8cd0d3;">last</span> keys))] | |
| 1272 | |
| 1273 (<span style="color: #8cd0d3;">apply</span> hash-map | |
| 1274 (<span style="color: #8cd0d3;">reduce</span> concat | |
| 1275 (<span style="color: #8cd0d3;">filter</span> (<span style="color: #8cd0d3;">comp</span> pivot-key first) dseq) | |
| 1276 )))) | |
| 1277 | |
| 1278 | |
| 1279 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">small-part</span> | |
| 1280 <span style="color: #8fb28f;">"Returns the 'infinitesimal' part of the differential sequence."</span> | |
| 1281 [dseq] | |
| 1282 (<span style="color: #f0dfaf; font-weight: bold;">let</span> | |
| 1283 [keys (<span style="color: #8cd0d3;">sort-by</span> count (<span style="color: #8cd0d3;">keys</span> dseq)) | |
| 1284 pivot-key (<span style="color: #8cd0d3;">first</span> (<span style="color: #8cd0d3;">last</span> keys))] | |
| 1285 | |
| 1286 (<span style="color: #8cd0d3;">apply</span> hash-map | |
| 1287 (<span style="color: #8cd0d3;">reduce</span> concat | |
| 1288 (<span style="color: #8cd0d3;">remove</span> (<span style="color: #8cd0d3;">comp</span> pivot-key first) dseq) | |
| 1289 )))) | |
| 1290 | |
| 1291 | |
| 1292 | |
| 1293 | |
| 1294 | |
| 1295 | |
| 1296 | |
| 1297 | |
| 1298 | |
| 1299 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">;; A differential term consists of a numerical coefficient and a</span> | |
| 1300 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">;; sorted </span> | |
| 1301 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(defrecord DifferentialTerm [coefficient variables])</span> | |
| 1302 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(defmethod print-method DifferentialTerm</span> | |
| 1303 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">[o w]</span> | |
| 1304 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(print-simple</span> | |
| 1305 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(apply str (.coefficient o)(map (comp (partial str "d") name) (.variables o)))</span> | |
| 1306 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">w))</span> | |
| 1307 | |
| 1308 | |
| 1309 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(defn differential-seq</span> | |
| 1310 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">"Constructs a sequence of differential terms from a numerical</span> | |
| 1311 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">coefficient and a list of keywords for variables. If no coefficient is</span> | |
| 1312 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">supplied, uses 1."</span> | |
| 1313 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">([variables] (differential-seq 1 variables))</span> | |
| 1314 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">([coefficient variables]</span> | |
| 1315 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(list</span> | |
| 1316 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(DifferentialTerm. coefficient (apply sorted-set variables))))</span> | |
| 1317 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">([coefficient variables & cvs]</span> | |
| 1318 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(sort-by</span> | |
| 1319 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">#(vec(.variables %))</span> | |
| 1320 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">(concat (differential-seq coefficient variables) (apply differential-seq cvs)))))</span> | |
| 1321 | |
| 1322 </pre> | |
| 1323 | |
| 1324 | |
| 1325 | |
| 1326 | |
| 1327 | |
| 1328 | |
| 1329 | |
| 1330 | |
| 1331 | |
| 1332 | |
| 1333 | |
| 1334 | |
| 1335 | |
| 1336 | |
| 1337 | |
| 1338 | |
| 1339 </div> | |
| 1340 | |
| 1341 </div> | |
| 1342 | |
| 1343 <div id="outline-container-6" class="outline-2"> | |
