+ iRt^RIHt^RIHt^RIHt^RIHt H t H t HtHtRRltRRltRRlt RR lt RR ltR RR R /RltR#)z,Functions returning normal forms of matrices)ZZ)Poly) DomainMatrix)smith_normal_formis_smith_normal_formsmith_normal_decompinvariant_factorshermite_normal_formNc\VRR4pVPR4p\P!V4pT;'gTpVeVP V4pV#)zConvert Matrix to DomainMatrixringNcR\V\4'dVP4#T#N) isinstanceras_expr)es&X/var/www/html/photoedit/myenv/lib/python3.14/site-packages/sympy/matrices/normalforms.py_to_domain..sZ4-@-@aiikGaG)getattr applyfuncr from_Matrix convert_to)mdomainr dMs&& r _to_domainrsU 1fd #D GHA  ! !! $B ^^tF  ]]6 " IrcJ\W4p\V4P4#)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_Matrixrrrs&& rrrs! A B 8   rc.\W4p\V4#)z0 Checks that the matrix is in Smith Normal Form )r_is_snfr s&& rrr0s A B 2;rc\W4p\V4wr4pVP4VP4VP43#)a Return the Smith Normal Decomposition 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_decomp >>> m = Matrix([[12, 6, 4], [3, 9, 6], [2, 16, 14]]) >>> a, s, t = smith_normal_decomp(m, domain=ZZ) >>> assert a == s * m * t )r_sndr)rrrasts&& rrr8s: A B2hGA! ;;=!++- 66rcaaa\W4o\S4p\;QJd.V3RlV4FNK 5M!V3RlV44p\VR4'daVPP 'dEVPoV3Rlo\;QJd.V3RlV4FNK 5M!V3RlV44pV#)a 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 c3Z<"TF pSPPV4xK" R#5ir )rto_sympy).0frs& r $invariant_factors..Ys#;7aBII&&q))7s(+r cH<\VSPSPR7#))r)rsymbolsr)r,Ks&rr#invariant_factors..^sQ !(( Crc34<"TF pS!V4xK R#5ir )r+r,to_polys& rr-r._s81GAJJs)r_invftuplehasattrr is_PolynomialRing)rrfactorsr1rr5s&& @@@rrrKs A BBiGe;7;ee;7;;Gq& 66 # # #ACGe88ee888G NrD check_rankFcVe1\P!V4'g\\V44p\VPWR7P 4#)aw 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.) )r;r<)rof_typeint_hnf_repr)Ar;r<s&$$rr r cs=n }RZZ]] s1vJ ! 3 = = ??rr )__doc__sympy.polys.domains.integerringrsympy.polys.polytoolsrsympy.polys.matricesr sympy.polys.matrices.normalformsrrrr"rr$rr6r r@rr4rrrHsJ2.&-  &7&09@9@9@r