Цi\3SrSSKJr SSKJr SSKJrJr SSKJ r SSK J r J r Sr S rS rS rS rS rSSS.Sjrg)z,Functions returning normal forms of matrices) defaultdict) DomainMatrix) DMDomainError DMShapeError)symmetric_residue)QQZZcr[U5n[R"XRUR5nU$)a Return the Smith Normal Form of a matrix `m` over the ring `domain`. This will only work if the ring is a principal ideal domain. Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> from sympy.polys.matrices.normalforms import smith_normal_form >>> m = DomainMatrix([[ZZ(12), ZZ(6), ZZ(4)], ... [ZZ(3), ZZ(9), ZZ(6)], ... [ZZ(2), ZZ(16), ZZ(14)]], (3, 3), ZZ) >>> print(smith_normal_form(m).to_Matrix()) Matrix([[1, 0, 0], [0, 10, 0], [0, 0, -30]]) )invariant_factorsrdiagdomainshape)minvssmfs _/var/www/html/ai-image-ml/venv/lib/python3.13/site-packages/sympy/polys/matrices/normalforms.pysmith_normal_formrs-$ Q D   D((AGG 4C Jc[[U55H2nXUnX8-X@UU--XU'XX-X`UU--XU'M4 gN)rangelen) rijabcdkes r add_columnsr"(sW3q6] DG#A$q' /Q#A$q' /Qrc^^^^^TRmTR(d Sn[U5eSTR;agTR=ummn[ TR 5R R55mU4SjmUUU4SjnUU4Sjn[T5Vs/sHnTUSS:wdMUPM nnU(a!USS:waTUSTSsTS'TUS'OW[T5Vs/sHnTSUS:wdMUPM nnU(a'USS:waTHnXSUSsUS'XS'M [U4Sj[ST555(d$[U4S j[ST555(a\U"T5mU"T5m[U4Sj[ST555(aM6[U4S j[ST555(aM\SU;aSn O6[TSS V s/sHoSS PM sn TS- TS- 4T5n [U 5n TSS(aTSS/n U RU 5 [[U 5S- 5HonX(acTRXS-X5SS:waCTRXS-X5n TRXU 5SXS--XS-'XU'Mo O O U TSS4-n [!U 5$s snfs snfs sn f) 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 z8The matrix entries must be over a principal ideal domainrc>[T 5H2nXUnX8-X@UU--XU'XX-X`UU--XU'M4 gr)r) rrrrrrrr r!colss radd_rows#invariant_factors..add_rowsHsTtAQAcAd1gIoADGcAd1gIoADGrc >USSS:XaU$USSn[ST 5HnXSS:XaMT RXSU5up4US:XaT "USUSSU*S5 M@T RXUS5upVnT RXSU5SnT RX5Sn T "USX%XhU *5 UnM U$Nrr)rdivgcdex) rpivotrrrrrgd_0d_jr'rrowss r clear_column'invariant_factors..clear_columnPs Q47a<H!Qq$AtAw!|::ad1gu-DAAvAq!QA. ,,ud1g6ajja!,Q/jj*1-AqQcT2 rc >USSS:XaU$USSn[ST 5HnUSUS:XaMT RUSUU5up4US:Xa[USUSSU*S5 MET RXSU5upVnT RUSUU5SnT RX5Sn [USX%XhU *5 UnM U$r*)rr+r"r,) rr-rrr.rrr/r0r1r&rs r clear_row$invariant_factors..clear_rowcs Q47a<H!Qq$AtAw!|::ad1gu-DAAvAq!QA2q1 ,,ud1g6ajj1a!,Q/jj*1-Aq!5 rc3:># UHnTSUS:gv M g7frNr$.0rrs r $invariant_factors..3]qtAw!