+ iBRt^RIHt^RIHt^RIHtHt^RIH t ^RI H t H t Rt RtRtR tR tR tR tR tRtRRRR/RltR#)z,Functions returning normal forms of matrices) defaultdict) DomainMatrix) DMDomainError DMShapeError)symmetric_residue)QQZZcr\V4p\P!WPVP4pV#)a 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/photoedit/myenv/lib/python3.14/site-packages/sympy/polys/matrices/normalforms.pysmith_normal_formrs-$ Q D   D((AGG 4C JcvVPpVPpVPpVP4p\ V^,4F<p\ V^,4F#pWE8XdK W,V,V8XdK!R# K> \ V^,V^,4p\ ^V4FpW^, ,V^, ,V8XdW,V,V8wdR#K>VP W,V,W^, ,V^, ,4^,pWs8wgKR# R#)z0 Checks that the matrix is in Smith Normal Form FT)r r zeroto_listrangemindiv)rr r rijupperrs& ris_smith_normal_formr(sXXF GGE ;;D A 58_uQxAv47d? ! a%( #E 1e_ qS6!A#;$ tAw$ 147AcF1Q3K03Ay rc \\V44FjpW,V,pW8,W@V,V,,,W,V&WX,W`V,V,,,W,V&Kl R#Nrlen rrrabcdkes &&&&&&& r add_columnsr*EsW3q6] DG#A$q' /Q#A$q' /QrcnVPpVPpVP4p\WVRR7#)a 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 s& rr r Ns0XXF GGE A U CCrcVPpVP;wr#pVP4p\WVRR7wrVp\P !WQV4P 4p\ WaW"3R7p\ WqW33R7pWV3#)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 rstrs & rsmith_normal_decompr5`s{ XXF JD A%au4HJDQ   D% 0 9 9 ;CQd\:AQd\:A 19rc  aaaaa a!a"a#a$a%SP'gRS 2p\V4hVwo"o SPo%SPo!V!V%3Rlp^V9dS'dR V!S"4V!S 43#R #S'dV!S"4o#V!S 4o$RoVVVVV"V#V%3RlpV VVVV$V%3Rlp\ S"4Uu.uFpSV,^,S%8wgKVNK p pV 'dfV ^,S%8wdXSV ^,,S^,uS^&SV ^,&S'd(S#V ^,,S#^,uS#^&S#V ^,&M\ S 4U u.uFp S^,V ,S%8wgKV NK p p V 'dsV ^,S%8wdeSF(p W^,,V ^,uV ^&W^,&K* S'd/S$F(p W^,,V ^,uV ^&W^,&K* \ ;QJd*VV%3Rl\ ^S 44F 'gK RM RM!VV%3Rl\ ^S 444'gW\ ;QJd*VV%3R l\ ^S"44F 'gK RM RM!VV%3R l\ ^S"444'dV!4V!4KV3R lp S^,^,^8wdSP S^,^,4p SP'd^S^,^,, p V SP8wdGS^,^;;,V ,uu&S'd#S#^,Uu.uF qV ,NK upS#^&^V9dR pEMSR ,Uu.uF pVR ,NK pp\VSS"^, S ^, 3SR 7pS'dVwppp^.^.S"^, ,,.VU u.uF p ^.V ,NK up ,p^.^.S ^, ,,.VU u.uF p ^.V ,NK up ,p\\V S#VS$V.44wo#po$pVS#,o#S$V,o$S#P4o#S$P4o$MTpS^,^,'EdyS^,^,.pVPV4\ \V4^, 4EF5pVV,VV^,,ppV'EdSPVV4^,S%8wdS'dSPVV4wpppMSP!VV4pSPVV4^,pS'dSPVV4^,pS!S#W^,^^X^4\#S$W^,^X^^4S!S#W^,^V)^^4\#S$W^,^^V)^4S!S#W^,^^R^4VV,VV^,&VVV&EK6Mp MmS'dNS"^8dS#R ,S#^,.,o#S ^8d(S$U u.uFqR ,V ^,.,NK up o$VS^,^,3,pS'd\%V4S#S$3#\%V4#uupiuup iuupiuupiuup iuup iuup i)z Return the