sympy_py.py

   1 BASIC   = 0
   2 SYMBOL  = 1
   3 ADD     = 2
   4 MUL     = 3
   5 POW     = 4
   6 INTEGER = 5
   7
   8 def hash_seq(args):
   9     """
  10     Hash of a sequence, that *depends* on the order of elements.
  11     """
  12     # make this more robust:
  13     m = 2
  14     for x in args:
  15         m = hash(m + 1001 ^ hash(x))
  16     return m
  17
  18 class Basic(object):
  19
  20     def __new__(cls, type, args):
  21         obj = object.__new__(cls)
  22         obj.type = type
  23         obj._args = tuple(args)
  24         obj.mhash = None
  25         return obj
  26
  27     def __repr__(self):
  28         return str(self)
  29
  30     def __hash__(self):
  31         if self.mhash is None:
  32             h = hash_seq(self.args)
  33             self.mhash = h
  34             return h
  35         else:
  36             return self.mhash
  37
  38     @property
  39     def args(self):
  40         return self._args
  41
  42     def as_coeff_rest(self):
  43         return (Integer(1), self)
  44
  45     def as_base_exp(self):
  46         return (self, Integer(1))
  47
  48     def expand(self):
  49         return self
  50
  51     def __add__(x, y):
  52         return Add((x, y))
  53
  54     def __radd__(x, y):
  55         return x.__add__(y)
  56
  57     def __sub__(x, y):
  58         return Add((x, -y))
  59
  60     def __rsub__(x, y):
  61         return Add((y, -x))
  62
  63     def __mul__(x, y):
  64         return Mul((x, y))
  65
  66     def __rmul__(x, y):
  67         return Mul((y, x))
  68
  69     def __div__(x, y):
  70         return Mul((x, Pow((y, Integer(-1)))))
  71
  72     def __rdiv__(x, y):
  73         return Mul((y, Pow((x, Integer(-1)))))
  74
  75     def __pow__(x, y):
  76         return Pow((x, y))
  77
  78     def __rpow__(x, y):
  79         return Pow((y, x))
  80
  81     def __neg__(x):
  82         return Mul((Integer(-1), x))
  83
  84     def __pos__(x):
  85         return x
  86
  87     def __ne__(self, x):
  88         return not self.__eq__(x)
  89
  90     def __eq__(self, o):
  91         o = sympify(o)
  92         if o.type == self.type:
  93             return self.args == o.args
  94         else:
  95             return False
  96
  97
  98 class Integer(Basic):
  99
 100     def __new__(cls, i):
 101         obj = Basic.__new__(cls, INTEGER, [])
 102         obj.i = i
 103         return obj
 104
 105     def __hash__(self):
 106         if self.mhash is None:
 107             h = hash(self.i)
 108             self.mhash = h
 109             return h
 110         else:
 111             return self.mhash
 112
 113     def __eq__(self, o):
 114         o = sympify(o)
 115         if o.type == INTEGER:
 116             return self.i == o.i
 117         else:
 118             return False
 119
 120     def __str__(self):
 121         return str(self.i)
 122
 123     def __add__(self, o):
 124         o = sympify(o)
 125         if o.type == INTEGER:
 126             return Integer(self.i+o.i)
 127         return Basic.__add__(self, o)
 128
 129     def __mul__(self, o):
 130         o = sympify(o)
 131         if o.type == INTEGER:
 132             return Integer(self.i*o.i)
 133         return Basic.__mul__(self, o)
 134
 135
 136 class Symbol(Basic):
 137
 138     def __new__(cls, name):
 139         obj = Basic.__new__(cls, SYMBOL, [])
 140         obj.name = name
 141         return obj
 142
 143     def __hash__(self):
 144         if self.mhash is None:
 145             h = hash(self.name)
 146             self.mhash = h
 147             return h
 148         else:
 149             return self.mhash
 150
 151     def __eq__(self, o):
 152         o = sympify(o)
 153         if o.type == SYMBOL:
 154             return self.name == o.name
 155         return False
 156
 157     def __str__(self):
 158         return self.name
 159
 160
 161 class Add(Basic):
 162
 163     def __new__(cls, args, canonicalize=True):
 164         if canonicalize == False:
 165             obj = Basic.__new__(cls, ADD, args)
 166             obj._args_set = None
 167             return obj
 168         args = [sympify(x) for x in args]
 169         return Add.canonicalize(args)
 170
 171     @classmethod
 172     def canonicalize(cls, args):
 173         use_glib = 0
 174         if use_glib:
 175             from csympy import HashTable
 176             d = HashTable()
 177         else:
 178             d = {}
 179         num = Integer(0)
 180         for a in args:
 181             if a.type == INTEGER:
 182                 num += a
 183             elif a.type == ADD:
 184                 for b in a.args:
 185                     if b.type == INTEGER:
 186                         num += b
 187                     else:
 188                         coeff, key = b.as_coeff_rest()
 189                         if key in d:
 190                             d[key] += coeff
 191                         else:
 192                             d[key] = coeff
 193             else:
 194                 coeff, key = a.as_coeff_rest()
 195                 if key in d:
 196                     d[key] += coeff
 197                 else:
 198                     d[key] = coeff
