+ i3Rt^RIHt^RIHt^RIHt^RIHt^RI H t ^RI H t ^RI Ht^RIHt^R IHt^R IHtR tR tR t]R4t]!RR44t]R4tRt]RRl4tR#)z'Utilities for algebraic number theory. )sympify) factorint)QQ)ZZ) DMRankError)minpoly)IntervalPrinter)public)lambdify)mpc\V\4;'g5\P!V4;'g\P!V4#)z Test whether an argument is of an acceptable type to be used as a rational number. Explanation =========== Returns ``True`` on any argument of type ``int``, :ref:`ZZ`, or :ref:`QQ`. See Also ======== is_int ) isinstanceintrof_typercs&ڂ/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/polys/numberfields/utilities.pyis_ratrs1, a  ? ?A ? ?"**Q-?c^\V\4;'g\P!V4#)z Test whether an argument is of an acceptable type to be used as an integer. Explanation =========== Returns ``True`` on any argument of type ``int`` or :ref:`ZZ`. See Also ======== is_rat )r rrrrs&ris_intr)s!$ a  . .A.rcH\V4pVPVP3#)z Given any argument on which :py:func:`~.is_rat` is ``True``, return the numerator and denominator of this number. See Also ======== is_rat )r numerator denominator)rrs& r get_num_denomr>s  1A ;; %%rcV^,R9d \R4hV^8Xd/^^/3#V^8Xd//3#\V4p/p/p^pVP4FVwrVV^,^8Xd9^W%&V^,^8Xd V^, pV^8dV^, ^,W5&KIKKV^,W5&KX ^V9pV'gV^,^8Xd9V^,pV^8gQhV^8XdV^M V^, V^&V'd^M^V^&W#3#)a Extract a fundamental discriminant from an integer *a*. Explanation =========== Given any rational integer *a* that is 0 or 1 mod 4, write $a = d f^2$, where $d$ is either 1 or a fundamental discriminant, and return a pair of dictionaries ``(D, F)`` giving the prime factorizations of $d$ and $f$ respectively, in the same format returned by :py:func:`~.factorint`. A fundamental discriminant $d$ is different from unity, and is either 1 mod 4 and squarefree, or is 0 mod 4 and such that $d/4$ is squarefree and 2 or 3 mod 4. This is the same as being the discriminant of some quadratic field. Examples ======== >>> from sympy.polys.numberfields.utilities import extract_fundamental_discriminant >>> print(extract_fundamental_discriminant(-432)) ({3: 1, -1: 1}, {2: 2, 3: 1}) For comparison: >>> from sympy import factorint >>> print(factorint(-432)) {2: 4, 3: 3, -1: 1} Parameters ========== a: int, must be 0 or 1 mod 4 Returns ======= Pair ``(D, F)`` of dictionaries. Raises ====== ValueError If *a* is not 0 or 1 mod 4. References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* (See Prop. 5.1.3) zATo extract fundamental discriminant, number must be 0 or 1 mod 4.)) ValueErrorritems) a a_factorsDF num_3_mod_4peevene2s & r extract_fundamental_discriminantr*Msl 1uF\]]AvAq6zAv2v ! I A AK! q5A:AD1uzq AvA!|6AD" 6D {Q!# qTAv v 7!6AaDqa! 4KrcLa]tRt^toRtR RltRtRtRtRt Rt R t Vt R#) AlgIntPowersa Compute the powers of an algebraic integer. Explanation =========== Given an algebraic integer $\theta$ by its monic irreducible polynomial ``T`` over :ref:`ZZ`, this class computes representations of arbitrarily high powers of $\theta$, as :ref:`ZZ`-linear combinations over $\{1, \theta, \ldots, \theta^{n-1}\}$, where $n = \deg(T)$. The representations are computed using the linear recurrence relations for powers of $\theta$, derived from the polynomial ``T``. See [1], Sec. 4.2.2. Optionally, the representations