NZMATH  1.2.0 About: NZMATH is a Python based number theory oriented calculation system. Fossies Dox: NZMATH-1.2.0.tar.gz  ("inofficial" and yet experimental doxygen-generated source code documentation) multiutil.py
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1 """
2 Multivariate polynomial extenders.
3 """
4
5 from __future__ import division
6 import nzmath.rational as rational
7 import nzmath.ring as ring
8 import nzmath.poly.termorder as termorder
9 import nzmath.poly.ring as poly_ring
10 import nzmath.poly.uniutil as uniutil
11 import nzmath.poly.multivar as multivar
12 import nzmath.poly.ratfunc as ratfunc
13
14
15 _MIXIN_MSG = "%s is mix-in"
16
17
18 class OrderProvider (object):
19  """
20  OrderProvider provides order and related operations.
21  """
22  def __init__(self, order):
23  """
24  Do not instantiate OrderProvider.
25  This initializer should be called from descendant:
26  OrderProvider.__init__(self, order)
27  """
28  if type(self) is OrderProvider:
29  raise NotImplementedError(_MIXIN_MSG % self.__class__.__name__)
30  self.order = order
31
32
33 class NestProvider (object):
34  """
35  Provide nest/unnest pair to convert a multivar polynomial to a
36  univar polynomial of polynomial coefficient and opposite
37  direction.
38  """
40  """
42  among all total degree one terms).
43
44  The leading term varies with term orders, so does the result.
45  The term order can be specified via the attribute 'order'.
46  """
47  if hasattr(self, 'order'):
48  order = self.order
49  pivot, pvar = (1,) + (0,)*(self.number_of_variables - 1), 0
50  for var in range(1, self.number_of_variables):
51  vindex = (0,)*var + (1,) + (0,)*(self.number_of_variables - var - 1)
52  if order.cmp(pivot, vindex) < 0:
53  pivot, pvar = vindex, var
54  else:
55  pvar = 0
56  return pvar
57
58  def nest(self, outer, coeffring):
59  """
60  Nest the polynomial by extracting outer variable at the given
61  position.
62  """
63  combined = {}
64  if self.number_of_variables == 2:
65  itercoeff = lambda coeff: [(i, c) for (i, c) in coeff]
66  poly = uniutil.polynomial
67  polyring = poly_ring.PolynomialRing.getInstance(coeffring)
68  elif self.number_of_variables >= 3:
69  itercoeff = lambda coeff: coeff
70  poly = self.__class__
71  polyring = poly_ring.PolynomialRing.getInstance(coeffring, self.number_of_variables - 1)
72  else:
73  raise TypeError("number of variable is not multiple")
74  for index, coeff in self.combine_similar_terms(outer):
75  combined[index] = poly(itercoeff(coeff), coeffring=coeffring)
76  return uniutil.polynomial(combined, coeffring=polyring)
77
78  def unnest(self, q, outer, coeffring):
79  """
80  Unnest the nested polynomial q by inserting outer variable at
81  the given position.
82  """
83  q_coeff = {}
84  for d, cpoly in q:
85  for inner_d, inner_c in cpoly:
86  if isinstance(inner_d, (int, long)):
87  inner_d = [inner_d]
88  else:
89  inner_d = list(inner_d)
90  inner_d.insert(outer, d)
91  q_coeff[tuple(inner_d)] = inner_c
92  return self.__class__(q_coeff, coeffring=coeffring)
93
94
96  """
97  Provides interfaces for ring.CommutativeRingElement.
98  """
99  def __init__(self):
100  """
101  Do not instantiate RingElementProvider.
102  This initializer should be called from descendant:
103  RingElementProvider.__init__(self)
104  """
105  if type(self) is RingElementProvider:
106  raise NotImplementedError(_MIXIN_MSG % self.__class__.__name__)
107  ring.CommutativeRingElement.__init__(self)
108  self._coefficient_ring = None
109  self._ring = None
110
111  def getRing(self):
112  """
113  Return an object of a subclass of Ring, to which the element
114  belongs.