| 1344 <h2 id="sec-6"><span class="section-number-2">6</span> Symbolic Manipulation </h2> | |
| 1345 <div class="outline-text-2" id="text-6"> | |
| 1346 | |
| 1347 | |
| 1348 | |
| 1349 | |
| 1350 | |
| 1351 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'sicm.utils) | |
| 1352 | |
| 1353 (<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">Symbolic</span> [type expression] | |
| 1354 Object | |
| 1355 (equals [this that] | |
| 1356 (<span style="color: #f0dfaf; font-weight: bold;">cond</span> | |
| 1357 (<span style="color: #8cd0d3;">=</span> (.expression this) (.expression that)) true | |
| 1358 <span style="color: #8cd0d3;">:else</span> | |
| 1359 (Symbolic. | |
| 1360 java.lang.Boolean | |
| 1361 (<span style="color: #8cd0d3;">list</span> '= (.expression this) (.expression that))) | |
| 1362 ))) | |
| 1363 | |
| 1364 | |
| 1365 | |
| 1366 | |
| 1367 (<span style="color: #f0dfaf; font-weight: bold;">deftype</span> <span style="color: #f0dfaf;">AbstractSet</span> [glyph membership-test]) | |
| 1368 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">member?</span> [abstract-set x] | |
| 1369 ((.membership-test abstract-set) x)) | |
| 1370 | |
| 1371 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">------------ Some important AbstractSets</span> | |
| 1372 | |
| 1373 | |
| 1374 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">Real</span> | |
| 1375 (AbstractSet. | |
| 1376 'R | |
| 1377 (<span style="color: #8cd0d3;">fn</span>[x](<span style="color: #8cd0d3;">number?</span> x)))) | |
| 1378 | |
| 1379 | |
| 1380 <span style="color: #708070;">;; </span><span style="color: #7f9f7f;">------------ Create new AbstractSets from existing ones</span> | |
| 1381 | |
| 1382 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">abstract-product</span> | |
| 1383 <span style="color: #8fb28f;">"Gives the cartesian product of abstract sets."</span> | |
| 1384 ([sets] | |
| 1385 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">count</span> sets) 1) (<span style="color: #8cd0d3;">first</span> sets) | |
| 1386 (AbstractSet. | |
| 1387 (<span style="color: #8cd0d3;">symbol</span> | |
| 1388 (<span style="color: #8cd0d3;">apply</span> str | |
| 1389 (<span style="color: #8cd0d3;">interpose</span> 'x (<span style="color: #8cd0d3;">map</span> #(.glyph %) sets)))) | |
| 1390 (<span style="color: #8cd0d3;">fn</span> [x] | |
| 1391 (<span style="color: #f0dfaf; font-weight: bold;">and</span> | |
| 1392 (<span style="color: #8cd0d3;">coll?</span> x) | |
| 1393 (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">count</span> sets) (<span style="color: #8cd0d3;">count</span> x)) | |
| 1394 (<span style="color: #8cd0d3;">reduce</span> #(<span style="color: #f0dfaf; font-weight: bold;">and</span> %1 %2) | |
| 1395 true | |
| 1396 (<span style="color: #8cd0d3;">map</span> #(member? %1 %2) sets x))))))) | |
| 1397 ([abstract-set n] | |
| 1398 (abstract-product (<span style="color: #8cd0d3;">repeat</span> n abstract-set)))) | |
| 1399 | |
| 1400 | |
| 1401 | |
| 1402 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">abstract-up</span> | |
| 1403 <span style="color: #8fb28f;">"Returns the abstract set of all up-tuples whose items belong to the</span> | |
| 1404 <span style="color: #8fb28f;"> corresponding abstract sets in coll."</span> | |
| 1405 ([coll] | |
| 1406 (AbstractSet. | |
| 1407 (<span style="color: #8cd0d3;">symbol</span> (<span style="color: #8cd0d3;">str</span> <span style="color: #cc9393;">"u["</span> | |
| 1408 (<span style="color: #8cd0d3;">apply</span> str | |
| 1409 (<span style="color: #8cd0d3;">interpose</span> <span style="color: #cc9393;">" "</span> | |
| 1410 (<span style="color: #8cd0d3;">map</span> #(.glyph %) coll))) | |
| 1411 <span style="color: #cc9393;">"]"</span>)) | |
