| 1 | r""" |
| 2 | Let $F(x) = \sum_{\nu \in \NN^d} F_{\nu} x^\nu$ be a multivariate power series with complex coefficients that converges in a neighborhood of the origin. Assume that $F = G/H$ for some functions $G$ and $H$ holomorphic in a neighborhood of the origin. |
| 3 | Assume also that $H$ is a polynomial. |
| 4 | |
| 5 | This Python module for use within `Sage <http://www.sagemath.org>`_ computes asymptotics for the coefficients $F_{r \alpha}$ as $r \to \infty$ with $r \alpha \in \NN^d$ for $\alpha$ in a permissible subset of $d$-tuples of positive reals. |
| 6 | More specifically, it computes arbitrary terms of the asymptotic expansion for $F_{r \alpha}$ when the asymptotics are controlled by a strictly minimal multiple point of the alegbraic variety $H = 0$. |
| 7 | |
| 8 | The algorithms and formulas implemented here come from [RaWi2008a]_ |
| 9 | and [RaWi2012]_. |
| 10 | |
| 11 | .. [AiYu1983] I.A. Aizenberg and A.P. Yuzhakov, "Integral representations and residues in multidimensional complex analysis", Translations of Mathematical Monographs, 58. American Mathematical Society, Providence, RI, 1983. x+283 pp. ISBN: 0-8218-4511-X. |
| 12 | |
| 13 | .. [Raic2012] Alexander Raichev, "Leinartas's partial fraction decomposition", `<http://arxiv.org/abs/1206.4740>`_. |
| 14 | |
| 15 | .. [RaWi2008a] Alexander Raichev and Mark C. Wilson, "Asymptotics of coefficients of multivariate generating functions: improvements for smooth points", Electronic Journal of Combinatorics, Vol. 15 (2008), R89, `<http://arxiv.org/pdf/0803.2914.pdf>`_. |
| 16 | |
| 17 | .. [RaWi2012] Alexander Raichev and Mark C. Wilson, "Asymptotics of coefficients of multivariate generating functions: improvements for smooth points", To appear in 2012 in the Online Journal of Analytic Combinatorics, `<http://arxiv.org/pdf/1009.5715.pdf>`_. |
| 18 | |
| 19 | AUTHORS: |
| 20 | |
| 21 | - Alexander Raichev (2008-10-01): Initial version |
| 22 | - Alexander Raichev (2010-09-28): Corrected many functions |
| 23 | - Alexander Raichev (2010-12-15): Updated documentation |
| 24 | - Alexander Raichev (2011-03-09): Fixed a division by zero bug in relative_error() |
| 25 | - Alexander Raichev (2011-04-26): Rewrote in object-oreinted style |
| 26 | - Alexander Raichev (2011-05-06): Fixed bug in cohomologous_integrand() and fixed _crit_cone_combo() to work in SR |
| 27 | - Alexander Raichev (2012-08-06): Major rewrite. Created class FFPD and moved functions around. |
| 28 | - Alexander Raichev (2012-10-03): Fixed whitespace errors, added examples to those six functions missing them (which i overlooked), changed package name to a more descriptive title, made asymptotics methods work for univariate functions. |
| 29 | |
| 30 | EXAMPLES:: |
| 31 | |
| 32 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 33 | |
| 34 | A univariate smooth point example:: |
| 35 | |
| 36 | sage: R.<x> = PolynomialRing(QQ) |
| 37 | sage: H = (x - 1/2)^3 |
| 38 | sage: Hfac = H.factor() |
| 39 | sage: G = -1/(x + 3)/Hfac.unit() |
| 40 | sage: F = FFPD(G, Hfac) |
| 41 | sage: print F |
| 42 | (-1/(x + 3), [(x - 1/2, 3)]) |
| 43 | sage: alpha = [1] |
| 44 | sage: decomp = F.asymptotic_decomposition(alpha) |
| 45 | sage: print decomp |
| 46 | [(0, []), (-1/2*(x^2 + 6*x + 9)*r^2/(x^5 + 9*x^4 + 27*x^3 + 27*x^2) - 1/2*(5*x^2 + 24*x + 27)*r/(x^5 + 9*x^4 + 27*x^3 + 27*x^2) - 3*(x^2 + 3*x + 3)/(x^5 + 9*x^4 + 27*x^3 + 27*x^2), [(x - 1/2, 1)])] |
| 47 | sage: F1 = decomp[1] |
| 48 | sage: p = {x: 1/2} |
| 49 | sage: asy = F1.asymptotics(p, alpha, 3) |
| 50 | sage: print asy |
| 51 | (8/343*(49*r^2 + 161*r + 114)*2^r, 2, 8/7*r^2 + 184/49*r + 912/343) |
| 52 | sage: print F.relative_error(asy[0], alpha, [1, 2, 4, 8, 16], asy[1]) |
| 53 | Calculating errors table in the form |
| 54 | exponent, scaled Maclaurin coefficient, scaled asymptotic values, relative errors... |
| 55 | [((1,), 7.555555556, [7.556851312], [-0.0001714971672]), ((2,), 14.74074074, [14.74052478], [0.00001465051901]), ((4,), 35.96502058, [35.96501458], [1.667911934e-7]), ((8,), 105.8425656, [105.8425656], [4.399565380e-11]), ((16,), 355.3119534, [355.3119534], [0.0000000000])] |
| 56 | |
| 57 | Another smooth point example (Example 5.4 of [RaWi2008a]_):: |
| 58 | |
| 59 | sage: R.<x,y> = PolynomialRing(QQ) |
| 60 | sage: q = 1/2 |
| 61 | sage: qq = q.denominator() |
| 62 | sage: H = 1 - q*x + q*x*y - x^2*y |
| 63 | sage: Hfac = H.factor() |
| 64 | sage: G = (1 - q*x)/Hfac.unit() |
| 65 | sage: F = FFPD(G, Hfac) |
| 66 | sage: alpha = list(qq*vector([2, 1 - q])) |
| 67 | sage: print alpha |
| 68 | [4, 1] |
| 69 | sage: I = F.smooth_critical_ideal(alpha) |
| 70 | sage: print I |
| 71 | Ideal (y^2 - 2*y + 1, x + 1/4*y - 5/4) of Multivariate Polynomial Ring |
| 72 | in x, y over Rational Field |
| 73 | sage: s = solve(I.gens(), [SR(x) for x in R.gens()], solution_dict=true) |
| 74 | sage: print s |
| 75 | [{y: 1, x: 1}] |
| 76 | sage: p = s[0] |
| 77 | sage: asy = F.asymptotics(p, alpha, 1) # long time |
| 78 | Creating auxiliary functions... |
| 79 | Computing derivatives of auxiallary functions... |
| 80 | Computing derivatives of more auxiliary functions... |
| 81 | Computing second order differential operator actions... |
| 82 | sage: print asy # long time |
| 83 | (1/12*2^(2/3)*sqrt(3)*gamma(1/3)/(pi*r^(1/3)), 1, |
| 84 | 1/12*2^(2/3)*sqrt(3)*gamma(1/3)/(pi*r^(1/3))) |
| 85 | sage: print F.relative_error(asy[0], alpha, [1, 2, 4, 8, 16], asy[1]) # long time |
| 86 | Calculating errors table in the form |
| 87 | exponent, scaled Maclaurin coefficient, scaled asymptotic values, |
| 88 | relative errors... |
| 89 | [((4, 1), 0.1875000000, [0.1953794675], [-0.04202382689]), ((8, 2), |
| 90 | 0.1523437500, [0.1550727862], [-0.01791367323]), ((16, 4), 0.1221771240, |
| 91 | [0.1230813519], [-0.007400959228]), ((32, 8), 0.09739671811, |
| 92 | [0.09768973377], [-0.003008475766]), ((64, 16), 0.07744253816, |
| 93 | [0.07753639308], [-0.001211929722])] |
| 94 | |
| 95 | A multiple point example (Example 6.5 of [RaWi2012]_):: |
| 96 | |
| 97 | sage: R.<x,y>= PolynomialRing(QQ) |
| 98 | sage: H = (1 - 2*x - y)**2 * (1 - x - 2*y)**2 |
| 99 | sage: Hfac = H.factor() |
| 100 | sage: G = 1/Hfac.unit() |
| 101 | sage: F = FFPD(G, Hfac) |
| 102 | sage: print F |
| 103 | (1, [(x + 2*y - 1, 2), (2*x + y - 1, 2)]) |
| 104 | sage: I = F.singular_ideal() |
| 105 | sage: print I |
| 106 | Ideal (x - 1/3, y - 1/3) of Multivariate Polynomial Ring in x, y over |
| 107 | Rational Field |
| 108 | sage: p = {x: 1/3, y: 1/3} |
| 109 | sage: print F.is_convenient_multiple_point(p) |
| 110 | (True, 'convenient in variables [x, y]') |
| 111 | sage: alpha = (var('a'), var('b')) |
| 112 | sage: decomp = F.asymptotic_decomposition(alpha); print decomp # long time |
| 113 | [(0, []), (-1/9*(2*a^2*y^2 - 5*a*b*x*y + 2*b^2*x^2)*r^2/(x^2*y^2) + |
| 114 | 1/9*(5*(a + b)*x*y - 6*a*y^2 - 6*b*x^2)*r/(x^2*y^2) - 1/9*(4*x^2 - 5*x*y |
| 115 | + 4*y^2)/(x^2*y^2), [(x + 2*y - 1, 1), (2*x + y - 1, 1)])] |
| 116 | sage: F1 = decomp[1] |
| 117 | sage: print F1.asymptotics(p, alpha, 2) # long time |
| 118 | (-3*((2*a^2 - 5*a*b + 2*b^2)*r^2 + (a + b)*r + |
| 119 | 3)*((1/3)^(-b)*(1/3)^(-a))^r, (1/3)^(-b)*(1/3)^(-a), -3*(2*a^2 - 5*a*b + |
| 120 | 2*b^2)*r^2 - 3*(a + b)*r - 9) |
| 121 | sage: alpha = [4, 3] |
| 122 | sage: decomp = F.asymptotic_decomposition(alpha) |
| 123 | sage: F1 = decomp[1] |
| 124 | sage: asy = F1.asymptotics(p, alpha, 2) # long time |
| 125 | sage: print asy # long time |
| 126 | (3*(10*r^2 - 7*r - 3)*2187^r, 2187, 30*r^2 - 21*r - 9) |
| 127 | sage: print F.relative_error(asy[0], alpha, [1, 2, 4, 8], asy[1]) # long time |
| 128 | Calculating errors table in the form |
| 129 | exponent, scaled Maclaurin coefficient, scaled asymptotic values, |
| 130 | relative errors... |
| 131 | [((4, 3), 30.72702332, [0.0000000000], [1.000000000]), ((8, 6), |
| 132 | 111.9315678, [69.00000000], [0.3835519207]), ((16, 12), 442.7813138, |
| 133 | [387.0000000], [0.1259793763]), ((32, 24), 1799.879232, [1743.000000], |
| 134 | [0.03160169385])] |
| 135 | |
| 136 | """ |
| 137 | #***************************************************************************** |
| 138 | # Copyright (C) 2008 Alexander Raichev <tortoise.said@gmail.com> |
| 139 | # |
| 140 | # Distributed under the terms of the GNU General Public License (GPL) |
| 141 | # http://www.gnu.org/licenses/ |
| 142 | #***************************************************************************** |
| 143 | |
| 144 | from functools import total_ordering |
| 145 | |
| 146 | # Sage libraries |
| 147 | from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing |
| 148 | from sage.rings.polynomial.polynomial_ring import is_PolynomialRing |
| 149 | from sage.rings.polynomial.multi_polynomial_ring_generic import is_MPolynomialRing |
| 150 | from sage.symbolic.ring import SR |
| 151 | from sage.geometry.cone import Cone |
| 152 | from sage.calculus.functional import diff |
| 153 | from sage.calculus.functions import jacobian |
| 154 | from sage.calculus.var import function, var |
| 155 | from sage.combinat.combinat import stirling_number1 |
| 156 | from sage.combinat.permutation import Permutation |
| 157 | from sage.combinat.tuple import UnorderedTuples |
| 158 | from sage.functions.log import exp, log |
| 159 | from sage.functions.other import factorial, gamma, sqrt |
| 160 | from sage.matrix.constructor import matrix |
| 161 | from sage.misc.misc import add |
| 162 | from sage.misc.misc_c import prod |
| 163 | from sage.misc.mrange import cartesian_product_iterator, mrange |
| 164 | from sage.modules.free_module_element import vector |
| 165 | from sage.rings.arith import binomial, xgcd |
| 166 | from sage.rings.all import CC |
| 167 | from sage.rings.fraction_field import FractionField |
| 168 | from sage.rings.integer import Integer |
| 169 | from sage.rings.rational_field import QQ |
| 170 | from sage.sets.set import Set |
| 171 | from sage.symbolic.constants import pi |
| 172 | from sage.symbolic.relation import solve |
| 173 | from sage.combinat.subset import Subsets |
| 174 | |
| 175 | @total_ordering |
| 176 | class FFPD(object): |
| 177 | r""" |
| 178 | Represents a fraction with factored polynomial denominator (FFPD) |
| 179 | $p/(q_1^{e_1} \cdots q_n^{e_n})$ by storing the parts $p$ and |
| 180 | $[(q_1, e_1), \ldots, (q_n, e_n)]$. |
| 181 | Here $q_1, \ldots, q_n$ are elements of a 0- or multi-variate factorial |
| 182 | polynomial ring $R$ , $q_1, \ldots, q_n$ are distinct irreducible elements |
| 183 | of $R$ , $e_1, \ldots, e_n$ are positive integers, and $p$ is a function |
| 184 | of the indeterminates of $R$ (a Sage Symbolic Expression). |
| 185 | An element $r$ with no polynomial denominator is represented as $[r, (,)]$. |
| 186 | |
| 187 | AUTHORS: |
| 188 | |
| 189 | - Alexander Raichev (2012-07-26) |
| 190 | """ |
| 191 | |
| 192 | def __init__(self, numerator=None, denominator_factored=None, |
| 193 | quotient=None, reduce_=True): |
| 194 | r""" |
| 195 | Create a FFPD instance. |
| 196 | |
| 197 | INPUT: |
| 198 | |
| 199 | - ``numerator`` - (Optional; default=None) An element $p$ of a |
| 200 | 0- or 1-variate factorial polynomial ring $R$. |
| 201 | - ``denominator_factored`` - (Optional; default=None) |
| 202 | A list of the form |
| 203 | $[(q_1, e_1), \ldots, (q_n, e_n)]$ where the $q_1, \ldots, q_n$ are |
| 204 | distinct irreducible elements of $R$ and the $e_i$ are positive |
| 205 | integers. |
| 206 | - ``quotient`` - (Optional; default=None) An element of a field of |
| 207 | fractions of a factorial ring. |
| 208 | - ``reduce_`` - (Optional; default=True) If True, then represent |
| 209 | $p/(q_1^{e_1} \cdots q_n^{e_n})$ in lowest terms. |
| 210 | If False, then won't attempt to divide $p$ by any of the $q_i$. |
| 211 | |
| 212 | OUTPUT: |
| 213 | |
| 214 | A FFPD instance representing the rational expression |
| 215 | $p/(q_1^{e_1} \cdots q_n^{e_n})$. |
| 216 | To get a non-None output, one of ``numerator`` or ``quotient`` must be |
| 217 | non-None. |
| 218 | |
| 219 | EXAMPLES:: |
| 220 | |
| 221 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 222 | sage: R.<x, y> = PolynomialRing(QQ) |
| 223 | sage: df = [x, 1], [y, 1], [x*y+1, 1] |
| 224 | sage: f = FFPD(x, df) |
| 225 | sage: print f |
| 226 | (1, [(y, 1), (x*y + 1, 1)]) |
| 227 | sage: ff = FFPD(x, df, reduce_=False) |
| 228 | sage: print ff |
| 229 | (x, [(y, 1), (x, 1), (x*y + 1, 1)]) |
| 230 | |
| 231 | :: |
| 232 | |
| 233 | sage: f = FFPD(x + y, [(x + y, 1)]) |
| 234 | sage: print f |
| 235 | (1, []) |
| 236 | |
| 237 | :: |
| 238 | |
| 239 | sage: R.<x> = PolynomialRing(QQ) |
| 240 | sage: f = 5*x^3 + 1/x + 1/(x-1) + 1/(3*x^2 + 1) |
| 241 | sage: print FFPD(quotient=f) |
| 242 | (5*x^7 - 5*x^6 + 5/3*x^5 - 5/3*x^4 + 2*x^3 - 2/3*x^2 + 1/3*x - 1/3, |
| 243 | [(x - 1, 1), (x, 1), (x^2 + 1/3, 1)]) |
| 244 | |
| 245 | :: |
| 246 | |
| 247 | sage: R.<x, y> = PolynomialRing(QQ) |
| 248 | sage: f = 2*y/(5*(x^3 - 1)*(y + 1)) |
| 249 | sage: print FFPD(quotient=f) |
| 250 | (2/5*y, [(y + 1, 1), (x - 1, 1), (x^2 + x + 1, 1)]) |
| 251 | |
| 252 | :: |
| 253 | |
| 254 | sage: R.<x, y>= PolynomialRing(QQ) |
| 255 | sage: p = 1/x^2 |
| 256 | sage: q = 3*x**2*y |
| 257 | sage: qs = q.factor() |
| 258 | sage: f = FFPD(p/qs.unit(), qs) |
| 259 | sage: print f |
| 260 | (1/(3*x^2), [(y, 1), (x, 2)]) |
| 261 | |
| 262 | :: |
| 263 | |
| 264 | sage: R.<x, y> = PolynomialRing(QQ) |
| 265 | sage: f = FFPD(cos(x)*x*y^2, [(x, 2), (y, 1)]) |
| 266 | sage: print f |
| 267 | (x*y^2*cos(x), [(y, 1), (x, 2)]) |
| 268 | |
| 269 | :: |
| 270 | |
| 271 | sage: R.<x, y> = PolynomialRing(QQ) |
| 272 | sage: G = exp(x + y) |
| 273 | sage: H = (1 - 2*x - y) * (1 - x - 2*y) |
| 274 | sage: a = FFPD(quotient=G/H) |
| 275 | sage: print a |
| 276 | (e^(x + y)/(2*x^2 + 5*x*y + 2*y^2 - 3*x - 3*y + 1), []) |
| 277 | sage: print a._ring |
| 278 | None |
| 279 | sage: b = FFPD(G, H.factor()) |
| 280 | sage: print b |
| 281 | (e^(x + y), [(x + 2*y - 1, 1), (2*x + y - 1, 1)]) |
| 282 | sage: print b._ring |
| 283 | Multivariate Polynomial Ring in x, y over Rational Field |
| 284 | |
| 285 | Singular throws a 'not implemented' error when trying to factor in |
| 286 | a multivariate polynomial ring over an inexact field :: |
| 287 | |
| 288 | sage: R.<x, y>= PolynomialRing(CC) |
| 289 | sage: f = (x + 1)/(x*y*(x*y + 1)^2) |
| 290 | sage: FFPD(quotient=f) |
| 291 | Traceback (most recent call last): |
| 292 | ... |
| 293 | TypeError: Singular error: |
| 294 | ? not implemented |
| 295 | ? error occurred in or before STDIN line 17: |
| 296 | `def sage9=factorize(sage8);` |
| 297 | |
| 298 | """ |
| 299 | # Attributes are |
| 300 | # self._numerator |
| 301 | # self._denominator_factored |
| 302 | # self._ring |
| 303 | if quotient is not None: |
| 304 | p = quotient.numerator() |
| 305 | q = quotient.denominator() |
| 306 | R = q.parent() |
| 307 | self._numerator = quotient |
| 308 | self._denominator_factored = [] |
| 309 | if is_PolynomialRing(R) or is_MPolynomialRing(R): |
| 310 | self._ring = R |
| 311 | if not R(q).is_unit(): |
| 312 | # Factor q |
| 313 | try: |
| 314 | df = q.factor() |
| 315 | except NotImplementedError: |
| 316 | # Singular's factor() needs 'proof=False'. |
| 317 | df = q.factor(proof=False) |
| 318 | self._numerator = p/df.unit() |
| 319 | df = sorted([tuple(t) for t in df]) # Sort for consitency. |
| 320 | self._denominator_factored = df |
| 321 | else: |
| 322 | self._ring = None |
| 323 | # Done. No reducing needed, as Sage reduced quotient beforehand. |
| 324 | return |
| 325 | |
| 326 | self._numerator = numerator |
| 327 | if denominator_factored: |
| 328 | self._denominator_factored = sorted([tuple(t) for t in |
| 329 | denominator_factored]) |
| 330 | self._ring = denominator_factored[0][0].parent() |
| 331 | else: |
| 332 | self._denominator_factored = [] |
| 333 | self._ring = None |
| 334 | R = self._ring |
| 335 | if R is not None and numerator in R and reduce_: |
| 336 | # Reduce fraction if possible. |
| 337 | numer = R(self._numerator) |
| 338 | df = self._denominator_factored |
| 339 | new_df = [] |
| 340 | for (q, e) in df: |
| 341 | ee = e |
| 342 | quo, rem = numer.quo_rem(q) |
| 343 | while rem == 0 and ee > 0: |
| 344 | ee -= 1 |
| 345 | numer = quo |
| 346 | quo, rem = numer.quo_rem(q) |
| 347 | if ee > 0: |
| 348 | new_df.append((q, ee)) |
| 349 | self._numerator = numer |
| 350 | self._denominator_factored = new_df |
| 351 | |
| 352 | def numerator(self): |
| 353 | r""" |
| 354 | Return the numerator of ``self``. |
| 355 | |
| 356 | EXAMPLES:: |
| 357 | |
| 358 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 359 | sage: R.<x,y>= PolynomialRing(QQ) |
| 360 | sage: H = (1 - x - y - x*y)**2*(1-x) |
| 361 | sage: Hfac = H.factor() |
| 362 | sage: G = exp(y)/Hfac.unit() |
| 363 | sage: F = FFPD(G, Hfac) |
| 364 | sage: print F.numerator() |
| 365 | -e^y |
| 366 | """ |
| 367 | return self._numerator |
| 368 | |
| 369 | def denominator(self): |
| 370 | r""" |
| 371 | Return the denominator of ``self``. |
| 372 | |
| 373 | EXAMPLES:: |
| 374 | |
| 375 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 376 | sage: R.<x,y>= PolynomialRing(QQ) |
| 377 | sage: H = (1 - x - y - x*y)**2*(1-x) |
| 378 | sage: Hfac = H.factor() |
| 379 | sage: G = exp(y)/Hfac.unit() |
| 380 | sage: F = FFPD(G, Hfac) |
| 381 | sage: print F.denominator() |
| 382 | x^3*y^2 + 2*x^3*y + x^2*y^2 + x^3 - 2*x^2*y - x*y^2 - 3*x^2 - 2*x*y |
| 383 | - y^2 + 3*x + 2*y - 1 |
| 384 | """ |
| 385 | return prod([q**e for q, e in self.denominator_factored()]) |
| 386 | |
| 387 | def denominator_factored(self): |
| 388 | r""" |
| 389 | Return the factorization in ``self.ring()`` of the denominator of |
| 390 | ``self`` but without the unit part. |
| 391 | |
| 392 | EXAMPLES:: |
| 393 | |
| 394 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 395 | sage: R.<x,y>= PolynomialRing(QQ) |
| 396 | sage: H = (1 - x - y - x*y)**2*(1-x) |
| 397 | sage: Hfac = H.factor() |
| 398 | sage: G = exp(y)/Hfac.unit() |
| 399 | sage: F = FFPD(G, Hfac) |
| 400 | sage: print F.denominator_factored() |
| 401 | [(x - 1, 1), (x*y + x + y - 1, 2)] |
| 402 | """ |
| 403 | return self._denominator_factored |
| 404 | |
| 405 | def ring(self): |
| 406 | r""" |
| 407 | Return the ring of the denominator of ``self``, which is |
| 408 | None in the case where ``self`` doesn't have a denominator. |
