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 293 <body>
 294 <div class="document" id="sympycore-release-0-1-demo">
 295 <h1 class="title">SympyCore Release 0.1 Demo</h1>
 296 <table class="docinfo" frame="void" rules="none">
 297 <col class="docinfo-name" />
 298 <col class="docinfo-content" />
 299 <tbody valign="top">
 300 <tr><th class="docinfo-name">Authors:</th>
 301 <td>Pearu Peterson &lt;pearu.peterson AT gmail DOT com&gt;</td></tr>
 302 <tr class="field"><th class="docinfo-name">Created:</th><td class="field-body">February 2008</td>
 303 </tr>
 304 <tr class="field"><th class="docinfo-name">Website:</th><td class="field-body"><a class="reference" href="http://sympycore.googlecode.com/">http://sympycore.googlecode.com/</a></td>
 305 </tr>
 306 </tbody>
 307 </table>
 308 <!-- -*- rest -*- -->
 309 <div class="sidebar">
 310 <p class="first sidebar-title">Table of contents</p>
 311 <div class="contents local last topic">
 312 <ul class="simple">
 313 <li><a class="reference" href="#getting-started" id="id2" name="id2">Getting started</a></li>
 314 <li><a class="reference" href="#constructing-symbolic-expressions" id="id3" name="id3">Constructing symbolic expressions</a></li>
 315 <li><a class="reference" href="#manipulations-with-symbolic-expressions" id="id4" name="id4">Manipulations with symbolic expressions</a></li>
 316 <li><a class="reference" href="#transformation-methods" id="id5" name="id5">Transformation methods</a></li>
 317 <li><a class="reference" href="#arithmetic-methods" id="id6" name="id6">Arithmetic methods</a></li>
 318 <li><a class="reference" href="#calculus-methods" id="id7" name="id7">Calculus methods</a></li>
 319 <li><a class="reference" href="#polynomial-rings" id="id8" name="id8">Polynomial rings</a></li>
 320 <li><a class="reference" href="#matrix-rings" id="id9" name="id9">Matrix rings</a></li>
 321 <li><a class="reference" href="#physical-units" id="id10" name="id10">Physical units</a></li>
 322 </ul>
 323 </div>
 324 </div>
 325 <div class="section">
 326 <h1><a class="toc-backref" href="#id2" id="getting-started" name="getting-started">Getting started</a></h1>
 327 <p>This document gives a short overview of SympyCore basic features. For
 328 more information see the <a class="reference" href="http://sympycore.googlecode.com/svn/trunk/doc/html/userguide.html">SympyCore User's Guide</a>.</p>
 329 <p>To use the <tt class="docutils literal"><span class="pre">sympycore</span></tt> package from Python, one must import it:</p>
 330 <pre class="doctest-block">
 331 &gt;&gt;&gt; from sympycore import *
 332 </pre>
 333 </div>
 334 <div class="section">
 335 <h1><a class="toc-backref" href="#id3" id="constructing-symbolic-expressions" name="constructing-symbolic-expressions">Constructing symbolic expressions</a></h1>
 336 <p>Generally speaking, symbolic expressions consist of symbols and
 337 operation between them. To create symbolic expressions using
 338 SympyCore, one can either create symbol objects and perform operations
 339 between them:</p>
 340 <pre class="doctest-block">
 341 &gt;&gt;&gt; x = Symbol('x')
 342 &gt;&gt;&gt; y = Symbol('y')
 343 &gt;&gt;&gt; z = Symbol('z')
 344 &gt;&gt;&gt; x + y
 345 Calculus('x + y')
 346 </pre>
 347 <p>or one can use symbolic expression parser to construct symbolic
 348 expressions from a string:</p>
 349 <pre class="doctest-block">
 350 &gt;&gt;&gt; Calculus('x + y')
 351 Calculus('x + y')
 352 </pre>
 353 <p>SympyCore converts symbolic expressions to a canonical form that is
 354 efficient for further manipulations and are also often obvious
 355 simplifications that users may expect:</p>
