o g\3@sdZddlmZddlmZddlmZmZddlm Z ddl m Z m Z dd Z d d Zd d ZddZddZddZdddddZdS)z,Functions returning normal forms of matrices) defaultdict) DomainMatrix) DMDomainError DMShapeError)symmetric_residue)QQZZcCst|}t||j|j}|S)aJ 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 ZZ >>> from sympy.polys.matrices import DomainMatrix >>> from sympy.polys.matrices.normalforms import smith_normal_form >>> m = DomainMatrix([[ZZ(12), ZZ(6), ZZ(4)], ... [ZZ(3), ZZ(9), ZZ(6)], ... [ZZ(2), ZZ(16), ZZ(14)]], (3, 3), ZZ) >>> print(smith_normal_form(m).to_Matrix()) Matrix([[1, 0, 0], [0, 10, 0], [0, 0, -30]]) )invariant_factorsrdiagdomainshape)minvssmfrW/var/www/visachat/venv/lib/python3.10/site-packages/sympy/polys/matrices/normalforms.pysmith_normal_formsrc Csbtt|D](}|||}|||||||||<|||||||||<qdSN)rangelen rijabcdkerrr add_columns(s   "r c sjjs d}t|djvrdSj\}tjfddfdd}fdd }fd d tD}|r]|ddkr]|ddd<|d<n)fd d tD}|r|ddkrD]}||d|d|d<||d<qrt fd dtdDst fddtdDr||t fd dtdDst fddtdDsd|vrd}nt dd ddDddf}t |}ddrHddg} | |tt | dD]B} | | r?| | d| | ddkr?| | d| | } | | | d| | d| | d<| | | <qt| St| S|ddf} t| S)a3 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 z8The matrix entries must be over a principal ideal domainrrc s^tD](}|||}|||||||||<|||||||||<qdSr)rr)colsrradd_rowsHs   "z#invariant_factors..add_rowsc s|dddkr |S|dd}tdD]U}||ddkr q||d|\}}|dkr<|d|dd| dq|||d\}}}||d|d}||d} |d||||| |}q|SNrr)rdivgcdex rpivotrrrrrgd_0d_j)r"r rowsrr clear_columnP z'invariant_factors..clear_columnc s|dddkr |S|dd}tdD]U}|d|dkr q|d||\}}|dkr.clear_rowcs g|] }|ddkr|qSrr.0rrrr w z%invariant_factors..cs g|] }d|dkr|qSr0r)r2rr3rrr4{r5c3s |] }d|dkVqdSrNrr1r3rr z$invariant_factors..rc3s |] }|ddkVqdSr6rr1r3rrr7r8cSsg|]}|ddqS)rNr)r2r(rrrr4sN)r is_PID ValueErrorr listto_denserepto_ddmranyrr extendrr$gcdtuple) rmsgr r-r/indrowr lower_rightresultrr)r)r"r!r rr,rr 1sT   $$* ,( r cCsDt||\}}}|dkr||dkrd}|dkrdnd}|||fS)a This supports the functions that compute Hermite Normal Form. Explanation =========== Let x, y be the coefficients returned by the extended Euclidean Algorithm, so that x*a + y*b = g. In the algorithms for computing HNF, it is critical that x, y not only satisfy the condition of being small in magnitude -- namely that |x| <= |b|/g, |y| <- |a|/g -- but also that y == 0 when a | b. rr)r r%)rrxyr)rrr_gcdexs  rKc Csp|jjstd|j\}}|j}|}t|dddD]}|dkr(n|d8}t|dddD]6}|||dkrjt ||||||\}}}||||||||} } t |||||| | q4|||} | dkrt |||dddd| } | dkr|d7}q t|d|D]}|||| } t |||d| ddqq t | dd|dfS)a Compute the Hermite Normal Form of DomainMatrix *A* over :ref:`ZZ`. Parameters ========== A : :py:class:`~.DomainMatrix` over domain :ref:`ZZ`. Returns ======= :py:class:`~.DomainMatrix` The HNF of matrix *A*. Raises ====== DMDomainError If the domain of the matrix is not :ref:`ZZ`. References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* (See Algorithm 2.4.5.) Matrix must be over domain ZZ.rrHrN)r is_ZZrr r<r=r>copyrrKr rfrom_rep to_dfm_or_ddm) Arnrrruvrr(srqrrr_hermite_normal_forms4  "  rWc Cs|jjstdt|r|dkrtddd}tt}|j\}}||kr*td| j }|}|}t |dddD]}|d8}t |dddD]7} ||| dkrt|||||| \} } } |||| ||| | } }||||| | | | | qM|||}|dkr||||<}t||\} } } t |D]}| |||||||<q|||dkr||||<t |d|D]} ||| |||}t|| |d| ddq|| }q?t|||ft S) a[ Perform the mod *D* Hermite Normal Form reduction algorithm on :py:class:`~.DomainMatrix` *A*. Explanation =========== If *A* is an $m \times n$ matrix of rank $m$, having Hermite Normal Form $W$, and if *D* is any positive integer known in advance to be a multiple of $\det(W)$, then the HNF of *A* can be computed by an algorithm that works mod *D* in order to prevent coefficient explosion. Parameters ========== A : :py:class:`~.DomainMatrix` over :ref:`ZZ` $m \times n$ matrix, having rank $m$. D : :ref:`ZZ` Positive integer, known to be a multiple of the determinant of the HNF of *A*. Returns ======= :py:class:`~.DomainMatrix` The HNF of matrix *A*. Raises ====== DMDomainError If the domain of the matrix is not :ref:`ZZ`, or if *D* is given but is not in :ref:`ZZ`. DMShapeError If the matrix has more rows than columns. References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* (See Algorithm 2.4.8.) rLrz0Modulus D must be positive element of domain ZZ.c Ssvtt|D]2}|||} t|| |||||||||<t|| |||||||||<qdSr)rrr) rRrrrrrrrrrrradd_columns_mod_R0s  *,z8_hermite_normal_form_modulo_D..add_columns_mod_Rz2Matrix must have at least as many columns as rows.rHr)r rMrr of_typerdictr rr<r=r>rNrrKr r)rQDrYWrrRrrXrrrSrTrr(rUriirVrrr_hermite_normal_form_modulo_DsB-  "    r_NF)r\ check_rankcCsF|jjstd|dur|r|t|jdkrt||St|S)a) Compute the Hermite Normal Form of :py:class:`~.DomainMatrix` *A* over :ref:`ZZ`. Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> from sympy.polys.matrices.normalforms import hermite_normal_form >>> m = DomainMatrix([[ZZ(12), ZZ(6), ZZ(4)], ... [ZZ(3), ZZ(9), ZZ(6)], ... [ZZ(2), ZZ(16), ZZ(14)]], (3, 3), ZZ) >>> print(hermite_normal_form(m).to_Matrix()) Matrix([[10, 0, 2], [0, 15, 3], [0, 0, 2]]) Parameters ========== A : $m \times n$ ``DomainMatrix`` over :ref:`ZZ`. D : :ref:`ZZ`, 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 ======= :py:class:`~.DomainMatrix` The HNF of matrix *A*. Raises ====== DMDomainError If the domain of the matrix is not :ref:`ZZ`, or if *D* is given but is not in :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.) rLNr) r rMr convert_torrankr r_rW)rQr\r`rrrhermite_normal_formVs ;$ rc)__doc__ collectionsr domainmatrixr exceptionsrrsympy.ntheory.modularrsympy.polys.domainsrr rr r rKrWr_rcrrrrs     kMX