| 1412 (<span style="color: #8cd0d3;">fn</span> [x] | |
| 1413 (<span style="color: #f0dfaf; font-weight: bold;">and</span> | |
| 1414 (<span style="color: #8cd0d3;">satisfies?</span> Spinning x) | |
| 1415 (up? x) | |
| 1416 (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">count</span> coll) (<span style="color: #8cd0d3;">count</span> x)) | |
| 1417 (<span style="color: #8cd0d3;">reduce</span> | |
| 1418 #(<span style="color: #f0dfaf; font-weight: bold;">and</span> %1 %2) | |
| 1419 true | |
| 1420 (<span style="color: #8cd0d3;">map</span> #(member? %1 %2) coll x)))))) | |
| 1421 ([abstract-set n] | |
| 1422 (abstract-up (<span style="color: #8cd0d3;">repeat</span> n abstract-set)))) | |
| 1423 | |
| 1424 | |
| 1425 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">abstract-down</span> | |
| 1426 <span style="color: #8fb28f;">"Returns the abstract set of all down-tuples whose items belong to the</span> | |
| 1427 <span style="color: #8fb28f;"> corresponding abstract sets in coll."</span> | |
| 1428 ([coll] | |
| 1429 (AbstractSet. | |
| 1430 (<span style="color: #8cd0d3;">symbol</span> (<span style="color: #8cd0d3;">str</span> <span style="color: #cc9393;">"d["</span> | |
| 1431 (<span style="color: #8cd0d3;">apply</span> str | |
| 1432 (<span style="color: #8cd0d3;">interpose</span> <span style="color: #cc9393;">" "</span> | |
| 1433 (<span style="color: #8cd0d3;">map</span> #(.glyph %) coll))) | |
| 1434 <span style="color: #cc9393;">"]"</span>)) | |
| 1435 (<span style="color: #8cd0d3;">fn</span> [x] | |
| 1436 (<span style="color: #f0dfaf; font-weight: bold;">and</span> | |
| 1437 (<span style="color: #8cd0d3;">satisfies?</span> Spinning x) | |
| 1438 (down? x) | |
| 1439 (<span style="color: #8cd0d3;">=</span> (<span style="color: #8cd0d3;">count</span> coll) (<span style="color: #8cd0d3;">count</span> x)) | |
| 1440 (<span style="color: #8cd0d3;">reduce</span> | |
| 1441 #(<span style="color: #f0dfaf; font-weight: bold;">and</span> %1 %2) | |
| 1442 true | |
| 1443 (<span style="color: #8cd0d3;">map</span> #(member? %1 %2) coll x)))))) | |
| 1444 ([abstract-set n] | |
| 1445 (abstract-down (<span style="color: #8cd0d3;">repeat</span> n abstract-set)))) | |
| 1446 | |
| 1447 | |
| 1448 | |
| 1449 | |
| 1450 | |
| 1451 <span style="color: #708070;">;</span><span style="color: #7f9f7f;">-------ABSTRACT FUNCTIONS</span> | |
| 1452 (<span style="color: #f0dfaf; font-weight: bold;">defrecord</span> <span style="color: #f0dfaf;">AbstractFn</span> | |
| 1453 [<span style="color: #dfdfbf; font-weight: bold;">#^AbstractSet</span> domain <span style="color: #dfdfbf; font-weight: bold;">#^AbstractSet</span> codomain]) | |
| 1454 | |
| 1455 | |
| 1456 (<span style="color: #f0dfaf; font-weight: bold;">defmethod</span> <span style="color: #f0dfaf;">print-method</span> AbstractFn | |
| 1457 [o w] | |
| 1458 (<span style="color: #8cd0d3;">print-simple</span> | |
| 1459 (<span style="color: #8cd0d3;">str</span> | |
| 1460 <span style="color: #cc9393;">"f:"</span> | |
| 1461 (.glyph (<span style="color: #8cd0d3;">:domain</span> o)) | |
| 1462 <span style="color: #cc9393;">"-->"</span> | |
| 1463 (.glyph (<span style="color: #8cd0d3;">:codomain</span> o))) w)) | |
| 1464 </pre> | |
| 1465 | |
| 1466 | |
| 1467 | |
| 1468 | |
| 1469 | |
| 1470 | |
| 1471 | |
| 1472 </div> | |
| 1473 </div> | |
| 1474 <div id="postamble"> | |
| 1475 <p class="date">Date: 2011-08-09 18:41:37 EDT</p> | |
| 1476 <p class="author">Author: Robert McIntyre & Dylan Holmes</p> | |
| 1477 <p class="creator">Org version 7.6 with Emacs version 23</p> | |
| 1478 <a href="http://validator.w3.org/check?uri=referer">Validate XHTML 1.0</a> | |
| 1479 </div> | |
| 1480 </div> | |
| 1481 </body> | |
| 1482 </html> |