 356 <pre class="doctest-block">
 357 &gt;&gt;&gt; x + x
 358 Calculus('2*x')
 359 </pre>
 360 <pre class="doctest-block">
 361 &gt;&gt;&gt; x - x
 362 Calculus('0')
 363 </pre>
 364 <p>General symbolic arithmetic expressions are instances of the Calculus
 365 class. Using the <tt class="docutils literal"><span class="pre">print</span></tt> statement (or <tt class="docutils literal"><span class="pre">str()</span></tt>) hides this information:</p>
 366 <pre class="doctest-block">
 367 &gt;&gt;&gt; print x + y
 368 x + y
 369 </pre>
 370 </div>
 371 <div class="section">
 372 <h1><a class="toc-backref" href="#id4" id="manipulations-with-symbolic-expressions" name="manipulations-with-symbolic-expressions">Manipulations with symbolic expressions</a></h1>
 373 <p>The most obvious manipulation task applied to symbolic expressions, is
 374 substitution -- replacing a sub-expression of a given expression with a
 375 new expression. For example,</p>
 376 <pre class="doctest-block">
 377 &gt;&gt;&gt; expr = x + y
 378 &gt;&gt;&gt; print expr.subs(y, sin(x))
 379 x + sin(x)
 380 </pre>
 381 <p>Other tasks include accessing parts of symbolic expressions:</p>
 382 <pre class="doctest-block">
 383 &gt;&gt;&gt; sorted(expr.args)
 384 [Calculus('x'), Calculus('y')]
 385 </pre>
 386 <p>and constructing new expressions:</p>
 387 <pre class="doctest-block">
 388 &gt;&gt;&gt; print Mul(*expr.args)
 389 x*y
 390 </pre>
 391 <p>An important presumption for implementing various algorithms is pattern
 392 matching. Pattern matching means that given a pattern expression, the
 393 pattern match method should first decide whether an expression can be
 394 expressed in a form that the pattern defines, and second. it should
 395 return information what sub-expression parts correspond to the pattern
 396 sub-expressions. For example, given a pattern</p>
 397 <pre class="doctest-block">
 398 &gt;&gt;&gt; w = Symbol('w')
 399 &gt;&gt;&gt; pattern = x * w ** 3
 400 </pre>
 401 <p>where symbol <tt class="docutils literal"><span class="pre">w</span></tt> is assumed to match any sub-expression, then expressions</p>
 402 <pre class="doctest-block">
 403 &gt;&gt;&gt; expr1 = x*sin(y)**3
 404 &gt;&gt;&gt; expr2 = x*(x+y)**3
 405 </pre>
 406 <p>do match the given pattern:</p>
 407 <pre class="doctest-block">
 408 &gt;&gt;&gt; d1 = expr1.match(pattern, w)
 409 &gt;&gt;&gt; print d1
 410 {Calculus('w'): Calculus('sin(y)')}
 411 </pre>
 412 <pre class="doctest-block">
 413 &gt;&gt;&gt; d2 = expr2.match(pattern, w)
 414 &gt;&gt;&gt; print d2
 415 {Calculus('w'): Calculus('x + y')}
 416 </pre>
 417 <p>The result of <tt class="docutils literal"><span class="pre">match</span></tt> method, when the match is found, is a dictionary
 418 with the property</p>
 419 <pre class="doctest-block">
 420 &gt;&gt;&gt; pattern.subs(d1.items())==expr1
 421 True
 422 &gt;&gt;&gt; pattern.subs(d2.items())==expr2
 423 True
 424 </pre>
 425 <p>If no match is found, then the <tt class="docutils literal"><span class="pre">match</span></tt> returns <tt class="docutils literal"><span class="pre">None</span></tt>:</p>
 426 <pre class="doctest-block">
 427 &gt;&gt;&gt; print (y*x**2).match(pattern, w)
 428 None
 429 </pre>
 430 </div>
 431 <div class="section">
 432 <h1><a class="toc-backref" href="#id5" id="transformation-methods" name="transformation-methods">Transformation methods</a></h1>
 433 <p>The most common transformation task is expansion of sub-expressions by
 434 opening parenthesis:</p>
 435 <pre class="doctest-block">
 436 &gt;&gt;&gt; expr = (x+y)*z
 437 &gt;&gt;&gt; print expr
 438 z*(x + y)
 439 &gt;&gt;&gt; print expr.expand()
 440 x*z + y*z