| 409 | |
| 410 | EXAMPLES:: |
| 411 | |
| 412 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 413 | sage: R.<x,y>= PolynomialRing(QQ) |
| 414 | sage: H = (1 - x - y - x*y)**2*(1-x) |
| 415 | sage: Hfac = H.factor() |
| 416 | sage: G = exp(y)/Hfac.unit() |
| 417 | sage: F = FFPD(G, Hfac) |
| 418 | sage: print F.ring() |
| 419 | Multivariate Polynomial Ring in x, y over Rational Field |
| 420 | sage: F = FFPD(quotient=G/H) |
| 421 | sage: print F |
| 422 | (e^y/(x^3*y^2 + 2*x^3*y + x^2*y^2 + x^3 - 2*x^2*y - x*y^2 - 3*x^2 - |
| 423 | 2*x*y - y^2 + 3*x + 2*y - 1), []) |
| 424 | sage: print F.ring() |
| 425 | None |
| 426 | """ |
| 427 | return self._ring |
| 428 | |
| 429 | def dimension(self): |
| 430 | r""" |
| 431 | Return the number of indeterminates of ``self.ring()``. |
| 432 | |
| 433 | EXAMPLES:: |
| 434 | |
| 435 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 436 | sage: R.<x,y>= PolynomialRing(QQ) |
| 437 | sage: H = (1 - x - y - x*y)**2*(1-x) |
| 438 | sage: Hfac = H.factor() |
| 439 | sage: G = exp(y)/Hfac.unit() |
| 440 | sage: F = FFPD(G, Hfac) |
| 441 | sage: print F.dimension() |
| 442 | 2 |
| 443 | """ |
| 444 | R = self.ring() |
| 445 | if is_PolynomialRing(R) or is_MPolynomialRing(R): |
| 446 | return R.ngens() |
| 447 | else: |
| 448 | return None |
| 449 | |
| 450 | def list(self): |
| 451 | r""" |
| 452 | Convert ``self`` into a list for printing. |
| 453 | |
| 454 | EXAMPLES:: |
| 455 | |
| 456 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 457 | sage: R.<x,y>= PolynomialRing(QQ) |
| 458 | sage: H = (1 - x - y - x*y)**2*(1-x) |
| 459 | sage: Hfac = H.factor() |
| 460 | sage: G = exp(y)/Hfac.unit() |
| 461 | sage: F = FFPD(G, Hfac) |
| 462 | sage: print F # indirect doctest |
| 463 | (-e^y, [(x - 1, 1), (x*y + x + y - 1, 2)]) |
| 464 | """ |
| 465 | return (self.numerator(), self.denominator_factored()) |
| 466 | |
| 467 | def quotient(self): |
| 468 | r""" |
| 469 | Convert ``self`` into a quotient. |
| 470 | |
| 471 | EXAMPLES:: |
| 472 | |
| 473 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 474 | sage: R.<x,y>= PolynomialRing(QQ) |
| 475 | sage: H = (1 - x - y - x*y)**2*(1-x) |
| 476 | sage: Hfac = H.factor() |
| 477 | sage: G = exp(y)/Hfac.unit() |
| 478 | sage: F = FFPD(G, Hfac) |
| 479 | sage: print F |
| 480 | (-e^y, [(x - 1, 1), (x*y + x + y - 1, 2)]) |
| 481 | sage: print F.quotient() |
| 482 | -e^y/(x^3*y^2 + 2*x^3*y + x^2*y^2 + x^3 - 2*x^2*y - x*y^2 - 3*x^2 - |
| 483 | 2*x*y - y^2 + 3*x + 2*y - 1) |
| 484 | """ |
| 485 | return self.numerator()/self.denominator() |
| 486 | |
| 487 | def __str__(self): |
| 488 | r""" |
| 489 | Returns a string representation of ``self`` |
| 490 | |
| 491 | EXAMPLES:: |
| 492 | |
| 493 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 494 | sage: R.<x,y> = PolynomialRing(QQ) |
| 495 | sage: f = FFPD(x + y, [(y, 1), (x, 1)]) |
| 496 | sage: print f |
| 497 | (x + y, [(y, 1), (x, 1)]) |
| 498 | |
| 499 | """ |
| 500 | return str(self.list()) |
| 501 | |
| 502 | def __eq__(self, other): |
| 503 | r""" |
| 504 | Two FFPD instances are equal iff they represent the same |
| 505 | fraction. |
| 506 | |
| 507 | EXAMPLES:: |
| 508 | |
| 509 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 510 | sage: R.<x, y>= PolynomialRing(QQ) |
| 511 | sage: df = [x, 1], [y, 1], [x*y+1, 1] |
| 512 | sage: f = FFPD(x, df) |
| 513 | sage: ff = FFPD(x, df, reduce_=False) |
| 514 | sage: f == ff |
| 515 | True |
| 516 | sage: g = FFPD(y, df) |
| 517 | sage: g == f |
| 518 | False |
| 519 | |
| 520 | :: |
| 521 | |
| 522 | sage: R.<x, y> = PolynomialRing(QQ) |
| 523 | sage: G = exp(x + y) |
| 524 | sage: H = (1 - 2*x - y) * (1 - x - 2*y) |
| 525 | sage: a = FFPD(quotient=G/H) |
| 526 | sage: b = FFPD(G, H.factor()) |
| 527 | sage: bool(a == b) |
| 528 | True |
| 529 | """ |
| 530 | return self.quotient() == other.quotient() |
| 531 | |
| 532 | def __ne__(self, other): |
| 533 | r""" |
| 534 | EXAMPLES:: |
| 535 | |
| 536 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 537 | sage: R.<x, y>= PolynomialRing(QQ) |
| 538 | sage: df = [x, 1], [y, 1], [x*y+1, 1] |
| 539 | sage: f = FFPD(x, df) |
| 540 | sage: ff = FFPD(x, df, reduce_=False) |
| 541 | sage: f != ff |
| 542 | False |
| 543 | sage: g = FFPD(y, df) |
| 544 | sage: g != f # indirect doctest |
| 545 | True |
| 546 | """ |
| 547 | return not (self == other) |
| 548 | |
| 549 | def __lt__(self, other): |
| 550 | r""" |
| 551 | FFPD A is less than FFPD B iff |
| 552 | (the denominator factorization of A is shorter than that of B) or |
| 553 | (the denominator factorization lengths are equal and |
| 554 | the denominator of A is less than that of B in their ring) or |
| 555 | (the denominator factorization lengths are equal and the |
| 556 | denominators are equal and the numerator of A is less than that of B |
| 557 | in their ring). |
| 558 | |
| 559 | EXAMPLES:: |
| 560 | |
| 561 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 562 | sage: R.<x, y>= PolynomialRing(QQ) |
| 563 | sage: df = [x, 1], [y, 1], [x*y+1, 1] |
| 564 | sage: f = FFPD(x, df) |
| 565 | sage: ff = FFPD(x, df, reduce_=False) |
| 566 | sage: g = FFPD(y, df) |
| 567 | sage: h = FFPD(exp(x), df) |
| 568 | sage: i = FFPD(sin(x + 2), df) |
| 569 | sage: print f, ff |
| 570 | (1, [(y, 1), (x*y + 1, 1)]) (x, [(y, 1), (x, 1), (x*y + 1, 1)]) |
| 571 | sage: print f < ff |
| 572 | True |
| 573 | sage: print f < g |
| 574 | True |
| 575 | sage: print g < h |
| 576 | True |
| 577 | sage: print h < i |
| 578 | False |
| 579 | """ |
| 580 | sn = self.numerator() |
| 581 | on = other.numerator() |
| 582 | sdf = self.denominator_factored() |
| 583 | odf = other.denominator_factored() |
| 584 | sd = self.denominator() |
| 585 | od = other.denominator() |
| 586 | |
| 587 | return bool(len(sdf) < len(odf) or\ |
| 588 | (len(sdf) == len(odf) and sd < od) or\ |
| 589 | (len(sdf) == len(odf) and sd == od and sn < on)) |
| 590 | |
| 591 | def univariate_decomposition(self): |
| 592 | r""" |
| 593 | Return the usual univariate partial fraction decomposition |
| 594 | of ``self`` as a FFPDSum instance. |
| 595 | Assume that ``self`` lies in the field of fractions |
| 596 | of a univariate factorial polynomial ring. |
| 597 | |
| 598 | EXAMPLES:: |
| 599 | |
| 600 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 601 | |
| 602 | One variable:: |
| 603 | |
| 604 | sage: R.<x> = PolynomialRing(QQ) |
| 605 | sage: f = 5*x^3 + 1/x + 1/(x-1) + 1/(3*x^2 + 1) |
| 606 | sage: print f |
| 607 | (15*x^7 - 15*x^6 + 5*x^5 - 5*x^4 + 6*x^3 - 2*x^2 + x - 1)/(3*x^4 - |
| 608 | 3*x^3 + x^2 - x) |
| 609 | sage: decomp = FFPD(quotient=f).univariate_decomposition() |
| 610 | sage: print decomp |
| 611 | [(5*x^3, []), (1, [(x - 1, 1)]), (1, [(x, 1)]), |
| 612 | (1/3, [(x^2 + 1/3, 1)])] |
| 613 | sage: print decomp.sum().quotient() == f |
| 614 | True |
| 615 | |
| 616 | One variable with numerator in symbolic ring:: |
| 617 | |
| 618 | sage: R.<x> = PolynomialRing(QQ) |
| 619 | sage: f = 5*x^3 + 1/x + 1/(x-1) + exp(x)/(3*x^2 + 1) |
| 620 | sage: print f |
| 621 | e^x/(3*x^2 + 1) + ((5*(x - 1)*x^3 + 2)*x - 1)/((x - 1)*x) |
| 622 | sage: decomp = FFPD(quotient=f).univariate_decomposition() |
| 623 | sage: print decomp |
| 624 | [(e^x/(3*x^2 + 1) + ((5*(x - 1)*x^3 + 2)*x - 1)/((x - 1)*x), [])] |
| 625 | |
| 626 | One variable over a finite field:: |
| 627 | |
| 628 | sage: R.<x> = PolynomialRing(GF(2)) |
| 629 | sage: f = 5*x^3 + 1/x + 1/(x-1) + 1/(3*x^2 + 1) |
| 630 | sage: print f |
| 631 | (x^6 + x^4 + 1)/(x^3 + x) |
| 632 | sage: decomp = FFPD(quotient=f).univariate_decomposition() |
| 633 | sage: print decomp |
| 634 | [(x^3, []), (1, [(x, 1)]), (x, [(x + 1, 2)])] |
| 635 | sage: print decomp.sum().quotient() == f |
| 636 | True |
| 637 | |
| 638 | One variable over an inexact field:: |
| 639 | |
| 640 | sage: R.<x> = PolynomialRing(CC) |
| 641 | sage: f = 5*x^3 + 1/x + 1/(x-1) + 1/(3*x^2 + 1) |
| 642 | sage: print f |
| 643 | (15.0000000000000*x^7 - 15.0000000000000*x^6 + 5.00000000000000*x^5 |
| 644 | - 5.00000000000000*x^4 + 6.00000000000000*x^3 - |
| 645 | 2.00000000000000*x^2 + x - 1.00000000000000)/(3.00000000000000*x^4 |
| 646 | - 3.00000000000000*x^3 + x^2 - x) |
| 647 | sage: decomp = FFPD(quotient=f).univariate_decomposition() |
| 648 | sage: print decomp |
| 649 | [(5.00000000000000*x^3, []), (1.00000000000000, |
| 650 | [(x - 1.00000000000000, 1)]), (-0.288675134594813*I, |
| 651 | [(x - 0.577350269189626*I, 1)]), (1.00000000000000, [(x, 1)]), |
| 652 | (0.288675134594813*I, [(x + 0.577350269189626*I, 1)])] |
| 653 | sage: print decomp.sum().quotient() == f # Rounding error coming |
| 654 | False |
| 655 | |
| 656 | NOTE:: |
| 657 | |
| 658 | Let $f = p/q$ be a rational expression where $p$ and $q$ lie in a |
| 659 | univariate factorial polynomial ring $R$. |
| 660 | Let $q_1^{e_1} \cdots q_n^{e_n}$ be the |
| 661 | unique factorization of $q$ in $R$ into irreducible factors. |
| 662 | Then $f$ can be written uniquely as |
| 663 | |
| 664 | (*) $p_0 + \sum_{i=1}^{m} p_i/ q_i^{e_i}$, |
| 665 | |
| 666 | for some $p_j \in R$. |
| 667 | I call (*) the *usual partial fraction decomposition* of f. |
| 668 | |
| 669 | AUTHORS: |
| 670 | |
| 671 | - Robert Bradshaw (2007-05-31) |
| 672 | - Alexander Raichev (2012-06-25) |
| 673 | """ |
| 674 | if self.dimension() is None or self.dimension() > 1: |
| 675 | return FFPDSum([self]) |
| 676 | |
| 677 | R = self.ring() |
| 678 | p = self.numerator() |
| 679 | q = self.denominator() |
| 680 | if p in R: |
| 681 | whole, p = p.quo_rem(q) |
| 682 | else: |
| 683 | whole = p |
| 684 | p = R(1) |
| 685 | df = self.denominator_factored() |
| 686 | decomp = [FFPD(whole, [])] |
| 687 | for (a, m) in df: |
| 688 | numer = p * prod([b**n for (b, n) in df if b != a]).\ |
| 689 | inverse_mod(a**m) % (a**m) |
| 690 | # The inverse exists because the product and a**m |
| 691 | # are relatively prime. |
| 692 | decomp.append(FFPD(numer, [(a, m)])) |
| 693 | return FFPDSum(decomp) |
| 694 | |
| 695 | def nullstellensatz_certificate(self): |
| 696 | r""" |
| 697 | Let $[(q_1, e_1), \ldots, (q_n, e_n)]$ be the denominator factorization |
| 698 | of ``self``. |
| 699 | Return a list of polynomials $h_1, \ldots, h_m$ in ``self.ring()`` |
| 700 | that satisfies $h_1 q_1 + \cdots + h_m q_n = 1$ if it exists. |
| 701 | Otherwise return None. |
| 702 | Only works for multivariate ``self``. |
| 703 | |
| 704 | EXAMPLES:: |
| 705 | |
| 706 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 707 | sage: R.<x, y> = PolynomialRing(QQ) |
| 708 | sage: G = sin(x) |
| 709 | sage: H = x^2 * (x*y + 1) |
| 710 | sage: f = FFPD(G, H.factor()) |
| 711 | sage: L = f.nullstellensatz_certificate() |
| 712 | sage: print L |
| 713 | [y^2, -x*y + 1] |
| 714 | sage: df = f.denominator_factored() |
| 715 | sage: sum([L[i]*df[i][0]**df[i][1] for i in xrange(len(df))]) == 1 |
| 716 | True |
| 717 | |
| 718 | :: |
| 719 | |
| 720 | sage: f = 1/(x*y) |
| 721 | sage: L = FFPD(quotient=f).nullstellensatz_certificate() |
| 722 | sage: L is None |
| 723 | True |
| 724 | |
| 725 | """ |
| 726 | |
| 727 | R = self.ring() |
| 728 | if R is None: |
| 729 | return None |
| 730 | |
| 731 | df = self.denominator_factored() |
| 732 | J = R.ideal([q**e for q, e in df]) |
| 733 | if R(1) in J: |
| 734 | return R(1).lift(J) |
| 735 | else: |
| 736 | return None |
| 737 | |
| 738 | def nullstellensatz_decomposition(self): |
| 739 | r""" |
| 740 | Return a Nullstellensatz decomposition of ``self`` as a FFPDSum |
| 741 | instance. |
| 742 | |
| 743 | Recursive. |
| 744 | Only works for multivariate ``self``. |
| 745 | |
| 746 | EXAMPLES:: |
| 747 | |
| 748 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 749 | sage: R.<x, y> = PolynomialRing(QQ) |
| 750 | sage: f = 1/(x*(x*y + 1)) |
| 751 | sage: decomp = FFPD(quotient=f).nullstellensatz_decomposition() |
| 752 | sage: print decomp |
| 753 | [(0, []), (1, [(x, 1)]), (-y, [(x*y + 1, 1)])] |
| 754 | sage: decomp.sum().quotient() == f |
| 755 | True |
| 756 | sage: for r in decomp: |
| 757 | ... L = r.nullstellensatz_certificate() |
| 758 | ... L is None |
| 759 | ... |
| 760 | True |
| 761 | True |
| 762 | True |
| 763 | |
| 764 | :: |
| 765 | |
| 766 | sage: R.<x, y> = PolynomialRing(QQ) |
| 767 | sage: G = sin(y) |
| 768 | sage: H = x*(x*y + 1) |
| 769 | sage: f = FFPD(G, H.factor()) |
| 770 | sage: decomp = f.nullstellensatz_decomposition() |
| 771 | sage: print decomp |
| 772 | [(0, []), (sin(y), [(x, 1)]), (-y*sin(y), [(x*y + 1, 1)])] |
| 773 | sage: bool(decomp.sum().quotient() == G/H) |
| 774 | True |
| 775 | sage: for r in decomp: |
| 776 | ... L = r.nullstellensatz_certificate() |
| 777 | ... L is None |
| 778 | ... |
| 779 | True |
| 780 | True |
| 781 | True |
| 782 | |
| 783 | NOTE:: |
| 784 | |
| 785 | Let $f = p/q$ where $q$ lies in a $d$ -variate polynomial ring $K[X]$ for some field $K$ and $d \ge 1$. |
| 786 | Let $q_1^{e_1} \cdots q_n^{e_n}$ be the |
| 787 | unique factorization of $q$ in $K[X]$ into irreducible factors and |
| 788 | let $V_i$ be the algebraic variety $\{x \in L^d: q_i(x) = 0\}$ of |
| 789 | $q_i$ over the algebraic closure $L$ of $K$. |
| 790 | By [Raic2012]_, $f$ can be written as |
| 791 | |
| 792 | (*) $\sum p_A/\prod_{i \in A} q_i^{e_i}$, |
| 793 | |
| 794 | where the $p_A$ are products of $p$ and elements in $K[X]$ and |
| 795 | the sum is taken over all subsets |
| 796 | $A \subseteq \{1, \ldots, m\}$ such that |
| 797 | $\cap_{i\in A} T_i \neq \emptyset$. |
| 798 | |
| 799 | I call (*) a *Nullstellensatz decomposition* of $f$. |
| 800 | Nullstellensatz decompositions are not unique. |
| 801 | |
| 802 | The algorithm used comes from [Raic2012]_. |
| 803 | """ |
| 804 | L = self.nullstellensatz_certificate() |
| 805 | if L is None: |
| 806 | # No decomposing possible. |
| 807 | return FFPDSum([self]) |
| 808 | |
| 809 | # Otherwise decompose recursively. |
| 810 | decomp = FFPDSum() |
| 811 | p = self.numerator() |
| 812 | df = self.denominator_factored() |
| 813 | m = len(df) |
| 814 | iteration1 = FFPDSum([FFPD(p*L[i],[df[j] |
| 815 | for j in xrange(m) if j != i]) |
| 816 | for i in xrange(m) if L[i] != 0]) |
| 817 | |
| 818 | # Now decompose each FFPD of iteration1. |
| 819 | for r in iteration1: |
| 820 | decomp.extend(r.nullstellensatz_decomposition()) |
| 821 | |
| 822 | # Simplify and return result. |
| 823 | return decomp.combine_like_terms().whole_and_parts() |
| 824 | |
| 825 | def algebraic_dependence_certificate(self): |
| 826 | r""" |
| 827 | Return the ideal $J$ of annihilating polynomials for the set |
| 828 | of polynomials ``[q**e for (q, e) in self.denominator_factored()]``, |
| 829 | which could be the zero ideal. |
| 830 | The ideal $J$ lies in a polynomial ring over the field |
| 831 | ``self.ring().base_ring()`` that has |
| 832 | ``m = len(self.denominator_factored())`` indeterminates. |
| 833 | Return None if ``self.ring()`` is None. |
| 834 | |
| 835 | EXAMPLES:: |
| 836 | |
| 837 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 838 | sage: R.<x, y> = PolynomialRing(QQ) |
| 839 | sage: f = 1/(x^2 * (x*y + 1) * y^3) |
| 840 | sage: ff = FFPD(quotient=f) |
| 841 | sage: J = ff.algebraic_dependence_certificate() |
| 842 | sage: print J |
| 843 | Ideal (1 - 6*T2 + 15*T2^2 - 20*T2^3 + 15*T2^4 - T0^2*T1^3 - |
| 844 | 6*T2^5 + T2^6) of Multivariate Polynomial Ring in |
| 845 | T0, T1, T2 over Rational Field |
| 846 | sage: g = J.gens()[0] |
| 847 | sage: df = ff.denominator_factored() |
| 848 | sage: g(*(q**e for q, e in df)) == 0 |
| 849 | True |
| 850 | |
| 851 | :: |
| 852 | |
| 853 | sage: R.<x, y> = PolynomialRing(QQ) |
| 854 | sage: G = exp(x + y) |
| 855 | sage: H = x^2 * (x*y + 1) * y^3 |
| 856 | sage: ff = FFPD(G, H.factor()) |
| 857 | sage: J = ff.algebraic_dependence_certificate() |
| 858 | sage: print J |
| 859 | Ideal (1 - 6*T2 + 15*T2^2 - 20*T2^3 + 15*T2^4 - T0^2*T1^3 - |
| 860 | 6*T2^5 + T2^6) of Multivariate Polynomial Ring in |
| 861 | T0, T1, T2 over Rational Field |
| 862 | sage: g = J.gens()[0] |
| 863 | sage: df = ff.denominator_factored() |
| 864 | sage: g(*(q**e for q, e in df)) == 0 |
| 865 | True |
| 866 | |
| 867 | :: |
| 868 | |
| 869 | sage: f = 1/(x^3 * y^2) |
| 870 | sage: J = FFPD(quotient=f).algebraic_dependence_certificate() |
| 871 | sage: print J |
| 872 | Ideal (0) of Multivariate Polynomial Ring in T0, T1 over |
| 873 | Rational Field |
| 874 | |
| 875 | :: |
| 876 | |
| 877 | sage: f = sin(1)/(x^3 * y^2) |
| 878 | sage: J = FFPD(quotient=f).algebraic_dependence_certificate() |
| 879 | sage: print J |
| 880 | None |
| 881 | """ |
| 882 | R = self.ring() |
| 883 | if R is None: |
| 884 | return None |
| 885 | |
| 886 | df = self.denominator_factored() |
| 887 | if not df: |
| 888 | return R.ideal() # The zero ideal. |
| 889 | m = len(df) |
| 890 | F = R.base_ring() |
| 891 | Xs = list(R.gens()) |
| 892 | d = len(Xs) |
| 893 | |
| 894 | # Expand R by 2*m new variables. |
| 895 | S = 'S' |
| 896 | while S in [str(x) for x in Xs]: |
| 897 | S = S + 'S' |
| 898 | Ss = [S + str(i) for i in xrange(m)] |
| 899 | T = 'T' |
| 900 | while T in [str(x) for x in Xs]: |
| 901 | T = T + 'T' |
| 902 | Ts = [T + str(i) for i in xrange(m)] |
| 903 | |
| 904 | Vs = [str(x) for x in Xs] + Ss + Ts |
| 905 | RR = PolynomialRing(F, Vs) |
| 906 | Xs = RR.gens()[:d] |
| 907 | Ss = RR.gens()[d: d + m] |
| 908 | Ts = RR.gens()[d + m: d + 2 * m] |
| 909 | |
| 910 | # Compute the appropriate elimination ideal. |
| 911 | J = RR.ideal([ Ss[j] - RR(df[j][0]) for j in xrange(m)] +\ |
| 912 | [ Ss[j]**df[j][1] - Ts[j] for j in xrange(m)]) |
| 913 | J = J.elimination_ideal(Xs + Ss) |
| 914 | |
| 915 | # Coerce J into the polynomial ring in the indeteminates Ts[m:]. |
| 916 | # I choose the negdeglex order because i find it useful in my work. |
| 917 | RRR = PolynomialRing(F, [str(t) for t in Ts], order ='negdeglex') |
| 918 | return RRR.ideal(J) |
| 919 | |
| 920 | def algebraic_dependence_decomposition(self, whole_and_parts=True): |
| 921 | r""" |
| 922 | Return an algebraic dependence decomposition of ``self`` as a FFPDSum |
| 923 | instance. |
| 924 | |
| 925 | Recursive. |
| 926 | |
| 927 | EXAMPLES:: |
| 928 | |
| 929 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 930 | sage: R.<x, y> = PolynomialRing(QQ) |
| 931 | sage: f = 1/(x^2 * (x*y + 1) * y^3) |
| 932 | sage: ff = FFPD(quotient=f) |
