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