115  """
116  if self._coefficient_ring is None or self._ring is None:
117  myring = None
118  for c in self.itercoefficients():
119  cring = ring.getRing(c)
120  if not myring or myring != cring and myring.issubring(cring):
121  myring = cring
122  elif not cring.issubring(myring):
123  myring = myring.getCommonSuperring(cring)
124  if not myring:
125  myring = rational.theIntegerRing
126  self.set_coefficient_ring(myring)
127  return self._ring
128
130  """
131  Return the coefficient ring.
132  """
133  return self._coefficient_ring
134
135  def set_coefficient_ring(self, coeffring):
136  if self._coefficient_ring is None:
137  self._coefficient_ring = coeffring
138  self._ring = poly_ring.PolynomialRing.getInstance(self._coefficient_ring, self.number_of_variables)
139
140
141 class PseudoDivisionProvider (object):
142  """
143  PseudoDivisionProvider provides pseudo divisions for multivariate
144  polynomials. It is assumed that the coefficient ring of the
145  polynomials is a domain.
146
147  The class should be used with NestProvider, RingElementProvider.
148  """
149  def pseudo_divmod(self, other):
150  """
151  self.pseudo_divmod(other) -> (Q, R)
152
153  Q, R are polynomials such that
154  d**(deg(self) - deg(other) + 1) * self == other * Q + R,
155  w.r.t. a fixed variable, where d is the leading coefficient of
156  other.
157
158  The leading coefficient varies with term orders, so does the
159  result. The term order can be specified via the attribute
160  'order'.
161  """
163  coeffring = self.getCoefficientRing()
164
165  s = self.nest(var, coeffring)
166  o = other.nest(var, coeffring)
167  q, r = s.pseudo_divmod(o)
168  qpoly = self.unnest(q, var, coeffring)
169  rpoly = self.unnest(r, var, coeffring)
170  return qpoly, rpoly
171
172  def pseudo_floordiv(self, other):
173  """
174  self.pseudo_floordiv(other) -> Q
175
176  Q is a polynomial such that
177  d**(deg(self) - deg(other) + 1) * self == other * Q + R,
178  where d is the leading coefficient of other and R is a
179  polynomial.
180
181  The leading coefficient varies with term orders, so does the
182  result. The term order can be specified via the attribute
183  'order'.
184  """
186  coeffring = self.getCoefficientRing()
187
188  s = self.nest(var, coeffring)
189  o = other.nest(var, coeffring)
190  q = s.pseudo_floordiv(o)
191  return self.unnest(q, var, coeffring)
192
193  def pseudo_mod(self, other):
194  """
195  self.pseudo_mod(other) -> R
196
197  R is a polynomial such that
198  d**(deg(self) - deg(other) + 1) * self == other * Q + R,
199  where d is the leading coefficient of other and Q a
200  polynomial.
201
202  The leading coefficient varies with term orders, so does the
203  result. The term order can be specified via the attribute
204  'order'.
205  """
207  coeffring = self.getCoefficientRing()
208
209  s = self.nest(var, coeffring)
210  o = other.nest(var, coeffring)
211  r = s.pseudo_mod(o)
212  return self.unnest(r, var, coeffring)
213
214  def __truediv__(self, other):
215  """
216  self / other
217
218  The result is a rational function.
219  """
220  return ratfunc.RationalFunction(self, other)
221
222  def exact_division(self, other):
223  """
224  Return quotient of exact division.
225  """
226  coeffring = self.getCoefficientRing()
227  if other in coeffring:
228  new_coeffs = []
229  keep_ring = True
230  for i, c in self:
231  ratio = c / other
232  if keep_ring and ratio not in coeffring:
233  keep_ring = False
234  new_coeffring = ratio.getRing()
235  new_coeffs.append((i, ratio))
236  if keep_ring:
237  return self.__class__(new_coeffs, coeffring=coeffring)
238  else:
239  return self.__class__(new_coeffs, coeffring=new_coeffring)
240
242  coeffring = self.getCoefficientRing()
243  s = self.nest(var, coeffring)
244  o = other.nest(var, coeffring)
245  q = s.exact_division(o)
246  return self.unnest(q, var, coeffring)
247
248
249 class GcdProvider (object):
250  """
251  Provides greatest common divisor for multivariate polynomial.
252
253  The class should be used with NestProvider, RingElementProvider.
254  """
255  def gcd(self, other):
256  """