 199         if len(d)==0:
 200             return num
 201         args = []
 202         for a, b in d.iteritems():
 203             args.append(Mul((a, b)))
 204         if num.i != 0:
 205             args.insert(0, num)
 206         if len(args) == 1:
 207             return args[0]
 208         else:
 209             return Add(args, False)
 210
 211     def freeze_args(self):
 212         #print "add is freezing"
 213         if self._args_set is None:
 214             self._args_set = frozenset(self.args)
 215         #print "done"
 216
 217     def __eq__(self, o):
 218         o = sympify(o)
 219         if o.type == ADD:
 220             self.freeze_args()
 221             o.freeze_args()
 222             return self._args_set == o._args_set
 223         else:
 224             return False
 225
 226     def __str__(self):
 227         s = str(self.args[0])
 228         if self.args[0].type == ADD:
 229             s = "(%s)" % str(s)
 230         for x in self.args[1:]:
 231             s = "%s + %s" % (s, str(x))
 232             if x.type == ADD:
 233                 s = "(%s)" % s
 234         return s
 235
 236     def __hash__(self):
 237         if self.mhash is None:
 238             # XXX: it is surprising, but this is *not* faster:
 239             #self.freeze_args()
 240             #h = hash(self._args_set)
 241
 242             # this is faster:
 243             a = list(self.args[:])
 244             a.sort(key=hash)
 245             h = hash_seq(a)
 246             self.mhash = h
 247             return h
 248         else:
 249             return self.mhash
 250
 251     def expand(self):
 252         r = []
 253         for term in self.args:
 254             r.append( term.expand() )
 255         return Add(r)
 256
 257 class Mul(Basic):
 258
 259     def __new__(cls, args, canonicalize=True):
 260         if canonicalize == False:
 261             obj = Basic.__new__(cls, MUL, args)
 262             obj._args_set = None
 263             return obj
 264         args = [sympify(x) for x in args]
 265         return Mul.canonicalize(args)
 266
 267     @classmethod
 268     def canonicalize(cls, args):
 269         use_glib = 0
 270         if use_glib:
 271             from csympy import HashTable
 272             d = HashTable()
 273         else:
 274             d = {}
 275         num = Integer(1)
 276         for a in args:
 277             if a.type == INTEGER:
 278                 num *= a
 279             elif a.type == MUL:
 280                 for b in a.args:
 281                     if b.type == INTEGER:
 282                         num *= b
 283                     else:
 284                         key, coeff = b.as_base_exp()
 285                         if key in d:
 286                             d[key] += coeff
 287                         else:
 288                             d[key] = coeff
 289             else:
 290                 key, coeff = a.as_base_exp()
 291                 if key in d:
 292                     d[key] += coeff
 293                 else:
 294                     d[key] = coeff
 295         if num.i == 0 or len(d)==0:
 296             return num
 297         args = []
 298         for a, b in d.iteritems():
 299             args.append(Pow((a, b)))
 300         if num.i != 1:
 301             args.insert(0, num)
 302         if len(args) == 1:
 303             return args[0]
 304         else:
 305             return Mul(args, False)
 306
 307     def __hash__(self):
 308         if self.mhash is None:
 309             # in contrast to Add, here it is faster:
 310             self.freeze_args()
 311             h = hash(self._args_set)
 312             # this is slower:
 313             #a = list(self.args[:])
 314             #a.sort(key=hash)
 315             #h = hash_seq(a)
 316             self.mhash = h
 317             return h
 318         else:
 319             return self.mhash
 320
 321     def freeze_args(self):
 322         #print "mul is freezing"
 323         if self._args_set is None:
 324             self._args_set = frozenset(self.args)
 325         #print "done"
 326
 327     def __eq__(self, o):
 328         o = sympify(o)
 329         if o.type == MUL:
 330             self.freeze_args()
 331             o.freeze_args()
 332             return self._args_set == o._args_set
 333         else:
 334             return False
 335
 336
 337     def as_coeff_rest(self):
 338         if self.args[0].type == INTEGER:
 339             return self.as_two_terms()
 340         return (Integer(1), self)
 341
 342     def as_two_terms(self):
 343         args = self.args
 344         a0   = args[0]
 345
 346         if len(args) == 2:
 347             return a0, args[1]
 348         else:
 349             return (a0, Mul(args[1:], False))
 350
 351
 352     def __str__(self):
 353         s = str(self.args[0])
 354         if self.args[0].type in [ADD, MUL]:
 355             s = "(%s)" % str(s)
 356         for x in self.args[1:]:
 357             if x.type in [ADD, MUL]:
 358                 s = "%s * (%s)" % (s, str(x))
 359             else:
 360                 s = "%s*%s" % (s, str(x))
 361         return s
 362
 363     @classmethod
 364     def expand_two(self, a, b):
 365         """
