g\3dZddlmZddlmZddlmZmZddlm Z ddl m Z m Z dZ d Zd Zd Zd Zd ZddddZdS)z,Functions returning normal forms of matrices) defaultdict) DomainMatrix) DMDomainError DMShapeError)symmetric_residue)QQZZcdt|}tj||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)minvssmfs \/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/polys/matrices/normalforms.pysmith_normal_formrs.$ Q  D  D!(AG 4 4C Jctt|D]P}|||}||z||||zz|||<||z||||zz|||<QdSN)rangelen) rijabcdkes r add_columnsr"(sz3q66]]"" aDGA#!A$q' /!QA#!A$q' /!Q""rcV jjsd}t|djvrdSjx\ }t j fd fd} fd}fdtD}|r0|ddkr$|ddcd<|d<nQfdt D}|r4|ddkr(D]%}||d|dc|d<||d<&tfd td Ds)tfd td Drh||tfd td D?tfd td Dhd |vrd}n:td d d Dd z d z f}t|}ddr݉ddg} | |tt| d z D]} | | r| | d z| | d dkr_| | d z| | } | | | d| | d zz| | d z<| | | <n|ddfz} t!| S)a3 Return the tuple of abelian invariants for a matrix `m` (as in the Smith-Normal form) References ========== [1] https://en.wikipedia.org/wiki/Smith_normal_form#Algorithm [2] https://web.archive.org/web/20200331143852/https://sierra.nmsu.edu/morandi/notes/SmithNormalForm.pdf z8The matrix entries must be over a principal ideal domainrct D]P}|||}||z||||zz|||<||z||||zz|||<QdSr)r) rrrrrrrr r!colss radd_rowsz#invariant_factors..add_rowsHsut & &A!QAcAad1gIoAaDGcAad1gIoAaDGG & &rc |dddkr|S|dd}td D]}||ddkr ||d|\}}|dkr |d|dd| dS |||d\}}} ||d|d} ||d}  |d||||| |}|SNrr)rdivgcdex) rpivotrrrrrgd_0d_jr'rrowss r clear_columnz'invariant_factors..clear_columnPs Q47a<<H!Qq$  AtAw!||::ad1gu--DAqAvvAq!QA.... ,,uad1g661ajj1a!,,Q/jj**1-Aq!QcT222rc  |dddkr|S|dd}td D]}|d|dkr |d||\}}|dkrt|d|dd| dW ||d|\}}} |d||d} ||d} t|d||||| |}|Sr))rr*r"r+) rr,rrr-rrr.r/r0r&rs r clear_rowz$invariant_factors..clear_rowcs Q47a<<H!Qq$  AtAw!||::ad1gu--DAqAvvAq!QA2q1111 ,,uad1g661ajj1a!,,Q/jj**1-Aq!Q3555rc8g|]}|ddk|Srr$.0rrs r z%invariant_factors..ws& 2 2 2QqT!W\\1\\\rc8g|]}d|dk|Sr6r$)r8rrs rr9z%invariant_factors..{s&666Q1aAqrc3<K|]}d|dkVdSrNr$r7s r z$invariant_factors../33qtAw!|333333rrc3<K|]}|ddkVdSr<r$r7s rr=z$invariant_factors..r>rc"g|] }|dd S)rNr$)r8r-s rr9z%invariant_factors..s #9#9#9aAabbE#9#9#9rN)ris_PID ValueErrorrlistto_denserepto_ddmranyrr extendrr*gcdtuple)rmsgrr2r4indrowr lower_rightresultrr.r'r&rr1s` @@@@rr r 1sXF =HooAG||r JD$ QZZ\\  $ $ & &''A&&&&&&( 3 2 2 2eDkk 2 2 2C :s1v{{CF)QqT!aAii6666%++666  :3q6Q;; : :&)#a&k3q6#ACF  3333U1T]]333 3 3 3333U1T]]333 3 3 LOO IaLL 3333U1T]]333 3 3 3333U1T]]333 3 3  Ezz"#9#91QRR5#9#9#9DFDF;KVTT  --tAw #A$q' ds6{{1}%%  Aay VZZqs VAY??BaGGJJvac{F1I66$jjA66q9&1+Eqs q 1a " ==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. rr)r r+)rrxyr.s r_gcdexrTsOhq!nnGAq!Avv!a%1** a%%BBQ a7Nrc |jjstd|j\}}|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.rrQrN)ris_ZZrrrDrErFcopyrrTr"rfrom_rep to_dfm_or_ddm) Arnr rruvrr-srqs r_hermite_normal_formras8 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 |jjstdtj|r|dkrtdd}t t }|j\}}||krtd| j }|}|}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!|||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.) rVrz0Modulus D must be positive element of domain ZZ.c$tt|D]r}|||} t|| z||||zz|z||||<t|| z||||zz|z||||<sdSr)rrr) rRrrrrrrr r!s radd_columns_mod_Rz8_hermite_normal_form_modulo_D..add_columns_mod_R0ss1vv F FA!QA'QQqT!W)<(A1EEAaDG'QQqT!W)<(A1EEAaDGG F Frz2Matrix must have at least as many columns as rows.rQr)rrWrr of_typerdictrrrDrErFrXrrTr"r)r[DreWrr\r rdrrr]r^rr-r_riir`s r_hermite_normal_form_modulo_DrksZ 8>><=== :a==PAEENOOOFFF DA 7DAq1uuOPPP !!&&((A A A 1q5"b ! ! Qq1ub"%% ; ;AtAw!|| 1a!A$q'221atAw!|QqT!W\1!!!Q1aQB::: aDG 66OAaDGaA,,1a(( & &B2qzA~AbE!HH Q47a<<AaDGq1ua . .A!Q1Q47"A 1aQB1 - - - - a Aq62 & & / / 1 11rNF)rh 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.) rVNr) rrWr convert_tor rankrrkra)r[rhrls rhermite_normal_formrpVsov 8>><===}j}ALL,<,<,A,A,C,Cqwqz,Q,Q,Q222#A&&&r)__doc__ collectionsr domainmatrixr exceptionsrrsympy.ntheory.modularrsympy.polys.domainsr r rr"r rTrarkrpr$rrrws22######&&&&&&33333333333333&&&&&&&&."""hhhV*J;J;J;ZU2U2U2p!%@'@'@'@'@'@'@'r