257  Return gcd.
258  The nested polynomials' gcd is used.
259  """
261  coeffring = self.getCoefficientRing()
262  s = self.nest(var, coeffring)
263  o = other.nest(var, coeffring)
264  if hasattr(s, "gcd"):
265  g = s.gcd(o)
266  elif hasattr(s, "subresultant_gcd"):
267  g = s.subresultant_gcd(o)
268  else:
269  raise TypeError("no gcd method available")
270  return self.unnest(g, var, coeffring)
271
272
274  NestProvider,
276  RingElementProvider):
277  """
278  General polynomial with commutative ring coefficients.
279  """
280  def __init__(self, coefficients, **kwds):
281  """
282  Initialize the polynomial.
283
284  Required argument:
285  - coefficients: initializer for polynomial coefficients
286
287  Keyword arguments should include:
288  - coeffring: domain
289  - number_of_variables: the number of variables
290  """
291  if "number_of_variables" not in kwds:
292  coefficients = dict(coefficients)
293  for i in coefficients.iterkeys():
294  kwds["number_of_variables"] = len(i)
295  break
296  multivar.BasicPolynomial.__init__(self, coefficients, **kwds)
297  NestProvider.__init__(self)
298  PseudoDivisionProvider.__init__(self)
299  RingElementProvider.__init__(self)
300  coeffring = None
301  if "coeffring" in kwds:
302  coeffring = kwds["coeffring"]
303  else:
304  coeffring = uniutil.init_coefficient_ring(self._coefficients)
305  if coeffring is not None:
306  self.set_coefficient_ring(coeffring)
307  else:
308  raise TypeError("argument `coeffring' is required")
309  if "order" in kwds:
310  order = kwds["order"]
311  else:
312  order = termorder.lexicographic_order
313  OrderProvider.__init__(self, order)
314
315  def getRing(self):
316  """
317  Return an object of a subclass of Ring, to which the element
318  belongs.
319  """
320  # short-cut self._ring is None case
321  return self._ring
322
324  """
325  Return an object of a subclass of Ring, to which the all
326  coefficients belong.
327  """
328  # short-cut self._coefficient_ring is None case
329  return self._coefficient_ring
330
331  def __repr__(self): # debug
332  return "%s(%s)" % (self.__class__.__name__, self._coefficients)
333
335  try:
337  except (AttributeError, TypeError):
338  one = self.getRing().one
339  try:
340  return multivar.BasicPolynomial.__add__(self, other * one)
341  except Exception:
342  return NotImplemented
343
345  one = self.getRing().one
346  try:
347  return other * one + self
348  except Exception:
349  return NotImplemented
350
351  def __sub__(self, other):
352  try:
353  return multivar.BasicPolynomial.__sub__(self, other)
354  except (AttributeError, TypeError):
355  one = self.getRing().one
356  try:
357  return multivar.BasicPolynomial.__sub__(self, other * one)
358  except Exception:
359  return NotImplemented
360
361  def __rsub__(self, other):
362  one = self.getRing().one
363  try:
364  return other * one - self
365  except Exception:
366  return NotImplemented
367
368
370  RingPolynomial):
371  """
372  Polynomial with domain coefficients.
373  """
374  def __init__(self, coefficients, **kwds):
375  """
376  Initialize the polynomial.
377
378  - coefficients: initializer for polynomial coefficients
379  - coeffring: domain
380  """
381  RingPolynomial.__init__(self, coefficients, **kwds)
382  if not self._coefficient_ring.isdomain():
383  raise TypeError("coefficient ring has to be a domain")
384  PseudoDivisionProvider.__init__(self)
385
386
388  DomainPolynomial):
389  """
390  Polynomial with unique factorization domain coefficients.
391  """
392  def __init__(self, coefficients, **kwds):
393  """
394  Initialize the polynomial.
395
396  - coefficients: initializer for polynomial coefficients
397  - coeffring: unique factorization domain
398  """
399  DomainPolynomial.__init__(self, coefficients, **kwds)
400  if not self._coefficient_ring.isufd():
401  raise TypeError("coefficient ring has to be a UFD")
402  GcdProvider.__init__(self)
403
404  def resultant(self, other, var):
405  """
406  Return resultant of two polynomials of the same ring, with
407  respect to the variable specified by its position var.
