OtjBdZddlmZddlmZddlmZmZddlm Z ddl m Z m Z dZ d Zd Zd Zd Zd ZdZdZdZddddZdS)z,Functions returning normal forms of matrices) defaultdict) DomainMatrix) DMDomainError DMShapeError)symmetric_residue)QQZZcdt|}tj||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)minvssmfs ^/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/polys/matrices/normalforms.pysmith_normal_formrs.$ Q  D  D!(AG 4 4C Jc2|j}|j}|j}|}t |dD]7}t |dD]}||kr ||||ksdS 8t |d|d}t d|D]s}||dz |dz |kr||||krdS0||||||dz |dz d}||krdStdS)z8 Checks that the matrix is in Smith Normal Form rrFT)rrzeroto_listrangemindiv)rrrrijupperrs ris_smith_normal_formr (s>XF GE ;D A 58__uQx  AAvvQ47d??uuu#  a%( # #E 1e__ QqS6!A#;$  tAw$uu 1Q47AacF1Q3K003ADyyuu 4rctt|D]P}|||}||z||||zz|||<||z||||zz|||<QdSNrlen rrrabcdkes r add_columnsr,Esz3q66]]"" aDGA#!A$q' /!QA#!A$q' /!Q""rcl|j}|j}|}t|||dS)a3 Return the tuple of abelian invariants for a matrix `m` (as in the Smith-Normal form) References ========== [1] https://en.wikipedia.org/wiki/Smith_normal_form#Algorithm [2] https://web.archive.org/web/20200331143852/https://sierra.nmsu.edu/morandi/notes/SmithNormalForm.pdf Frfull)rrr_smith_normal_decomp)rrrs rr r Ns6XF GE A 6U C C CCrc(|j}|jx\}}}|}t|||d\}}}t j|||}t ||||f}t ||||f}|||fS)a Return the Smith-Normal form decomposition of matrix `m`. Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> from sympy.polys.matrices.normalforms import smith_normal_decomp >>> m = DomainMatrix([[ZZ(12), ZZ(6), ZZ(4)], ... [ZZ(3), ZZ(9), ZZ(6)], ... [ZZ(2), ZZ(16), ZZ(14)]], (3, 3), ZZ) >>> a, s, t = smith_normal_decomp(m) >>> assert a == s * m * t Tr.)rr)rrrr0rr to_dense) rrrowscolsrrstrs rsmith_normal_decompr7`s XF JD$ A%au4HHHJD!Q  D&% 0 0 9 9 ; ;CQvdD\:::AQvdD\:::A 19rc   !"jsd}t||\j"j"fd}d|vrrd||fSdSr| |!d "fd}!"fd}"fdt D}|rU|d"krI|ddcd<|d<r# |d dc d< |d<n|"fd t D}|r^|d"krRD]%} | |d| dc| d<| |d<&r(!D]%} | |d| dc| d<| |d<&t "fd t d Ds*t "fd t d Drh||t "fd t d D>t "fd t d Dhfd } dddkrsddjrd ddz jkr/ddxxzcc<rfd dD d<d |vrd} ndd dD} t| d z d z f} r| \} }}d gdgd z zzgd|Dz}d gdgd z zzgd|Dz}tt| |!|g\ }!}| z !|z! ! !n| } ddrkddg}| | t t|d z D]%}||||d z}}|r ||d "krr||\}}}n||}||d}r||d} ||d zd d|d t#!||d zd |dd  ||d zd | dd t#!||d zd d| d  ||d zdd dd||z||d z<|||<&n@r,d kr d d dgz d kr d!D!| ddfz}rt%| !fSt%|S)z Return the tuple of abelian invariants for a matrix `m` (as in the Smith-Normal form). 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AaDG A  ?? aDGGGqLGGG 32222QqT222!Ezz,,ae,,, ";ax*777  %( "D'7#T!V $%(F(Fg(F(F(FFB#T!V $%(F(Fg(F(F(FFB$4q"an E EFFLAr1bQABA A AADtAw#A$q' ds6{{1}%%  A!9fQqSkqA VZZ1%%a(D00)$ll1a00GAq!! 1a((A 1a((+7!::a++A.DHQ1q5!Q15551a!eQ1a888HQ1q5!eVQ:::1a!eQD5!