ghdZddlmZddlmZddlmZddlmZ m Z m Z d dZd dZd d Z dd d d Z dS)z,Functions returning normal forms of matrices)ZZ)Poly) DomainMatrix)smith_normal_forminvariant_factorshermite_normal_formNct|dd}|d}tj|}|p|}|||}|S)zConvert Matrix to DomainMatrixringNcXt|tr|n|SN) isinstanceras_expr)es V/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/matrices/normalforms.pyz_to_domain..s!Z4-@-@Gaiikkka)getattr applyfuncr from_Matrix convert_to)mdomainr dMs r _to_domainr s_ 1fd # #D GGHHA  !! $ $B ^tF  ]]6 " " Ircdt||}t|S)a Return the Smith Normal Form of a matrix `m` over the ring `domain`. This will only work if the ring is a principal ideal domain. Examples ======== >>> from sympy import Matrix, ZZ >>> from sympy.matrices.normalforms import smith_normal_form >>> m = Matrix([[12, 6, 4], [3, 9, 6], [2, 16, 14]]) >>> print(smith_normal_form(m, domain=ZZ)) Matrix([[1, 0, 0], [0, 10, 0], [0, 0, -30]]) )r_snf to_Matrix)rrrs rrrs+ Av  B 88    rct||t}tfd|D}t|dr3|jjr'|jfdtfd|D}|S)a9 Return the tuple of abelian invariants for a matrix `m` (as in the Smith-Normal form) References ========== .. [1] https://en.wikipedia.org/wiki/Smith_normal_form#Algorithm .. [2] https://web.archive.org/web/20200331143852/https://sierra.nmsu.edu/morandi/notes/SmithNormalForm.pdf c3LK|]}j|VdSr )rto_sympy).0frs r z$invariant_factors..<s3;;aBI&&q));;;;;;rr c<t|jjS)N)r)rsymbolsr)r"Ks rrz#invariant_factors..AsQ !( C C Crc3.K|]}|VdSr )r!r"to_polys rr#z$invariant_factors..Bs+881GGAJJ888888r)r_invftuplehasattrr is_PolynomialRing)rrfactorsr&rr)s @@@rrr.s Av  BBiiG;;;;7;;;;;Gq&9 6 # 9ACCCCG888888888G NrFD check_rankc|0tj|stt|}t|j||S)a Compute the Hermite Normal Form of a Matrix *A* of integers. Examples ======== >>> from sympy import Matrix >>> from sympy.matrices.normalforms import hermite_normal_form >>> m = Matrix([[12, 6, 4], [3, 9, 6], [2, 16, 14]]) >>> print(hermite_normal_form(m)) Matrix([[10, 0, 2], [0, 15, 3], [0, 0, 2]]) Parameters ========== A : $m \times n$ ``Matrix`` of integers. D : int, optional Let $W$ be the HNF of *A*. If known in advance, a positive integer *D* being any multiple of $\det(W)$ may be provided. In this case, if *A* also has rank $m$, then we may use an alternative algorithm that works mod *D* in order to prevent coefficient explosion. check_rank : boolean, optional (default=False) The basic assumption is that, if you pass a value for *D*, then you already believe that *A* has rank $m$, so we do not waste time checking it for you. If you do want this to be checked (and the ordinary, non-modulo *D* algorithm to be used if the check fails), then set *check_rank* to ``True``. Returns ======= ``Matrix`` The HNF of matrix *A*. Raises ====== DMDomainError If the domain of the matrix is not :ref:`ZZ`. DMShapeError If the mod *D* algorithm is used but the matrix has more rows than columns. References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* (See Algorithms 2.4.5 and 2.4.8.) Nr/)rof_typeint_hnf_repr)Ar0r1s rrrFsJn }RZ]]} s1vvJJ ! 3 3 3 = = ? ??rr )__doc__sympy.polys.domains.integerringrsympy.polys.polytoolsrsympy.polys.matricesr sympy.polys.matrices.normalformsrrrr*rr5rr(rrr=s22......&&&&&&------        &0!%9@9@9@9@9@9@9@r