408  """
409  cring = self._coefficient_ring
410  return self.nest(var, cring).resultant(other.nest(var, cring))
411
412
414  """
415  The class of multivariate polynomial ring.
416  There's no need to specify the variable names.
417  """
418
419  _instances = {}
420
421  def __init__(self, coeffring, number_of_variables):
422  if not isinstance(coeffring, ring.Ring):
423  raise TypeError("%s should not be passed as ring" % coeffring.__class__)
424  ring.CommutativeRing.__init__(self)
425  self._coefficient_ring = coeffring
426  if self._coefficient_ring.isufd():
427  self.properties.setIsufd(True)
429  self.properties.setIsnoetherian(True)
430  elif self._coefficient_ring.isdomain() in (True, False):
431  self.properties.setIsdomain(self._coefficient_ring.isdomain())
432  self.number_of_variables = number_of_variables
433
435  """
436  Return the coefficient ring.
437  """
438  return self._coefficient_ring
439
440  def getQuotientField(self):
441  """
442  getQuotientField returns the quotient field of the ring
443  if coefficient ring has its quotient field. Otherwise,
444  an exception will be raised.
445  """
446  coefficientField = self._coefficient_ring.getQuotientField()
447  variables = ["x%d" % i for i in range(self.number_of_variables)]
448  return ratfunc.RationalFunctionField(coefficientField, variables)
449
450  def __eq__(self, other):
451  if self is other:
452  return True
453  if (isinstance(other, PolynomialRingAnonymousVariables) and
454  self._coefficient_ring == other._coefficient_ring and
455  self.number_of_variables == other.number_of_variables):
456  return True
457  return False
458
459  def __repr__(self):
460  """
461  Return 'PolynomialRingAnonymousVariables(Ring, #vars)'
462  """
463  return "%s(%s, %d)" % (self.__class__.__name__, repr(self._coefficient_ring), self.number_of_variables)
464
465  def __str__(self):
466  """
467  Return R[][]
468  """
469  return str(self._coefficient_ring) + "[]" * self.number_of_variables
470
471  def __hash__(self):
472  """
473  hash(self)
474  """
475  return (hash(self._coefficient_ring) ^ (self.number_of_variables * hash(self.__class__.__name__) + 1)) & 0x7fffffff
476
477  def __contains__(self, element):
478  """
479  `in' operator is provided for checking the element be in the
480  ring.
481  """
482  if element in self._coefficient_ring:
483  return True
484  elem_ring = ring.getRing(element)
485  if elem_ring is not None and elem_ring.issubring(self):
486  return True
487  return False
488
489  def issubring(self, other):
490  """
491  reports whether another ring contains this polynomial ring.
492  """
493  if isinstance(other, poly_ring.PolynomialRing):
494  if (self._coefficient_ring.issubring(other.getCoefficientRing()) and
495  self.number_of_variables <= other.number_of_variables):
496  return True
497  elif isinstance(other, poly_ring.RationalFunctionField):
498  if (len(other.vars) >= self.number_of_variables and
499  other.coefficientField.issuperring(self._coefficient_ring)):
500  return True
501  try:
502  return other.issuperring(self)
503  except RuntimeError:
504  # reach max recursion by calling each other
505  return False
506
507  def issuperring(self, other):
508  """
509  reports whether this polynomial ring contains another ring.
510  """
511  if self._coefficient_ring.issuperring(other):
512  return True
513  if isinstance(other, poly_ring.PolynomialRing):
514  return (self._coefficient_ring.issuperring(other.getCoefficientRing()) and
515  self.number_of_variables >= other.number_of_variables)
516  try:
517  return other.issubring(self)
518  except RuntimeError:
519  # reach max recursion by calling each other
520  return False
521
522  def getCommonSuperring(self, other):
523  """
524  Return common superring of two rings.
525  """
526  if self.issuperring(other):
527  return self
528  elif other.issuperring(self):
529  return other
530  elif (not isinstance(other, PolynomialRingAnonymousVariables) and
531  other.issuperring(self._coefficient_ring)):
532  return self.__class__(other, self.number_of_variables)
533  try:
534  if hasattr(other, "getCommonSuperring"):
535  return other.getCommonSuperring(self)
536  except RuntimeError:
537  # reached recursion limit by calling on each other
538  pass
539  raise TypeError("no common super ring")
540
541  def createElement(self, seed):
542  """
543  Return an element of the polynomial ring made from seed
544  overriding ring.createElement.
