o hB@sdZddlmZddlmZddlmZmZddlm Z ddl m Z m Z dd Z d d Zd d ZddZddZddZddZddZddZdddddZdS)z,Functions returning normal forms of matrices) defaultdict) DomainMatrix) DMDomainError DMShapeError)symmetric_residue)QQZZcCst|}t||j|j}|S)aI 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)minvssmfrV/var/www/html/ai/venv/lib/python3.10/site-packages/sympy/polys/matrices/normalforms.pysmith_normal_formsrcCs|j}|j}|j}|}t|dD]}t|dD]}||kr"q||||ks.dSqqt|d|d}td|D]5}||d|d|krX||||krWdSq>||||||d|dd}||krsdSq>dS)z8 Checks that the matrix is in Smith Normal Form rrFT)r r zeroto_listrangemindiv)rr r rijupperrrrris_smith_normal_form(s.(rc Csbtt|D](}|||}|||||||||<|||||||||<qdSNrlen rrrabcdkerrr add_columnsEs   "r(cCs$|j}|j}|}t|||ddS)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 Fr full)r r r_smith_normal_decomp)rr r rrrr Ns r c Csr|j}|j\}}}|}t|||dd\}}}t|||}t||||fd}t||||fd}|||fS)a Return the Smith-Normal form decomposition of matrix `m`. Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> from sympy.polys.matrices.normalforms import smith_normal_decomp >>> m = DomainMatrix([[ZZ(12), ZZ(6), ZZ(4)], ... [ZZ(3), ZZ(9), ZZ(6)], ... [ZZ(2), ZZ(16), ZZ(14)]], (3, 3), ZZ) >>> a, s, t = smith_normal_decomp(m) >>> assert a == s * m * t Tr))r r )r r rr+rr to_dense) rr rowscolsr rstrrrrsmith_normal_decomp`s r1c sHjs d}t||\j j fdd}d|vr.r,d||fSdSr8|| dd fdd } fd d } fd d tD}|r|d kr|ddd<|d<r|ddd<|d<nB fdd tD}|r|d krΈD]} | |d| d| d<| |d<qrΈ D]} | |d| d| d<| |d<qt fddtdDst fddtdDr||t fddtdDst fddtdDsfdd} dddkrUddjr5dddjkrUdd9<rUfdd dDd<d|vr]d} ngdd ddD} t| ddfd} r| \} }}dgdgdgdd |D}dgdgdgdd |D}t t | | |g\} }| |   n| } ddrnddg}| | tt |dD]}||||d}}|rl||d krlr||\}}}n||}||d}r^||d}||ddd|dt ||dd|dd||dd| ddt ||ddd| d||ddddd||||d<|||<qn(rdkrdddgdkrdd D | ddf}rt| fSt|S)z Return the tuple of abelian invariants for a matrix `m` (as in the Smith-Normal form). If `full=True` then invertible matrices ``s, t`` such that the product ``s, m, t`` is the Smith Normal Form are also returned. zBThe matrix entries must be over a principal ideal domain, but got csfddtDS)Ncs&g|]fddtDqS)csg|] }|kr nqSrr.0r)ronerrr sz@_smith_normal_decomp..eye...r)r3)nr4r)rrr5s&z5_smith_normal_decomp..eye..r6r7)r4rr8reyesz!_smith_normal_decomp..eyerrc Ssftt|dD](}|||}|||||||||<|||||||||<qdS)Nrrr!rrradd_rowss   "z&_smith_normal_decomp..add_rowsc sdd}tdD]k}|dkrq |d|\}}|kr?d|dd| dr>d|dd| dq ||d\}}}|d|}||}d||||| rtd||||| |}q dSNrr)rrgcdexexquo pivotrr%rr"r#gd_0d_j)r:r r*rr-r/rrr clear_column$  z*_smith_normal_decomp..clear_columnc sdd}tdD]k}d|krq d||\}}|kr?td|dd| dr>td|dd| dq |d|\}}}d||}||}td||||| rttd||||| |}q dSr;)rrr(r<r=r>)r.r r*rr0rrr clear_rowrDz'_smith_normal_decomp..clear_rowcs g|] }|dkr|qSrrr2rrrrr5 z(_smith_normal_decomp..cs g|] }d|kr|qSrFr)r3rrGrrr5rHc3s |] }d|kVqdSrNrr2rGrr z'_smith_normal_decomp..rc3s |] }|dkVqdSrIrr2rGrrrJrKcst|t|t|dfdS)Nr)r r )rr )r)r rrto_domain_matrixsz._smith_normal_decomp..to_domain_matrixcsg|]}|qSrr)r3elem)r$rrr5scSsg|]}|ddqS)rNr)r3rrrrr5sNr)cSg|]}dg|qSrFrr3rowrrrr5cSrNrFrrOrrrr5rQcSs"g|] }|dd|dgqS)rNrrrOrrrr5s")is_PID ValueErrorrr4ranycanonical_unitis_Fieldr+listmaprextendr rr<gcdr(tuple)rr r r*msgr9rCrEindrPrLr lower_rightrets_smallt_smalls2t2resultrr"r#xyr%alphabetar) r:r$r.r r*rr4r-r/r0rrr+|s ""$$    $$      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. rrRr)r r<)r"r#rfrgr@rrr_gcdex"s  rjc Csj|jjstd|j\}}|}|}t|dddD]}|dkr%n|d8}t|dddD]6}|||dkrgt||||||\}}}||||||||} } t|||||| | q1|||} | dkrt|||dddd| } | dkr|d7}qt|d|D]}|||| } t|||d| ddqqt | 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.rrRrN) r is_ZZrr to_ddmcopyrrjr(rfrom_rep to_dfm_or_ddm) Arr7r&rruvr%rr/r#qrrr_hermite_normal_form7s4   "  ruc Cs|jjstdt|r|dkrtddd}tt}|j\}}||kr*td| }|}|}t |dddD]}|d8}t |dddD]7} ||| dkrt |||||| \} } } |||| ||| | } }||||| | | | | qH|||}|dkr||||<}t ||\} } } t |D]}| |||||||<q|||dkr||||<t |d|D]} ||| |||}t || |d| ddq|| }q:t |||ftS) 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.) rkrz0Modulus D must be positive element of domain ZZ.c Ssvtt|D]2}|||} t|| |||||||||<t|| |||||||||<qdSr)rr r) rRrrr"r#r$r%r&r'rrradd_columns_mod_Rs  *,z8_hermite_normal_form_modulo_D..add_columns_mod_Rz2Matrix must have at least as many columns as rows.rRr)r rlrr of_typerdictr rrrrjr(rr,)rqDrwWrr7r&rvrrrrrsr%rr/r#iirtrrr_hermite_normal_form_modulo_DsB-  "    r}NF)rz 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.) rkNr) r rlr convert_torrankr r}ru)rqrzr~rrrhermite_normal_forms ;$ r)__doc__ collectionsr domainmatrixr exceptionsrrsympy.ntheory.modularrsympy.polys.domainsrr rrr(r r1r+rjrur}rrrrrs"     'MX