sg\3~dZddlmZddlmZddlmZmZddlm Z ddl m Z m Z dZ d Zd Zd Zd Zd ZddddZy)z,Functions returning normal forms of matrices) defaultdict) DomainMatrix) DMDomainError DMShapeError)symmetric_residue)QQZZcrt|}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 S/var/www/html/venv/lib/python3.12/site-packages/sympy/polys/matrices/normalforms.pysmith_normal_formrs/$ Q D   D!((AGG 4C Jctt|D]8}|||}||z||||zz|||<||z||||zz|||<:yN)rangelen) rijabcdkes r add_columnsr"(sj3q6]" aDGA#!A$q' /!QA#!A$q' /!Q"rcjjs d}t|djvryjx\}t j j jfdfd}fd}tDcgc]}|ddk7s|}}|r |ddk7r|ddcd<|d<nLtDcgc]}d|dk7s|}}|r&|ddk7rD]}||d|dc|d<||d<tfdtdDstfd tdDrN||tfdtdDr/tfd tdDrNd|vrd} n4tdd D cgc]} | dd  c} dz dz f} t| } ddrddg} | j| tt| dz D]{}| |rij| |dz| |ddk7rHj| |dz| |} j| || d| |dzz| |dz<| | |<qt!| St!| S| ddfz} t!| Scc}wcc}wcc} w) 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]8}|||}||z||||zz|||<||z||||zz|||<:yr)r) rrrrrrrr r!colss radd_rowsz#invariant_factors..add_rowsHsgt &A!QAcAad1gIoAaDGcAad1gIoAaDG &rc |dddk(r|S|dd}td D]}||ddk(r j||d|\}}|dk(r |d|dd| d? j|||d\}}} j||d|d} j||d}  |d||||| |}|SNrr)rdivgcdex) rpivotrrrrrgd_0d_jr'rrowss r clear_columnz'invariant_factors..clear_columnPs Q47a<H!Qq$ AtAw!|::ad1gu-DAqAvAq!QA. ,,uad1g61ajj1a!,Q/jj*1-Aq!QcT2 rc |dddk(r|S|dd}td D]}|d|dk(r j|d||\}}|dk(rt|d|dd| dB j||d|\}}} j|d||d} j||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-DAqAvAq!QA2q1 ,,uad1g61ajj1a!,Q/jj*1-Aq!Q35 rc34K|]}d|dk7ywrNr$.0rrs r z$invariant_factors..3qtAw!|3rc34K|]}|ddk7ywr6r$r7s rr9z$invariant_factors..r:r;N)ris_PID ValueErrorrlistto_denserepto_ddmranyrr extendrr*gcdtuple)rmsgrr2r4rindrrowrr- lower_rightresultr.r'r&rr1s` @@@@rr r 1sXXF ==HoAGG| JD$ QZZ\   $ $ &'A&&(Dk 2QqT!W\1 2C 2 s1v{CF)QqT!aAi+6Q1aAq66 3q6Q; :&)#a&k3q6#ACF  : 3U1T]3 3 3U1T]3 3 O aL 3U1T]3 3 3U1T]3 3 Ez"1QR5#9aAabE#9DFDF;KVT  -tAwA$q' ds6{1}% AayVZZqs VAY?BaGJJvac{F1I6$jjA6q9&1+Eqs q  =  =1a " =E 37$:sJ>,J>!K2K= Kcptj||\}}}|dk7r||zdk(r d}|dkrdnd}|||fS)a This supports the functions that compute Hermite Normal Form. Explanation =========== Let x, y be the coefficients returned by the extended Euclidean Algorithm, so that x*a + y*b = g. In the algorithms for computing HNF, it is critical that x, y not only satisfy the condition of being small in magnitude -- namely that |x| <= |b|/g, |y| <- |a|/g -- but also that y == 0 when a | b. rr)r r+)rrxyr.s r_gcdexrPsGhhq!nGAq!Av!a%1* a%BQ a7Nrc |jjs td|j\}}|j j j j}|}t|dz ddD]}|dk(rn|dz}t|dz ddD]R}|||dk7st||||||\}}}||||z||||z} } t|||||| | T|||} | dkrt|||dddd| } | dk(r|dz }t|dz|D]}|||| z} t|||d| dd!tj|jdd|dfS)a Compute the Hermite Normal Form of DomainMatrix *A* over :ref:`ZZ`. Parameters ========== A : :py:class:`~.DomainMatrix` over domain :ref:`ZZ`. Returns ======= :py:class:`~.DomainMatrix` The HNF of matrix *A*. Raises ====== DMDomainError If the domain of the matrix is not :ref:`ZZ`. References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* (See Algorithm 2.4.5.) Matrix must be over domain ZZ.rrMrN)ris_ZZrrr@rArBcopyrrPr"rfrom_rep to_dfm_or_ddm) Arnr rruvrr-srqs r_hermite_normal_formr]s8 88>><== 77DAq !&&(A A 1q5"b !!2 6  Qq1ub"% 2AtAw!|!1a!A$q'21atAw!|QqT!W\1Aq!QA2q1 2 aDG q5 1aQA .A 6 FA 1q5!_ 2aDGqLAq!QAq1 2?!2H  !2 3AqrE ::rc |jjs tdtj|r|dkr tdd}t t }|j\}}||kr td|jjjj}|}|}t|dz ddD]}|dz}t|dz ddD]P} ||| dk7st|||||| \} } } |||| z||| | z}} ||||| | | | | R|||}|dk(r |x|||<}t||\} } } t|D]}| |||z|z|||<|||dk(r||||<t|dz|D]%} ||| |||z}t|| |d| dd'|| z}t!|||ftjS)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.) rRrz0Modulus D must be positive element of domain ZZ.ctt|D]R}|||} t|| z||||zz|z||||<t|| z||||zz|z||||<Tyr)rrr) rRrrrrrrr r!s radd_columns_mod_Rz8_hermite_normal_form_modulo_D..add_columns_mod_R0ss1v FA!QA'QQqT!W)<(A1EAaDG'QQqT!W)<(A1EAaDG Frz2Matrix must have at least as many columns as rows.rMr)rrSrr of_typerdictrrr@rArBrTrrPr"r)rWDraWrrXr r`rrrYrZrr-r[riir\s r_hermite_normal_form_modulo_Drgs9Z 88>><== ::a=AENOOF DA 77DAq1uOPP !&&(A A A 1q5"b ! Qq1ub"% ;AtAw!| 1a!A$q'21atAw!|QqT!W\1!!Q1aQB:  ; aDG 6OAaDGaA,1a( &B2qzA~AbE!H & Q47a<AaDGq1ua .A!Q1Q47"A 1aQB1 - . a%& Aq62 & / / 11rNF)rd check_rankc|jjs td|A|r3|jtj |j dk(r t||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.) rRr) rrSr convert_tor rankrrgr])rWrdrhs rhermite_normal_formrlVs\v 88>><==}jALL,<,A,A,Cqwwqz,Q,Q22#A&&r)__doc__ collectionsr domainmatrixr exceptionsrrsympy.ntheory.modularrsympy.polys.domainsr r rr"r rPr]rgrlr$rrrssH2#&33&."hV*J;ZU2p!%@'r