 441 </pre>
 442 <p>In general, the <tt class="docutils literal"><span class="pre">expand</span></tt> method expands products of sums and
 443 integer powers of sums:</p>
 444 <pre class="doctest-block">
 445 &gt;&gt;&gt; expr = (x+y)*(1+x)**3
 446 &gt;&gt;&gt; print expr.expand()
 447 x + y + x**4 + 3*x**2 + 3*x**3 + 3*x*y + 3*y*x**2 + y*x**3
 448 </pre>
 449 </div>
 450 <div class="section">
 451 <h1><a class="toc-backref" href="#id6" id="arithmetic-methods" name="arithmetic-methods">Arithmetic methods</a></h1>
 452 <p>SympyCore provides exact rational and complex numbers:</p>
 453 <pre class="doctest-block">
 454 &gt;&gt;&gt; Calculus('3/12')
 455 Calculus('1/4')
 456 &gt;&gt;&gt; print (2 + 3*I/4)**4
 457 721/256 + 165/8*I
 458 </pre>
 459 <p>Fractional powers of integers are evaluated to simpler
 460 expressions when possible:</p>
 461 <pre class="doctest-block">
 462 &gt;&gt;&gt; Calculus('8**(1/3)')
 463 Calculus('2')
 464 &gt;&gt;&gt; Calculus('243**(1/5)')
 465 Calculus('3')
 466 </pre>
 467 <p>SympyCore supports converting symbolic expressions with exact numbers
 468 such as integers and rational numbers to expressions with arbitrary
 469 precision floating-point numbers:</p>
 470 <pre class="doctest-block">
 471 &gt;&gt;&gt; expr = 2*pi + E**x
 472 &gt;&gt;&gt; print expr
 473 E**x + 2*pi
 474 &gt;&gt;&gt; print expr.evalf(5)
 475 6.2832 + 2.7183**x
 476 &gt;&gt;&gt; print expr.evalf(25)
 477 6.283185307179586476925287 + 2.718281828459045235360287**x
 478 </pre>
 479 </div>
 480 <div class="section">
 481 <h1><a class="toc-backref" href="#id7" id="calculus-methods" name="calculus-methods">Calculus methods</a></h1>
 482 <p>SympyCore provides methods to differentiate symbolic expressions:</p>
 483 <pre class="doctest-block">
 484 &gt;&gt;&gt; expr = x+sin(x*y)*x
 485 &gt;&gt;&gt; print expr.diff(x)
 486 1 + sin(x*y) + x*y*cos(x*y)
 487 </pre>
 488 <p>as well as integrate symbolic expression representing polynomials:</p>
 489 <pre class="doctest-block">
 490 &gt;&gt;&gt; expr = x + 3*z*x**2
 491 &gt;&gt;&gt; print expr.integrate(x)
 492 1/2*x**2 + z*x**3
 493 &gt;&gt;&gt; print expr.integrate((x, 2, y))
 494 1/2*y**2 + z*(y**3 - 8) - 2
 495 </pre>
 496 <p>SympyCore implements the elementary functions <tt class="docutils literal"><span class="pre">exp</span></tt>, <tt class="docutils literal"><span class="pre">log</span></tt>,
 497 <tt class="docutils literal"><span class="pre">sqrt</span></tt>, <tt class="docutils literal"><span class="pre">cos</span></tt>, <tt class="docutils literal"><span class="pre">sin</span></tt>, <tt class="docutils literal"><span class="pre">tan</span></tt>, <tt class="docutils literal"><span class="pre">cot</span></tt>, and simplifies
 498 their values in basic cases:</p>
 499 <pre class="doctest-block">
 500 &gt;&gt;&gt; print log(10000,10)
 501 4
 502 &gt;&gt;&gt; print sin(5*pi/6)
 503 1/2
 504 &gt;&gt;&gt; print cos(x+pi/2)
 505 -sin(x)
 506 </pre>
 507 </div>
 508 <div class="section">
 509 <h1><a class="toc-backref" href="#id8" id="polynomial-rings" name="polynomial-rings">Polynomial rings</a></h1>
 510 <p>SympyCore provides efficient ways to represent univariate and
 511 multivariate polynomials. Currently there are two representation
 512 supported. The first one is suitable for univariate dense polynomials:</p>
 513 <pre class="doctest-block">
 514 &gt;&gt;&gt; poly1 = UnivariatePolynomial([2,0,3,4], symbol='x')
 515 &gt;&gt;&gt; poly2 = UnivariatePolynomial([0,1,0,5,6], symbol='x')
 516 &gt;&gt;&gt; poly1
 517 2 + 3*x**2 + 4*x**3
 518 &gt;&gt;&gt; poly2
 519 x + 5*x**3 + 6*x**4
 520 &gt;&gt;&gt; poly1 + poly2
 521 2 + x + 3*x**2 + 9*x**3 + 6*x**4
