\hBSrSSKJr SSKJr SSKJrJr SSKJ r SSK J r J r Sr S rS rS rS rS rSrSrSrSSS.Sjrg)z,Functions returning normal forms of matrices) defaultdict) DomainMatrix) DMDomainError DMShapeError)symmetric_residue)QQZZcr[U5n[R"XRUR5nU$)a Return the Smith Normal Form of a matrix `m` over the ring `domain`. This will only work if the ring is a principal ideal domain. Examples ======== >>> from sympy import ZZ >>> from sympy.polys.matrices import DomainMatrix >>> from sympy.polys.matrices.normalforms import smith_normal_form >>> m = DomainMatrix([[ZZ(12), ZZ(6), ZZ(4)], ... [ZZ(3), ZZ(9), ZZ(6)], ... [ZZ(2), ZZ(16), ZZ(14)]], (3, 3), ZZ) >>> print(smith_normal_form(m).to_Matrix()) Matrix([[1, 0, 0], [0, 10, 0], [0, 0, 30]]) )invariant_factorsrdiagdomainshape)minvssmfs X/var/www/auris/envauris/lib/python3.13/site-packages/sympy/polys/matrices/normalforms.pysmith_normal_formrs-$ Q D   D((AGG 4C JcURnURnURnUR5n[ US5H-n[ US5HnXE:XaM XUU:XaM g M/ [ USUS5n[ SU5HOnXS- US- U:XaXUU:wa gM#UR XUXS- US- 5SnXs:wdMO g g)z0 Checks that the matrix is in Smith Normal Form rrFT)rrzeroto_listrangemindiv)rrrrijupperrs ris_smith_normal_formr (sXXF GGE ;;D A 58_uQxAv47d? ! a%( #E 1e_ qS6!A#;$ tAw$ 147AcF1Q3K03Ay rc[[U55H2nXUnX8-X@UU--XU'XX-X`UU--XU'M4 gNrlen rrrabcdkes r add_columnsr,EsW3q6] DG#A$q' /Q#A$q' /QrchURnURnUR5n[XUSS9$)a Return the tuple of abelian invariants for a matrix `m` (as in the Smith-Normal form) References ========== [1] https://en.wikipedia.org/wiki/Smith_normal_form#Algorithm [2] https://web.archive.org/web/20200331143852/https://sierra.nmsu.edu/morandi/notes/SmithNormalForm.pdf Frfull)rrr_smith_normal_decomp)rrrs rr r Ns0XXF GGE A U CCrcURnUR=up#nUR5n[XUSS9upVn[R "XQU5R 5n[ XaX"4S9n[ XqX34S9nXU4$)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{ XXF JD A%au4HJDQ   D% 0 9 9 ;CQd\:AQd\:A 19rc  ^^^^^ ^!^"^#^$^%TR(dST3n[U5eUum"m TRm%TRm!U!U%4SjnSU;aT(aSU"T"5U"T 54$gT(aU"T"5m#U"T 5m$SmUUUUU"U#U%4SjnU UUUU$U%4Sjn[ T"5Vs/sHnTUST%:wdMUPM n nU (a?U ST%:wa6TU STSsTS'TU S'T(aT#U ST#SsT#S'T#U S'O|[ T 5V s/sHn TSU T%:wdMU PM n n U (aLU ST%:waCTHn XSU SsU S'XS'M T(aT$Hn XSU SsU S'XS'M [ UU%4Sj[ S T 555(d%[ UU%4S j[ S T"555(a\U"5 U"5 [ UU%4Sj[ S T 555(aM5[ UU%4S j[ S T"555(aM\U4S jn TSSS:wauTR TSS5n TR(a S TSS- n U TR:wa2TSS==U -ss'T(aT#SVs/sHoU -PM snT#S'S