| 933 | sage: decomp = ff.algebraic_dependence_decomposition() |
| 934 | sage: print decomp |
| 935 | [(0, []), (-x, [(x*y + 1, 1)]), (x^2*y^2 - x*y + 1, |
| 936 | [(y, 3), (x, 2)])] |
| 937 | sage: print decomp.sum().quotient() == f |
| 938 | True |
| 939 | sage: for r in decomp: |
| 940 | ... J = r.algebraic_dependence_certificate() |
| 941 | ... J is None or J == J.ring().ideal() # The zero ideal |
| 942 | ... |
| 943 | True |
| 944 | True |
| 945 | True |
| 946 | |
| 947 | :: |
| 948 | |
| 949 | sage: R.<x, y> = PolynomialRing(QQ) |
| 950 | sage: G = sin(x) |
| 951 | sage: H = x^2 * (x*y + 1) * y^3 |
| 952 | sage: f = FFPD(G, H.factor()) |
| 953 | sage: decomp = f.algebraic_dependence_decomposition() |
| 954 | sage: print decomp |
| 955 | [(0, []), (x^4*y^3*sin(x), [(x*y + 1, 1)]), |
| 956 | (-(x^5*y^5 - x^4*y^4 + x^3*y^3 - x^2*y^2 + x*y - 1)*sin(x), |
| 957 | [(y, 3), (x, 2)])] |
| 958 | sage: bool(decomp.sum().quotient() == G/H) |
| 959 | True |
| 960 | sage: for r in decomp: |
| 961 | ... J = r.algebraic_dependence_certificate() |
| 962 | ... J is None or J == J.ring().ideal() |
| 963 | ... |
| 964 | True |
| 965 | True |
| 966 | True |
| 967 | |
| 968 | NOTE:: |
| 969 | |
| 970 | Let $f = p/q$ where $q$ lies in a $d$ -variate polynomial ring |
| 971 | $K[X]$ for some field $K$. |
| 972 | Let $q_1^{e_1} \cdots q_n^{e_n}$ be the |
| 973 | unique factorization of $q$ in $K[X]$ into irreducible factors and |
| 974 | let $V_i$ be the algebraic variety $\{x\in L^d: q_i(x) = 0\}$ of |
| 975 | $q_i$ over the algebraic closure $L$ of $K$. |
| 976 | By [Raic2012]_, $f$ can be written as |
| 977 | |
| 978 | (*) $\sum p_A/\prod_{i \in A} q_i^{b_i}$, |
| 979 | |
| 980 | where the $b_i$ are positive integers, each $p_A$ is a products |
| 981 | of $p$ and an element in $K[X]$, |
| 982 | and the sum is taken over all subsets |
| 983 | $A \subseteq \{1, \ldots, m\}$ such that $|A| \le d$ and |
| 984 | $\{q_i : i\in A\}$ is algebraically independent. |
| 985 | |
| 986 | I call (*) an *algebraic dependence decomposition* of $f$. |
| 987 | Algebraic dependence decompositions are not unique. |
| 988 | |
| 989 | The algorithm used comes from [Raic2012]_. |
| 990 | """ |
| 991 | J = self.algebraic_dependence_certificate() |
| 992 | if not J: |
| 993 | # No decomposing possible. |
| 994 | return FFPDSum([self]) |
| 995 | |
| 996 | # Otherwise decompose recursively. |
| 997 | decomp = FFPDSum() |
| 998 | p = self.numerator() |
| 999 | df = self.denominator_factored() |
| 1000 | m = len(df) |
| 1001 | g = J.gens()[0] # An annihilating polynomial for df. |
| 1002 | new_vars = J.ring().gens() |
| 1003 | # Note that each new_vars[j] corresponds to df[j] such that |
| 1004 | # g([q**e for q, e in df]) = 0. |
| 1005 | # Assuming here that g.parent() has negdeglex term order |
| 1006 | # so that g.lt() is indeed the monomial we want below. |
| 1007 | # Use g to rewrite r into a sum of FFPDs, |
| 1008 | # each with < m distinct denominator factors. |
| 1009 | gg = (g.lt() - g)/(g.lc()) |
| 1010 | numers = map(prod, zip(gg.coefficients(), gg.monomials())) |
| 1011 | e = list(g.lt().exponents())[0:m] |
| 1012 | denoms = [(new_vars[j], e[0][j] + 1) for j in xrange(m)] |
| 1013 | # Write r in terms of new_vars, |
| 1014 | # cancel factors in the denominator, and combine like terms. |
| 1015 | iteration1_temp = FFPDSum([FFPD(a, denoms) for a in numers]).\ |
| 1016 | combine_like_terms() |
| 1017 | # Substitute in df. |
| 1018 | qpowsub = dict([(new_vars[j], df[j][0]**df[j][1]) |
| 1019 | for j in xrange(m)]) |
| 1020 | iteration1 = FFPDSum() |
| 1021 | for r in iteration1_temp: |
| 1022 | num1 = p*g.parent()(r.numerator()).subs(qpowsub) |
| 1023 | denoms1 = [] |
| 1024 | for q, e in r.denominator_factored(): |
| 1025 | j = new_vars.index(q) |
| 1026 | denoms1.append((df[j][0], df[j][1]*e)) |
| 1027 | iteration1.append(FFPD(num1, denoms1)) |
| 1028 | # Now decompose each FFPD of iteration1. |
| 1029 | for r in iteration1: |
| 1030 | decomp.extend(r.algebraic_dependence_decomposition()) |
| 1031 | |
| 1032 | # Simplify and return result. |
| 1033 | return decomp.combine_like_terms().whole_and_parts() |
| 1034 | |
| 1035 | def leinartas_decomposition(self): |
| 1036 | r""" |
| 1037 | Return a Leinartas decomposition of ``self`` as a FFPDSum instance. |
| 1038 | |
| 1039 | EXAMPLES:: |
| 1040 | |
| 1041 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 1042 | sage: R.<x> = PolynomialRing(QQ) |
| 1043 | sage: f = (x^2 + 1)/((x + 2)*(x - 1)*(x^2 + x + 1)) |
| 1044 | sage: decomp = FFPD(quotient=f).leinartas_decomposition() |
| 1045 | sage: print decomp |
| 1046 | [(0, []), (2/9, [(x - 1, 1)]), (-5/9, [(x + 2, 1)]), (1/3*x, [(x^2 + x + 1, 1)])] |
| 1047 | sage: print decomp.sum().quotient() == f |
| 1048 | True |
| 1049 | |
| 1050 | :: |
| 1051 | |
| 1052 | sage: R.<x, y> = PolynomialRing(QQ) |
| 1053 | sage: f = 1/x + 1/y + 1/(x*y + 1) |
| 1054 | sage: decomp = FFPD(quotient=f).leinartas_decomposition() |
| 1055 | sage: print decomp |
| 1056 | [(0, []), (1, [(x*y + 1, 1)]), (x + y, [(y, 1), (x, 1)])] |
| 1057 | sage: print decomp.sum().quotient() == f |
| 1058 | True |
| 1059 | sage: for r in decomp: |
| 1060 | ... L = r.nullstellensatz_certificate() |
| 1061 | ... print L is None |
| 1062 | ... J = r.algebraic_dependence_certificate() |
| 1063 | ... print J is None or J == J.ring().ideal() |
| 1064 | ... |
| 1065 | True |
| 1066 | True |
| 1067 | True |
| 1068 | True |
| 1069 | True |
| 1070 | True |
| 1071 | |
| 1072 | :: |
| 1073 | |
| 1074 | sage: R.<x, y> = PolynomialRing(QQ) |
| 1075 | sage: f = sin(x)/x + 1/y + 1/(x*y + 1) |
| 1076 | sage: G = f.numerator() |
| 1077 | sage: H = R(f.denominator()) |
| 1078 | sage: ff = FFPD(G, H.factor()) |
| 1079 | sage: decomp = ff.leinartas_decomposition() |
| 1080 | sage: print decomp |
| 1081 | [(0, []), (-(x*y^2*sin(x) + x^2*y + x*y + y*sin(x) + x)*y, |
| 1082 | [(y, 1)]), ((x*y^2*sin(x) + x^2*y + x*y + y*sin(x) + x)*x*y, |
| 1083 | [(x*y + 1, 1)]), (x*y^2*sin(x) + x^2*y + x*y + y*sin(x) + x, |
| 1084 | [(y, 1), (x, 1)])] |
| 1085 | sage: bool(decomp.sum().quotient() == f) |
| 1086 | True |
| 1087 | sage: for r in decomp: |
| 1088 | ... L = r.nullstellensatz_certificate() |
| 1089 | ... print L is None |
| 1090 | ... J = r.algebraic_dependence_certificate() |
| 1091 | ... print J is None or J == J.ring().ideal() |
| 1092 | ... |
| 1093 | True |
| 1094 | True |
| 1095 | True |
| 1096 | True |
| 1097 | True |
| 1098 | True |
| 1099 | True |
| 1100 | True |
| 1101 | |
| 1102 | :: |
| 1103 | |
| 1104 | sage: R.<x, y, z>= PolynomialRing(GF(2, 'a')) |
| 1105 | sage: f = 1/(x * y * z * (x*y + z)) |
| 1106 | sage: decomp = FFPD(quotient=f).leinartas_decomposition() |
| 1107 | sage: print decomp |
| 1108 | [(0, []), (1, [(z, 2), (x*y + z, 1)]), |
| 1109 | (1, [(z, 2), (y, 1), (x, 1)])] |
| 1110 | sage: print decomp.sum().quotient() == f |
| 1111 | True |
| 1112 | |
| 1113 | NOTE:: |
| 1114 | |
| 1115 | Let $f = p/q$ where $q$ lies in a $d$ -variate polynomial ring $K[X]$ |
| 1116 | for some field $K$. |
| 1117 | Let $q_1^{e_1} \cdots q_n^{e_n}$ be the |
| 1118 | unique factorization of $q$ in $K[X]$ into irreducible factors and |
| 1119 | let $V_i$ be the algebraic variety |
| 1120 | $\{x\in L^d: q_i(x) = 0\}$ of $q_i$ over the algebraic closure |
| 1121 | $L$ of $K$. |
| 1122 | By [Raic2012]_, $f$ can be written as |
| 1123 | |
| 1124 | (*) $\sum p_A/\prod_{i \in A} q_i^{b_i}$, |
| 1125 | |
| 1126 | where the $b_i$ are positive integers, each $p_A$ is a product of $p$ and an element of $K[X]$, |
| 1127 | and the sum is taken over all subsets $A \subseteq \{1, \ldots, m\}$ such that |
| 1128 | (1) $|A| \le d$, |
| 1129 | (2) $\cap_{i\in A} T_i \neq \emptyset$, and |
| 1130 | (3) $\{q_i : i\in A\}$ is algebraically independent. |
| 1131 | |
| 1132 | In particular, any rational expression in $d$ variables |
| 1133 | can be represented as a sum of rational expressions |
| 1134 | whose denominators each contain at most $d$ distinct irreducible |
| 1135 | factors. |
| 1136 | |
| 1137 | I call (*) a *Leinartas decomposition* of $f$. |
| 1138 | Leinartas decompositions are not unique. |
| 1139 | |
| 1140 | The algorithm used comes from [Raic2012]_. |
| 1141 | """ |
| 1142 | if self.dimension() == 1: |
| 1143 | # Sage's lift() function doesn't work in |
| 1144 | # univariate polynomial rings. |
| 1145 | # So nullstellensatz_decomposition() won't work. |
| 1146 | # Can use algebraic_dependence_decomposition(), |
| 1147 | # which is sufficient. |
| 1148 | # temp = FFPDSum([self]) |
| 1149 | # Alternatively can use univariate_decomposition(), |
| 1150 | # which is more efficient. |
| 1151 | return self.univariate_decomposition() |
| 1152 | temp = self.nullstellensatz_decomposition() |
| 1153 | decomp = FFPDSum() |
| 1154 | for r in temp: |
| 1155 | decomp.extend(r.algebraic_dependence_decomposition()) |
| 1156 | |
| 1157 | # Simplify and return result. |
| 1158 | return decomp.combine_like_terms().whole_and_parts() |
| 1159 | |
| 1160 | def cohomology_decomposition(self): |
| 1161 | r""" |
| 1162 | Let $p/(q_1^{e_1} \cdots q_n^{e_n})$ be the fraction represented |
| 1163 | by ``self`` and let $K[x_1, \ldots, x_d]$ be the polynomial ring |
| 1164 | in which the $q_i$ lie. |
| 1165 | Assume that $n \le d$ and that the gradients of the $q_i$ are linearly |
| 1166 | independent at all points in the intersection |
| 1167 | $V_1 \cap \ldots \cap V_n$ of the algebraic varieties |
| 1168 | $V_i = \{x \in L^d: q_i(x) = 0 \}$, where $L$ is the algebraic closure |
| 1169 | of the field $K$. |
| 1170 | Return a FFPDSum $f$ such that the differential form |
| 1171 | $f dx_1 \wedge \cdots \wedge dx_d$ is de Rham cohomologous to the |
| 1172 | differential form |
| 1173 | $p/(q_1^{e_1} \cdots q_n^{e_n}) dx_1 \wedge \cdots \wedge dx_d$ |
| 1174 | and such that the denominator of each summand of $f$ contains |
| 1175 | no repeated irreducible factors. |
| 1176 | |
| 1177 | EXAMPLES:: |
| 1178 | |
| 1179 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 1180 | sage: R.<x> = PolynomialRing(QQ) |
| 1181 | sage: f = 1/(x^2 + x + 1)^3 |
| 1182 | sage: decomp = FFPD(quotient=f).cohomology_decomposition() |
| 1183 | sage: print decomp |
| 1184 | [(0, []), (2/3, [(x^2 + x + 1, 1)])] |
| 1185 | |
| 1186 | :: |
| 1187 | |
| 1188 | sage: R.<x, y>= PolynomialRing(QQ) |
| 1189 | sage: print FFPD(1, [(x, 1), (y, 2)]).cohomology_decomposition() |
| 1190 | [(0, [])] |
| 1191 | |
| 1192 | :: |
| 1193 | |
| 1194 | sage: R.<x, y>= PolynomialRing(QQ) |
| 1195 | sage: p = 1 |
| 1196 | sage: qs = [(x*y - 1, 1), (x**2 + y**2 - 1, 2)] |
| 1197 | sage: f = FFPD(p, qs) |
| 1198 | sage: print f.cohomology_decomposition() |
| 1199 | [(0, []), (4/3*x*y + 4/3, [(x^2 + y^2 - 1, 1)]), |
| 1200 | (1/3, [(x*y - 1, 1), (x^2 + y^2 - 1, 1)])] |
| 1201 | |
| 1202 | NOTES: |
| 1203 | |
| 1204 | The algorithm used here comes from the proof of Theorem 17.4 of |
| 1205 | [AiYu1983]_. |
| 1206 | |
| 1207 | AUTHORS: |
| 1208 | |
| 1209 | - Alexander Raichev (2008-10-01, 2011-01-15, 2012-07-31) |
| 1210 | """ |
| 1211 | R = self.ring() |
| 1212 | df = self.denominator_factored() |
| 1213 | n = len(df) |
| 1214 | if R is None or sum([e for (q, e) in df]) <= n: |
| 1215 | # No decomposing possible. |
| 1216 | return FFPDSum([self]) |
| 1217 | |
| 1218 | # Otherwise decompose recursively. |
| 1219 | decomp = FFPDSum() |
| 1220 | p = self.numerator() |
| 1221 | qs = [q for (q, e) in df] |
| 1222 | X = sorted(R.gens()) |
| 1223 | var_sets_n = Set(X).subsets(n) |
| 1224 | |
| 1225 | # Compute Jacobian determinants for qs. |
| 1226 | dets = [] |
| 1227 | for v in var_sets_n: |
| 1228 | # Sort v according to the term order of R. |
| 1229 | x = sorted(v) |
| 1230 | jac = jacobian(qs, x) |
| 1231 | dets.append(R(jac.determinant())) |
| 1232 | |
| 1233 | # Get a Nullstellensatz certificate for qs and dets. |
| 1234 | if self.dimension() == 1: |
| 1235 | # Sage's lift() function doesn't work in |
| 1236 | # univariate polynomial rings. |
| 1237 | # So use xgcd(), which does the same thing in this case. |
| 1238 | # Note that by assumption qs and dets have length 1. |
| 1239 | L = xgcd(qs[0], dets[0])[1:] |
| 1240 | else: |
| 1241 | L = R(1).lift(R.ideal(qs + dets)) |
| 1242 | |
| 1243 | # Do first iteration of decomposition. |
| 1244 | iteration1 = FFPDSum() |
| 1245 | # Contributions from qs. |
| 1246 | for i in xrange(n): |
| 1247 | if L[i] == 0: |
| 1248 | continue |
| 1249 | # Cancel one df[i] from denominator. |
| 1250 | new_df = [list(t) for t in df] |
| 1251 | if new_df[i][1] > 1: |
| 1252 | new_df[i][1] -= 1 |
| 1253 | else: |
| 1254 | del(new_df[i]) |
| 1255 | iteration1.append(FFPD(p*L[i], new_df)) |
| 1256 | |
| 1257 | # Contributions from dets. |
| 1258 | # Compute each contribution's cohomologous form using |
| 1259 | # the least index j such that new_df[j][1] > 1. |
| 1260 | # Know such an index exists by first 'if' statement at |
| 1261 | # the top. |
| 1262 | for j in xrange(n): |
| 1263 | if df[j][1] > 1: |
| 1264 | J = j |
| 1265 | break |
| 1266 | new_df = [list(t) for t in df] |
| 1267 | new_df[J][1] -= 1 |
| 1268 | for k in xrange(var_sets_n.cardinality()): |
| 1269 | if L[n + k] == 0: |
| 1270 | continue |
| 1271 | # Sort variables according to the term order of R. |
| 1272 | x = sorted(var_sets_n[k]) |
| 1273 | # Compute Jacobian in the Symbolic Ring. |
| 1274 | jac = jacobian([SR(p*L[n + k])] + |
| 1275 | [SR(qs[j]) for j in xrange(n) if j != J], |
| 1276 | [SR(xx) for xx in x]) |
| 1277 | det = jac.determinant() |
| 1278 | psign = FFPD.permutation_sign(x, X) |
| 1279 | iteration1.append(FFPD((-1)**J*det/\ |
| 1280 | (psign*new_df[J][1]), |
| 1281 | new_df)) |
| 1282 | |
| 1283 | # Now decompose each FFPD of iteration1. |
| 1284 | for r in iteration1: |
| 1285 | decomp.extend(r.cohomology_decomposition()) |
| 1286 | |
| 1287 | # Simplify and return result. |
| 1288 | return decomp.combine_like_terms().whole_and_parts() |
| 1289 | |
| 1290 | @staticmethod |
| 1291 | def permutation_sign(s, u): |
| 1292 | r""" |
| 1293 | This function returns the sign of the permutation on |
| 1294 | ``1, ..., len(u)`` that is induced by the sublist |
| 1295 | ``s`` of ``u``. |
| 1296 | For internal use by cohomology_decomposition(). |
| 1297 | |
| 1298 | INPUT: |
| 1299 | |
| 1300 | - ``s`` - A sublist of ``u``. |
| 1301 | - ``u`` - A list. |
| 1302 | |
| 1303 | OUTPUT: |
| 1304 | |
| 1305 | The sign of the permutation obtained by taking indices |
| 1306 | within ``u`` of the list ``s + sc``, where ``sc`` is ``u`` |
| 1307 | with the elements of ``s`` removed. |
| 1308 | |
| 1309 | EXAMPLES:: |
| 1310 | |
| 1311 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 1312 | sage: u = ['a','b','c','d','e'] |
| 1313 | sage: s = ['b','d'] |
| 1314 | sage: FFPD.permutation_sign(s, u) |
| 1315 | -1 |
| 1316 | sage: s = ['d','b'] |
| 1317 | sage: FFPD.permutation_sign(s, u) |
| 1318 | 1 |
| 1319 | |
| 1320 | AUTHORS: |
| 1321 | |
| 1322 | - Alexander Raichev (2008-10-01, 2012-07-31) |
| 1323 | """ |
| 1324 | # Convert lists to lists of numbers in {1,..., len(u)} |
| 1325 | A = [i + 1 for i in xrange(len(u))] |
| 1326 | B = [u.index(x) + 1 for x in s] |
| 1327 | |
| 1328 | C = sorted(list(Set(A).difference(Set(B)))) |
| 1329 | P = Permutation(B + C) |
| 1330 | return P.signature() |
| 1331 | |
| 1332 | def asymptotic_decomposition(self, alpha, asy_var=None): |
| 1333 | r""" |
| 1334 | Return a FFPDSum that has the same asymptotic expansion as ``self`` |
| 1335 | in the direction ``alpha`` but each of whose FFPDs has a |
| 1336 | denominator factorization of the form $[(q_1, 1), \ldots, (q_n, 1)]$, |
| 1337 | where ``n`` is at most ``d = self.dimension()``. |
| 1338 | The output results from a Leinartas decomposition followed by a |
| 1339 | cohomology decomposition. |
| 1340 | |
| 1341 | INPUT: |
| 1342 | |
| 1343 | - ``alpha`` - A d-tuple of positive integers or symbolic variables. |
| 1344 | - ``asy_var`` - (Optional; default=None) A symbolic variable with |
| 1345 | respect to which to compute asymptotics. |
| 1346 | If None is given the set ``asy_var = var('r')``. |
| 1347 | |
| 1348 | EXAMPLES:: |
| 1349 | |
| 1350 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 1351 | sage: R.<x> = PolynomialRing(QQ) |
| 1352 | sage: f = (x^2 + 1)/((x - 1)^3*(x + 2)) |
| 1353 | sage: F = FFPD(quotient=f) |
| 1354 | sage: alpha = [var('a')] |
| 1355 | sage: print F.asymptotic_decomposition(alpha) |
| 1356 | [(0, []), (1/54*(5*a^2*x^2 + 2*a^2*x + 11*a^2)*r^2/x^2 - 1/54*(5*a*x^2 - 2*a*x - 33*a)*r/x^2 + 11/27/x^2, [(x - 1, 1)]), (-5/27, [(x + 2, 1)])] |
| 1357 | |
| 1358 | :: |
| 1359 | |
| 1360 | sage: R.<x, y>= PolynomialRing(QQ) |
| 1361 | sage: H = (1 - 2*x -y)*(1 - x -2*y)**2 |
| 1362 | sage: Hfac = H.factor() |
| 1363 | sage: G = 1/Hfac.unit() |
| 1364 | sage: F = FFPD(G, Hfac) |
| 1365 | sage: alpha = var('a, b') |
| 1366 | sage: print F.asymptotic_decomposition(alpha) # long time |
| 1367 | [(0, []), (-1/3*(a*y - 2*b*x)*r/(x*y) + 1/3*(2*x - y)/(x*y), |
| 1368 | [(x + 2*y - 1, 1), (2*x + y - 1, 1)])] |
| 1369 | |
| 1370 | AUTHORS: |
| 1371 | |
| 1372 | - Alexander Raichev (2012-08-01) |
| 1373 | """ |
| 1374 | R = self.ring() |
| 1375 | if R is None: |
| 1376 | return None |
| 1377 | |
| 1378 | d = self.dimension() |
| 1379 | n = len(self.denominator_factored()) |
| 1380 | X = [SR(x) for x in R.gens()] |
| 1381 | |
| 1382 | # Reduce number of distinct factors in denominator of self |
| 1383 | # down to at most d. |
| 1384 | decomp1 = FFPDSum([self]) |
| 1385 | if n > d: |
| 1386 | decomp1 = decomp1[0].leinartas_decomposition() |
| 1387 | |
| 1388 | # Reduce to no repeated factors in denominator of each element |
| 1389 | # of decomp1. |
| 1390 | # Compute the cohomology decomposition for each |
| 1391 | # Cauchy differential form generated by each element of decomp. |
| 1392 | if asy_var is None: |
| 1393 | asy_var = var('r') |
| 1394 | cauchy_stuff = prod([X[j]**(-alpha[j]*asy_var - 1) for j in xrange(d)]) |
| 1395 | decomp2 = FFPDSum() |
| 1396 | for f in decomp1: |
| 1397 | ff = FFPD(f.numerator()*cauchy_stuff, |
| 1398 | f.denominator_factored()) |
| 1399 | decomp2.extend(ff.cohomology_decomposition()) |
| 1400 | decomp2 = decomp2.combine_like_terms() |
| 1401 | |
| 1402 | # Divide out cauchy_stuff from integrands. |
| 1403 | decomp3 = FFPDSum() |
| 1404 | for f in decomp2: |
| 1405 | ff = FFPD((f.numerator()/cauchy_stuff).\ |