 522 </pre>
 523 <p>And the other representation is suitable for multivariate sparse
 524 polynomials:</p>
 525 <pre class="doctest-block">
 526 &gt;&gt;&gt; P = PolynomialRing[(x,y)]
 527 &gt;&gt;&gt; poly1 = P({(1,2):7, (300,4):5})
 528 &gt;&gt;&gt; poly2 = P({(3,4):-7, (2,500):12})
 529 &gt;&gt;&gt; poly1
 530 PolynomialRing[(x, y), Calculus]('5*x**300*y**4 + 7*x*y**2')
 531 &gt;&gt;&gt; print poly2
 532 ((-7))*x**3*y**4 + 12*x**2*y**500
 533 &gt;&gt;&gt; print poly1 + poly2
 534 5*x**300*y**4 + ((-7))*x**3*y**4 + 12*x**2*y**500 + 7*x*y**2
 535 </pre>
 536 <p>Here the <tt class="docutils literal"><span class="pre">PolynomialRing[symbols,</span> <span class="pre">Algebra]</span></tt> represents a factory of
 537 a polynomial ring over <tt class="docutils literal"><span class="pre">Algebra</span></tt> with <tt class="docutils literal"><span class="pre">symbols</span></tt>.</p>
 538 </div>
 539 <div class="section">
 540 <h1><a class="toc-backref" href="#id9" id="matrix-rings" name="matrix-rings">Matrix rings</a></h1>
 541 <p>SympyCore supports representing rectangular matrix ring elements using
 542 similar idea of ring factory:</p>
 543 <pre class="doctest-block">
 544 &gt;&gt;&gt; M = MatrixRing[(3,4)]
 545 &gt;&gt;&gt; matrix = M({(1,2):x+y, (0,0):x+z})
 546 &gt;&gt;&gt; print matrix
 547  x + z  0      0  0
 548      0  0  x + y  0
 549      0  0      0  0
 550 </pre>
 551 <p>Note that matrices are mutable in SympyCore and indexes start from 0:</p>
 552 <pre class="doctest-block">
 553 &gt;&gt;&gt; matrix[1,0] = 5
 554 &gt;&gt;&gt; print matrix
 555  x + z  0      0  0
 556      5  0  x + y  0
 557      0  0      0  0
 558 </pre>
 559 <p>SympyCore provides <tt class="docutils literal"><span class="pre">SquareMatrix</span></tt> and <tt class="docutils literal"><span class="pre">PermutationMatrix</span></tt>
 560 factories for convenience:</p>
 561 <pre class="doctest-block">
 562 &gt;&gt;&gt; SqM = SquareMatrix[3]
 563 &gt;&gt;&gt; m = SqM({(0,0): 1, (2,1): 3, (2,2):6, (1,2):-2, (2,0): -1})
 564 &gt;&gt;&gt; print m
 565   1  0   0
 566   0  0  -2
 567  -1  3   6
 568 &gt;&gt;&gt; print PermutationMatrix[4]([2,1,3,0])
 569  0  0  1  0
 570  0  1  0  0
 571  0  0  0  1
 572  1  0  0  0
 573 </pre>
 574 <p>One can perform LU factorization on any rectangular matrix:</p>
 575 <pre class="doctest-block">
 576 &gt;&gt;&gt; p, l, u = m.lu()
 577 &gt;&gt;&gt; print p
 578  1  0  0
 579  0  0  1
 580  0  1  0
 581 &gt;&gt;&gt; print l
 582   1  0  0
 583  -1  1  0
 584   0  0  1
 585 </pre>
 586 <pre class="doctest-block">
 587 &gt;&gt;&gt; print u
 588  1  0   0
 589  0  3   6
 590  0  0  -2
 591 </pre>
 592 <p>The <tt class="docutils literal"><span class="pre">*</span></tt> denotes matrix multiplication:</p>
 593 <pre class="doctest-block">
 594 &gt;&gt;&gt; print p * l * u == m
 595 True
 596 </pre>
 597 <p>SympyCore supports computing inverses of square
 598 matrices:</p>
 599 <pre class="doctest-block">
 600 &gt;&gt;&gt; print m.inv()
 601    1     0    0
 602  1/3     1  1/3
 603    0  -1/2    0
 604 </pre>
 605 <pre class="doctest-block">
 606 &gt;&gt;&gt; m.inv() * m == SqM.one
 607 True
 608 </pre>
 609 </div>
 610 <div class="section">
 611 <h1><a class="toc-backref" href="#id10" id="physical-units" name="physical-units">Physical units</a></h1>
 612 <p>SympyCore has a basic support for dealing with symbolic expressions with
 613 units:</p>
 614 <pre class="doctest-block">
 615 &gt;&gt;&gt; mass1 = 5 * kilogram
 616 &gt;&gt;&gt; mass2 = x * kilogram
 617 &gt;&gt;&gt; print mass1 + mass2
 618 (5 + x)*kg
 619 </pre>
 620 </div>
 621 </div>
 622 </body>
 623 </html>