|]rc3:># UHnTUSS:gv M g7fr9r$r:s rr<r=r>r?N)ris_PID ValueErrorrlistto_denserepto_ddmranyrr extendrr+gcdtuple)rmsgrr3r6rindrrowrr. lower_rightresultr/r'r&rr2s` @@@@rr r 1sXXF ==HoAGG| JD$ QZZ\   $ $ &'A&&(Dk 2kQqT!W\1kC 2 s1v{CF)QqT!aAi+6+Q1aAq+6 3q6Q;&)a&k3q6#AF  3U1T]3 3 3 3U1T]3 3 3 O aL 3U1T]3 3 3 3U1T]3 3 3 Ez"1QR5#95aabE5#9DFDF;KVT  -tAwA$q' ds6{1}%AyVZZs VY?BaGJJvc{FI6$jjA6q9&1+Es q  &1a " =E 37$:s%K,9K,6K1 K19K6cp[R"X5up#nUS:waX-S:Xa SnUS:aSOSnX#U4$)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_gcdexrTs?hhqnGA!Av!%1* a%BQ 7Nrc URR(d [S5eURupUR 5R R 5R5nUn[US- SS5HnUS:Xa OUS-n[US- SS5HKnXUS:wdM[XUXU5upgnXUU-XUU-p[XXVXz*U 5 MM XUn U S:a[XUSSSS5 U *n U S:XaUS- nM[US-U5HnXUU -n [XUSU *SS5 M M [R"UR55SS2US24$)a Compute the Hermite Normal Form of DomainMatrix *A* over :ref:`ZZ`. Parameters ========== A : :py:class:`~.DomainMatrix` over domain :ref:`ZZ`. Returns ======= :py:class:`~.DomainMatrix` The HNF of matrix *A*. Raises ====== DMDomainError If the domain of the matrix is not :ref:`ZZ`. References ========== .. [1] Cohen, H. *A Course in Computational Algebraic Number Theory.* (See Algorithm 2.4.5.) Matrix must be over domain ZZ.rrQrN)ris_ZZrrrDrErFcopyrrTr"rfrom_rep to_dfm_or_ddm) Arnr rruvrr.srqs r_hermite_normal_formras8 88>><== 77DA !&&(A A 1q5"b ! 6  Qq1ub"%AtAw!|!a!$q'2atAw!|QT!W\1A!2q1& DG q5 aQA .A 6 FA 1q5!_DGqLA!QAq1%?"H  !2 3AqrE ::rc URR(d [S5e[R"U5(aUS:a [S5eSn[ [ 5nURupEXT:a [S5eUR5RR5R5nUnUn[US- SS5HnUS-n[US- SS5HIn XU S:wdM[XUXU 5upn XUU -XU U -pU"XXiXU*U 5 MK XUnUS:Xa U=XU'n[X5upn [U5HnXUU-U-UUU'M X8US:XaXsUU'[US-U5H#n X8U X8U-n[X9USU*SS5 M% X|-nM [!X4U4[5R5$)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[[U55HLnXUn [XI-XPUU--U-U5XU'[Xi-XpUU--U-U5XU'MN gr)rrr) rRrrrrrrr r!s radd_columns_mod_R8_hermite_normal_form_modulo_D..add_columns_mod_R0sqs1vAQA'qT!W)<(A1EADG'qT!W)<(A1EADGrz2Matrix 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_DrlsZ 88>><== ::a==AENOOF DA 77DAuOPP !&&(A A A 1q5"b ! Qq1ub"%AtAw!| a!$q'2atAw!|QT!W\1!!aQB: & DG 6OADGa,a(B2qzA~AbE!H 47a<aDGq1uaAQ147"A aQB1 -! %"& q62 & / / 11rNF)ri check_rankcURR(d [S5eUbFU(a4UR[5R 5UR S:Xa [X5$[U5$)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.) rVr) rrWr convert_tor rankrrlra)r[rirms rhermite_normal_formrqVsZv 88>><==}jALL,<,A,A,Cqwwqz,Q,Q22#A&&r)__doc__ collectionsr domainmatrixr exceptionsrrsympy.ntheory.modularrsympy.polys.domainsr r rr"r rTrarlrqr$rrrxsH2#&33&."hV*J;ZU2p!%@'r