U Zʹj\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_factorsrZdiagdomainshape)minvsZsmfr_/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/polys/matrices/normalforms.pysmith_normal_formsrc Csbtt|D]P}|||}|||||||||<|||||||||<q dSN)rangelen r ijabcdkerrr add_columns(s  rc sxjjsd}t|djkr&dSj\}tjfddfdd}fdd }fd d tD}|r|ddkr|ddd<|d<nVfd d tD}|r|ddkrD]&}||d|d|d<||d<qt fd dtdDsLt fddtdDr`||qd|krpd}n2t dd ddDddf}t |}ddr^ddg} | |tt | dD]} | | rR| | d| | ddkrR| | d| | } | | | d| | d| | d<| | | <nqpqn|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]P}|||}|||||||||<|||||||||<qdSr)rr)colsrradd_rowsHs   z#invariant_factors..add_rowsc s|dddkr|S|dd}tdD]}||ddkr@q*||d|\}}|dkrx|d|dd| dq*|||d\}}}||d|d}||d} |d||||| |}q*|SNrr)rdivgcdex r ZpivotrrrrrgZd_0Zd_j)r r rowsrr clear_columnPs z'invariant_factors..clear_columnc s|dddkr|S|dd}tdD]}|d|dkr@q*|d||\}}|dkrxt|d|dd| dq*||d|\}}}|d||d}||d} t|d||||| |}q*|Sr!)rr"rr#r$)rr rr clear_rowcs z$invariant_factors..clear_rowcs g|]}|ddkr|qSrr.0rr rr wsz%invariant_factors..cs g|]}d|dkr|qSr*r)r,rr-rrr.{sc3s|]}d|dkVqdSrNrr+r-rr sz$invariant_factors..rc3s|]}|ddkVqdSr/rr+r-rrr0scSsg|]}|ddqS)rNr)r,r%rrrr.sN)r Zis_PID ValueErrorr listto_denserepto_ddmranyrr extendrr"gcdtuple) r msgr r(r)indrowrZ lower_rightresultrr&r)r rr r r'rr 1sL   $$  * ,(  r cCsDt||\}}}|dkr:||dkr:d}|dkr6dnd}|||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 rAc Csz|jjstd|j\}}|j}|}t|dddD]}|dkrTq\|d8}t|dddD]l}|||dkrlt ||||||\}}}||||||||} } t |||||| | ql|||} | dkr t |||dddd| } | dkr|d7}q@t|d|D],}|||| } t |||d| ddq,q@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.rr>rN)r is_ZZrr r3r4r5copyrrArrZfrom_repZ to_dfm_or_ddm) Ar nrrruvrr%srqrrr_hermite_normal_forms0  "    rKc Cs|jjstdt|r"|dkr*tddd}tt}|j\}}||krTtd| j }|}|}t |dddD]X}|d8}t |dddD]n} ||| dkrt|||||| \} } } |||| ||| | } }||||| | | | | q|||}|dkr2||||<}t||\} } } t |D]"}| |||||||<qJ|||dkr||||<t |d|D]4} ||| |||}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.) rBrz0Modulus D must be positive element of domain ZZ.c Ssvtt|D]d}|||} t|| |||||||||<t|| |||||||||<q dSr)rrr) r Rrrrrrrrrrrradd_columns_mod_R0s *z8_hermite_normal_form_modulo_D..add_columns_mod_Rz2Matrix must have at least as many columns as rows.r>r)r rCrr Zof_typerdictr rr3r4r5rDrrArr)rEDrMWr rFrrLrrrGrHrr%rIriirJrrr_hermite_normal_form_modulo_Ds@-  "      rRNF)rO check_rankcCsJ|jjstd|dk r>|r4|t|jdkr>t||St|SdS)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.) rBNr) r rCrZ convert_torZrankr rRrK)rErOrSrrrhermite_normal_formVs ;$ rT)__doc__ collectionsrZ domainmatrixr exceptionsrrZsympy.ntheory.modularrZsympy.polys.domainsrr rrr rArKrRrTrrrrs     kMX