U;aSnOTS S Vs/sHnUS S PM nn[UTT"S - T S - 4TS 9nT(aUunnnS /S/T"S - --/UV s/sH n S/U -PM sn -nS /S/T S - --/UV s/sH n S/U -PM sn -n[[U T#UT$U/55um#nm$nUT#-m#T$U-m$T#R5m#T$R5m$OUnTSS(Ga1TSS/nURU5 [ [U5S - 5HnUUUUS -nnU(aTRUU5S T%:waT(aTRUU5unnnOTR!UU5nTRUU5SnT(amTRUU5SnT"T#XS -S SWS 5 [#T$XS -S WSS 5 T"T#XS -S U*SS 5 [#T$XS -S SU*S 5 T"T#XS -SS SS5 UU-UUS -'UUU'M OJ OGT(a4T"S :a T#S S T#S/-m#T S :aT$V s/sHoS S U S/-PM sn m$UTSS4-nT(a[%U5T#T$4$[%U5$s snfs sn fs snfs snfs sn fs sn fs sn f)z Return the tuple of abelian invariants for a matrix `m` (as in the Smith-Normal form). If `full=True` then invertible matrices ``s, t`` such that the product ``s, m, t`` is the Smith Normal Form are also returned. zBThe matrix entries must be over a principal ideal domain, but got c >[U5VVs/sH&n[U5Vs/sH o"U:XaTOTPM snPM( snn$s snfs snnfr")r)nrroners reye!_smith_normal_decomp..eyes>EJ1XNX%(;(QQD((;XNN;NsAA AArc[[US55H2nXUnX8-X@UU--XU'XX-X`UU--XU'M4 g)Nrr#r%s radd_rows&_smith_normal_decomp..add_rowss\s1Q4y!AQAcAd1gIoADGcAd1gIoADG"rc >T SSn[ST 5HnT UST:XaMT RT USU5up#UT:Xa)T "T SUSSU*S5 T (aT "TSUSSU*S5 MXMZT RUT US5upEnT RT USU5nT RX5nT "T SXXWU*5 T (a T "TSXXWU*5 UnM gNrr)rrgcdexexquo)pivotrr)rr&r'gd_0d_jr@rr/rr3r5rs r clear_column*_smith_normal_decomp..clear_columns!Qq$AtAw$::ad1gu-DADyAq!QA.Q1aQB2!,,uad1g6all1Q47A.ll5,AqQcT2Q1#6 rc >T SSn[ST 5HnT SUT:XaMT RT SUU5up#UT:Xa/[T SUSSU*S5 T (a[T SUSSU*S5 M^M`T RUT SU5upEnT R T SUU5nT R X5n[T SXXWU*5 T (a[T SXXWU*5 UnM grC)rrr,rDrE)rFrr)rr&r'rGrHrIr4rr/rr6rs r clear_row'_smith_normal_decomp..clear_rows!Qq$AtAw$::ad1gu-DADyAq!QA2q11aAr15!,,uad1g6all1Q47A.ll5,Aq!51aASD9 rc3:># UHnTSUT:gv M g7frNr>.0rrrs r '_smith_normal_decomp..6 1qtAw$ rc3:># UHnTUST:gv M g7frPr>rQs rrSrTrUrVcH>[U[U5[US54TS9$)Nr)rr)rr$)rrs rto_domain_matrix._smith_normal_decomp..to_domain_matrixs#Ac!fc!A$i%8HHrNr.)is_PID ValueErrorrr;ranycanonical_unitis_Fieldr0listmaprextendr$rrDgcdr,tuple)&rrrr/msgr<rJrMrindrrowrYr(elemrr lower_rightrets_smallt_smalls2t2resultr&r'xyr)alphabetar@r4r;r3r5r6rs&`` ` @@@@@@@rr0r0|s ==RSYRZ[oJD$ ;;D **CO Ez s4y#d)+ + I I&(*Dk 5kQqT!W_1kC 5 s1v~CF)QqT!aAi Ai1OAaD!CF)+9+Q1aDq+9 3q6T>&)a&k3q6#AF C*-!f+s1v'CFCAK 6a 6 6 6 6a 6 6 6  6a 6 6 6 6a 6 6 6I tAw!