| 1406 | simplify_full().collect(asy_var), |
| 1407 | f.denominator_factored()) |
| 1408 | decomp3.append(ff) |
| 1409 | |
| 1410 | return decomp3 |
| 1411 | |
| 1412 | def asymptotics(self, p, alpha, N, asy_var=None, numerical=0): |
| 1413 | r""" |
| 1414 | Return the first $N$ terms (some of which could be zero) |
| 1415 | of the asymptotic expansion of the Maclaurin ray coefficients |
| 1416 | $F_{r\alpha}$ of the function $F$ represented by ``self`` |
| 1417 | as $r\to\infty$, where $r$ = ``asy_var`` and ``alpha`` is a tuple of |
| 1418 | positive integers of length ``d = self.dimension()``. |
| 1419 | Assume that |
| 1420 | |
| 1421 | - $F$ is holomorphic in a neighborhood of the origin; |
| 1422 | - the unique factorization of the denominator $H$ of $F$ in the local |
| 1423 | algebraic ring at $p$ equals its unique factorization in the local |
| 1424 | analytic ring at $p$; |
| 1425 | - the unique factorization of $H$ in the local algebraic ring at $p$ |
| 1426 | has $<= d$ irreducible factors, none of which are repeated |
| 1427 | (one can reduce to this case via asymptotic_decomposition()); |
| 1428 | - $p$ is a convenient strictly minimal smooth or multiple point |
| 1429 | with all nonzero coordinates that is critical and nondegenerate |
| 1430 | for ``alpha``. |
| 1431 | |
| 1432 | INPUT: |
| 1433 | |
| 1434 | - ``p`` - A dictionary with keys that can be coerced to equal |
| 1435 | ``self.ring().gens()``. |
| 1436 | - ``alpha`` - A tuple of length ``self.dimension()`` of |
| 1437 | positive integers or, if $p$ is a smooth point, |
| 1438 | possibly of symbolic variables. |
| 1439 | - ``N`` - A positive integer. |
| 1440 | - ``numerical`` - (Optional; default=0) A natural number. |
| 1441 | If numerical > 0, then return a numerical approximation |
| 1442 | of $F_{r \alpha}$ with ``numerical`` digits of precision. |
| 1443 | Otherwise return exact values. |
| 1444 | - ``asy_var`` - (Optional; default=None) A symbolic variable. |
| 1445 | The variable of the asymptotic expansion. |
| 1446 | If none is given, ``var('r')`` will be assigned. |
| 1447 | |
| 1448 | OUTPUT: |
| 1449 | |
| 1450 | The tuple (asy, exp_scale, subexp_part). |
| 1451 | Here asy is the sum of the first $N$ terms (some of which might be 0) |
| 1452 | of the asymptotic expansion of $F_{r\alpha}$ as $r\to\infty$; |
| 1453 | exp_scale**r is the exponential factor of asy; |
| 1454 | subexp_part is the subexponential factor of asy. |
| 1455 | |
| 1456 | EXAMPLES:: |
| 1457 | |
| 1458 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 1459 | |
| 1460 | A smooth point example:: |
| 1461 | |
| 1462 | sage: R.<x,y>= PolynomialRing(QQ) |
| 1463 | sage: H = (1 - x - y - x*y)**2 |
| 1464 | sage: Hfac = H.factor() |
| 1465 | sage: G = 1/Hfac.unit() |
| 1466 | sage: F = FFPD(G, Hfac); print(F) |
| 1467 | (1, [(x*y + x + y - 1, 2)]) |
| 1468 | sage: alpha = [4, 3] |
| 1469 | sage: decomp = F.asymptotic_decomposition(alpha); print decomp |
| 1470 | [(0, []), (-3/2*(y + 1)*r/y - 1/2*(y + 1)/y, [(x*y + x + y - 1, 1)])] |
| 1471 | sage: F1 = decomp[1] |
| 1472 | sage: p = {y: 1/3, x: 1/2} |
| 1473 | sage: asy = F1.asymptotics(p, alpha, 2) # long time |
| 1474 | Creating auxiliary functions... |
| 1475 | Computing derivatives of auxiallary functions... |
| 1476 | Computing derivatives of more auxiliary functions... |
| 1477 | Computing second order differential operator actions... |
| 1478 | sage: print asy # long time |
| 1479 | (1/6000*(3600*sqrt(2)*sqrt(3)*sqrt(5)*sqrt(r)/sqrt(pi) + 463*sqrt(2)*sqrt(3)*sqrt(5)/(sqrt(pi)*sqrt(r)))*432^r, 432, 3/5*sqrt(2)*sqrt(3)*sqrt(5)*sqrt(r)/sqrt(pi) + 463/6000*sqrt(2)*sqrt(3)*sqrt(5)/(sqrt(pi)*sqrt(r))) |
| 1480 | sage: print F.relative_error(asy[0], alpha, [1, 2, 4, 8, 16], asy[1]) # long time |
| 1481 | Calculating errors table in the form |
| 1482 | exponent, scaled Maclaurin coefficient, scaled asymptotic |
| 1483 | values, relative errors... |
| 1484 | [((4, 3), 2.083333333, [2.092576110], [-0.004436533009]), |
| 1485 | ((8, 6), 2.787374614, [2.790732875], [-0.001204811281]), |
| 1486 | ((16, 12), 3.826259447, [3.827462310], [-0.0003143703383]), |
| 1487 | ((32, 24), 5.328112821, [5.328540787], [-0.00008032229296]), |
| 1488 | ((64, 48), 7.475927885, [7.476079664], [-0.00002030233658])] |
| 1489 | |
| 1490 | A multiple point example:: |
| 1491 | |
| 1492 | sage: R.<x,y,z>= PolynomialRing(QQ) |
| 1493 | sage: H = (4 - 2*x - y - z)**2*(4 - x - 2*y - z) |
| 1494 | sage: Hfac = H.factor() |
| 1495 | sage: G = 16/Hfac.unit() |
| 1496 | sage: F = FFPD(G, Hfac) |
| 1497 | sage: print F |
| 1498 | (-16, [(x + 2*y + z - 4, 1), (2*x + y + z - 4, 2)]) |
| 1499 | sage: alpha = [3, 3, 2] |
| 1500 | sage: decomp = F.asymptotic_decomposition(alpha); print decomp |
| 1501 | [(0, []), (16*(4*y - 3*z)*r/(y*z) + 16*(2*y - z)/(y*z), [(x + 2*y + z - 4, 1), (2*x + y + z - 4, 1)])] |
| 1502 | sage: F1 = decomp[1] |
| 1503 | sage: p = {x: 1, y: 1, z: 1} |
| 1504 | sage: asy = F1.asymptotics(p, alpha, 2) |
| 1505 | Creating auxiliary functions... |
| 1506 | Computing derivatives of auxiliary functions... |
| 1507 | Computing derivatives of more auxiliary functions... |
| 1508 | Computing second-order differential operator actions... |
| 1509 | sage: print asy |
| 1510 | (4/3*sqrt(3)*sqrt(r)/sqrt(pi) + 47/216*sqrt(3)/(sqrt(pi)*sqrt(r)), 1, 4/3*sqrt(3)*sqrt(r)/sqrt(pi) + 47/216*sqrt(3)/(sqrt(pi)*sqrt(r))) |
| 1511 | sage: print F.relative_error(asy[0], alpha, [1, 2, 4, 8], asy[1]) |
| 1512 | Calculating errors table in the form |
| 1513 | exponent, scaled Maclaurin coefficient, scaled asymptotic values, |
| 1514 | relative errors... |
| 1515 | [((3, 3, 2), 0.9812164307, [1.515572606], [-0.5445854340]), |
| 1516 | ((6, 6, 4), |
| 1517 | 1.576181132, [1.992989399], [-0.2644418580]), |
| 1518 | ((12, 12, 8), 2.485286378, |
| 1519 | [2.712196351], [-0.09130133851]), ((24, 24, 16), 3.700576827, |
| 1520 | [3.760447895], [-0.01617884750])] |
| 1521 | |
| 1522 | NOTES: |
| 1523 | |
| 1524 | The algorithms used here come from [RaWi2008a]_ and [RaWi2012]_. |
| 1525 | |
| 1526 | AUTHORS: |
| 1527 | |
| 1528 | - Alexander Raichev (2008-10-01, 2010-09-28, 2011-04-27, 2012-08-03) |
| 1529 | """ |
| 1530 | R = self.ring() |
| 1531 | if R is None: |
| 1532 | return None |
| 1533 | |
| 1534 | # Coerce keys of p into R. |
| 1535 | p = FFPD.coerce_point(R, p) |
| 1536 | |
| 1537 | if asy_var is None: |
| 1538 | asy_var = var('r') |
| 1539 | d = self.dimension() |
| 1540 | X = list(R.gens()) |
| 1541 | alpha = list(alpha) |
| 1542 | df = self.denominator_factored() |
| 1543 | n = len(df) # Number of smooth factors |
| 1544 | |
| 1545 | # Find greatest i such that X[i] is a convenient coordinate, |
| 1546 | # that is, such that for all (h, e) in df, we have |
| 1547 | # (X[i]*diff(h, X[i])).subs(p) != 0. |
| 1548 | # Assuming such an i exists. |
| 1549 | i = d - 1 |
| 1550 | while 0 in [(X[i]*diff(h, X[i])).subs(p) for (h, e) in df]: |
| 1551 | i -= 1 |
| 1552 | coordinate = i |
| 1553 | |
| 1554 | if n == 1: |
| 1555 | # Smooth point. |
| 1556 | return self.asymptotics_smooth(p, alpha, N, asy_var, |
| 1557 | coordinate, numerical) |
| 1558 | else: |
| 1559 | # Multiple point. |
| 1560 | return self.asymptotics_multiple(p, alpha, N, asy_var, |
| 1561 | coordinate, numerical) |
| 1562 | |
| 1563 | def asymptotics_smooth(self, p, alpha, N, asy_var, coordinate=None, |
| 1564 | numerical=0): |
| 1565 | r""" |
| 1566 | Does what asymptotics() does but only in the case of a |
| 1567 | convenient smooth point. |
| 1568 | |
| 1569 | INPUT: |
| 1570 | |
| 1571 | - ``p`` - A dictionary with keys that can be coerced to equal |
| 1572 | ``self.ring().gens()``. |
| 1573 | - ``alpha`` - A tuple of length ``d = self.dimension()`` of |
| 1574 | positive integers or, if $p$ is a smooth point, |
| 1575 | possibly of symbolic variables. |
| 1576 | - ``N`` - A positive integer. |
| 1577 | - ``asy_var`` - (Optional; default=None) A symbolic variable. |
| 1578 | The variable of the asymptotic expansion. |
| 1579 | If none is given, ``var('r')`` will be assigned. |
| 1580 | - ``coordinate``- (Optional; default=None) An integer in |
| 1581 | {0, ..., d-1} indicating a convenient coordinate to base |
| 1582 | the asymptotic calculations on. |
| 1583 | If None is assigned, then choose ``coordinate=d-1``. |
| 1584 | - ``numerical`` - (Optional; default=0) A natural number. |
| 1585 | If numerical > 0, then return a numerical approximation of the |
| 1586 | Maclaurin ray coefficients of ``self`` with ``numerical`` digits |
| 1587 | of precision. |
| 1588 | Otherwise return exact values. |
| 1589 | |
| 1590 | NOTES: |
| 1591 | |
| 1592 | The formulas used for computing the asymptotic expansions are |
| 1593 | Theorems 3.2 and 3.3 [RaWi2008a]_ with the exponent of H equal to 1. |
| 1594 | Theorem 3.2 is a specialization of Theorem 3.4 of [RaWi2012]_ |
| 1595 | with $n=1$. |
| 1596 | |
| 1597 | EXAMPLES:: |
| 1598 | |
| 1599 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 1600 | sage: R.<x> = PolynomialRing(QQ) |
| 1601 | sage: H = 2 - 3*x |
| 1602 | sage: Hfac = H.factor() |
| 1603 | sage: G = 1/Hfac.unit() |
| 1604 | sage: F = FFPD(G, Hfac) |
| 1605 | sage: print F |
| 1606 | (-1/3, [(x - 2/3, 1)]) |
| 1607 | sage: alpha = [2] |
| 1608 | sage: p = {x: 2/3} |
| 1609 | sage: asy = F.asymptotics_smooth(p, alpha, 3, asy_var=var('r')) |
| 1610 | sage: print asy |
| 1611 | (1/2*(9/4)^r, 9/4, 1/2) |
| 1612 | |
| 1613 | :: |
| 1614 | |
| 1615 | sage: R.<x, y>= PolynomialRing(QQ) |
| 1616 | sage: H = 1-x-y-x*y |
| 1617 | sage: Hfac = H.factor() |
| 1618 | sage: G = 1/Hfac.unit() |
| 1619 | sage: F = FFPD(G, Hfac) |
| 1620 | sage: alpha = [3, 2] |
| 1621 | sage: p = {y: 1/2*sqrt(13) - 3/2, x: 1/3*sqrt(13) - 2/3} |
| 1622 | sage: print F.asymptotics_smooth(p, alpha, 2, var('r'), numerical=3) # long time |
| 1623 | Creating auxiliary functions... |
| 1624 | Computing derivatives of auxiallary functions... |
| 1625 | Computing derivatives of more auxiliary functions... |
| 1626 | Computing second order differential operator actions... |
| 1627 | ((0.369/sqrt(r) - 0.0186/r^(3/2))*71.2^r, 71.2, |
| 1628 | 0.369/sqrt(r) - 0.0186/r^(3/2)) |
| 1629 | |
| 1630 | :: |
| 1631 | |
| 1632 | sage: R.<x, y> = PolynomialRing(QQ) |
| 1633 | sage: q = 1/2 |
| 1634 | sage: qq = q.denominator() |
| 1635 | sage: H = 1 - q*x + q*x*y - x^2*y |
| 1636 | sage: Hfac = H.factor() |
| 1637 | sage: G = (1 - q*x)/Hfac.unit() |
| 1638 | sage: F = FFPD(G, Hfac) |
| 1639 | sage: alpha = list(qq*vector([2, 1 - q])) |
| 1640 | sage: print alpha |
| 1641 | [4, 1] |
| 1642 | sage: p = {x: 1, y: 1} |
| 1643 | sage: print F.asymptotics_smooth(p, alpha, 5, var('r')) # long time |
| 1644 | Creating auxiliary functions... |
| 1645 | Computing derivatives of auxiallary functions... |
| 1646 | Computing derivatives of more auxiliary functions... |
| 1647 | Computing second order differential operator actions... |
| 1648 | (1/12*2^(2/3)*sqrt(3)*gamma(1/3)/(pi*r^(1/3)) - |
| 1649 | 1/96*2^(1/3)*sqrt(3)*gamma(2/3)/(pi*r^(5/3)), 1, |
| 1650 | 1/12*2^(2/3)*sqrt(3)*gamma(1/3)/(pi*r^(1/3)) - |
| 1651 | 1/96*2^(1/3)*sqrt(3)*gamma(2/3)/(pi*r^(5/3))) |
| 1652 | |
| 1653 | AUTHORS: |
| 1654 | |
| 1655 | - Alexander Raichev (2008-10-01, 2010-09-28, 2012-08-01) |
| 1656 | """ |
| 1657 | R = self.ring() |
| 1658 | if R is None: |
| 1659 | return None |
| 1660 | |
| 1661 | d = self.dimension() |
| 1662 | I = sqrt(-Integer(1)) |
| 1663 | # Coerce everything into the Symbolic Ring. |
| 1664 | X = [SR(x) for x in R.gens()] |
| 1665 | G = SR(self.numerator()) |
| 1666 | H = SR(self.denominator()) |
| 1667 | p = dict([(SR(x), p[x]) for x in R.gens()]) |
| 1668 | alpha = [SR(a) for a in alpha] |
| 1669 | |
| 1670 | # Put given convenient coordinate at end of variable list. |
| 1671 | if coordinate is not None: |
| 1672 | x = X.pop(coordinate) |
| 1673 | X.append(x) |
| 1674 | a = alpha.pop(coordinate) |
| 1675 | alpha.append(a) |
| 1676 | |
| 1677 | # Deal with the simple univariate case first. |
| 1678 | # Same as the multiple point case with n == d. |
| 1679 | # but with a negative sign. |
| 1680 | # I'll just past the code from the multiple point case. |
| 1681 | if d == 1: |
| 1682 | det = jacobian(H, X).subs(p).determinant().abs() |
| 1683 | exp_scale = prod([(p[X[i]]**(-alpha[i])).subs(p) |
| 1684 | for i in xrange(d)] ) |
| 1685 | subexp_part = -G.subs(p)/(det*prod(p.values())) |
| 1686 | if numerical: |
| 1687 | exp_scale = exp_scale.n(digits=numerical) |
| 1688 | subexp_part = subexp_part.n(digits=numerical) |
| 1689 | return (exp_scale**asy_var*subexp_part, exp_scale, subexp_part) |
| 1690 | |
| 1691 | # If p is a tuple of rationals, then compute with it directly. |
| 1692 | # Otherwise, compute symbolically and plug in p at the end. |
| 1693 | if vector(p.values()) in QQ**d: |
| 1694 | P = p |
| 1695 | else: |
| 1696 | sP = [var('p' + str(j)) for j in xrange(d)] |
| 1697 | P = dict( [(X[j], sP[j]) for j in xrange(d)] ) |
| 1698 | p = dict( [(sP[j], p[X[j]]) for j in xrange(d)] ) |
| 1699 | |
| 1700 | # Setup. |
| 1701 | print "Creating auxiliary functions..." |
| 1702 | # Implicit functions. |
| 1703 | h = function('h', *tuple(X[:d - 1])) |
| 1704 | U = function('U', *tuple(X)) |
| 1705 | # All other functions are defined in terms of h, U, and |
| 1706 | # explicit functions. |
| 1707 | Gcheck = -G/U * (h/X[d - 1]) |
| 1708 | A = Gcheck.subs({X[d - 1]: Integer(1)/h})/h |
| 1709 | t = 't' |
| 1710 | while t in [str(x) for x in X]: |
| 1711 | t = t + 't' |
| 1712 | T = [var(t + str(i)) for i in xrange(d - 1)] |
| 1713 | e = dict([(X[i], P[X[i]]*exp(I*T[i])) for i in xrange(d - 1)]) |
| 1714 | ht = h.subs(e) |
| 1715 | At = A.subs(e) |
| 1716 | Phit = -log(P[X[d - 1]]*ht) + \ |
| 1717 | I * add([alpha[i]/alpha[d - 1]*T[i] for i in xrange(d - 1)]) |
| 1718 | Tstar = dict([(t, Integer(0)) for t in T]) |
| 1719 | # Store h and U and all their derivatives evaluated at P. |
| 1720 | atP = P.copy() |
| 1721 | atP.update({h.subs(P): Integer(1)/P[X[d - 1]]}) |
| 1722 | |
| 1723 | # Compute the derivatives of h up to order 2*N, evaluate at P, |
| 1724 | # and store in atP. |
| 1725 | # Keep a copy of unevaluated h derivatives for use in the case |
| 1726 | # d = 2 and v > 2 below. |
| 1727 | hderivs1 = {} # First derivatives of h. |
| 1728 | for i in xrange(d - 1): |
| 1729 | s = solve( diff(H.subs({X[d - 1]: Integer(1)/h}), X[i]), |
| 1730 | diff(h, X[i]))[0].rhs().simplify() |
| 1731 | hderivs1.update({diff(h, X[i]): s}) |
| 1732 | atP.update({diff(h, X[i]).subs(P): s.subs(P).subs(atP)}) |
| 1733 | hderivs = FFPD.diff_all(h, X[0: d - 1], 2*N, sub=hderivs1, rekey=h) |
| 1734 | for k in hderivs.keys(): |
| 1735 | atP.update({k.subs(P):hderivs[k].subs(atP)}) |
| 1736 | |
| 1737 | # Compute the derivatives of U up to order 2*N and evaluate at P. |
| 1738 | # To do this, differentiate H = U*Hcheck over and over, evaluate at P, |
| 1739 | # and solve for the derivatives of U at P. |
| 1740 | # Need the derivatives of H with short keys to pass on |
| 1741 | # to diff_prod later. |
| 1742 | Hderivs = FFPD.diff_all(H, X, 2*N, ending=[X[d - 1]], sub_final=P) |
| 1743 | print "Computing derivatives of auxiallary functions..." |
| 1744 | # For convenience in checking if all the nontrivial derivatives of U |
| 1745 | # at p are zero a few line below, store the value of U(p) in atP |
| 1746 | # instead of in Uderivs. |
| 1747 | Uderivs ={} |
| 1748 | atP.update({U.subs(P): diff(H, X[d - 1]).subs(P)}) |
| 1749 | end = [X[d - 1]] |
| 1750 | Hcheck = X[d - 1] - Integer(1)/h |
| 1751 | k = H.polynomial(CC).degree() - 1 |
| 1752 | if k == 0: |
| 1753 | # Then we can conclude that all higher derivatives of U are zero. |
| 1754 | for l in xrange(1, 2*N + 1): |
| 1755 | for s in UnorderedTuples(X, l): |
| 1756 | Uderivs[diff(U, s).subs(P)] = Integer(0) |
| 1757 | elif k > 0 and k < 2*N: |
| 1758 | all_zero = True |
| 1759 | Uderivs = FFPD.diff_prod(Hderivs, U, Hcheck, X, |
| 1760 | range(1, k + 1), end, Uderivs, atP) |
| 1761 | # Check for a nonzero U derivative. |
| 1762 | if Uderivs.values() != [Integer(0) for i in xrange(len(Uderivs))]: |
| 1763 | all_zero = False |
| 1764 | if all_zero: |
| 1765 | # Then, using a proposition at the end of [RaWi2012], we can |
| 1766 | # conclude that all higher derivatives of U are zero. |
| 1767 | for l in xrange(k + 1, 2*N +1): |
| 1768 | for s in UnorderedTuples(X, l): |
| 1769 | Uderivs.update({diff(U, s).subs(P): Integer(0)}) |
| 1770 | else: |
| 1771 | # Have to compute the rest of the derivatives. |
| 1772 | Uderivs = FFPD.diff_prod(Hderivs, U, Hcheck, X, |
| 1773 | range(k + 1, 2*N + 1), end, Uderivs, |
| 1774 | atP) |
| 1775 | else: |
| 1776 | Uderivs = FFPD.diff_prod(Hderivs, U, Hcheck, X, |
| 1777 | range(1, 2*N + 1), end, Uderivs, atP) |
| 1778 | atP.update(Uderivs) |
| 1779 | |
| 1780 | # In general, this algorithm is not designed to handle the case of a |
| 1781 | # singular Phit''(Tstar). |
| 1782 | # However, when d = 2 the algorithm can cope. |
| 1783 | if d == 2: |
| 1784 | # Compute v, the order of vanishing at Tstar of Phit. |
| 1785 | # It is at least 2. |
| 1786 | v = Integer(2) |
| 1787 | Phitderiv = diff(Phit, T[0], 2) |
| 1788 | splat = Phitderiv.subs(Tstar).subs(atP).subs(p).simplify() |
| 1789 | while splat == 0: |
| 1790 | v += 1 |
| 1791 | if v > 2*N: |
| 1792 | # Then need to compute more derivatives of h for atP. |
| 1793 | hderivs.update({diff(h, X[0], v): |
| 1794 | diff(hderivs[diff(h, X[0], v - 1)], |
| 1795 | X[0]).subs(hderivs1)}) |
| 1796 | atP.update({diff(h, X[0], v).subs(P): |
| 1797 | hderivs[diff(h, X[0], v)].subs(atP)}) |
| 1798 | Phitderiv = diff(Phitderiv, T[0]) |
| 1799 | splat = Phitderiv.subs(Tstar).subs(atP).subs(p).simplify() |
| 1800 | if d == 2 and v > 2: |
| 1801 | t = T[0] # Simplify variable names. |
| 1802 | a = splat/factorial(v) |
| 1803 | Phitu = Phit - a*t**v |
| 1804 | |
| 1805 | # Compute all partial derivatives of At and Phitu |
| 1806 | # up to orders 2*(N - 1) and 2*(N - 1) + v, respectively, |
| 1807 | # in case v is even. |
| 1808 | # Otherwise, compute up to orders N - 1 and N - 1 + v, |
| 1809 | # respectively. |
| 1810 | # To speed up later computations, |
| 1811 | # create symbolic functions AA and BB |
| 1812 | # to stand in for the expressions At and Phitu, respectively. |
| 1813 | print "Computing derivatives of more auxiliary functions..." |