545  """
546  if not seed:
547  return polynomial((), coeffring=self._coefficient_ring, number_of_variables=self.number_of_variables)
548  elif seed in self._coefficient_ring:
549  return polynomial([((0,)*self.number_of_variables, seed)], coeffring=self._coefficient_ring)
550  # implementation should be replaced later
551  raise NotImplementedError("unclear which type of polynomial be chosen")
552
553  def _getOne(self):
554  "getter for one"
555  if self._one is None:
556  self._one = self.createElement(self._coefficient_ring.one)
557  return self._one
558
559  one = property(_getOne, None, None, "multiplicative unit")
560
561  def _getZero(self):
562  "getter for zero"
563  if self._zero is None:
564  self._zero = self.createElement(self._coefficient_ring.zero)
565  return self._zero
566
567  zero = property(_getZero, None, None, "additive unit")
568
569  def gcd(self, a, b):
570  """
571  Return the greatest common divisor of given polynomials.
572  The polynomials must be in the polynomial ring.
573  If the coefficient ring is a field, the result is monic.
574  """
575  if hasattr(a, "gcd"):
576  return a.gcd(b)
577  elif hasattr(a, "subresultant_gcd"):
578  return a.subresultant_gcd(b)
579  raise NotImplementedError("no gcd")
580
581  def extgcd(self, a, b):
582  """
583  Return the tuple (u, v, d): d is the greatest common divisor
584  of given polynomials, and they satisfy d = u*a + v*b. The
585  polynomials must be in the polynomial ring. If the
586  coefficient ring is a field, the result is monic.
587  """
588  if hasattr(a, "extgcd"):
589  return a.extgcd(b)
590  raise NotImplementedError("no extgcd")
591
592  @classmethod
593  def getInstance(cls, coeffring, number_of_variables):
594  """
595  Return an instance of the class with specified coefficient ring
596  and number of variables.
597  """
598  if (coeffring, number_of_variables) not in cls._instances:
599  cls._instances[coeffring, number_of_variables] = cls(coeffring, number_of_variables)
600  return cls._instances[coeffring, number_of_variables]
601
602
604  """
605  Multivariate polynomial ideal.
606  """
607  def __init__(self, generators, aring):
608  """
609  Initialize a polynomial ideal.
610  """
611  ring.Ideal.__init__(self, generators, aring)
612
613  def __contains__(self, elem):
614  """
615  Return whether elem is in the ideal or not.
616  """
617  if not elem.getRing().issubring(self.ring):
618  return False
619  if self.generators == [self.ring.zero]:
620  return elem == self.ring.zero
621  return not self.reduce(elem)
622
623  def __nonzero__(self):
624  """
625  Report whether the ideal is zero ideal or not. Of course,
626  False is for zero ideal.
627  """
628  return self.generators and self.generators != [self.ring.zero]
629
630  def __repr__(self):
631  """
632  Return repr string.
633  """
634  return "%s(%r, %r)" % (self.__class__.__name__, self.generators, self.ring)
635
636  def __str__(self):
637  """
638  Return str string.
639  """
640  return "(%s)%s" % (", ".join([str(g) for g in self.generators]), self.ring)
641
642
643 # factories
644
645 special_ring_table = {}
646
647 def polynomial(coefficients, coeffring, number_of_variables=None):
648  """
649  Return a polynomial.
650  - coefficients has to be a initializer for dict, whose keys are
651  variable indices and values are coefficients at the indices.
652  - coeffring has to be an object inheriting ring.Ring.
653  - number_of_variables has to be the number of variables.
654
655  One can override the way to choose a polynomial type from a
656  coefficient ring, by setting:
657  special_ring_table[coeffring_type] = polynomial_type
658  before the function call.
659  """
660  if type(coeffring) in special_ring_table:
661  poly_type = special_ring_table[type(coeffring)]
662  elif coeffring.isufd():
663  poly_type = UniqueFactorizationDomainPolynomial
664  elif coeffring.isdomain():
665  poly_type = DomainPolynomial
666  else:
667  poly_type = multivar.BasicPolynomial
668  if number_of_variables is None:
669  coefficients = dict(coefficients)
670  for k in coefficients:
671  number_of_variables = len(k)
672  break
673  return poly_type(coefficients, coeffring=coeffring, number_of_variables=number_of_variables)
674
675 def MultiVariableSparsePolynomial(coefficient, variable, coeffring=None):
676  """
677  MultiVariableSparsePolynomial(coefficient, variable [,coeffring])
678
679  - coefficient has to be a dictionary of form {(i1,...,ik): c}
680  - variable has to be a list of character strings.