|  ! !!A$q' * ??AaDG A  ? aDGqLG-.qT2TTqT2!Ez&'e,eque ,";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!9fQqSkqAVZZ1%a(D0$ll1a0GAq! 1a(A 1a(+!::a+A.DQq5!Q151!eQ1a8Qq5!eVQ:1!eQD5!<Qq5!QA6%iqs q )&, axabEQqTFNax3451CWAx'151a " V}a""V}m 6 :03 - )G(FN6s6-T7T7T<0T<$UU U 1U3Ucp[R"X5up#nUS:waX-S:Xa SnUS:aSOSnX#U4$)a This supports the functions that compute Hermite Normal Form. Explanation =========== Let x, y be the coefficients returned by the extended Euclidean Algorithm, so that x*a + y*b = g. In the algorithms for computing HNF, it is critical that x, y not only satisfy the condition of being small in magnitude -- namely that |x| <= |b|/g, |y| <- |a|/g -- but also that y == 0 when a | b. rr[r)r rD)r&r'rqrrrGs r_gcdexrv"s?hhqnGA!Av!%1* a%BQ 7Nrc URR(d [S5eURupUR 5R 5nUn[ US- SS5HnUS:Xa OUS-n[ US- SS5HKnXUS:wdM[XUXU5upgnXUU-XUU-p[XXVXz*U 5 MM XUn U S:a[XUSSSS5 U *n U S:XaUS- nM[ US-U5HnXUU -n [XUSU *SS5 M M [R"UR55SS2US24$)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.rr[rN) ris_ZZrrto_ddmcopyrrvr,rfrom_rep to_dfm_or_ddm) Arr:r*rruvr)rr5r'qs r_hermite_normal_formr7sz8 88>><== 77DA  A A 1q5"b ! 6  Qq1ub"%AtAw!|!a!$q'2atAw!|QT!W\1A!2q1& DG q5 aQA .A 6 FA 1q5!_DGqLA!QAq1%?"H  !2 3AqrE ::rc URR(d [S5e[R"U5(aUS:a [S5eSn[ [ 5nURupEXT:a [S5eUR5nUnUn[US- SS5HnUS-n[US- SS5HIn XU S:wdM[XUXU 5upn XUU -XU U -pU"XXiXU*U 5 MK XUnUS:Xa U=XU'n[X5upn [U5HnXUU-U-UUU'M X8US:XaXsUU'[US-U5H#n X8U X8U-n[X9USU*SS5 M% X|-nM [X4U4[5R5$)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.) rxrz0Modulus D must be positive element of domain ZZ.c[[U55HLnXUn [XI-XPUU--U-U5XU'[Xi-XpUU--U-U5XU'MN gr")rr$r) rRrrr&r'r(r)r*r+s radd_columns_mod_R8_hermite_normal_form_modulo_D..add_columns_mod_Rsqs1vAQA'qT!W)<(A1EADG'qT!W)<(A1EADGrz2Matrix must have at least as many columns as rows.r[r)rryrr of_typerdictrrrrrvr,rr2)r~DrWrr:r*rrrrrr)rr5r'iirs r_hermite_normal_form_modulo_DrsZ 88>><== ::a==AENOOF DA 77DAuOPP A A A 1q5"b ! Qq1ub"%AtAw!| a!$q'2atAw!|QT!W\1!!aQB: & DG 6OADGa,a(B2qzA~AbE!H 47a<aDGq1uaAQ147"A aQB1 -! %"& q62 & / / 11rNF)r 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.) rxr) rryr convert_tor rankrrr)r~rrs rhermite_normal_formrsZv 88>><==}jALL,<,A,A,Cqwwqz,Q,Q22#A&&r)__doc__ collectionsr domainmatrixr exceptionsrrsympy.ntheory.modularrsympy.polys.domainsr r rr r,r r7r0rvrrrr>rrrsX2#&33&.:"D$8cL*J;ZU2p!%@'r