may be reduced with respect to a modulus. Examples ======== >>> from sympy import Poly, cyclotomic_poly >>> from sympy.polys.numberfields.utilities import AlgIntPowers >>> T = Poly(cyclotomic_poly(5)) >>> zeta_pow = AlgIntPowers(T) >>> print(zeta_pow[0]) [1, 0, 0, 0] >>> print(zeta_pow[1]) [0, 1, 0, 0] >>> print(zeta_pow[4]) # doctest: +SKIP [-1, -1, -1, -1] >>> print(zeta_pow[24]) # doctest: +SKIP [-1, -1, -1, -1] References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* Nc WnW nVP4Vn\ VP P 44Uu.uF q3)V,NK upRR.VnVPVnR#uupi)z Parameters ========== T : :py:class:`~.Poly` The monic irreducible polynomial over :ref:`ZZ` defining the algebraic integer. modulus : int, None, optional If not ``None``, all representations will be reduced w.r.t. this. N) Tmodulusdegreenreversedrepto_listpowers_n_and_up max_so_far)selfr/r0rs&&& r__init__AlgIntPowers.__init__se 4)r8others&&r__rmod__AlgIntPowers.__rmod__sxxrc VPpW8:dR#VPpVPpV^,p\V^,V^,4FpWF^, V, ,V^, ,pTP V^,V,V,.\^V4Uu.uFBqV^, V, ,V^, ,WX,V,,V,NKD up,4K WnR#uupir<)r7r2r6rangeappend) r8r'mr2rrkbis && rcompute_up_throughAlgIntPowers.compute_up_throughs OO 66 FF   aDqsAaCAA#a%1 A HH1a$=B1a[#=Hqs1uXac]QT!V+t33[# ! #s'AD cVPpV^8d \R4hW8d$\V4Uu.uF q3V8Xd^M^NK up#VPV4VPW, ,#uupi)rzExponent must be non-negative.)r2rrErKr6)r8r'r2rJs&& rgetAlgIntPowers.getsk FF q5=> > U05a91aAQ&9 9  # #A &''. .:sA2c$VPV4#r<)rN)r8items&&r __getitem__AlgIntPowers.__getitem__sxx~r)r/r7r0r2r6r<) __name__ __module__ __qualname____firstlineno____doc__r9r>rBrKrNrR__static_attributes____classdictcell__) __classdict__s@rr,r,s/%N!&C /rr,c#x"TpV.V,pW8XgW9g V)V9d VR,xV^, pW4,V)8Xd V^,pKW4;;,^,uu&\V^,V4FpWV&K \V4FpW4,^8wgKK V^, pV.V,pK5i)a Generate coefficients for searching through polynomials. Explanation =========== Lead coeff is always non-negative. Explore all combinations with coeffs bounded in absolute value before increasing the bound. Skip the all-zero list, and skip any repeats. See examples. Examples ======== >>> from sympy.polys.numberfields.utilities import coeff_search >>> cs = coeff_search(2, 1) >>> C = [next(cs) for i in range(13)] >>> print(C) [[1, 1], [1, 0], [1, -1], [0, 1], [2, 2], [2, 1], [2, 0], [2, -1], [2, -2], [1, 2], [1, -2], [0, 2], [3, 3]] Parameters ========== m : int Length of coeff list. R : int Initial max abs val for coeffs (will increase as search proceeds). Returns ======= generator Infinite generator of lists of coefficients. NNN)rE)rGRR0rjrJs&& r coeff_searchrasJ B aA  7afaA$J Edqbj FA  q1uaAaD!qAtqy FAaAs BB: B:c"VPwrVPVPWP44pVP 4wrEVRV\ \ V448wd \R4hVRVR13,pVP4pV#)a Extend a basis for a subspace to a basis for the whole space. Explanation =========== Given an $n \times r$ matrix *M* of rank $r$ (so $r \leq n$), this function computes an invertible $n \times n$ matrix $B$ such that the first $r$ columns of $B$ equal *M*. This operation can be interpreted as a way of extending a basis for a subspace, to give a basis for the whole space. To be precise, suppose you have an $n$-dimensional vector space $V$, with basis $\{v_1, v_2, \ldots, v_n\}$, and an $r$-dimensional subspace $W$ of $V$, spanned by a basis $\{w_1, w_2, \ldots, w_r\}$, where