681  - coeffring has to be, if specified, an object inheriting ring.Ring.
682
683  This function is provided for backward compatible way of defining
684  multivariate polynomial. The variable names are ignored, but
685  their number is used.
686  """
687  if not isinstance(variable, list) or not isinstance(coefficient, dict):
688  raise ValueError("You must input MultiVariableSparsePolynomial(dict, list) but (%s, %s)." % (coefficient.__class__, variable.__class__))
689  if coeffring is None:
690  coeffring = uniutil.init_coefficient_ring(coefficient)
691  return polynomial(coefficient, coeffring=coeffring, number_of_variables=len(variable))
692
693 def prepare_indeterminates(names, ctx, coeffring=None):
694  """
695  From space separated names of indeterminates, prepare variables
696  representing the indeterminates. The result will be stored in ctx
697  dictionary.
698
699  The variables should be prepared at once, otherwise wrong aliases
700  of variables may confuse you in later calculation.
701
702  If an optional coeffring is not given, indeterminates will be
703  initialized as integer coefficient polynomials.
704
705  Example:
706  >>> prepare_indeterminates("X Y Z", globals())
707  >>> Y
708  UniqueFactorizationDomainPolynomial({(0, 1, 0): 1})
709
710  """
711  split_names = names.split()
712  number_of_variables = len(split_names)
713  if coeffring is None:
714  coeffring = uniutil.init_coefficient_ring({1:1})
715  for i, name in enumerate(split_names):
716  e_i = tuple( * i +  +  * (number_of_variables - i - 1))
717  ctx[name] = polynomial({e_i: 1}, coeffring, number_of_variables)
718
nzmath.poly.multiutil.RingElementProvider.__init__
def __init__(self)
Definition: multiutil.py:99
nzmath.poly.multiutil.PseudoDivisionProvider.pseudo_divmod
def pseudo_divmod(self, other)
Definition: multiutil.py:149
nzmath.bigrange.range
def range(start, stop=None, step=None)
Definition: bigrange.py:19
nzmath.ring
Definition: ring.py:1
nzmath.poly.multiutil.PolynomialRingAnonymousVariables.getCommonSuperring
def getCommonSuperring(self, other)
Definition: multiutil.py:522
nzmath.poly.multiutil.PolynomialRingAnonymousVariables._getZero
def _getZero(self)
Definition: multiutil.py:561
nzmath.ring.CommutativeRing.isdomain
def isdomain(self)
Definition: ring.py:114
nzmath.poly.multiutil.PolynomialRingAnonymousVariables.createElement
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Definition: multiutil.py:541
nzmath.poly.multiutil.PseudoDivisionProvider.pseudo_mod
def pseudo_mod(self, other)
Definition: multiutil.py:193
nzmath.poly.multiutil.PolynomialRingAnonymousVariables.__hash__
def __hash__(self)
Definition: multiutil.py:471
nzmath.poly.multiutil.RingPolynomial.__repr__
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Definition: multiutil.py:331
nzmath.poly.multiutil.PolynomialRingAnonymousVariables.number_of_variables
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Definition: multiutil.py:432
Definition: multiutil.py:334
nzmath.poly.multiutil.PseudoDivisionProvider.exact_division
def exact_division(self, other)
Definition: multiutil.py:222
nzmath.poly.multiutil.RingPolynomial.__rsub__
def __rsub__(self, other)
Definition: multiutil.py:361
nzmath.poly.multiutil.NestProvider.number_of_variables
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Definition: multiutil.py:64
nzmath.poly.multiutil.MultiVariableSparsePolynomial
def MultiVariableSparsePolynomial(coefficient, variable, coeffring=None)
Definition: multiutil.py:675
nzmath.poly.multiutil.RingPolynomial.__sub__
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Definition: multiutil.py:351
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Definition: termorder.py:1
nzmath.poly.ring
Definition: ring.py:1
nzmath.poly.multiutil.PolynomialRingAnonymousVariables.__repr__
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Definition: multiutil.py:459
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Definition: ring.py:121
nzmath.poly.multiutil.PolynomialIdeal.__contains__
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Definition: multiutil.py:613
nzmath.poly.multiutil.PolynomialRingAnonymousVariables.issuperring
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Definition: multiutil.py:507
nzmath.poly.multiutil.PolynomialIdeal.__init__
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Definition: multiutil.py:607
nzmath.poly.multiutil.PolynomialRingAnonymousVariables._instances
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Definition: multiutil.py:419
nzmath.poly.multiutil.UniqueFactorizationDomainPolynomial.resultant
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Definition: multiutil.py:404
nzmath.ring.Ring
Definition: ring.py:8
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Definition: multiutil.py:78
nzmath.poly.multiutil.GcdProvider.gcd
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Definition: multiutil.py:255
nzmath.poly.multiutil.PseudoDivisionProvider.pseudo_floordiv