| 1814 | AA = function('AA', t) |
| 1815 | BB = function('BB', t) |
| 1816 | if v.mod(2) == 0: |
| 1817 | At_derivs = FFPD.diff_all(At, T, 2*N - 2, |
| 1818 | sub=hderivs1, sub_final=[Tstar, atP], |
| 1819 | rekey=AA) |
| 1820 | Phitu_derivs = FFPD.diff_all(Phitu, T, 2*N - 2 +v, |
| 1821 | sub=hderivs1, |
| 1822 | sub_final=[Tstar, atP], |
| 1823 | zero_order=v + 1, rekey=BB) |
| 1824 | else: |
| 1825 | At_derivs = FFPD.diff_all(At, T, N - 1, sub=hderivs1, |
| 1826 | sub_final=[Tstar, atP], rekey=AA) |
| 1827 | Phitu_derivs = FFPD.diff_all(Phitu, T, N - 1 + v, |
| 1828 | sub=hderivs1, |
| 1829 | sub_final=[Tstar, atP], |
| 1830 | zero_order=v + 1 , rekey=BB) |
| 1831 | AABB_derivs = At_derivs |
| 1832 | AABB_derivs.update(Phitu_derivs) |
| 1833 | AABB_derivs[AA] = At.subs(Tstar).subs(atP) |
| 1834 | AABB_derivs[BB] = Phitu.subs(Tstar).subs(atP) |
| 1835 | print "Computing second order differential operator actions..." |
| 1836 | DD = FFPD.diff_op_simple(AA, BB, AABB_derivs, t, v, a, N) |
| 1837 | |
| 1838 | # Plug above into asymptotic formula. |
| 1839 | L = [] |
| 1840 | if v.mod(2) == 0: |
| 1841 | for k in xrange(N): |
| 1842 | L.append(add([ |
| 1843 | (-1)**l * gamma((2*k + v*l + 1)/v)/\ |
| 1844 | (factorial(l) * factorial(2*k + v*l))*\ |
| 1845 | DD[(k, l)] for l in xrange(0, 2*k + 1) ])) |
| 1846 | chunk = a**(-1/v)/(pi*v)*add([ |
| 1847 | alpha[d - 1 ]**(-(2*k + 1)/v)*\ |
| 1848 | L[k]*asy_var**(-(2*k + 1)/v) for k in xrange(N) ]) |
| 1849 | else: |
| 1850 | zeta = exp(I*pi/(2*v)) |
| 1851 | for k in xrange(N): |
| 1852 | L.append(add([ |
| 1853 | (-1)**l*gamma((k + v*l + 1)/v)/\ |
| 1854 | (factorial(l)*factorial(k + v*l))*\ |
| 1855 | (zeta**(k + v*l + 1) +\ |
| 1856 | (-1)**(k + v*l)*zeta**(-(k + v*l + 1)))*\ |
| 1857 | DD[(k, l)] for l in xrange(0, k + 1) ])) |
| 1858 | chunk = abs(a)**(-1/v)/(2*pi*v)*add([ |
| 1859 | alpha[d - 1]**(-(k + 1)/v)*\ |
| 1860 | L[k] *asy_var**(-(k + 1)/v) for k in xrange(N) ]) |
| 1861 | |
| 1862 | # Asymptotics for d >= 2 case. |
| 1863 | # A singular Phit''(Tstar) will cause a crash in this case. |
| 1864 | else: |
| 1865 | Phit1 = jacobian(Phit, T).subs(hderivs1) |
| 1866 | a = jacobian(Phit1, T).subs(hderivs1).subs(Tstar).subs(atP) |
| 1867 | a_inv = a.inverse() |
| 1868 | Phitu = Phit - (Integer(1)/Integer(2))*matrix([T])*\ |
| 1869 | a*matrix([T]).transpose() |
| 1870 | Phitu = Phitu[0][0] |
| 1871 | # Compute all partial derivatives of At and Phitu up to |
| 1872 | # orders 2*N-2 and 2*N, respectively. |
| 1873 | # Take advantage of the fact that At and Phitu |
| 1874 | # are sufficiently differentiable functions so that mixed partials |
| 1875 | # are equal. Thus only need to compute representative partials. |
| 1876 | # Choose nondecreasing sequences as representative differentiation- |
| 1877 | # order sequences. |
| 1878 | # To speed up later computations, |
| 1879 | # create symbolic functions AA and BB |
| 1880 | # to stand in for the expressions At and Phitu, respectively. |
| 1881 | print "Computing derivatives of more auxiliary functions..." |
| 1882 | AA = function('AA', *tuple(T)) |
| 1883 | At_derivs = FFPD.diff_all(At, T, 2*N - 2, sub=hderivs1, |
| 1884 | sub_final =[Tstar, atP], rekey=AA) |
| 1885 | BB = function('BB', *tuple(T)) |
| 1886 | Phitu_derivs = FFPD.diff_all(Phitu, T, 2*N, sub=hderivs1, |
| 1887 | sub_final =[Tstar, atP], rekey=BB, |
| 1888 | zero_order=3) |
| 1889 | AABB_derivs = At_derivs |
| 1890 | AABB_derivs.update(Phitu_derivs) |
| 1891 | AABB_derivs[AA] = At.subs(Tstar).subs(atP) |
| 1892 | AABB_derivs[BB] = Phitu.subs(Tstar).subs(atP) |
| 1893 | print "Computing second order differential operator actions..." |
| 1894 | DD = FFPD.diff_op(AA, BB, AABB_derivs, T, a_inv, 1 , N) |
| 1895 | |
| 1896 | # Plug above into asymptotic formula. |
| 1897 | L =[] |
| 1898 | for k in xrange(N): |
| 1899 | L.append(add([ |
| 1900 | DD[(0, k, l)]/((-1)**k*2**(l+k)*\ |
| 1901 | factorial(l)*factorial(l+k)) |
| 1902 | for l in xrange(0, 2*k + 1) ])) |
| 1903 | chunk = add([ (2*pi)**((1 - d)/Integer(2))*\ |
| 1904 | a.determinant()**(-Integer(1)/Integer(2))*\ |
| 1905 | alpha[d - 1]**((Integer(1) - d)/Integer(2) - k)*L[k]*\ |
| 1906 | asy_var**((Integer(1) - d)/Integer(2) - k) |
| 1907 | for k in xrange(N) ]) |
| 1908 | |
| 1909 | chunk = chunk.subs(p).simplify() |
| 1910 | coeffs = chunk.coefficients(asy_var) |
| 1911 | coeffs.reverse() |
| 1912 | coeffs = coeffs[:N] |
| 1913 | if numerical: |
| 1914 | subexp_part = add([co[0].subs(p).n(digits=numerical)*\ |
| 1915 | asy_var**co[1] for co in coeffs]) |
| 1916 | exp_scale = prod([(P[X[i]]**(-alpha[i])).subs(p) |
| 1917 | for i in xrange(d)]).n(digits=numerical) |
| 1918 | else: |
| 1919 | subexp_part = add([co[0].subs(p)*asy_var**co[1] for co in coeffs]) |
| 1920 | exp_scale = prod([(P[X[i]]**(-alpha[i])).subs(p) |
| 1921 | for i in xrange(d)]) |
| 1922 | return (exp_scale**asy_var*subexp_part, exp_scale, subexp_part) |
| 1923 | |
| 1924 | def asymptotics_multiple(self, p, alpha, N, asy_var, coordinate=None, |
| 1925 | numerical=0): |
| 1926 | r""" |
| 1927 | Does what asymptotics() does but only in the case of a |
| 1928 | convenient multiple point nondegenerate for alpha. |
| 1929 | Assume also that ``self.dimension >= 2`` and that the |
| 1930 | ``p.values()`` are not symbolic variables. |
| 1931 | |
| 1932 | INPUT: |
| 1933 | |
| 1934 | - ``p`` - A dictionary with keys that can be coerced to equal |
| 1935 | ``self.ring().gens()``. |
| 1936 | - ``alpha`` - A tuple of length ``d = self.dimension()`` of |
| 1937 | positive integers or, if $p$ is a smooth point, |
| 1938 | possibly of symbolic variables. |
| 1939 | - ``N`` - A positive integer. |
| 1940 | - ``asy_var`` - (Optional; default=None) A symbolic variable. |
| 1941 | The variable of the asymptotic expansion. |
| 1942 | If none is given, ``var('r')`` will be assigned. |
| 1943 | - ``coordinate``- (Optional; default=None) An integer in |
| 1944 | {0, ..., d-1} indicating a convenient coordinate to base |
| 1945 | the asymptotic calculations on. |
| 1946 | If None is assigned, then choose ``coordinate=d-1``. |
| 1947 | - ``numerical`` - (Optional; default=0) A natural number. |
| 1948 | If numerical > 0, then return a numerical approximation of the |
| 1949 | Maclaurin ray coefficients of ``self`` with ``numerical`` digits |
| 1950 | of precision. |
| 1951 | Otherwise return exact values. |
| 1952 | |
| 1953 | NOTES: |
| 1954 | |
| 1955 | The formulas used for computing the asymptotic expansion are |
| 1956 | Theorem 3.4 and Theorem 3.7 of [RaWi2012]_. |
| 1957 | |
| 1958 | EXAMPLES:: |
| 1959 | |
| 1960 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 1961 | sage: R.<x, y, z>= PolynomialRing(QQ) |
| 1962 | sage: H = (4 - 2*x - y - z)*(4 - x -2*y - z) |
| 1963 | sage: Hfac = H.factor() |
| 1964 | sage: G = 16/Hfac.unit() |
| 1965 | sage: F = FFPD(G, Hfac) |
| 1966 | sage: print F |
| 1967 | (16, [(x + 2*y + z - 4, 1), (2*x + y + z - 4, 1)]) |
| 1968 | sage: p = {x: 1, y: 1, z: 1} |
| 1969 | sage: alpha = [3, 3, 2] |
| 1970 | sage: print F.asymptotics_multiple(p, alpha, 2, var('r')) # long time |
| 1971 | Creating auxiliary functions... |
| 1972 | Computing derivatives of auxiliary functions... |
| 1973 | Computing derivatives of more auxiliary functions... |
| 1974 | Computing second-order differential operator actions... |
| 1975 | (4/3*sqrt(3)/(sqrt(pi)*sqrt(r)) - |
| 1976 | 25/216*sqrt(3)/(sqrt(pi)*r^(3/2)), 1, |
| 1977 | 4/3*sqrt(3)/(sqrt(pi)*sqrt(r)) - 25/216*sqrt(3)/(sqrt(pi)*r^(3/2))) |
| 1978 | |
| 1979 | :: |
| 1980 | |
| 1981 | sage: R.<x, y, z>= PolynomialRing(QQ) |
| 1982 | sage: H = (1 - x*(1 + y))*(1 - z*x**2*(1 + 2*y)) |
| 1983 | sage: Hfac = H.factor() |
| 1984 | sage: G = 1/Hfac.unit() |
| 1985 | sage: F = FFPD(G, Hfac) |
| 1986 | sage: print F |
| 1987 | (1, [(x*y + x - 1, 1), (2*x^2*y*z + x^2*z - 1, 1)]) |
| 1988 | sage: p = {x: 1/2, z: 4/3, y: 1} |
| 1989 | sage: alpha = [8, 3, 3] |
| 1990 | sage: print F.asymptotics_multiple(p, alpha, 2, var('r'), coordinate=1) # long time |
| 1991 | Creating auxiliary functions... |
| 1992 | Computing derivatives of auxiliary functions... |
| 1993 | Computing derivatives of more auxiliary functions... |
| 1994 | Computing second-order differential operator actions... |
| 1995 | (1/172872*(24696*sqrt(3)*sqrt(7)/(sqrt(pi)*sqrt(r)) - |
| 1996 | 1231*sqrt(3)*sqrt(7)/(sqrt(pi)*r^(3/2)))*108^r, 108, |
| 1997 | 1/7*sqrt(3)*sqrt(7)/(sqrt(pi)*sqrt(r)) - |
| 1998 | 1231/172872*sqrt(3)*sqrt(7)/(sqrt(pi)*r^(3/2))) |
| 1999 | |
| 2000 | :: |
| 2001 | |
| 2002 | sage: R.<x, y>= PolynomialRing(QQ) |
| 2003 | sage: H = (1 - 2*x - y) * (1 - x - 2*y) |
| 2004 | sage: Hfac = H.factor() |
| 2005 | sage: G = exp(x + y)/Hfac.unit() |
| 2006 | sage: F = FFPD(G, Hfac) |
| 2007 | sage: print F |
| 2008 | (e^(x + y), [(x + 2*y - 1, 1), (2*x + y - 1, 1)]) |
| 2009 | sage: p = {x: 1/3, y: 1/3} |
| 2010 | sage: alpha = (var('a'), var('b')) |
| 2011 | sage: print F.asymptotics_multiple(p, alpha, 2, var('r')) # long time |
| 2012 | (3*((1/3)^(-b)*(1/3)^(-a))^r*e^(2/3), (1/3)^(-b)*(1/3)^(-a), |
| 2013 | 3*e^(2/3)) |
| 2014 | |
| 2015 | AUTHORS: |
| 2016 | |
| 2017 | - Alexander Raichev (2008-10-01, 2010-09-28, 2012-08-02) |
| 2018 | """ |
| 2019 | from itertools import product |
| 2020 | |
| 2021 | R = self.ring() |
| 2022 | if R is None: |
| 2023 | return None |
| 2024 | |
| 2025 | # Coerce keys of p into R. |
| 2026 | p = FFPD.coerce_point(R, p) |
| 2027 | |
| 2028 | d = self.dimension() |
| 2029 | I = sqrt(-Integer(1)) |
| 2030 | # Coerce everything into the Symbolic Ring. |
| 2031 | X = [SR(x) for x in R.gens()] |
| 2032 | G = SR(self.numerator()) |
| 2033 | H = [SR(h) for (h, e) in self.denominator_factored()] |
| 2034 | Hprod = prod(H) |
| 2035 | n = len(H) |
| 2036 | P = dict([(SR(x), p[x]) for x in R.gens()]) |
| 2037 | Sstar = self.crit_cone_combo(p, alpha, coordinate) |
| 2038 | |
| 2039 | # Put the given convenient variable at end of variable list. |
| 2040 | if coordinate is not None: |
| 2041 | x = X.pop(coordinate) |
| 2042 | X.append(x) |
| 2043 | a = alpha.pop(coordinate) |
| 2044 | alpha.append(a) |
| 2045 | |
| 2046 | |
| 2047 | # Case n = d. |
| 2048 | if n == d: |
| 2049 | det = jacobian(H, X).subs(P).determinant().abs() |
| 2050 | exp_scale = prod([(P[X[i]]**(-alpha[i])).subs(P) |
| 2051 | for i in xrange(d)] ) |
| 2052 | subexp_part = G.subs(P)/(det*prod(P.values())) |
| 2053 | if numerical: |
| 2054 | exp_scale = exp_scale.n(digits=numerical) |
| 2055 | subexp_part = subexp_part.n(digits=numerical) |
| 2056 | return (exp_scale**asy_var*subexp_part, exp_scale, subexp_part) |
| 2057 | |
| 2058 | # Case n < d. |
| 2059 | # If P is a tuple of rationals, then compute with it directly. |
| 2060 | # Otherwise, compute symbolically and plug in P at the end. |
| 2061 | if vector(P.values()) not in QQ**d: |
| 2062 | sP = [var('p' + str(j)) for j in xrange(d)] |
| 2063 | P = dict( [(X[j], sP[j]) for j in xrange(d)] ) |
| 2064 | p = dict( [(sP[j], p[X[j]]) for j in xrange(d)] ) |
| 2065 | |
| 2066 | # Setup. |
| 2067 | print "Creating auxiliary functions..." |
| 2068 | # Create T and S variables. |
| 2069 | t = 't' |
| 2070 | while t in [str(x) for x in X]: |
| 2071 | t = t + 't' |
| 2072 | T = [var(t + str(i)) for i in xrange(d - 1)] |
| 2073 | s = 's' |
| 2074 | while s in [str(x) for x in X]: |
| 2075 | s = s + 't' |
| 2076 | S = [var(s + str(i)) for i in xrange(n - 1)] |
| 2077 | Sstar = dict([(S[j], Sstar[j]) for j in xrange(n - 1)]) |
| 2078 | thetastar = dict([(t, Integer(0)) for t in T]) |
| 2079 | thetastar.update(Sstar) |
| 2080 | # Create implicit functions. |
| 2081 | h = [function('h' + str(j), *tuple(X[:d - 1])) for j in xrange(n)] |
| 2082 | U = function('U', *tuple(X)) |
| 2083 | # All other functions are defined in terms of h, U, and |
| 2084 | # explicit functions. |
| 2085 | Hcheck = prod([X[d - 1] - Integer(1)/h[j] for j in xrange(n)]) |
| 2086 | Gcheck = -G/U * prod([-h[j]/X[d - 1] for j in xrange(n)]) |
| 2087 | A = [(-1)**(n - 1)*X[d - 1]**(-n + j)*\ |
| 2088 | diff(Gcheck.subs({X[d - 1]: Integer(1)/X[d - 1]}), X[d - 1], j) |
| 2089 | for j in xrange(n)] |
| 2090 | e = dict([(X[i], P[X[i]]*exp(I*T[i])) for i in xrange(d - 1)]) |
| 2091 | ht = [hh.subs(e) for hh in h] |
| 2092 | hsumt = add([S[j]*ht[j] for j in xrange(n - 1)]) +\ |
| 2093 | (Integer(1) - add(S))*ht[n - 1] |
| 2094 | At = [AA.subs(e).subs({X[d - 1]: hsumt}) for AA in A] |
| 2095 | Phit = -log(P[X[d - 1]]*hsumt) +\ |
| 2096 | I*add([alpha[i]/alpha[d - 1]*T[i] for i in xrange(d - 1)]) |
| 2097 | # atP Stores h and U and all their derivatives evaluated at C. |
| 2098 | atP = P.copy() |
| 2099 | atP.update(dict([(hh.subs(P), Integer(1)/P[X[d - 1]]) for hh in h])) |
| 2100 | |
| 2101 | # Compute the derivatives of h up to order 2*N and evaluate at P. |
| 2102 | hderivs1 = {} # First derivatives of h. |
| 2103 | for (i, j) in mrange([d - 1, n]): |
| 2104 | s = solve(diff(H[j].subs({X[d - 1]: Integer(1)/h[j]}), X[i]), |
| 2105 | diff(h[j], X[i]))[0].rhs().simplify() |
| 2106 | hderivs1.update({diff(h[j], X[i]): s}) |
| 2107 | atP.update({diff(h[j], X[i]).subs(P): s.subs(P).subs(atP)}) |
| 2108 | hderivs = FFPD.diff_all(h, X[0:d - 1], 2*N, sub=hderivs1, rekey=h) |
| 2109 | for k in hderivs.keys(): |
| 2110 | atP.update({k.subs(P): hderivs[k].subs(atP)}) |
| 2111 | |
| 2112 | # Compute the derivatives of U up to order 2*N - 2 + min{n, N} - 1 and |
| 2113 | # evaluate at P. |
| 2114 | # To do this, differentiate H = U*Hcheck over and over, evaluate at P, |
| 2115 | # and solve for the derivatives of U at P. |
| 2116 | # Need the derivatives of H with short keys to pass on to |
| 2117 | # diff_prod later. |
| 2118 | print "Computing derivatives of auxiliary functions..." |
| 2119 | m = min(n, N) |
| 2120 | end = [X[d-1] for j in xrange(n)] |
| 2121 | Hprodderivs = FFPD.diff_all(Hprod, X, 2*N - 2 + n, ending=end, |
| 2122 | sub_final=P) |
| 2123 | atP.update({U.subs(P): diff(Hprod, X[d - 1], n).subs(P)/factorial(n)}) |
| 2124 | Uderivs ={} |
| 2125 | k = Hprod.polynomial(CC).degree() - n |
| 2126 | if k == 0: |
| 2127 | # Then we can conclude that all higher derivatives of U are zero. |
| 2128 | for l in xrange(1, 2*N - 2 + m): |
| 2129 | for s in UnorderedTuples(X, l): |
| 2130 | Uderivs[diff(U, s).subs(P)] = Integer(0) |
| 2131 | elif k > 0 and k < 2*N - 2 + m - 1: |
| 2132 | all_zero = True |
| 2133 | Uderivs = FFPD.diff_prod(Hprodderivs, U, Hcheck, X, |
| 2134 | range(1, k + 1), end, Uderivs, atP) |
| 2135 | # Check for a nonzero U derivative. |
| 2136 | if Uderivs.values() != [Integer(0) for i in xrange(len(Uderivs))]: |
| 2137 | all_zero = False |
| 2138 | if all_zero: |
| 2139 | # Then all higher derivatives of U are zero. |
| 2140 | for l in xrange(k + 1, 2*N - 2 + m): |
| 2141 | for s in UnorderedTuples(X, l): |
| 2142 | Uderivs.update({diff(U, s).subs(P): Integer(0)}) |
| 2143 | else: |
| 2144 | # Have to compute the rest of the derivatives. |
| 2145 | Uderivs = FFPD.diff_prod(Hprodderivs, U, Hcheck, X, |
| 2146 | range(k + 1, 2*N - 2 + m), end, |
| 2147 | Uderivs, atP) |
| 2148 | else: |
| 2149 | Uderivs = FFPD.diff_prod(Hprodderivs, U, Hcheck, X, |
| 2150 | range(1, 2*N - 2 + m), end, Uderivs, atP) |
| 2151 | atP.update(Uderivs) |
| 2152 | Phit1 = jacobian(Phit, T + S).subs(hderivs1) |
| 2153 | a = jacobian(Phit1, T + S).subs(hderivs1).subs(thetastar).subs(atP) |
| 2154 | a_inv = a.inverse() |
| 2155 | Phitu = Phit - (Integer(1)/Integer(2))*matrix([T + S])*a*\ |
| 2156 | matrix([T + S]).transpose() |
| 2157 | Phitu = Phitu[0][0] |
| 2158 | |
| 2159 | # Compute all partial derivatives of At and Phitu up to orders 2*N - 2 |
| 2160 | # and 2*N, respectively. Take advantage of the fact that At and Phitu |
| 2161 | # are sufficiently differentiable functions so that mixed partials |
| 2162 | # are equal. Thus only need to compute representative partials. |
| 2163 | # Choose nondecreasing sequences as representative differentiation- |
| 2164 | # order sequences. |
| 2165 | # To speed up later computations, create symbolic functions AA and BB |
| 2166 | # to stand in for the expressions At and Phitu respectively. |
| 2167 | print "Computing derivatives of more auxiliary functions..." |
| 2168 | AA = [function('A' + str(j), *tuple(T + S)) for j in xrange(n)] |
| 2169 | At_derivs = FFPD.diff_all(At, T + S, 2*N - 2, sub=hderivs1, |
| 2170 | sub_final =[thetastar, atP], rekey=AA) |
| 2171 | BB = function('BB', *tuple(T + S)) |
| 2172 | Phitu_derivs = FFPD.diff_all(Phitu, T + S, 2*N, sub=hderivs1, |
| 2173 | sub_final =[thetastar, atP], rekey=BB, |
| 2174 | zero_order=3) |
| 2175 | AABB_derivs = At_derivs |
| 2176 | AABB_derivs.update(Phitu_derivs) |
| 2177 | for j in xrange(n): |
| 2178 | AABB_derivs[AA[j]] = At[j].subs(thetastar).subs(atP) |
| 2179 | AABB_derivs[BB] = Phitu.subs(thetastar).subs(atP) |
| 2180 | |
| 2181 | print "Computing second-order differential operator actions..." |
| 2182 | DD = FFPD.diff_op(AA, BB, AABB_derivs, T + S, a_inv, n, N) |
| 2183 | L = {} |
| 2184 | for (j, k) in product(xrange(min(n, N)), xrange(max(0, N - 1 - n), N)): |
| 2185 | if j + k <= N - 1: |
| 2186 | L[(j, k)] = add([DD[(j, k, l)]/((-1)**k*2**(k + l)*\ |
| 2187 | factorial(l)*factorial(k + l)) |
| 2188 | for l in xrange(2*k + 1)]) |
| 2189 | det = a.determinant()**(-Integer(1)/Integer(2))*\ |
| 2190 | (2*pi)**((n - d)/Integer(2)) |
| 2191 | chunk = det*add([ |
| 2192 | (alpha[d - 1]*asy_var)**((n - d)/Integer(2) - q)*\ |
| 2193 | add([L[(j, k)]*binomial(n - 1, j)*\ |
| 2194 | stirling_number1(n - j, n + k - q)*(-1)**(q - j - k) |
| 2195 | for (j, k) in product(xrange(min(n - 1, q) + 1), |
| 2196 | xrange(max(0, q - n), q + 1)) |
| 2197 | if j + k <= q]) |
| 2198 | for q in xrange(N)]) |
| 2199 | chunk = chunk.subs(P).simplify() |
| 2200 | coeffs = chunk.coefficients(asy_var) |
| 2201 | coeffs.reverse() |
| 2202 | coeffs = coeffs[:N] |
| 2203 | if numerical: |
| 2204 | subexp_part = add([co[0].subs(p).n(digits=numerical)*asy_var**co[1] |
| 2205 | for co in coeffs]) |
| 2206 | exp_scale = prod([(P[X[i]]**(-alpha[i])).subs(p) |
| 2207 | for i in xrange(d)]).n(digits=numerical) |
| 2208 | else: |
| 2209 | subexp_part = add([co[0].subs(p)*asy_var**co[1] for co in coeffs]) |
| 2210 | exp_scale = prod([(P[X[i]]**(-alpha[i])).subs(p) |
| 2211 | for i in xrange(d)]) |
| 2212 | return (exp_scale**asy_var*subexp_part, exp_scale, subexp_part) |