<<<HQ1q5!QA666%iqs q   6axxabbEQqTFNaxx5515551a " V}}a""V}}rcttj||\}}}|dkr||zdkr d}|dkrdnd}|||fS)a This supports the functions that compute Hermite Normal Form. Explanation =========== Let x, y be the coefficients returned by the extended Euclidean Algorithm, so that x*a + y*b = g. In the algorithms for computing HNF, it is critical that x, y not only satisfy the condition of being small in magnitude -- namely that |x| <= |b|/g, |y| <- |a|/g -- but also that y == 0 when a | b. rrcr)r rG)r&r'rxryrJs r_gcdexr}"sOhq!nnGAq!Avv!a%1** a%%BBQ a7Nrc R|jjstd|j\}}|}|}t |dz ddD]}|dkrn |dz}t |dz ddD]x}|||dkrdt||||||\}}}||||z||||z} } t|||||| | y|||} | dkrt|||dddd| } | dkr|dz }t |dz|D])}|||| z} t|||d| dd*tj | dd|dfS)a Compute the Hermite Normal Form of DomainMatrix *A* over :ref:`ZZ`. Parameters ========== A : :py:class:`~.DomainMatrix` over domain :ref:`ZZ`. Returns ======= :py:class:`~.DomainMatrix` The HNF of matrix *A*. Raises ====== DMDomainError If the domain of the matrix is not :ref:`ZZ`. References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* (See Algorithm 2.4.5.) Matrix must be over domain ZZ.rrcrN) ris_ZZrrto_ddmcopyrr}r,rfrom_rep to_dfm_or_ddm) ArrAr*rruvr)rr5r'qs r_hermite_normal_formr7s8 8>><=== 7DAq  A A 1q5"b ! !!2!2 66 E Qq1ub"%% 2 2AtAw!||!1a!A$q'221atAw!|QqT!W\1Aq!QA2q111 aDG q55 1aQA . . .A 66 FAA 1q5!__ 2 2aDGqLAq!QAq1111 2  !2!2 3 3AAAqrrE ::rc h|jjstdtj|r|dkrtdd}t t }|j\}}||krtd| }|}|}t|dz ddD]a}|dz}t|dz ddD]u} ||| dkrat|||||| \} } } |||| z||| | z}} ||||| | | | | v|||}|dkr |x|||<}t||\} } } t|D]}| |||z|z|||< |||dkr ||||<t|dz|D]5} ||| |||z}t|| |d| dd6|| z}ct|||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.) rrz0Modulus D must be positive element of domain ZZ.c$tt|D]r}|||} t|| z||||zz|z||||<t|| z||||zz|z||||<sdSr")rr$r) rRrrr&r'r(r)r*r+s radd_columns_mod_Rz8_hermite_normal_form_modulo_D..add_columns_mod_Rss1vv F FA!QA'QQqT!W)<(A1EEAaDG'QQqT!W)<(A1EEAaDGG F Frz2Matrix must have at least as many columns as rows.rcr)rrrr of_typerdictrrrrr}r,rr2)rDrWrrAr*rrrrrr)rr5r'iirs r_hermite_normal_form_modulo_DrspZ 8>><=== :a==PAEENOOOFFF DA 7DAq1uuOPPP A A A 1q5"b ! ! Qq1ub"%% ; ;AtAw!|| 1a!A$q'221atAw!|QqT!W\1!!!Q1aQB::: aDG 66OAaDGaA,,1a(( & &B2qzA~AbE!HH Q47a<<AaDGq1ua . .A!Q1Q47"A 1aQB1 - - - - a Aq62 & & / / 1 11rNF)r check_rankc|jjstd|M|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.) rNr) rrr convert_tor rankrrr)rrrs rhermite_normal_formrsov 8>><===}j}ALL,<,<,A,A,C,Cqwqz,Q,Q,Q222#A&&&r)__doc__ collectionsr domainmatrixr exceptionsrrsympy.ntheory.modularrsympy.polys.domainsr r rr r,r r7r0r}rrrr<rrrs822######&&&&&&33333333333333&&&&&&&&.:"""DDD$8cccL*J;J;J;ZU2U2U2p!%@'@'@'@'@'@'@'r