tuple of abelian invariants for a matrix `m` (as in the Smith-Normal form). If `full=True` then invertible matrices ``s, t`` such that the product ``s, m, t`` is the Smith Normal Form are also returned. zBThe matrix entries must be over a principal ideal domain, but got c <\V4UUu.uF&p\V4Uu.uF q"V8XdSMSNK upNK( upp#uupiuuppir )r)nrroners& reye!_smith_normal_decomp..eyes>EJ1XNX%(;(QQD((;XNN;NsAA AAc\\V^,44FjpW,V,pW8,W@V,V,,,W,V&WX,W`V,V,,,W,V&Kl R#Nr!r#s &&&&&&& radd_rows&_smith_normal_decomp..add_rowss\s1Q4y!AQAcAd1gIoADGcAd1gIoADG"rc <S ^,^,p\^S 4FpS V,^,S8XdKS PS V,^,V4wr#VS8Xd+S !S ^V^^V)^4S 'dS !S^V^^V)^4KkKmS PVS V,^,4wrEpS PS V,^,V4pS PW4pS !S ^WWWV)4S 'dS !S^WWWV)4TpK R#r=)rrgcdexexquo)pivotrr'rr$r%gd_0d_jr?r r-rr1r3rs r clear_column*_smith_normal_decomp..clear_columns!Qq$AtAw$::ad1gu-DADyAq!QA.Q1aQB2!,,uad1g6all1Q47A.ll5,AqQcT2Q1#6 rc *<S ^,^,p\^S 4FpS ^,V,S8XdKS PS ^,V,V4wr#VS8Xd1\S ^V^^V)^4S 'd\S ^V^^V)^4KqKsS PVS ^,V,4wrEpS P S ^,V,V4pS P W4p\S ^WWWV)4S 'd\S ^WWWV)4TpK R#r=)rrr*rBrC)rDrr'rr$r%rErFrGr2r r-rr4rs r clear_row'_smith_normal_decomp..clear_rows!Qq$AtAw$::ad1gu-DADyAq!QA2q11aAr15!,,uad1g6all1Q47A.ll5,Aq!51aASD9 rc3J<"TFpS^,V,S8gxK R#5ir=.0rrrs& r '_smith_normal_decomp..6 1qtAw$  #TFc3J<"TFpSV,^,S8gxK R#5ir=rNrOs& rrQrRrSrTcV<\V\V4\V^,43SR7#)r>)r r )rr")rr s&rto_domain_matrix._smith_normal_decomp..to_domain_matrixs#Ac!fc!A$i%8HHr:NNr,rN)is_PID ValueErrorrr9ranycanonical_unitis_Fieldr.listmaprextendr"rrBgcdr*tuple)&rr r r-msgr:rHrKrindrrowrWr&elemrr lower_rightrets_smallt_smalls2t2resultr$r%xyr'alphabetar?r2r9r1r3r4rs&ff&f @@@@@@@rr.r.|s ===RSYRZ[oJD$ ;;D **CO Ez s4y#d)+ +I I I&(*Dk 5kQqT!W_11kC 5 s1v~CF)QqT!aAi Ai1OAaD!CF)+9+Q1aDqq+9 3q6T>&)a&k3q6#AF C*-!f+s1v'CFCAK 36a 63336a 6 6 6 36a 63336a 6 6 6 I tAw!|  ! !!A$q' * ???AaDG A  ? aDGqLG-.qT2TTqT2!Ez&'e,equue ,";ax*7 %( "D'7#T!V $%g(Fgs!sg(FFB#T!V $%g(Fgs!sg(FFB$4q"an EFLAr1bQABA A ADtAwwA$q' ds6{1}%A!9fQqSkqAqVZZ1%a(D0$ll1a0GAq! 1a(A 1a(+!::a+A.DQq5!Q151!eQ1a8Qq5!eVQ:1!eQD5!