| 2213 | |
| 2214 | @staticmethod |
| 2215 | def subs_all(f, sub, simplify=False): |
| 2216 | r""" |
| 2217 | Return the items of $f$ substituted by the dictionaries |
| 2218 | of $sub$ in order of their appearance in $sub$. |
| 2219 | |
| 2220 | INPUT: |
| 2221 | |
| 2222 | - ``f`` - An individual or list of symbolic expressions or dictionaries |
| 2223 | - ``sub`` - An individual or list of dictionaries. |
| 2224 | - ``simplify`` - Boolean (default: False). |
| 2225 | |
| 2226 | OUTPUT: |
| 2227 | |
| 2228 | The items of $f$ substituted by the dictionaries of $sub$ in order of |
| 2229 | their appearance in $sub$. |
| 2230 | The subs() command is used. |
| 2231 | If simplify is True, then simplify() is used after substitution. |
| 2232 | |
| 2233 | EXAMPLES:: |
| 2234 | |
| 2235 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 2236 | sage: var('x, y, z') |
| 2237 | (x, y, z) |
| 2238 | sage: a = {x:1} |
| 2239 | sage: b = {y:2} |
| 2240 | sage: c = {z:3} |
| 2241 | sage: FFPD.subs_all(x + y + z, a) |
| 2242 | y + z + 1 |
| 2243 | sage: FFPD.subs_all(x + y + z, [c, a]) |
| 2244 | y + 4 |
| 2245 | sage: FFPD.subs_all([x + y + z, y^2], b) |
| 2246 | [x + z + 2, 4] |
| 2247 | sage: FFPD.subs_all([x + y + z, y^2], [b, c]) |
| 2248 | [x + 5, 4] |
| 2249 | |
| 2250 | :: |
| 2251 | |
| 2252 | sage: var('x, y') |
| 2253 | (x, y) |
| 2254 | sage: a = {'foo': x**2 + y**2, 'bar': x - y} |
| 2255 | sage: b = {x: 1 , y: 2} |
| 2256 | sage: FFPD.subs_all(a, b) |
| 2257 | {'foo': 5, 'bar': -1} |
| 2258 | |
| 2259 | AUTHORS: |
| 2260 | |
| 2261 | - Alexander Raichev (2009-05-05) |
| 2262 | """ |
| 2263 | singleton = False |
| 2264 | if not isinstance(f, list): |
| 2265 | f = [f] |
| 2266 | singleton = True |
| 2267 | if not isinstance(sub, list): |
| 2268 | sub = [sub] |
| 2269 | g = [] |
| 2270 | for ff in f: |
| 2271 | for D in sub: |
| 2272 | if isinstance(ff, dict): |
| 2273 | ff = dict( [(k, ff[k].subs(D)) for k in ff.keys()] ) |
| 2274 | else: |
| 2275 | ff = ff.subs(D) |
| 2276 | g.append(ff) |
| 2277 | if singleton and simplify: |
| 2278 | if isinstance(g[Integer(0) ], dict): |
| 2279 | return g[Integer(0) ] |
| 2280 | else: |
| 2281 | return g[Integer(0) ].simplify() |
| 2282 | elif singleton and not simplify: |
| 2283 | return g[Integer(0) ] |
| 2284 | elif not singleton and simplify: |
| 2285 | G = [] |
| 2286 | for gg in g: |
| 2287 | if isinstance(gg, dict): |
| 2288 | G.append(gg) |
| 2289 | else: |
| 2290 | G.append(gg.simplify()) |
| 2291 | return G |
| 2292 | else: |
| 2293 | return g |
| 2294 | |
| 2295 | @staticmethod |
| 2296 | def diff_all(f, V, n, ending=[], sub=None, sub_final=None, |
| 2297 | zero_order=0, rekey=None): |
| 2298 | r""" |
| 2299 | Return a dictionary of representative mixed partial |
| 2300 | derivatives of $f$ from order 1 up to order $n$ with respect to the |
| 2301 | variables in $V$. |
| 2302 | The default is to key the dictionary by all nondecreasing sequences |
| 2303 | in $V$ of length 1 up to length $n$. |
| 2304 | For internal use. |
| 2305 | |
| 2306 | INPUT: |
| 2307 | |
| 2308 | - ``f`` - An individual or list of $\mathcal{C}^{n+1}$ functions. |
| 2309 | - ``V`` - A list of variables occurring in $f$. |
| 2310 | - ``n`` - A natural number. |
| 2311 | - ``ending`` - A list of variables in $V$. |
| 2312 | - ``sub`` - An individual or list of dictionaries. |
| 2313 | - ``sub_final`` - An individual or list of dictionaries. |
| 2314 | - ``rekey`` - A callable symbolic function in $V$ or list thereof. |
| 2315 | - ``zero_order`` - A natural number. |
| 2316 | |
| 2317 | OUTPUT: |
| 2318 | |
| 2319 | The dictionary ${s_1:deriv_1,..., s_r:deriv_r}$. |
| 2320 | Here $s_1,\ldots, s_r$ is a listing of |
| 2321 | all nondecreasing sequences of length 1 up to length $n$ over the |
| 2322 | alphabet $V$, where $w > v$ in $X$ iff $str(w) > str(v)$, and |
| 2323 | $deriv_j$ is the derivative of $f$ with respect to the derivative |
| 2324 | sequence $s_j$ and simplified with respect to the substitutions in |
| 2325 | $sub$ and evaluated at $sub_final$. |
| 2326 | Moreover, all derivatives with respect to sequences of length less than |
| 2327 | $zero_order$ (derivatives of order less than $zero_order$ ) |
| 2328 | will be made zero. |
| 2329 | |
| 2330 | If $rekey$ is nonempty, then $s_1,\ldots, s_r$ will be replaced by the |
| 2331 | symbolic derivatives of the functions in $rekey$. |
| 2332 | |
| 2333 | If $ending$ is nonempty, then every derivative sequence $s_j$ will be |
| 2334 | suffixed by $ending$. |
| 2335 | |
| 2336 | EXAMPLES:: |
| 2337 | |
| 2338 | I'd like to print the entire dictionaries, but that doesn't yield |
| 2339 | consistent output order for doctesting. |
| 2340 | Order of keys changes.:: |
| 2341 | |
| 2342 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 2343 | sage: f = function('f', x) |
| 2344 | sage: dd = FFPD.diff_all(f, [x], 3) |
| 2345 | sage: dd[(x, x, x)] |
| 2346 | D[0, 0, 0](f)(x) |
| 2347 | |
| 2348 | :: |
| 2349 | |
| 2350 | sage: d1 = {diff(f, x): 4*x^3} |
| 2351 | sage: dd = FFPD.diff_all(f,[x], 3, sub=d1) |
| 2352 | sage: dd[(x, x, x)] |
| 2353 | 24*x |
| 2354 | |
| 2355 | :: |
| 2356 | |
| 2357 | sage: dd = FFPD.diff_all(f,[x], 3, sub=d1, rekey=f) |
| 2358 | sage: dd[diff(f, x, 3)] |
| 2359 | 24*x |
| 2360 | |
| 2361 | :: |
| 2362 | |
| 2363 | sage: a = {x:1} |
| 2364 | sage: dd = FFPD.diff_all(f,[x], 3, sub=d1, rekey=f, sub_final=a) |
| 2365 | sage: dd[diff(f, x, 3)] |
| 2366 | 24 |
| 2367 | |
| 2368 | :: |
| 2369 | |
| 2370 | sage: X = var('x, y, z') |
| 2371 | sage: f = function('f',*X) |
| 2372 | sage: dd = FFPD.diff_all(f, X, 2, ending=[y, y, y]) |
| 2373 | sage: dd[(z, y, y, y)] |
| 2374 | D[1, 1, 1, 2](f)(x, y, z) |
| 2375 | |
| 2376 | :: |
| 2377 | |
| 2378 | sage: g = function('g',*X) |
| 2379 | sage: dd = FFPD.diff_all([f, g], X, 2) |
| 2380 | sage: dd[(0, y, z)] |
| 2381 | D[1, 2](f)(x, y, z) |
| 2382 | |
| 2383 | :: |
| 2384 | |
| 2385 | sage: dd[(1, z, z)] |
| 2386 | D[2, 2](g)(x, y, z) |
| 2387 | |
| 2388 | :: |
| 2389 | |
| 2390 | sage: f = exp(x*y*z) |
| 2391 | sage: ff = function('ff',*X) |
| 2392 | sage: dd = FFPD.diff_all(f, X, 2, rekey=ff) |
| 2393 | sage: dd[diff(ff, x, z)] |
| 2394 | x*y^2*z*e^(x*y*z) + y*e^(x*y*z) |
| 2395 | |
| 2396 | AUTHORS: |
| 2397 | |
| 2398 | - Alexander Raichev (2008-10-01, 2009-04-01, 2010-02-01) |
| 2399 | """ |
| 2400 | singleton=False |
| 2401 | if not isinstance(f, list): |
| 2402 | f = [f] |
| 2403 | singleton=True |
| 2404 | |
| 2405 | # Build the dictionary of derivatives iteratively from a list |
| 2406 | # of nondecreasing derivative-order sequences. |
| 2407 | derivs = {} |
| 2408 | r = len(f) |
| 2409 | if ending: |
| 2410 | seeds = [ending] |
| 2411 | start = Integer(1) |
| 2412 | else: |
| 2413 | seeds = [[v] for v in V] |
| 2414 | start = Integer(2) |
| 2415 | if singleton: |
| 2416 | for s in seeds: |
| 2417 | derivs[tuple(s)] = FFPD.subs_all(diff(f[0], s), sub) |
| 2418 | for l in xrange(start, n + 1): |
| 2419 | for t in UnorderedTuples(V, l): |
| 2420 | s = tuple(t + ending) |
| 2421 | derivs[s] = FFPD.subs_all(diff(derivs[s[1:]], s[0]), sub) |
| 2422 | else: |
| 2423 | # Make the dictionary keys of the form (j, sequence of variables), |
| 2424 | # where j in range(r). |
| 2425 | for s in seeds: |
| 2426 | value = FFPD.subs_all([diff(f[j], s) for j in xrange(r)], sub) |
| 2427 | derivs.update(dict([(tuple([j]+s), value[j]) |
| 2428 | for j in xrange(r)])) |
| 2429 | for l in xrange(start, n + 1): |
| 2430 | for t in UnorderedTuples(V, l): |
| 2431 | s = tuple(t + ending) |
| 2432 | value = FFPD.subs_all([diff(derivs[(j,) + s[1:]], |
| 2433 | s[0]) for j in xrange(r)], sub) |
| 2434 | derivs.update(dict([((j,) + s, value[j]) |
| 2435 | for j in xrange(r)])) |
| 2436 | if zero_order: |
| 2437 | # Zero out all the derivatives of order < zero_order |
| 2438 | if singleton: |
| 2439 | for k in derivs.keys(): |
| 2440 | if len(k) < zero_order: |
| 2441 | derivs[k] = Integer(0) |
| 2442 | else: |
| 2443 | # Ignore the first of element of k, which is an index. |
| 2444 | for k in derivs.keys(): |
| 2445 | if len(k) - 1 < zero_order: |
| 2446 | derivs[k] = Integer(0) |
| 2447 | if sub_final: |
| 2448 | # Substitute sub_final into the values of derivs. |
| 2449 | for k in derivs.keys(): |
| 2450 | derivs[k] = FFPD.subs_all(derivs[k], sub_final) |
| 2451 | if rekey: |
| 2452 | # Rekey the derivs dictionary by the value of rekey. |
| 2453 | F = rekey |
| 2454 | if singleton: |
| 2455 | # F must be a singleton. |
| 2456 | derivs = dict( [(diff(F, list(k)), derivs[k]) |
| 2457 | for k in derivs.keys()] ) |
| 2458 | else: |
| 2459 | # F must be a list. |
| 2460 | derivs = dict( [(diff(F[k[0]], list(k)[1:]), derivs[k]) |
| 2461 | for k in derivs.keys()] ) |
| 2462 | return derivs |
| 2463 | |
| 2464 | @staticmethod |
| 2465 | def diff_op(A, B, AB_derivs, V, M, r, N): |
| 2466 | r""" |
| 2467 | Return the derivatives $DD^(l+k)(A[j] B^l)$ evaluated at a point |
| 2468 | $p$ for various natural numbers $j, k, l$ which depend on $r$ and $N$. |
| 2469 | Here $DD$ is a specific second-order linear differential operator |
| 2470 | that depends on $M$ , $A$ is a list of symbolic functions, |
| 2471 | $B$ is symbolic function, and $AB_derivs$ contains all the derivatives |
| 2472 | of $A$ and $B$ evaluated at $p$ that are necessary for the computation. |
| 2473 | For internal use by the functions asymptotics_smooth() and |
| 2474 | asymptotics_multiple(). |
| 2475 | |
| 2476 | INPUT: |
| 2477 | |
| 2478 | - ``A`` - A single or length ``r`` list of symbolic functions in the |
| 2479 | variables ``V``. |
| 2480 | - ``B`` - A symbolic function in the variables ``V``. |
| 2481 | - ``AB_derivs`` - A dictionary whose keys are the (symbolic) |
| 2482 | derivatives of ``A[0], ..., A[r-1]`` up to order ``2*N-2`` and |
| 2483 | the (symbolic) derivatives of ``B`` up to order ``2*N``. |
| 2484 | The values of the dictionary are complex numbers that are |
| 2485 | the keys evaluated at a common point $p$. |
| 2486 | - ``V`` - The variables of the ``A[j]`` and ``B``. |
| 2487 | - ``M`` - A symmetric $l \times l$ matrix, where $l$ is the |
| 2488 | length of ``V``. |
| 2489 | - ``r, N`` - Natural numbers. |
| 2490 | |
| 2491 | OUTPUT: |
| 2492 | |
| 2493 | A dictionary whose keys are natural number tuples of the form |
| 2494 | $(j, k, l)$, where $l \le 2k$, $j \le r-1$, and $j+k \le N-1$, |
| 2495 | and whose values are $DD^(l+k)(A[j] B^l)$ evaluated at a point |
| 2496 | $p$, where $DD$ is the linear second-order differential operator |
| 2497 | $-\sum_{i=0}^{l-1} \sum_{j=0}^{l-1} M[i][j] |
| 2498 | \partial^2 /(\partial V[j] \partial V[i])$. |
| 2499 | |
| 2500 | EXAMPLES:: |
| 2501 | |
| 2502 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 2503 | sage: T = var('x, y') |
| 2504 | sage: A = function('A',*tuple(T)) |
| 2505 | sage: B = function('B',*tuple(T)) |
| 2506 | sage: AB_derivs = {} |
| 2507 | sage: M = matrix([[1, 2],[2, 1]]) |
| 2508 | sage: DD = FFPD.diff_op(A, B, AB_derivs, T, M, 1, 2) |
| 2509 | sage: print sorted(DD.keys()) |
| 2510 | [(0, 0, 0), (0, 1, 0), (0, 1, 1), (0, 1, 2)] |
| 2511 | sage: len(DD[(0, 1, 2)]) |
| 2512 | 246 |
| 2513 | |
| 2514 | AUTHORS: |
| 2515 | |
| 2516 | - Alexander Raichev (2008-10-01, 2010-01-12) |
| 2517 | """ |
| 2518 | if not isinstance(A, list): |
| 2519 | A = [A] |
| 2520 | |
| 2521 | # First, compute the necessary product derivatives of A and B. |
| 2522 | product_derivs = {} |
| 2523 | for (j, k) in mrange([r, N]): |
| 2524 | if j + k < N: |
| 2525 | for l in xrange(2*k + 1): |
| 2526 | for s in UnorderedTuples(V, 2*(k + l)): |
| 2527 | product_derivs[tuple([j, k, l] + s)] = \ |
| 2528 | diff(A[j]*B**l, s).subs(AB_derivs) |
| 2529 | |
| 2530 | # Second, compute DD^(k+l)(A[j]*B^l)(p) and store values in dictionary. |
| 2531 | DD = {} |
| 2532 | rows = M.nrows() |
| 2533 | for (j, k) in mrange([r, N]): |
| 2534 | if j + k < N: |
| 2535 | for l in xrange(2*k + 1): |
| 2536 | # Take advantage of the symmetry of M by ignoring |
| 2537 | # the upper-diagonal entries of M and multiplying by |
| 2538 | # appropriate powers of 2. |
| 2539 | if k + l == 0 : |
| 2540 | DD[(j, k, l)] = product_derivs[(j, k, l)] |
| 2541 | continue |
| 2542 | S = [(a, b) for (a, b) in mrange([rows, rows]) if b <= a] |
| 2543 | P = cartesian_product_iterator([S for i in range(k+l)]) |
| 2544 | diffo = Integer(0) |
| 2545 | for t in P: |
| 2546 | if product_derivs[(j, k, l) + FFPD.diff_seq(V, t)] !=\ |
| 2547 | Integer(0): |
| 2548 | MM = Integer(1) |
| 2549 | for (a, b) in t: |
| 2550 | MM = MM * M[a][b] |
| 2551 | if a != b: |
| 2552 | MM = Integer(2) *MM |
| 2553 | diffo = diffo + MM*product_derivs[(j, k, l) +\ |
| 2554 | FFPD.diff_seq(V, t)] |
| 2555 | DD[(j, k, l)] = (-Integer(1) )**(k+l)*diffo |
| 2556 | return DD |
| 2557 | |
| 2558 | @staticmethod |
| 2559 | def diff_seq(V, s): |
| 2560 | r""" |
| 2561 | Given a list ``s`` of tuples of natural numbers, return the |
| 2562 | list of elements of ``V`` with indices the elements of the elements |
| 2563 | of ``s``. |
| 2564 | This function is for internal use by the function diff_op(). |
| 2565 | |
| 2566 | INPUT: |
| 2567 | |
| 2568 | - ``V`` - A list. |
| 2569 | - ``s`` - A list of tuples of natural numbers in the interval |
| 2570 | ``range(len(V))``. |
| 2571 | |
| 2572 | OUTPUT: |
| 2573 | |
| 2574 | The tuple ``tuple([V[tt] for tt in sorted(t)])``, where ``t`` is the |
| 2575 | list of elements of the elements of ``s``. |
| 2576 | |
| 2577 | EXAMPLES:: |
| 2578 | |
| 2579 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 2580 | sage: V = list(var('x, t, z')) |
| 2581 | sage: FFPD.diff_seq(V,([0, 1],[0, 2, 1],[0, 0])) |
| 2582 | (x, x, x, x, t, t, z) |
| 2583 | |
| 2584 | AUTHORS: |
| 2585 | |
| 2586 | - Alexander Raichev (2009-05-19) |
| 2587 | """ |
| 2588 | t = [] |
| 2589 | for ss in s: |
| 2590 | t.extend(ss) |
| 2591 | return tuple([V[tt] for tt in sorted(t)]) |
| 2592 | |
| 2593 | @staticmethod |
| 2594 | def diff_op_simple(A, B, AB_derivs, x, v, a, N): |
| 2595 | r""" |
| 2596 | Return $DD^(e k + v l)(A B^l)$ evaluated at a point $p$ for |
| 2597 | various natural numbers $e, k, l$ that depend on $v$ and $N$. |
| 2598 | Here $DD$ is a specific linear differential operator that depends |
| 2599 | on $a$ and $v$ , $A$ and $B$ are symbolic functions, and $AB_derivs$ |
| 2600 | contains all the derivatives of $A$ and $B$ evaluated at $p$ that are |
| 2601 | necessary for the computation. |
| 2602 | For internal use by the function asymptotics_smooth(). |
| 2603 | |
| 2604 | INPUT: |
| 2605 | |
| 2606 | - ``A, B`` - Symbolic functions in the variable ``x``. |
| 2607 | - ``AB_derivs`` - A dictionary whose keys are the (symbolic) |
| 2608 | derivatives of ``A`` up to order ``2*N`` if ``v`` is even or |
| 2609 | ``N`` if ``v`` is odd and the (symbolic) derivatives of ``B`` |
| 2610 | up to order ``2*N + v`` if ``v`` is even or ``N + v`` |
| 2611 | if ``v`` is odd. |
| 2612 | The values of the dictionary are complex numbers that are |
| 2613 | the keys evaluated at a common point $p$. |
| 2614 | - ``x`` - Symbolic variable. |
| 2615 | - ``a`` - A complex number. |
| 2616 | - ``v, N`` - Natural numbers. |
| 2617 | |
| 2618 | OUTPUT: |
| 2619 | |
| 2620 | A dictionary whose keys are natural number pairs of the form $(k, l)$, |
| 2621 | where $k < N$ and $l \le 2k$ and whose values are |
| 2622 | $DD^(e k + v l)(A B^l)$ evaluated at a point $p$. |
| 2623 | Here $e=2$ if $v$ is even, $e=1$ if $v$ is odd, and $DD$ is the |
| 2624 | linear differential operator |
| 2625 | $(a^{-1/v} d/dt)$ if $v$ is even and |
| 2626 | $(|a|^{-1/v} i \text{sgn}(a) d/dt)$ if $v$ is odd. |
| 2627 | |
| 2628 | EXAMPLES:: |
| 2629 | |
| 2630 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 2631 | sage: A = function('A', x) |
| 2632 | sage: B = function('B', x) |
| 2633 | sage: AB_derivs = {} |
| 2634 | sage: FFPD.diff_op_simple(A, B, AB_derivs, x, 3, 2, 2) |
| 2635 | {(1, 0): 1/2*I*2^(2/3)*D[0](A)(x), (0, 0): A(x), (1, 1): |
| 2636 | 1/4*(A(x)*D[0, 0, 0, 0](B)(x) + B(x)*D[0, 0, 0, 0](A)(x) + |
| 2637 | 4*D[0](A)(x)*D[0, 0, 0](B)(x) + 4*D[0](B)(x)*D[0, 0, 0](A)(x) + |
| 2638 | 6*D[0, 0](A)(x)*D[0, 0](B)(x))*2^(2/3)} |
| 2639 | |
| 2640 | AUTHORS: |
| 2641 | |
| 2642 | - Alexander Raichev (2010-01-15) |
| 2643 | """ |
| 2644 | I = sqrt(-Integer(1)) |
| 2645 | DD = {} |
| 2646 | if v.mod(Integer(2)) == Integer(0) : |
| 2647 | for k in xrange(N): |
| 2648 | for l in xrange(2*k + 1): |
| 2649 | DD[(k, l)] = (a**(-Integer(1)/v))**(2*k + v*l)*\ |
| 2650 | diff(A*B**l, x, 2*k + v*l).subs(AB_derivs) |
| 2651 | else: |
| 2652 | for k in xrange(N): |
| 2653 | for l in xrange(k + 1): |
| 2654 | DD[(k, l)] = (abs(a)**(-Integer(1)/v)*I*\ |
| 2655 | a/abs(a))**(k+v*l)*\ |
| 2656 | diff(A*B**l, x, k + v*l).subs(AB_derivs) |
| 2657 | return DD |
| 2658 | |
| 2659 | @staticmethod |
| 2660 | def diff_prod(f_derivs, u, g, X, interval, end, uderivs, atc): |
| 2661 | r""" |
| 2662 | Take various derivatives of the equation $f = ug$, |
| 2663 | evaluate them at a point $c$, and solve for the derivatives of $u$. |
| 2664 | For internal use by the function asymptotics_multiple(). |
| 2665 | |
| 2666 | INPUT: |
| 2667 | |
| 2668 | - ``f_derivs`` - A dictionary whose keys are all tuples of the form |
| 2669 | ``s + end``, where ``s`` is a sequence of variables from ``X`` whose |
| 2670 | length lies in ``interval``, and whose values are the derivatives |
| 2671 | of a function $f$ evaluated at $c$. |
| 2672 | - ``u`` - A callable symbolic function. |
| 2673 | - ``g`` - An expression or callable symbolic function. |
| 2674 | - ``X`` - A list of symbolic variables. |
| 2675 | - ``interval`` - A list of positive integers. |
| 2676 | Call the first and last values $n$ and $nn$, respectively. |
| 2677 | - ``end`` - A possibly empty list of repetitions of the |
| 2678 | variable ``z``, where ``z`` is the last element of ``X``. |
| 2679 | - ``uderivs`` - A dictionary whose keys are the symbolic |
| 2680 | derivatives of order 0 to order $n-1$ of ``u`` evaluated at $c$ |
| 2681 | and whose values are the corresponding derivatives evaluated |
| 2682 | at $c$. |
| 2683 | - ``atc`` - A dictionary whose keys are the keys of $c$ and all |
| 2684 | the symbolic derivatives of order 0 to order $nn$ of ``g`` |
| 2685 | evaluated $c$ and whose values are the corresponding |
