\hˀSSKJr SSKJr SSKJr SSKJrJr SSK J r J r J r J r JrJrJr \S:waS/r\"S5rS /r\"S/S 9"S S 55rSS KJr SS KJr g)) GROUND_TYPES) import_module)doctest_depends_on)ZZQQ)DMBadInputError DMDomainErrorDMNonSquareMatrixErrorDMNonInvertibleMatrixError DMRankError DMShapeError DMValueErrorflint*DFM ground_typesc\rSrSrSrSrSrSrSr\ S5r Sr \ S 5r \ S 5r \ S 5r\S 5rS rSrSr\ S5rSrSrSrSrSrSr\ S5r\ S5rSrSr\ S5rSr \ S5r!Sr"\ S5r#S r$S!r%S"r&S#r'S$r(S%r)S&r*S'r+S(r,S)r-S*r.S+r/S,r0S-r1S.r2S/r3\ S05r4\ S15r5\ S25r6\ S35r7S4r8S5r9S6r:S7r;S8rS;r?S<r@S=rAS>rB\C"S?S@9SA5rD\C"S?S@9SB5rE\C"S?S@9SC5rFSDrGSErH\C"S?S@9SF5rISGrJSHrKSPSJjrLSKrMSQSLjrN\C"S?S@9SRSMj5rO\C"S?S@9SRSNj5rPSOrQgI)SrEa Dense FLINT matrix. This class is a wrapper for matrices from python-flint. >>> from sympy.polys.domains import ZZ >>> from sympy.polys.matrices.dfm import DFM >>> dfm = DFM([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ) >>> dfm [[1, 2], [3, 4]] >>> dfm.rep [1, 2] [3, 4] >>> type(dfm.rep) # doctest: +SKIP Usually, the DFM class is not instantiated directly, but is created as the internal representation of :class:`~.DomainMatrix`. When `SYMPY_GROUND_TYPES` is set to `flint` and `python-flint` is installed, the :class:`DFM` class is used automatically as the internal representation of :class:`~.DomainMatrix` in dense format if the domain is supported by python-flint. >>> from sympy.polys.matrices.domainmatrix import DM >>> dM = DM([[1, 2], [3, 4]], ZZ) >>> dM.rep [[1, 2], [3, 4]] A :class:`~.DomainMatrix` can be converted to :class:`DFM` by calling the :meth:`to_dfm` method: >>> dM.to_dfm() [[1, 2], [3, 4]] denseTFcURU5nSU;a U"U5nOU"U6nUR XRU5$![[4a [SU35ef=f)Construct from a nested list.rz"Input should be a list of list of )_get_flint_func ValueError TypeErrorr _new)clsrowslistshapedomain flint_matreps Q/var/www/auris/envauris/lib/python3.13/site-packages/sympy/polys/matrices/_dfm.py__new__ DFM.__new__msq''/ E> U)U#CxxF++  * U%(J6(&STT Us 9AcURXU5 [RU5nXlU=UluUlUlX4lU$)z)Internal constructor from a flint matrix.)_checkobjectr%r#r rowscolsr!)rr#r r!objs r$rDFM._new{sD 3v&nnS!).. &CHch  cNURXRUR5$)z>Create a new DFM with the same shape and domain but a new rep.)rr r!)selfr#s r$_new_rep DFM._new_repsyyjj$++66r.c$UR5UR54nXB:wa [S5eU[:Xa*[ U[ R 5(d [S5eU[:Xa*[ U[ R5(d [S5eUR(a:[ U[ R[ R45(d [S5eU[[4;aUR(d [S5egg)Nz(Shape of rep does not match shape of DFMzRep is not a flint.fmpz_matzRep is not a flint.fmpq_matz1Rep is not a flint.fmpz_mod_mat or flint.nmod_mat#Only ZZ and QQ are supported by DFM)nrowsncolsr r isinstancerfmpz_mat RuntimeErrorrfmpq_matis_FF fmpz_mod_matnmod_matNotImplementedError)rr#r r!repshapes r$r( DFM._checksIIK-  !"LM M R< 3 ? ?<= = r\*S%.."A"A<= = \\*S53E3Eu~~2V"W"WRS S B8 #FLL%&KL L-9 #r.clU[[4;=(d UR=(a UR$)z4Return True if the given domain is supported by DFM.)rrr; _is_flint)rr!s r$_supports_domainDFM._supports_domains'"b!FV\\%Ff6F6FFr.c^^^U[:Xa[R$U[:Xa[R$UR (aoUR 5m[UR[R5(a[RmUU4SjnU$[RT5mU4SjnU$[S5e)z3Return the flint matrix class for the given domain.c>[U5S:Xa-[US[R5(a T"US5$T"/UQTP76$)Nrr)lenr7rr=)e_clscs r$_func"DFM._get_flint_func.._funcs@1v{z!A$'G'G#AaDz)#{Q{{*r.c4>[R"/UQTP76$N)rr<)rHms r$%DFM._get_flint_func..s5#5#5#)rr!rKrIrJrOs @@@r$rDFM._get_flint_funcs R<>> ! r\>> ! \\%%'A&**ejj11~~+L&&q)<L%&KL Lr.c8URUR5$)z5Callable to create a flint matrix of the same domain.)rr!r0s r$rK DFM._funcs##DKK00r.c4[UR55$)zReturn ``str(self)``.)strto_ddmrXs r$__str__ DFM.