Otj3dZddlmZddlmZddlmZddlmZddl m Z ddl m Z ddl mZdd lmZdd lmZdd lmZd Zd ZdZedZeGddZedZdZeddZdS)z'Utilities for algebraic number theory. )sympify) factorint)QQ)ZZ) DMRankError)minpoly)IntervalPrinter)public)lambdify)mpc|t|tp'tj|pt j|S)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 `/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/polys/numberfields/utilities.pyis_ratrs., a   ?A ?"*Q--?cTt|tptj|S)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 )rrrrrs ris_intr)s!$ a   .A.rc<t|}|j|jfS)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 ; %%rc|dzdvrtd|dkriddifS|dkriifSt|}i}i}d}|D];\}}|dzdkr%d||<|dzdkr|dz }|dkr |dz dz||<3|dz||<>> 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) )rzATo extract fundamental discriminant, number must be 0 or 1 mod 4.rr) ValueErrorritems) a a_factorsDF num_3_mod_4peevene2s r extract_fundamental_discriminantr-MsMl 1uF\]]]AvvAq6zAvv2v ! I A AK!!1 q5A::AaD1uzzq AvvA!|!6AaDD 6D  {Q!## qTAvvvv 77!6AaDqqa! a4Krc8eZdZdZd dZdZdZdZdZdZ dS) 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|_|_|_fdt |jDddg_j_dS)a- 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. cg|]}| z Sr2).0rselfs r z)AlgIntPowers.__init__..s N N Nq!d N N NrN) Tmodulusdegreenreversedrepto_listpowers_n_and_up max_so_far)r4r7r8s` r__init__zAlgIntPowers.__init__sj  N N N NHQU]]__4M4M N N NsPRs ST&rc(|j|n ||jzSN)r8)r4exps rredzAlgIntPowers.redsl*ssdl0BBrc,||SrB)rD)r4others r__rmod__zAlgIntPowers.__rmod__sxxrc Rj}||krdSjjdt|dz|dzD]]dz z dz dzzgfdtdDz^|_dS)Nrrc\g|](}dz z |dz |zzz)S)rr2)r3ibrkr:rr4s rr5z3AlgIntPowers.compute_up_through..sL###89Qqs1uXac]QqT!V+t3###r)r?r:r>rangeappend)r4r*mrKrrLr:rs` @@@@@rcompute_up_throughzAlgIntPowers.compute_up_throughs O 6666 F   aDqsAaC  A!A#a%1 A HH1a$#########=B1a[[###     rc|j}dkrtd|krfdt|DS||j|z S)NrzExponent must be non-negative.c$g|] }|krdnd S)rrr2)r3rJr*s rr5z$AlgIntPowers.get..s%9991aAAQ999r)r:r"rMrPr>)r4r*r:s ` rgetzAlgIntPowers.getsq F q55=>> > UU9999a999 9  # #A & & &'A. .rc,||SrB)rS)r4items r __getitem__zAlgIntPowers.__getitem__sxx~~rrB) __name__ __module__ __qualname____doc__r@rDrGrPrSrVr2rrr/r/s%%N!!!!&CCC   ///rr/c#DK|}|g|z} ||ks ||vs| |vr |ddV|dz }||| kr|dz}||| k||xxdzcc<t|dz|D]}|||<t|D]}||dkrn |dz }|g|z})a[ Generate coefficients for searching through polynomials. Explanation =========== Lead coeff is always non-negative. Explore all combinations with coeffs bounded in absolute value before increasing the bound. Skip the all-zero list, and skip any repeats. See examples. Examples ======== >>> from sympy.polys.numberfields.utilities import coeff_search >>> cs = coeff_search(2, 1) >>> C = [next(cs) for i in range(13)] >>> print(C) [[1, 1], [1, 0], [1, -1], [0, 1], [2, 2], [2, 1], [2, 0], [2, -1], [2, -2], [1, 2], [1, -2], [0, 2], [3, 3]] Parameters ========== m : int Length of coeff list. R : int Initial max abs val for coeffs (will increase as search proceeds). Returns ======= generator Infinite generator of lists of coefficients. TNrr)rM)rORR0rjrJs r coeff_searchr_sJ B aA 77a1ffaAAA$JJJ Edqbjj FAdqbjj ! q1ua  AAaDDq  Atqyy FAaArcV|j\}}||||j}|\}}|d|t t |krtd|dd|df}|}|S)ax 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 rank) shapehstackeyedomainrreftuplerMrinv)Mr:rMaugr\pivotsABs rsupplement_a_subspacerm=sF 7DAq 88AEE!QX&& ' 'D IAv bqbzU588__$$5666 !!!QRR%A A HrNFct|}|jr||fS|jstdt d|dt }t |d}|d}tj d}} |sC|}|D]\}} ||j kr|j | krd}n txj d zc_ |C|t_ n#|t_ wxYw|| || || \}} || fS) 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 supportedr2mpmath)modulesprinterT)polys)sqfFr N)epsfast) r is_Rationalis_realNotImplementedErrorr r r intervalsr dpsr$rK refine_root) algrtrufuncpolyryrzdoner$rKs risolaters=N #,,C ;Sz [;! 9;; ; BX7H7H I I ID 3d # # #D4((IC  $&&C!  1::#%1**DE!    1#D991 q6Ms ACC&)NF)rZsympy.core.sympifyrsympy.ntheory.factor_r!sympy.polys.domains.rationalfieldrsympy.polys.domains.integerringrsympy.polys.matrices.exceptionsr sympy.polys.numberfields.minpolyrsympy.printing.lambdareprr sympy.utilities.decoratorr sympy.utilities.lambdifyr ror rrrr-r/r_rmrr2rrrs--&&&&&&++++++000000......777777444444555555,,,,,,------@@@2///* & & &UUUp[[[[[[[[|444nT T T nEEEEEEr