| 2686 | derivatives evaluated at $c$. |
| 2687 | |
| 2688 | OUTPUT: |
| 2689 | |
| 2690 | A dictionary whose keys are the derivatives of ``u`` up to order |
| 2691 | $nn$ and whose values are those derivatives evaluated at $c$. |
| 2692 | |
| 2693 | EXAMPLES:: |
| 2694 | |
| 2695 | I'd like to print out the entire dictionary, but that does not give |
| 2696 | consistent output for doctesting. |
| 2697 | Order of keys changes :: |
| 2698 | |
| 2699 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 2700 | sage: u = function('u', x) |
| 2701 | sage: g = function('g', x) |
| 2702 | sage: fd = {(x,):1,(x, x):1} |
| 2703 | sage: ud = {u(x=2): 1} |
| 2704 | sage: atc = {x: 2, g(x=2): 3, diff(g, x)(x=2): 5} |
| 2705 | sage: atc[diff(g, x, x)(x=2)] = 7 |
| 2706 | sage: dd = FFPD.diff_prod(fd, u, g, [x], [1, 2], [], ud, atc) |
| 2707 | sage: dd[diff(u, x, 2)(x=2)] |
| 2708 | 22/9 |
| 2709 | |
| 2710 | NOTES: |
| 2711 | |
| 2712 | This function works by differentiating the equation $f = ug$ |
| 2713 | with respect to the variable sequence ``s + end``, |
| 2714 | for all tuples ``s`` of ``X`` of lengths in ``interval``, |
| 2715 | evaluating at the point $c$ , |
| 2716 | and solving for the remaining derivatives of ``u``. |
| 2717 | This function assumes that ``u`` never appears in the |
| 2718 | differentiations of $f = ug$ after evaluating at $c$. |
| 2719 | |
| 2720 | AUTHORS: |
| 2721 | |
| 2722 | - Alexander Raichev (2009-05-14, 2010-01-21) |
| 2723 | """ |
| 2724 | for l in interval: |
| 2725 | D = {} |
| 2726 | rhs = [] |
| 2727 | lhs = [] |
| 2728 | for t in UnorderedTuples(X, l): |
| 2729 | s = t + end |
| 2730 | lhs.append(f_derivs[tuple(s)]) |
| 2731 | rhs.append(diff(u*g, s).subs(atc).subs(uderivs)) |
| 2732 | # Since Sage's solve command can't take derivatives as variable |
| 2733 | # names, make new variables based on t to stand in for |
| 2734 | # diff(u, t) and store them in D. |
| 2735 | D[diff(u, t).subs(atc)] = var('zing' +\ |
| 2736 | ''.join([str(x) for x in t])) |
| 2737 | eqns = [lhs[i] == rhs[i].subs(uderivs).subs(D) |
| 2738 | for i in xrange(len(lhs))] |
| 2739 | variables = D.values() |
| 2740 | sol = solve(eqns,*variables, solution_dict=True) |
| 2741 | uderivs.update(FFPD.subs_all(D, sol[Integer(0) ])) |
| 2742 | return uderivs |
| 2743 | |
| 2744 | def crit_cone_combo(self, p, alpha, coordinate=None): |
| 2745 | r""" |
| 2746 | Return an auxiliary point associated to the multiple |
| 2747 | point ``p`` of the factors ``self``. |
| 2748 | For internal use by asymptotics_multiple(). |
| 2749 | |
| 2750 | INPUT: |
| 2751 | |
| 2752 | - ``p`` - A dictionary with keys that can be coerced to equal |
| 2753 | ``self.ring().gens()``. |
| 2754 | - ``alpha`` - A list of rationals. |
| 2755 | |
| 2756 | OUTPUT: |
| 2757 | |
| 2758 | A solution of the matrix equation $y \Gamma = \alpha'$ for $y$ , |
| 2759 | where $\Gamma$ is the matrix given by |
| 2760 | ``[FFPD.direction(v) for v in self.log_grads(p)]`` and $\alpha'$ |
| 2761 | is ``FFPD.direction(alpha)`` |
| 2762 | |
| 2763 | EXAMPLES:: |
| 2764 | |
| 2765 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 2766 | sage: R.<x, y>= PolynomialRing(QQ) |
| 2767 | sage: p = exp(x) |
| 2768 | sage: df = [(1 - 2*x - y, 1), (1 - x - 2*y, 1)] |
| 2769 | sage: f = FFPD(p, df) |
| 2770 | sage: p = {x: 1/3, y: 1/3} |
| 2771 | sage: alpha = (var('a'), var('b')) |
| 2772 | sage: print f.crit_cone_combo(p, alpha) |
| 2773 | [1/3*(2*a - b)/b, -2/3*(a - 2*b)/b] |
| 2774 | |
| 2775 | NOTES: |
| 2776 | |
| 2777 | Use this function only when $\Gamma$ is well-defined and |
| 2778 | there is a unique solution to the matrix equation |
| 2779 | $y \Gamma = \alpha'$. |
| 2780 | Fails otherwise. |
| 2781 | |
| 2782 | AUTHORS: |
| 2783 | |
| 2784 | - Alexander Raichev (2008-10-01, 2008-11-25, 2009-03-04, 2010-09-08, |
| 2785 | 2010-12-02, 2012-08-02) |
| 2786 | """ |
| 2787 | # Assuming here that each log_grads(f) has nonzero final component. |
| 2788 | # Then 'direction' will not throw a division by zero error. |
| 2789 | R = self.ring() |
| 2790 | if R is None: |
| 2791 | return None |
| 2792 | |
| 2793 | # Coerce keys of p into R. |
| 2794 | p = FFPD.coerce_point(R, p) |
| 2795 | |
| 2796 | d = self.dimension() |
| 2797 | n = len(self.denominator_factored()) |
| 2798 | Gamma = matrix([FFPD.direction(v, coordinate) |
| 2799 | for v in self.log_grads(p)]) |
| 2800 | beta = FFPD.direction(alpha, coordinate) |
| 2801 | # solve_left() fails when working in SR :-(. |
| 2802 | # So use solve() instead. |
| 2803 | # Gamma.solve_left(vector(beta)) |
| 2804 | V = [var('sss'+str(i)) for i in range(n)] |
| 2805 | M = matrix(V)*Gamma |
| 2806 | eqns = [M[0][j] == beta[j] for j in range(d)] |
| 2807 | s = solve(eqns, V, solution_dict=True)[0] # Assume a unique solution. |
| 2808 | return [s[v] for v in V] |
| 2809 | |
| 2810 | @staticmethod |
| 2811 | def direction(v, coordinate=None): |
| 2812 | r""" |
| 2813 | Returns ``[vv/v[coordinate] for vv in v]`` where |
| 2814 | ``coordinate`` is the last index of v if not specified otherwise. |
| 2815 | |
| 2816 | INPUT: |
| 2817 | |
| 2818 | - ``v`` - A vector. |
| 2819 | - ``coordinate`` - (Optional; default=None) An index for ``v``. |
| 2820 | |
| 2821 | EXAMPLES:: |
| 2822 | |
| 2823 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 2824 | sage: FFPD.direction([2, 3, 5]) |
| 2825 | (2/5, 3/5, 1) |
| 2826 | sage: FFPD.direction([2, 3, 5], 0) |
| 2827 | (1, 3/2, 5/2) |
| 2828 | |
| 2829 | AUTHORS: |
| 2830 | |
| 2831 | - Alexander Raichev (2010-08-25) |
| 2832 | """ |
| 2833 | if coordinate is None: |
| 2834 | coordinate = len(v) - 1 |
| 2835 | return tuple([vv/v[coordinate] for vv in v]) |
| 2836 | |
| 2837 | def grads(self, p): |
| 2838 | r""" |
| 2839 | Return a list of the gradients of the polynomials |
| 2840 | ``[q for (q, e) in self.denominator_factored()]`` evalutated at ``p``. |
| 2841 | |
| 2842 | INPUT: |
| 2843 | |
| 2844 | - ``p`` - (Optional: default=None) A dictionary whose keys are the |
| 2845 | generators of ``self.ring()``. |
| 2846 | |
| 2847 | |
| 2848 | EXAMPLES:: |
| 2849 | |
| 2850 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 2851 | sage: R.<x, y>= PolynomialRing(QQ) |
| 2852 | sage: p = exp(x) |
| 2853 | sage: df = [(x**3 + 3*y^2, 5), (x*y, 2), (y, 1)] |
| 2854 | sage: f = FFPD(p, df) |
| 2855 | sage: print f |
| 2856 | (e^x, [(y, 1), (x*y, 2), (x^3 + 3*y^2, 5)]) |
| 2857 | sage: print R.gens() |
| 2858 | (x, y) |
| 2859 | sage: p = None |
| 2860 | sage: print f.grads(p) |
| 2861 | [(0, 1), (y, x), (3*x^2, 6*y)] |
| 2862 | |
| 2863 | :: |
| 2864 | |
| 2865 | sage: p = {x: sqrt(2), y: var('a')} |
| 2866 | sage: print f.grads(p) |
| 2867 | [(0, 1), (a, sqrt(2)), (6, 6*a)] |
| 2868 | |
| 2869 | AUTHORS: |
| 2870 | |
| 2871 | - Alexander Raichev (2009-03-06) |
| 2872 | """ |
| 2873 | R = self.ring() |
| 2874 | if R is None: |
| 2875 | return |
| 2876 | # Coerce keys of p into R. |
| 2877 | p = FFPD.coerce_point(R, p) |
| 2878 | |
| 2879 | X = R.gens() |
| 2880 | d = self.dimension() |
| 2881 | H = [h for (h, e) in self.denominator_factored()] |
| 2882 | n = len(H) |
| 2883 | return [tuple([diff(H[i], X[j]).subs(p) for j in xrange(d)]) |
| 2884 | for i in xrange(n)] |
| 2885 | |
| 2886 | def log_grads(self, p): |
| 2887 | r""" |
| 2888 | Return a list of the logarithmic gradients of the polynomials |
| 2889 | ``[q for (q, e) in self.denominator_factored()]`` evalutated at ``p``. |
| 2890 | |
| 2891 | INPUT: |
| 2892 | |
| 2893 | - ``p`` - (Optional: default=None) A dictionary whose keys are the |
| 2894 | generators of ``self.ring()``. |
| 2895 | |
| 2896 | NOTE: |
| 2897 | |
| 2898 | The logarithmic gradient of a function $f$ at point $p$ is the vector |
| 2899 | $(x_1 \partial_1 f(x), \ldots, x_d \partial_d f(x) )$ evaluated at |
| 2900 | $p$. |
| 2901 | |
| 2902 | |
| 2903 | EXAMPLES:: |
| 2904 | |
| 2905 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 2906 | sage: R.<x, y>= PolynomialRing(QQ) |
| 2907 | sage: p = exp(x) |
| 2908 | sage: df = [(x**3 + 3*y^2, 5), (x*y, 2), (y, 1)] |
| 2909 | sage: f = FFPD(p, df) |
| 2910 | sage: print f |
| 2911 | (e^x, [(y, 1), (x*y, 2), (x^3 + 3*y^2, 5)]) |
| 2912 | sage: print R.gens() |
| 2913 | (x, y) |
| 2914 | sage: p = None |
| 2915 | sage: print f.log_grads(p) |
| 2916 | [(0, y), (x*y, x*y), (3*x^3, 6*y^2)] |
| 2917 | |
| 2918 | :: |
| 2919 | |
| 2920 | sage: p = {x: sqrt(2), y: var('a')} |
| 2921 | sage: print f.log_grads(p) |
| 2922 | [(0, a), (sqrt(2)*a, sqrt(2)*a), (6*sqrt(2), 6*a^2)] |
| 2923 | |
| 2924 | AUTHORS: |
| 2925 | |
| 2926 | - Alexander Raichev (2009-03-06) |
| 2927 | """ |
| 2928 | R = self.ring() |
| 2929 | if R is None: |
| 2930 | return None |
| 2931 | |
| 2932 | # Coerce keys of p into R. |
| 2933 | p = FFPD.coerce_point(R, p) |
| 2934 | |
| 2935 | X = R.gens() |
| 2936 | d = self.dimension() |
| 2937 | H = [h for (h, e) in self.denominator_factored()] |
| 2938 | n = len(H) |
| 2939 | return [tuple([(X[j]*diff(H[i], X[j])).subs(p) for j in xrange(d)]) |
| 2940 | for i in xrange(n)] |
| 2941 | |
| 2942 | def critical_cone(self, p, coordinate=None): |
| 2943 | r""" |
| 2944 | Return the critical cone of the convenient multiple point ``p``. |
| 2945 | |
| 2946 | INPUT: |
| 2947 | |
| 2948 | - ``p`` - A dictionary with keys that can be coerced to equal |
| 2949 | ``self.ring().gens()`` and values in a field. |
| 2950 | - ``coordinate`` - (Optional; default=None) A natural number. |
| 2951 | |
| 2952 | OUTPUT: |
| 2953 | |
| 2954 | A list of vectors that generate the critical cone of ``p`` and |
| 2955 | the cone itself, which is None if the values of ``p`` don't lie in QQ. |
| 2956 | Divide logarithmic gradients by their component ``coordinate`` entries. |
| 2957 | If ``coordinate=None``, then search from d-1 down to 0 for the |
| 2958 | first index j such that for all i ``self.log_grads()[i][j] != 0`` |
| 2959 | and set ``coordinate=j``. |
| 2960 | |
| 2961 | EXAMPLES:: |
| 2962 | |
| 2963 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 2964 | sage: R.<x, y, z>= PolynomialRing(QQ) |
| 2965 | sage: G = 1 |
| 2966 | sage: H = (1 - x*(1 + y))*(1 - z*x**2*(1 + 2*y)) |
| 2967 | sage: Hfac = H.factor() |
| 2968 | sage: G = 1/Hfac.unit() |
| 2969 | sage: F = FFPD(G, Hfac) |
| 2970 | sage: p = {x: 1/2, y: 1, z: 4/3} |
| 2971 | sage: print F.critical_cone(p) |
| 2972 | ([(2, 1, 0), (3, 1, 3/2)], 2-d cone in 3-d lattice N) |
| 2973 | |
| 2974 | AUTHORS: |
| 2975 | |
| 2976 | - Alexander Raichev (2010-08-25, 2012-08-02) |
| 2977 | """ |
| 2978 | R = self.ring() |
| 2979 | if R is None: |
| 2980 | return |
| 2981 | |
| 2982 | # Coerce keys of p into R. |
| 2983 | p = FFPD.coerce_point(R, p) |
| 2984 | |
| 2985 | d = self.dimension() |
| 2986 | lg = self.log_grads(p) |
| 2987 | n = len(lg) |
| 2988 | if coordinate not in xrange(d): |
| 2989 | # Search from d-1 down to 0 for a coordinate j such that |
| 2990 | # for all i we have lg[i][j] != 0. |
| 2991 | # One is guaranteed to exist in the case of a convenient multiple |
| 2992 | # point. |
| 2993 | for j in reversed(xrange(d)): |
| 2994 | if 0 not in [lg[i][j] for i in xrange(n)]: |
| 2995 | coordinate = j |
| 2996 | break |
| 2997 | Gamma = [FFPD.direction(v, coordinate) for v in lg] |
| 2998 | try: |
| 2999 | cone = Cone(Gamma) |
| 3000 | except TypeError: |
| 3001 | cone = None |
| 3002 | return (Gamma, cone) |
| 3003 | |
| 3004 | def is_convenient_multiple_point(self, p): |
| 3005 | r""" |
| 3006 | Return True if ``p`` is a convenient multiple point of ``self`` and |
| 3007 | False otherwise. |
| 3008 | Also return a short comment. |
| 3009 | |
| 3010 | INPUT: |
| 3011 | |
| 3012 | - ``p`` - A dictionary with keys that can be coerced to equal |
| 3013 | ``self.ring().gens()``. |
| 3014 | |
| 3015 | OUTPUT: |
| 3016 | |
| 3017 | A pair (verdict, comment). |
| 3018 | In case ``p`` is a convenient multiple point, verdict=True and |
| 3019 | comment ='No problem'. |
| 3020 | In case ``p`` is not, verdict=False and comment is string explaining |
| 3021 | why ``p`` fails to be a convenient multiple point. |
| 3022 | |
| 3023 | EXAMPLES:: |
| 3024 | |
| 3025 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 3026 | sage: R.<x, y, z>= PolynomialRing(QQ) |
| 3027 | sage: H = (1 - x*(1 + y))*(1 - z*x**2*(1 + 2*y)) |
| 3028 | sage: df = H.factor() |
| 3029 | sage: G = 1/df.unit() |
| 3030 | sage: F = FFPD(G, df) |
| 3031 | sage: p1 = {x: 1/2, y: 1, z: 4/3} |
| 3032 | sage: p2 = {x: 1, y: 2, z: 1/2} |
| 3033 | sage: print F.is_convenient_multiple_point(p1) |
| 3034 | (True, 'convenient in variables [x, y]') |
| 3035 | sage: print F.is_convenient_multiple_point(p2) |
| 3036 | (False, 'not a singular point') |
| 3037 | |
| 3038 | NOTES: |
| 3039 | |
| 3040 | See [RaWi2012]_ for more details. |
| 3041 | |
| 3042 | AUTHORS: |
| 3043 | |
| 3044 | - Alexander Raichev (2011-04-18, 2012-08-02) |
| 3045 | """ |
| 3046 | R = self.ring() |
| 3047 | if R is None: |
| 3048 | return |
| 3049 | |
| 3050 | # Coerce keys of p into R. |
| 3051 | p = FFPD.coerce_point(R, p) |
| 3052 | |
| 3053 | H = [h for (h, e) in self.denominator_factored()] |
| 3054 | n = len(H) |
| 3055 | d = self.dimension() |
| 3056 | |
| 3057 | # Test 1: Are the factors in H zero at p? |
| 3058 | if [h.subs(p) for h in H] != [0 for h in H]: |
| 3059 | # Failed test 1. Move on to next point. |
| 3060 | return (False, 'not a singular point') |
| 3061 | |
| 3062 | # Test 2: Are the factors in H smooth at p? |
| 3063 | grads = self.grads(p) |
| 3064 | for v in grads: |
| 3065 | if v == [0 for i in xrange(d)]: |
| 3066 | return (False, 'not smooth point of factors') |
| 3067 | |
| 3068 | # Test 3: Do the factors in H intersect transversely at p? |
| 3069 | if n <= d: |
| 3070 | M = matrix(grads) |
| 3071 | if M.rank() != n: |
| 3072 | return (False, 'not a transverse intersection') |
| 3073 | else: |
| 3074 | # Check all sub-multisets of grads of size d. |
| 3075 | for S in Subsets(grads, d, submultiset=True): |
| 3076 | M = matrix(S) |
| 3077 | if M.rank() != d: |
| 3078 | return (False, 'not a transverse intersection') |
| 3079 | |
| 3080 | # Test 4: Is p convenient? |
| 3081 | M = matrix(self.log_grads(p)) |
| 3082 | convenient_coordinates = [] |
| 3083 | for j in xrange(d): |
| 3084 | if 0 not in M.columns()[j]: |
| 3085 | convenient_coordinates.append(j) |
| 3086 | if not convenient_coordinates: |
| 3087 | return (False, 'multiple point but not convenient') |
| 3088 | |
| 3089 | # Tests all passed |
| 3090 | X = R.gens() |
| 3091 | return (True, 'convenient in variables %s' %\ |
| 3092 | [X[i] for i in convenient_coordinates]) |
| 3093 | |
| 3094 | def singular_ideal(self): |
| 3095 | r""" |
| 3096 | Let $R$ be the ring of ``self`` and $H$ its denominator. |
| 3097 | Let $Hred$ be the reduction (square-free part) of $H$. |
| 3098 | Return the ideal in $R$ generated by $Hred$ and |
| 3099 | its partial derivatives. |
| 3100 | If the coefficient field of $R$ is algebraically closed, |
| 3101 | then the output is the ideal of the singular locus (which is a variety) |
| 3102 | of the variety of $H$. |
| 3103 | |
| 3104 | EXAMPLES:: |
| 3105 | |
| 3106 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 3107 | sage: R.<x, y, z>= PolynomialRing(QQ) |
| 3108 | sage: H = (1 - x*(1 + y))**3*(1 - z*x**2*(1 + 2*y)) |
| 3109 | sage: df = H.factor() |
| 3110 | sage: G = 1/df.unit() |
| 3111 | sage: F = FFPD(G, df) |
| 3112 | sage: F.singular_ideal() |
| 3113 | Ideal (x*y + x - 1, y^2 - 2*y*z + 2*y - z + 1, x*z + y - 2*z + 1) |
| 3114 | of Multivariate Polynomial Ring in x, y, z over Rational Field |
| 3115 | |
| 3116 | AUTHORS: |
| 3117 | |
| 3118 | - Alexander Raichev (2008-10-01, 2008-11-20, 2010-12-03, 2011-04-18, |
| 3119 | 2012-08-03) |
| 3120 | """ |
| 3121 | R = self.ring() |
| 3122 | if R is None: |
| 3123 | return |
| 3124 | |
| 3125 | Hred = prod([h for (h, e) in self.denominator_factored()]) |
| 3126 | J = R.ideal([Hred] + Hred.gradient()) |
| 3127 | return R.ideal(J.groebner_basis()) |
| 3128 | |
| 3129 | def smooth_critical_ideal(self, alpha): |
| 3130 | r""" |
| 3131 | Let $R$ be the ring of ``self`` and $H$ its denominator. |
| 3132 | Return the ideal in $R$ of smooth critical points of the variety |
| 3133 | of $H$ for the direction ``alpha``. |
| 3134 | If the variety $V$ of $H$ has no smooth points, then return the ideal |
| 3135 | in $R$ of $V$. |
| 3136 | |
| 3137 | INPUT: |
| 3138 | |
| 3139 | - ``alpha`` - A d-tuple of positive integers and/or symbolic entries, |
| 3140 | where d = ``self.ring().ngens()``. |
| 3141 | |
| 3142 | EXAMPLES:: |
| 3143 | |
| 3144 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 3145 | sage: R.<x, y> = PolynomialRing(QQ) |
| 3146 | sage: H = (1-x-y-x*y)^2 |
| 3147 | sage: Hfac = H.factor() |
| 3148 | sage: G = 1/Hfac.unit() |
| 3149 | sage: F = FFPD(G, Hfac) |
| 3150 | sage: alpha = var('a1, a2') |
| 3151 | sage: F.smooth_critical_ideal(alpha) |
| 3152 | Ideal (y^2 + ((-2*a1)/(-a2))*y - 1, x + (a2/(-a1))*y + (a1 - a2)/(-a1)) of Multivariate Polynomial Ring in x, y over Fraction Field of Multivariate Polynomial Ring in a1, a2 over Rational Field |
| 3153 | |
| 3154 | :: |
| 3155 | |
| 3156 | sage: R.<x, y> = PolynomialRing(QQ) |
| 3157 | sage: H = (1-x-y-x*y)^2 |
| 3158 | sage: Hfac = H.factor() |
| 3159 | sage: G = 1/Hfac.unit() |
| 3160 | sage: F = FFPD(G, Hfac) |
| 3161 | sage: alpha = [7/3, var('a')] |
| 3162 | sage: F.smooth_critical_ideal(alpha) |
| 3163 | Ideal (y^2 + (-14/(-3*a))*y - 1, x + (-3/7*a)*y + 3/7*a - 1) of |
| 3164 | Multivariate Polynomial Ring in x, y over Fraction Field of |
| 3165 | Univariate Polynomial Ring in a over Rational Field |
| 3166 | |
| 3167 | NOTES: |
| 3168 | |
| 3169 | See [RaWi2012]_ for more details. |
| 3170 | |
| 3171 | AUTHORS: |
| 3172 | |
| 3173 | - Alexander Raichev (2008-10-01, 2008-11-20, 2009-03-09, 2010-12-02, |
| 3174 | 2011-04-18, 2012-08-03) |
| 3175 | """ |
| 3176 | R = self.ring() |
| 3177 | if R is None: |
| 3178 | return |
| 3179 | |
| 3180 | Hred = prod([h for (h, e) in self.denominator_factored()]) |
| 3181 | K = R.base_ring() |
| 3182 | d = self.dimension() |
| 3183 | |
| 3184 | # Expand K by the variables of alpha if there are any. |
| 3185 | indets = [] |
| 3186 | for a in alpha: |
| 3187 | if a not in K and a in SR: |
| 3188 | indets.append(a) |
| 3189 | indets = list(Set(indets)) # Delete duplicates in indets. |
| 3190 | if indets: |