 366         Both a and b are assumed to be expanded.
 367         """
 368         if a.type == ADD and b.type == ADD:
 369             terms = []
 370             for x in a.args:
 371                 for y in b.args:
 372                     terms.append(x*y)
 373             return Add(terms)
 374         if a.type == ADD:
 375             terms = []
 376             for x in a.args:
 377                 terms.append(x*b)
 378             return Add(terms)
 379         if b.type == ADD:
 380             terms = []
 381             for y in b.args:
 382                 terms.append(a*y)
 383             return Add(terms)
 384         return a*b
 385
 386     def expand(self):
 387         a, b = self.as_two_terms()
 388         r = Mul.expand_two(a, b)
 389         if r == self:
 390             a = a.expand()
 391             b = b.expand()
 392             return Mul.expand_two(a, b)
 393         else:
 394             return r.expand()
 395
 396 class Pow(Basic):
 397
 398     def __new__(cls, args, canonicalize=True):
 399         if canonicalize == False:
 400             obj = Basic.__new__(cls, POW, args)
 401             return obj
 402         args = [sympify(x) for x in args]
 403         return Pow.canonicalize(args)
 404
 405     @classmethod
 406     def canonicalize(cls, args):
 407         base, exp = args
 408         if base.type == INTEGER:
 409             if base.i == 0:
 410                 return Integer(0)
 411             if base.i == 1:
 412                 return Integer(1)
 413         if exp.type == INTEGER:
 414             if exp.i == 0:
 415                 return Integer(1)
 416             if exp.i == 1:
 417                 return base
 418         if base.type == POW:
 419             return Pow((base.args[0], base.args[1]*exp))
 420         return Pow(args, False)
 421
 422     def __str__(self):
 423         s = str(self.args[0])
 424         if self.args[0].type == ADD:
 425             s = "(%s)" % s
 426         if self.args[1].type == ADD:
 427             s = "%s^(%s)" % (s, str(self.args[1]))
 428         else:
 429             s = "%s^%s" % (s, str(self.args[1]))
 430         return s
 431
 432     def as_base_exp(self):
 433         return self.args
 434
 435     def expand(self):
 436         base, exp = self.args
 437         if base.type == ADD and exp.type == INTEGER:
 438             n = exp.i
 439             m = len(base.args)
 440             #print "multi"
 441             d = multinomial_coefficients(m, n)
 442             #print "assembly"
 443             r = []
 444             for powers, coeff in d.iteritems():
 445                 if coeff == 1:
 446                     t = []
 447                 else:
 448                     t = [Integer(coeff)]
 449                 for x, p in zip(base.args, powers):
 450                     if p != 0:
 451                         t.append(Pow((x, p)))
 452                 assert len(t) != 0
 453                 if len(t) == 1:
 454                     t = t[0]
 455                 else:
 456                     t = Mul(t, False)
 457                 r.append(t)
 458             r = Add(r, False)
 459             #print "done"
 460             return r
 461         return self
 462
 463 def sympify(x):
 464     if isinstance(x, int):
 465         return Integer(x)
 466     return x
 467
 468 def var(s):
 469     """
 470         Create a symbolic variable with the name *s*.
 471
 472         INPUT:
 473             s -- a string, either a single variable name, or
 474                  a space separated list of variable names, or
 475                  a list of variable names.
 476
 477         NOTE: The new variable is both returned and automatically injected into
 478         the parent's *global* namespace.  It's recommended not to use "var" in
 479         library code, it is better to use symbols() instead.