the $w_j$ are given as linear combinations of the $v_i$. If the columns of *M* represent the $w_j$ as such linear combinations, then the columns of the matrix $B$ computed by this function give a new basis $\{u_1, u_2, \ldots, u_n\}$ for $V$, again relative to the $\{v_i\}$ basis, and such that $u_j = w_j$ for $1 \leq j \leq r$. Examples ======== Note: The function works in terms of columns, so in these examples we print matrix transposes in order to make the columns easier to inspect. >>> from sympy.polys.matrices import DM >>> from sympy import QQ, FF >>> from sympy.polys.numberfields.utilities import supplement_a_subspace >>> M = DM([[1, 7, 0], [2, 3, 4]], QQ).transpose() >>> print(supplement_a_subspace(M).to_Matrix().transpose()) Matrix([[1, 7, 0], [2, 3, 4], [1, 0, 0]]) >>> M2 = M.convert_to(FF(7)) >>> print(M2.to_Matrix().transpose()) Matrix([[1, 0, 0], [2, 3, -3]]) >>> print(supplement_a_subspace(M2).to_Matrix().transpose()) Matrix([[1, 0, 0], [2, 3, -3], [0, 1, 0]]) Parameters ========== M : :py:class:`~.DomainMatrix` The columns give the basis for the subspace. Returns ======= :py:class:`~.DomainMatrix` This matrix is invertible and its first $r$ columns equal *M*. Raises ====== DMRankError If *M* was not of maximal rank. References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory* (See Sec. 2.3.2.) NzM was not of maximal rankr]) shapehstackeyedomainrreftuplerErinv)Mr2rMaugr^pivotsABs& rsupplement_a_subspacero=sF 77DA 88AEE!XX& 'D IA bqzU58_$566 !QR%A A HrNcB\V4pVP'dW3#VP'g \R4h\ R VR\ 4R7p\ VRR7pVPRR7p\PRrvV'gZV!4pVF-wrWP8:gKVPV 8:gK*RpK@ \;P^,un KaV\n VeVPXX WR7wrXX 3# T\n i;i) a Find a rational isolating interval for a real algebraic number. Examples ======== >>> from sympy import isolate, sqrt, Rational >>> print(isolate(sqrt(2))) # doctest: +SKIP (1, 2) >>> print(isolate(sqrt(2), eps=Rational(1, 100))) (24/17, 17/12) Parameters ========== alg : str, int, :py:class:`~.Expr` The algebraic number to be isolated. Must be a real number, to use this particular function. However, see also :py:meth:`.Poly.intervals`, which isolates complex roots when you pass ``all=True``. eps : positive element of :ref:`QQ`, None, optional (default=None) Precision to be passed to :py:meth:`.Poly.refine_root` fast : boolean, optional (default=False) Say whether fast refinement procedure should be used. (Will be passed to :py:meth:`.Poly.refine_root`.) Returns ======= Pair of rational numbers defining an isolating interval for the given algebraic number. See Also ======== .Poly.intervals z+complex algebraic numbers are not supportedmpmath)modulesprinterT)polys)sqfF)epsfast) r is_Rationalis_realNotImplementedErrorr rr intervalsr dpsr!rI refine_root) algrvrwfuncpolyr|r}doner!rIs &&& risolatersN #,C z [[[! 9; ; BX7H ID 3d #D4(I &C!:#%%1*D" !  1#9 q6M sDD.D&D D)NF)rXsympy.core.sympifyrsympy.ntheory.factor_r!sympy.polys.domains.rationalfieldrsympy.polys.domains.integerringrsympy.polys.matrices.exceptionsr sympy.polys.numberfields.minpolyrsympy.printing.lambdareprrsympy.utilities.decoratorr sympy.utilities.lambdifyr rqr rrrr*r,rarorrxrrrs-&+0.745,-@2/* &UUp[[[|44nT nEEr