| 3191 | L = FractionField(PolynomialRing(K, indets)) |
| 3192 | S = R.change_ring(L) |
| 3193 | # Coerce alpha into L. |
| 3194 | alpha = [L(a) for a in alpha] |
| 3195 | else: |
| 3196 | S = R |
| 3197 | |
| 3198 | # Find smooth, critical points for alpha. |
| 3199 | X = S.gens() |
| 3200 | Hred = S(Hred) |
| 3201 | J = S.ideal([Hred] +\ |
| 3202 | [alpha[d - 1]*X[i]*diff(Hred, X[i]) -\ |
| 3203 | alpha[i]*X[d - 1]*diff(Hred, X[d - 1]) |
| 3204 | for i in xrange(d - 1)]) |
| 3205 | return S.ideal(J.groebner_basis()) |
| 3206 | |
| 3207 | def maclaurin_coefficients(self, multi_indices, numerical=0): |
| 3208 | r""" |
| 3209 | Returns the Maclaurin coefficients of self that have multi-indices |
| 3210 | in ``multi_indices``. |
| 3211 | |
| 3212 | INPUT: |
| 3213 | |
| 3214 | - ``multi_indices`` - A list of tuples of positive integers, where |
| 3215 | each tuple has length ``self.dimension()``. |
| 3216 | - ``numerical`` - (Optional; default=0) A natural number. |
| 3217 | If positive, return numerical |
| 3218 | approximations of coefficients with ``numerical`` digits of |
| 3219 | accuracy. |
| 3220 | |
| 3221 | OUTPUT: |
| 3222 | |
| 3223 | A dictionary of the form |
| 3224 | (nu, Maclaurin coefficient of index nu of self). |
| 3225 | |
| 3226 | EXAMPLES:: |
| 3227 | |
| 3228 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 3229 | sage: R.<x> = PolynomialRing(QQ) |
| 3230 | sage: H = 2 - 3*x |
| 3231 | sage: Hfac = H.factor() |
| 3232 | sage: G = 1/Hfac.unit() |
| 3233 | sage: F = FFPD(G, Hfac) |
| 3234 | sage: print F |
| 3235 | (-1/3, [(x - 2/3, 1)]) |
| 3236 | sage: print F.maclaurin_coefficients([(2*k,) for k in range(6)]) |
| 3237 | {(0,): 1/2, (2,): 9/8, (8,): 6561/512, (4,): 81/32, (10,): 59049/2048, (6,): 729/128} |
| 3238 | |
| 3239 | :: |
| 3240 | |
| 3241 | sage: R.<x, y, z> = PolynomialRing(QQ) |
| 3242 | sage: H = (4 - 2*x - y - z) * (4 - x - 2*y - z) |
| 3243 | sage: Hfac = H.factor() |
| 3244 | sage: G = 16/Hfac.unit() |
| 3245 | sage: F = FFPD(G, Hfac) |
| 3246 | sage: alpha = vector([3, 3, 2]) |
| 3247 | sage: interval = [1, 2, 4] |
| 3248 | sage: S = [r*alpha for r in interval] |
| 3249 | sage: print F.maclaurin_coefficients(S, numerical=10) |
| 3250 | {(6, 6, 4): 0.7005249476, (12, 12, 8): 0.5847732654, |
| 3251 | (3, 3, 2): 0.7849731445} |
| 3252 | |
| 3253 | NOTES: |
| 3254 | |
| 3255 | Uses iterated univariate Maclaurin expansions. |
| 3256 | Slow. |
| 3257 | |
| 3258 | AUTHORS: |
| 3259 | |
| 3260 | - Alexander Raichev (2011-04-08, 2012-08-03) |
| 3261 | """ |
| 3262 | R = self.ring() |
| 3263 | if R is None: |
| 3264 | return |
| 3265 | |
| 3266 | d = self.dimension() |
| 3267 | coeffs = {} |
| 3268 | |
| 3269 | # Deal with the simple univariate case first. |
| 3270 | if d == 1: |
| 3271 | f = SR(self.quotient()) |
| 3272 | x = SR(R.gens()[0]) |
| 3273 | m = max(multi_indices)[0] |
| 3274 | f = f.taylor(x, 0, m) |
| 3275 | F = R(f) |
| 3276 | tmp = F.coefficients() |
| 3277 | for nu in multi_indices: |
| 3278 | val = tmp[nu[0]] |
| 3279 | if numerical: |
| 3280 | val = val.n(digits=numerical) |
| 3281 | coeffs[tuple(nu)] = val |
| 3282 | return coeffs |
| 3283 | |
| 3284 | # Create biggest multi-index needed. |
| 3285 | alpha = [] |
| 3286 | for i in xrange(d): |
| 3287 | alpha.append(max((nu[i] for nu in multi_indices))) |
| 3288 | |
| 3289 | # Compute Maclaurin expansion of self up to index alpha. |
| 3290 | # Use iterated univariate expansions. |
| 3291 | # Slow! |
| 3292 | f = SR(self.quotient()) |
| 3293 | X = [SR(x) for x in R.gens()] |
| 3294 | for i in xrange(d): |
| 3295 | f = f.taylor(X[i], 0, alpha[i]) |
| 3296 | F = R(f) |
| 3297 | |
| 3298 | # Collect coefficients. |
| 3299 | X = R.gens() |
| 3300 | for nu in multi_indices: |
| 3301 | monomial = prod([X[i]**nu[i] for i in xrange(d)]) |
| 3302 | val = F.monomial_coefficient(monomial) |
| 3303 | if numerical: |
| 3304 | val = val.n(digits=numerical) |
| 3305 | coeffs[tuple(nu)] = val |
| 3306 | return coeffs |
| 3307 | |
| 3308 | def relative_error(self, approx, alpha, interval, exp_scale=Integer(1), |
| 3309 | digits=10): |
| 3310 | r""" |
| 3311 | Returns the relative error between the values of the Maclaurin |
| 3312 | coefficients of ``self`` with multi-indices ``r alpha`` for ``r`` in |
| 3313 | ``interval`` and the values of the functions (of the variable ``r``) |
| 3314 | in ``approx``. |
| 3315 | |
| 3316 | INPUT: |
| 3317 | |
| 3318 | - ``approx`` - An individual or list of symbolic expressions in |
| 3319 | one variable. |
| 3320 | - ``alpha`` - A list of positive integers of length |
| 3321 | ``self.ring().ngens()`` |
| 3322 | - ``interval`` - A list of positive integers. |
| 3323 | - ``exp_scale`` - (Optional; default=1) A number. |
| 3324 | |
| 3325 | OUTPUT: |
| 3326 | |
| 3327 | A list whose entries are of the form |
| 3328 | ``[r*alpha, a_r, b_r, err_r]`` for ``r`` in ``interval``. |
| 3329 | Here ``r*alpha`` is a tuple; ``a_r`` is the ``r*alpha`` (multi-index) |
| 3330 | coefficient of the Maclaurin series for ``self`` divided by |
| 3331 | ``exp_scale**r``; |
| 3332 | ``b_r`` is a list of the values of the functions in ``approx`` |
| 3333 | evaluated at ``r`` and divided by ``exp_scale**m``; |
| 3334 | ``err_r`` is the list of relative errors |
| 3335 | ``(a_r - f)/a_r`` for ``f`` in ``b_r``. |
| 3336 | All outputs are decimal approximations. |
| 3337 | |
| 3338 | EXAMPLES:: |
| 3339 | |
| 3340 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 3341 | sage: R.<x, y>= PolynomialRing(QQ) |
| 3342 | sage: H = 1 - x - y - x*y |
| 3343 | sage: Hfac = H.factor() |
| 3344 | sage: G = 1/Hfac.unit() |
| 3345 | sage: F = FFPD(G, Hfac) |
| 3346 | sage: alpha = [1, 1] |
| 3347 | sage: r = var('r') |
| 3348 | sage: a1 = (0.573/sqrt(r))*5.83^r |
| 3349 | sage: a2 = (0.573/sqrt(r) - 0.0674/r^(3/2))*5.83^r |
| 3350 | sage: es = 5.83 |
| 3351 | sage: F.relative_error([a1, a2], alpha, [1, 2, 4, 8], es) # long time |
| 3352 | Calculating errors table in the form |
| 3353 | exponent, scaled Maclaurin coefficient, scaled asymptotic values, |
| 3354 | relative errors... |
| 3355 | [((1, 1), 0.5145797599, [0.5730000000, 0.5056000000], |
| 3356 | [-0.1135300000, 0.01745066667]), ((2, 2), 0.3824778089, |
| 3357 | [0.4051721856, 0.3813426871], [-0.05933514614, 0.002967810973]), |
| 3358 | ((4, 4), 0.2778630595, [0.2865000000, 0.2780750000], |
| 3359 | [-0.03108344267, -0.0007627515584]), ((8, 8), 0.1991088276, |
| 3360 | [0.2025860928, 0.1996074055], [-0.01746414394, -0.002504047242])] |
| 3361 | |
| 3362 | AUTHORS: |
| 3363 | |
| 3364 | - Alexander Raichev (2009-05-18, 2011-04-18, 2012-08-03) |
| 3365 | """ |
| 3366 | |
| 3367 | if not isinstance(approx, list): |
| 3368 | approx = [approx] |
| 3369 | av = approx[0].variables()[0] |
| 3370 | |
| 3371 | print "Calculating errors table in the form" |
| 3372 | print "exponent, scaled Maclaurin coefficient, scaled asymptotic values, relative errors..." |
| 3373 | |
| 3374 | # Get Maclaurin coefficients of self. |
| 3375 | multi_indices = [r*vector(alpha) for r in interval] |
| 3376 | mac = self.maclaurin_coefficients(multi_indices, numerical=digits) |
| 3377 | #mac = self.old_maclaurin_coefficients(alpha, max(interval)) |
| 3378 | mac_approx = {} |
| 3379 | stats = [] |
| 3380 | for r in interval: |
| 3381 | beta = tuple(r*vector(alpha)) |
| 3382 | mac[beta] = (mac[beta]/exp_scale**r).n(digits=digits) |
| 3383 | mac_approx[beta] = [(f.subs({av:r})/exp_scale**r).n(digits=digits) |
| 3384 | for f in approx] |
| 3385 | stats_row = [beta, mac[beta], mac_approx[beta]] |
| 3386 | if mac[beta] == 0: |
| 3387 | stats_row.extend([None for a in mac_approx[beta]]) |
| 3388 | else: |
| 3389 | stats_row.append([(mac[beta] - a)/mac[beta] |
| 3390 | for a in mac_approx[beta]]) |
| 3391 | stats.append(tuple(stats_row)) |
| 3392 | return stats |
| 3393 | |
| 3394 | @staticmethod |
| 3395 | def coerce_point(R, p): |
| 3396 | r""" |
| 3397 | Coerce the keys of the dictionary ``p`` into the ring ``R``. |
| 3398 | |
| 3399 | Assume that it is possible. |
| 3400 | |
| 3401 | EXAMPLES:: |
| 3402 | |
| 3403 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 3404 | sage: R.<x,y> = PolynomialRing(QQ) |
| 3405 | sage: f = FFPD() |
| 3406 | sage: p = {SR(x): 1, SR(y): 7/8} |
| 3407 | sage: print p |
| 3408 | {y: 7/8, x: 1} |
| 3409 | sage: for k in sorted(p.keys()): |
| 3410 | ... print k, k.parent() |
| 3411 | ... |
| 3412 | y Symbolic Ring |
| 3413 | x Symbolic Ring |
| 3414 | sage: q = f.coerce_point(R, p) |
| 3415 | sage: print q |
| 3416 | {y: 7/8, x: 1} |
| 3417 | sage: for k in sorted(q.keys()): |
| 3418 | ... print k, k.parent() |
| 3419 | ... |
| 3420 | y Multivariate Polynomial Ring in x, y over Rational Field |
| 3421 | x Multivariate Polynomial Ring in x, y over Rational Field |
| 3422 | |
| 3423 | AUTHORS: |
| 3424 | |
| 3425 | - Alexander Raichev (2009-05-18, 2011-04-18, 2012-08-03) |
| 3426 | """ |
| 3427 | result = p |
| 3428 | if p is not None and p.keys() and p.keys()[0].parent() != R: |
| 3429 | try: |
| 3430 | result = dict([(x, p[SR(x)]) for x in R.gens()]) |
| 3431 | except TypeError: |
| 3432 | pass |
| 3433 | return result |
| 3434 | |
| 3435 | |
| 3436 | class FFPDSum(list): |
| 3437 | r""" |
| 3438 | A list representing the sum of FFPD objects with distinct |
| 3439 | denominator factorizations. |
| 3440 | |
| 3441 | AUTHORS: |
| 3442 | |
| 3443 | - Alexander Raichev (2012-06-25) |
| 3444 | """ |
| 3445 | def __str__(self): |
| 3446 | r""" |
| 3447 | Returns a string representation of ``self`` |
| 3448 | |
| 3449 | EXAMPLES:: |
| 3450 | |
| 3451 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 3452 | sage: R.<x,y> = PolynomialRing(QQ) |
| 3453 | sage: f = FFPD(x + y, [(y, 1), (x, 1)]) |
| 3454 | sage: g = FFPD(x**2 + y, [(y, 1), (x, 2)]) |
| 3455 | sage: print FFPDSum([f, g]) |
| 3456 | [(x + y, [(y, 1), (x, 1)]), (x^2 + y, [(y, 1), (x, 2)])] |
| 3457 | |
| 3458 | """ |
| 3459 | return str([r.list() for r in self]) |
| 3460 | |
| 3461 | def __eq__(self, other): |
| 3462 | r""" |
| 3463 | Returns True if ``self`` is equal to ``other`` |
| 3464 | |
| 3465 | EXAMPLES:: |
| 3466 | |
| 3467 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 3468 | sage: R.<x,y> = PolynomialRing(QQ) |
| 3469 | sage: f = FFPD(x + y, [(y, 1), (x, 1)]) |
| 3470 | sage: g = FFPD(x*(x + y), [(y, 1), (x, 2)]) |
| 3471 | sage: s = FFPDSum([f]) |
| 3472 | sage: t = FFPDSum([g]) |
| 3473 | sage: print s, t |
| 3474 | [(x + y, [(y, 1), (x, 1)])] [(x + y, [(y, 1), (x, 1)])] |
| 3475 | sage: print s == t |
| 3476 | True |
| 3477 | |
| 3478 | """ |
| 3479 | return sorted(self) == sorted(other) |
| 3480 | |
| 3481 | def __ne__(self, other): |
| 3482 | r""" |
| 3483 | Returns True if ``self`` is not equal to ``other`` |
| 3484 | |
| 3485 | EXAMPLES:: |
| 3486 | |
| 3487 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 3488 | sage: R.<x,y> = PolynomialRing(QQ) |
| 3489 | sage: f = FFPD(x + y, [(y, 1), (x, 1)]) |
| 3490 | sage: g = FFPD(x + y, [(y, 1), (x, 2)]) |
| 3491 | sage: s = FFPDSum([f]) |
| 3492 | sage: t = FFPDSum([g]) |
| 3493 | sage: print s, t |
| 3494 | [(x + y, [(y, 1), (x, 1)])] [(x + y, [(y, 1), (x, 2)])] |
| 3495 | sage: print s != t |
| 3496 | True |
| 3497 | |
| 3498 | """ |
| 3499 | return not (self == other) |
| 3500 | |
| 3501 | def ring(self): |
| 3502 | r""" |
| 3503 | Return the polynomial ring of the denominators of ``self``. |
| 3504 | |
| 3505 | If ``self`` does not have any denominators, then return None. |
| 3506 | |
| 3507 | EXAMPLES:: |
| 3508 | |
| 3509 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 3510 | sage: R.<x,y> = PolynomialRing(QQ) |
| 3511 | sage: f = FFPD(x + y, [(y, 1), (x, 1)]) |
| 3512 | sage: s = FFPDSum([f]) |
| 3513 | sage: print s.ring() |
| 3514 | Multivariate Polynomial Ring in x, y over Rational Field |
| 3515 | sage: g = FFPD(x + y, []) |
| 3516 | sage: t = FFPDSum([g]) |
| 3517 | sage: print t.ring() |
| 3518 | None |
| 3519 | |
| 3520 | """ |
| 3521 | for r in self: |
| 3522 | R = r.ring() |
| 3523 | if R is not None: |
| 3524 | return R |
| 3525 | return None |
| 3526 | |
| 3527 | def whole_and_parts(self): |
| 3528 | r""" |
| 3529 | Rewrite ``self`` as a FFPDSum of a (possibly zero) polynomial |
| 3530 | FFPD followed by reduced rational expression FFPDs. |
| 3531 | |
| 3532 | Only useful for multivariate decompositions. |
| 3533 | |
| 3534 | EXAMPLES:: |
| 3535 | |
| 3536 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 3537 | sage: R.<x, y> = PolynomialRing(QQ, 'x, y') |
| 3538 | sage: f = x**2 + 3*y + 1/x + 1/y |
| 3539 | sage: f = FFPD(quotient=f) |
| 3540 | sage: print f |
| 3541 | (x^3*y + 3*x*y^2 + x + y, [(y, 1), (x, 1)]) |
| 3542 | sage: print FFPDSum([f]).whole_and_parts() |
| 3543 | [(x^2 + 3*y, []), (x + y, [(y, 1), (x, 1)])] |
| 3544 | |
| 3545 | :: |
| 3546 | |
| 3547 | sage: R.<x, y> = PolynomialRing(QQ) |
| 3548 | sage: f = cos(x)**2 + 3*y + 1/x + 1/y |
| 3549 | sage: print f |
| 3550 | 1/x + 1/y + cos(x)^2 + 3*y |
| 3551 | sage: G = f.numerator() |
| 3552 | sage: H = R(f.denominator()) |
| 3553 | sage: f = FFPD(G, H.factor()) |
| 3554 | sage: print f |
| 3555 | (x*y*cos(x)^2 + 3*x*y^2 + x + y, [(y, 1), (x, 1)]) |
| 3556 | sage: print FFPDSum([f]).whole_and_parts() |
| 3557 | [(0, []), (x*y*cos(x)^2 + 3*x*y^2 + x + y, [(y, 1), (x, 1)])] |
| 3558 | """ |
| 3559 | whole = 0 |
| 3560 | parts = [] |
| 3561 | R = self.ring() |
| 3562 | for r in self: |
| 3563 | # Since r has already passed through FFPD.__init__()'s reducing |
| 3564 | # procedure, r is already in lowest terms. |
| 3565 | # Check if can write r as a mixed fraction: whole + fraction. |
| 3566 | p = r.numerator() |
| 3567 | q = r.denominator() |
| 3568 | if q == 1: |
| 3569 | # r is already whole |
| 3570 | whole += p |
| 3571 | else: |
| 3572 | try: |
| 3573 | # Coerce p into R and divide p by q |
| 3574 | p = R(p) |
| 3575 | a, b = p.quo_rem(q) |
| 3576 | except TypeError: |
| 3577 | # p is not in R and so can't divide p by q |
| 3578 | a = 0 |
| 3579 | b = p |
| 3580 | whole += a |
| 3581 | parts.append(FFPD(b, r.denominator_factored(), reduce_=False)) |
| 3582 | return FFPDSum([FFPD(whole, ())] + parts) |
| 3583 | |
| 3584 | def combine_like_terms(self): |
| 3585 | r""" |
| 3586 | Combine terms in ``self`` with the same denominator. |
| 3587 | Only useful for multivariate decompositions. |
| 3588 | |
| 3589 | EXAMPLES:: |
| 3590 | |
| 3591 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 3592 | sage: R.<x, y>= PolynomialRing(QQ) |
| 3593 | sage: f = FFPD(quotient=1/(x * y * (x*y + 1))) |
| 3594 | sage: g = FFPD(quotient=x/(x * y * (x*y + 1))) |
| 3595 | sage: s = FFPDSum([f, g, f]) |
| 3596 | sage: t = s.combine_like_terms() |
| 3597 | sage: print s |
| 3598 | [(1, [(y, 1), (x, 1), (x*y + 1, 1)]), (1, [(y, 1), (x*y + 1, 1)]), |
| 3599 | (1, [(y, 1), (x, 1), (x*y + 1, 1)])] |
| 3600 | sage: print t |
| 3601 | [(1, [(y, 1), (x*y + 1, 1)]), (2, [(y, 1), (x, 1), (x*y + 1, 1)])] |
| 3602 | |
| 3603 | :: |
| 3604 | |
| 3605 | sage: R.<x, y>= PolynomialRing(QQ) |
| 3606 | sage: H = x * y * (x*y + 1) |
| 3607 | sage: f = FFPD(1, H.factor()) |
| 3608 | sage: g = FFPD(exp(x + y), H.factor()) |
| 3609 | sage: s = FFPDSum([f, g]) |
| 3610 | sage: print s |
| 3611 | [(1, [(y, 1), (x, 1), (x*y + 1, 1)]), (e^(x + y), [(y, 1), (x, 1), |
| 3612 | (x*y + 1, 1)])] |
| 3613 | sage: t = s.combine_like_terms() |
| 3614 | sage: print t |
| 3615 | [(e^(x + y) + 1, [(y, 1), (x, 1), (x*y + 1, 1)])] |
| 3616 | """ |
| 3617 | if not self: |
| 3618 | return self |
| 3619 | |
| 3620 | # Combine like terms. |
| 3621 | FFPDs = sorted(self) |
| 3622 | new_FFPDs = [] |
| 3623 | temp = FFPDs[0] |
| 3624 | for f in FFPDs[1:]: |
| 3625 | if temp.denominator_factored() == f.denominator_factored(): |
| 3626 | # Add f to temp. |
| 3627 | num = temp.numerator() + f.numerator() |
| 3628 | temp = FFPD(num, temp.denominator_factored()) |
| 3629 | else: |
| 3630 | # Append temp to new_FFPDs and update temp. |
| 3631 | new_FFPDs.append(temp) |
| 3632 | temp = f |
| 3633 | new_FFPDs.append(temp) |
| 3634 | return FFPDSum(new_FFPDs) |
| 3635 | |
| 3636 | def sum(self): |
| 3637 | r""" |
| 3638 | Return the sum of the FFPDs in ``self`` as a FFPD. |
| 3639 | |
| 3640 | EXAMPLES:: |
| 3641 | |
| 3642 | sage: from sage.combinat.asymptotics_multivariate_generating_functions import * |
| 3643 | sage: R.<x, y> = PolynomialRing(QQ) |
| 3644 | sage: df = (x, 1), (y, 1), (x*y + 1, 1) |
| 3645 | sage: f = FFPD(2, df) |
| 3646 | sage: g = FFPD(2*x*y, df) |
| 3647 | sage: print FFPDSum([f, g]) |
| 3648 | [(2, [(y, 1), (x, 1), (x*y + 1, 1)]), (2, [(x*y + 1, 1)])] |
| 3649 | sage: print FFPDSum([f, g]).sum() |
| 3650 | (2, [(y, 1), (x, 1)]) |
| 3651 | |
| 3652 | :: |
| 3653 | |
| 3654 | sage: R.<x, y> = PolynomialRing(QQ) |
| 3655 | sage: f = FFPD(cos(x), [(x, 2)]) |
| 3656 | sage: g = FFPD(cos(y), [(x, 1), (y, 2)]) |
| 3657 | sage: print FFPDSum([f, g]) |
| 3658 | [(cos(x), [(x, 2)]), (cos(y), [(y, 2), (x, 1)])] |
| 3659 | sage: print FFPDSum([f, g]).sum() |
| 3660 | (y^2*cos(x) + x*cos(y), [(y, 2), (x, 2)]) |
| 3661 | |
| 3662 | |
| 3663 | """ |
| 3664 | if not self: |
| 3665 | return self |
| 3666 | |
| 3667 | # Compute the sum's numerator and denominator. |
| 3668 | R = self.ring() |
| 3669 | summy = sum((f.quotient() for f in self)) |
| 3670 | numer = summy.numerator() |
| 3671 | denom = R(summy.denominator()) |
| 3672 | |
| 3673 | # Compute the sum's denominator factorization. |
| 3674 | # Could use the factor() command, but it's probably faster to use |
| 3675 | # the irreducible factors of the denominators of self. |
| 3676 | df = [] # The denominator factorization for the sum. |
| 3677 | if denom == 1: |
| 3678 | # Done |
| 3679 | return FFPD(numer, df, reduce_=False) |
| 3680 | |
| 3681 | factors = [] |
| 3682 | for f in self: |
| 3683 | factors.extend([q for (q, e) in f.denominator_factored()]) |
| 3684 | |
| 3685 | # Eliminate repeats from factors and sort. |
| 3686 | factors = sorted(list(set(factors))) |
| 3687 | |
| 3688 | # The irreducible factors of denom lie in factors. |
| 3689 | # Use this fact to build df. |
| 3690 | for q in factors: |
| 3691 | e = 0 |
| 3692 | quo, rem = denom.quo_rem(q) |
| 3693 | while rem == 0: |
| 3694 | e += 1 |
| 3695 | denom = quo |
| 3696 | quo, rem = denom.quo_rem(q) |
| 3697 | if e > 0: |
| 3698 | df.append((q, e)) |
| 3699 | return FFPD(numer, df, reduce_=False) |