 480
 481         EXAMPLES:
 482         We define some symbolic variables:
 483             >>> var('m')
 484             m
 485             >>> var('n xx yy zz')
 486             (n, xx, yy, zz)
 487             >>> n
 488             n
 489
 490     """
 491     import re
 492     import inspect
 493     frame = inspect.currentframe().f_back
 494
 495     try:
 496         if not isinstance(s, list):
 497             s = re.split('\s|,', s)
 498
 499         res = []
 500
 501         for t in s:
 502             # skip empty strings
 503             if not t:
 504                 continue
 505             sym = Symbol(t)
 506             frame.f_globals[t] = sym
 507             res.append(sym)
 508
 509         res = tuple(res)
 510         if len(res) == 0:   # var('')
 511             res = None
 512         elif len(res) == 1: # var('x')
 513             res = res[0]
 514                             # otherwise var('a b ...')
 515         return res
 516
 517     finally:
 518         # we should explicitly break cyclic dependencies as stated in inspect
 519         # doc
 520         del frame
 521
 522 def binomial_coefficients(n):
 523     """Return a dictionary containing pairs {(k1,k2) : C_kn} where
 524     C_kn are binomial coefficients and n=k1+k2."""
 525     d = {(0, n):1, (n, 0):1}
 526     a = 1
 527     for k in xrange(1, n//2+1):
 528         a = (a * (n-k+1))//k
 529         d[k, n-k] = d[n-k, k] = a
 530     return d
 531
 532 def binomial_coefficients_list(n):
 533     """ Return a list of binomial coefficients as rows of the Pascal's
 534     triangle.
 535     """
 536     d = [1] * (n+1)
 537     a = 1
 538     for k in xrange(1, n//2+1):
 539         a = (a * (n-k+1))//k
 540         d[k] = d[n-k] = a
 541     return d
 542
 543 def multinomial_coefficients(m, n, _tuple=tuple, _zip=zip):
 544     """Return a dictionary containing pairs ``{(k1,k2,..,km) : C_kn}``
 545     where ``C_kn`` are multinomial coefficients such that
 546     ``n=k1+k2+..+km``.
 547
 548     For example:
 549
 550     >>> print multinomial_coefficients(2,5)
 551     {(3, 2): 10, (1, 4): 5, (2, 3): 10, (5, 0): 1, (0, 5): 1, (4, 1): 5}
 552
 553     The algorithm is based on the following result:
 554
 555        Consider a polynomial and it's ``m``-th exponent::
 556
 557          P(x) = sum_{i=0}^m p_i x^k
 558          P(x)^n = sum_{k=0}^{m n} a(n,k) x^k
 559
 560        The coefficients ``a(n,k)`` can be computed using the
 561        J.C.P. Miller Pure Recurrence [see D.E.Knuth, Seminumerical
 562        Algorithms, The art of Computer Programming v.2, Addison
 563        Wesley, Reading, 1981;]::
 564
 565          a(n,k) = 1/(k p_0) sum_{i=1}^m p_i ((n+1)i-k) a(n,k-i),
 566
 567        where ``a(n,0) = p_0^n``.
 568     """
 569
 570     if m==2:
 571         return binomial_coefficients(n)
 572     symbols = [(0,)*i + (1,) + (0,)*(m-i-1) for i in range(m)]
 573     s0 = symbols[0]
 574     p0 = [_tuple(aa-bb for aa,bb in _zip(s,s0)) for s in symbols]
 575     r = {_tuple(aa*n for aa in s0):1}
 576     r_get = r.get
 577     r_update = r.update
 578     l = [0] * (n*(m-1)+1)
 579     l[0] = r.items()
 580     for k in xrange(1, n*(m-1)+1):
 581         d = {}
 582         d_get = d.get
 583         for i in xrange(1, min(m,k+1)):
 584             nn = (n+1)*i-k
 585             if not nn:
 586                 continue
 587             t = p0[i]
 588             for t2, c2 in l[k-i]:
 589                 tt = _tuple([aa+bb for aa,bb in _zip(t2,t)])
 590                 cc = nn * c2
 591                 b = d_get(tt)
 592                 if b is None:
 593                     d[tt] = cc
 594                 else:
 595                     cc = b + cc
 596                     if cc:
 597                         d[tt] = cc
 598                     else:
 599                         del d[tt]
 600         r1 = [(t, c//k) for (t, c) in d.iteritems()]
 601         l[k] = r1
 602         r_update(r1)
 603     return r