<Qq5!QA6%iqs q )&, axbEQqTFNax3451CWAx''151a " V}a""V}m 6 :03 - )G(FN6s62ZZZ",Z" Z':Z,Z1Z6 Z;c~\P!W4wr#pV^8wdW,^8Xd^pV^8dRM^pW#V3#)a This supports the functions that compute Hermite Normal Form. Explanation =========== Let x, y be the coefficients returned by the extended Euclidean Algorithm, so that x*a + y*b = g. In the algorithms for computing HNF, it is critical that x, y not only satisfy the condition of being small in magnitude -- namely that |x| <= |b|/g, |y| <- |a|/g -- but also that y == 0 when a | b. rZ)rrB)r$r%rprqrEs&& r_gcdexru"s?hhqnGA!Av!%1* a%BQ 7Nrc VPP'g \R4hVPwrVP 4P 4pTp\ V^, RR4EF'pV^8XdEMV^,p\ V^, RR4F|pW,V,^8wgK\W,V,W,V,4wrgpW,V,V,W,V,V,r\WWVWz)V 4K~ W,V,p V ^8d\WVR^R^4V )p V ^8Xd V^, pK\ V^,V4F*pW,V,V ,p \WV^V )^^4K, EK* \P!VP44RVR13,#)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.:NNNNrZ) r is_ZZrr to_ddmcopyrrur*rfrom_rep to_dfm_or_ddm) Arr8r(rruvr'rr3r%qs & r_hermite_normal_formr7s|8 88>>><== 77DA  A A 1q5"b ! 6  Qq1ub"%AtAw!|!a!$q'2atAw!|QT!W\1A!2q1& DG q5 aQA .A 6 FA 1q5!_DGqLA!QAq1%?"H  !2 3AqrE ::rc VPP'g \R4h\P!V4'dV^8d \R4hRp\ \ 4pVPwrEWT8d \R4hVP4pTpTp\V^, RR4EF~pV^,p\V^, RR4Fzp W,V ,^8wgK\W,V,W,V ,4wrp W,V,V ,W,V ,V ,rV!WWiWV)V 4K| W,V,pV^8XdT;W,V&p\W4wrp \V4F*pWV,V,,V,VV,V&K, W8,V,^8Xd WsV,V&\V^,V4F7p W8,V ,W8,V,,p\W9V^V)^^4K9 W|,pEK \W4V3\4P4#)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.) rwz0Modulus D must be positive element of domain ZZ.cN\\V44FpW,V,p \WI,WPV,V,,,V,V4W,V&\Wi,WpV,V,,,V,V4W,V&K R#r )rr"r) rRrrr$r%r&r'r(r)s &&&&&&&& radd_columns_mod_R8_hermite_normal_form_modulo_D..add_columns_mod_Rsqs1vAQA'qT!W)<(A1EADG'qT!W)<(A1EADGrz2Matrix must have at least as many columns as rows.rZ)r rxrrof_typerdictr rrrrur*rr0)r}DrWrr8r(rrrr~rr'rr3r%iirs&& r_hermite_normal_form_modulo_DrsZ 88>>><== ::a==AENOOF DA 77DAuOPP A A A 1q5"b ! Qq1ub"%AtAw!| a!$q'2atAw!|QT!W\1!!aQB: & DG 6OADGa,a(B2qzA~AbE!H 47a<aDGq1uaAQ147"A aQB1 -! %"& q62 & / / 11rrN check_rankFcVPP'g \R4hVeMV'd:VP\4P 4VP ^,8Xd \W4#\V4#)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.) rw) r rxr convert_torrankr rr)r}rrs&$$rhermite_normal_formrs\v 88>>><==}jALL,<,A,A,Cqwwqz,Q,Q22#A&&r)__doc__ collectionsr domainmatrixr exceptionsrrsympy.ntheory.modularrsympy.polys.domainsrrrrr*r r5r.rurrrrNrrrsa2#&33&.:"D$8cL*J;ZU2p@'@'@'r