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Definition: multiutil.py:172
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Definition: rational.py:1
nzmath.poly.multiutil.PolynomialRingAnonymousVariables.gcd
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Definition: multiutil.py:569
nzmath.poly.multiutil.PolynomialRingAnonymousVariables.__str__
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Definition: multiutil.py:465
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Definition: multiutil.py:323
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Definition: ring.py:88
nzmath.poly.multiutil.PolynomialRingAnonymousVariables.getInstance
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Definition: multiutil.py:593
nzmath.poly.multiutil.RingElementProvider._ring
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Definition: multiutil.py:109
Definition: multiutil.py:344
nzmath.poly.multiutil.PolynomialIdeal.__nonzero__
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Definition: multiutil.py:623
nzmath.poly.multiutil.PolynomialRingAnonymousVariables.__contains__
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Definition: multiutil.py:477
nzmath.poly.multiutil.PolynomialRingAnonymousVariables.extgcd
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Definition: multiutil.py:581
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Definition: multiutil.py:630
nzmath.poly.multiutil.UniqueFactorizationDomainPolynomial.__init__
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Definition: multiutil.py:392
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Definition: multiutil.py:619
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Definition: ring.py:128
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Definition: multiutil.py:434
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Definition: ring.py:552
nzmath.poly.multiutil.RingPolynomial.__init__
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Definition: multiutil.py:280
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Definition: ratfunc.py:1
nzmath.poly.multiutil.RingPolynomial.getRing
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Definition: multiutil.py:315
nzmath.poly.multiutil.PolynomialRingAnonymousVariables.__init__
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Definition: multiutil.py:421
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Definition: uniutil.py:1
nzmath.ring.CommutativeRingElement
Definition: ring.py:291
nzmath.poly.multiutil.PolynomialRingAnonymousVariables._getOne
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Definition: multiutil.py:553
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Definition: multiutil.py:111
nzmath.poly.multiutil.RingElementProvider._coefficient_ring
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Definition: multiutil.py:108
nzmath.poly.multiutil.RingElementProvider.set_coefficient_ring
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Definition: multiutil.py:135
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Definition: multivar.py:1
nzmath.poly.multiutil.OrderProvider.order
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Definition: multiutil.py:30
Definition: multiutil.py:39
nzmath.ring.Ideal.ring
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Definition: ring.py:476
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Definition: multiutil.py:693
nzmath.poly.multiutil.PseudoDivisionProvider.__truediv__
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Definition: multiutil.py:214
nzmath.poly.multiutil.polynomial
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Definition: multiutil.py:647
nzmath.poly.multiutil.PolynomialRingAnonymousVariables._coefficient_ring
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Definition: multiutil.py:425
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Definition: multiutil.py:489
nzmath.poly.multiutil.OrderProvider.__init__
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Definition: multiutil.py:22
nzmath.ring.CommutativeRing.properties
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Definition: ring.py:103
nzmath.poly.multivar.BasicPolynomial._coefficients
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Definition: multivar.py:191
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Definition: ring.py:460
nzmath.poly.multiutil.DomainPolynomial.__init__
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Definition: multiutil.py:374
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Definition: multiutil.py:58
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Definition: multiutil.py:129
nzmath.poly.multiutil.PolynomialIdeal.__str__
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Definition: multiutil.py:636
nzmath.poly.multiutil.PolynomialRingAnonymousVariables.__eq__
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Definition: multiutil.py:450
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Definition: multiutil.py:440