__str__s4;;=!!r.c@S[UR55SS3$)zReturn ``repr(self)``.rN)reprr\rXs r$__repr__ DFM.__repr__s"T$++-(,-..r.c[U[5(d[$URUR:H=(a URUR:H$)zReturn ``self == other``.)r7rNotImplementedr!r#r0others r$__eq__ DFM.__eq__s<%%%! !{{ell*Dtxx599/DDr.cU"XU5$)r)rrr r!s r$ from_list DFM.from_lists8F++r.c6URR5$)zConvert to a nested list.)r#tolistrXs r$to_list DFM.to_listsxx  r.cVURURUR55$)zReturn a copy of self.)r1rKr#rXs r$copyDFM.copys}}TZZ122r.cv[R"UR5URUR5$)zConvert to a DDM.)DDMrlrpr r!rXs r$r\ DFM.to_ddm#}}T\\^TZZEEr.cv[R"UR5URUR5$)zConvert to a SDM.)SDMrlrpr r!rXs r$to_sdm DFM.to_sdmrxr.cU$)z Return self.rkrXs r$to_dfm DFM.to_dfms r.cU$)a Convert to a :class:`DFM`. This :class:`DFM` method exists to parallel the :class:`~.DDM` and :class:`~.SDM` methods. For :class:`DFM` it will always return self. See Also ======== to_ddm to_sdm sympy.polys.matrices.domainmatrix.DomainMatrix.to_dfm_or_ddm rkrXs r$ to_dfm_or_ddmDFM.to_dfm_or_ddms  r.clURUR5URUR5$)zConvert from a DDM.)rlrpr r!)rddms r$from_ddm DFM.from_ddms%}}S[[]CIIszzBBr.cURU5nU"/UQUP76nU"XRU5$![a [SU35e[a [SU35ef=f)z Inverse of :meth:`to_list_flat`.z'Incorrect number of elements for shape zInput should be a list of )rrr r)relementsr r!funcr#s r$from_list_flatDFM.from_list_flats}""6* I((x(C 3v&&  U!$KE7"ST T I!$>vh"GH H Is &0Ac6URR5$)zConvert to a flat list.)r#entriesrXs r$ to_list_flatDFM.to_list_flatsxx!!r.c>UR5R5$)z$Convert to a flat list of non-zeros.)r\ to_flat_nzrXs r$rDFM.to_flat_nzs{{}''))r.cL[R"XU5R5$)zInverse of :meth:`to_flat_nz`.)rv from_flat_nzr~)rrdatar!s r$rDFM.from_flat_nz s 7>>@@r.c>UR5R5$)zConvert to a DOD.)r\to_dodrXs r$r DFM.to_dod{{}##%%r.cL[R"XU5R5$)zInverse of :meth:`to_dod`.)rvfrom_dodr~)rdodr r!s r$r DFM.from_dod||C/6688r.c>UR5R5$)zConvert to a DOK.)r\to_dokrXs r$r DFM.to_dokrr.cL[R"XU5R5$)zInverse of :math:`to_dod`.)rvfrom_dokr~)rdokr r!s r$r DFM.from_dokrr.c## URupURn[U5H,n[U5HnX4U4nU(dMX4U4v M M. g7f)z/Iterate over the non-zero values of the matrix.Nr r#ranger0rOnr#ijrepijs r$ iter_valuesDFM.iter_values"sPzzhhqA1XqD 5d)Os AAAc## URupURn[U5H+n[U5HnX4U4nU(dMXE4U4v M M- g7f)zBIterate over indices and values of nonzero elements of the matrix.Nrrs r$ iter_itemsDFM.iter_items,sQzzhhqA1XqD 565/)s AAAcXR:XaUR5$U[:XaNUR[:Xa:UR [ R UR5URU5$URU5(a-UR5RU5R5$[S5e)zConvert to a new domain.r4)r!rsrrrrr:r#r rCr\ convert_tor~r>)r0r!s r$rDFM.convert_to6s [[ 99;  r\dkkR/99U^^DHH5tzz6J J  " "6 * *;;=++F3::< <&&KL Lr.c URup4US:aX- nUS:aX$- nURX4$![a [SUSUSUR35ef=f)zGet the ``(i, j)``-th entry.rInvalid indices (, ) for Matrix of shape r r#r IndexError)r0rrrOrs r$getitem DFM.getitemDsvzz q5 FA q5 FA ]88AD> ! ]02aS8Ntzzl[\ \ ] 4)Ac URupEUS:aX- nUS:aX%- nX0RX4'g![a [SUSUSUR35ef=f)zSet the ``(i, j)``-th entry.rrrrNr)r0rrvaluerOrs r$setitem DFM.setitemRsszz q5 FA q5 FA ]"HHQTN ]02aS8Ntzzl[\ \ ]rc URnUVVs/sHoBVs/sHoSXE4PM snPM nnn[U5[U54nURXgUR5$s snfs snnf)z%Extract a submatrix with no checking.)r#rGrlr!)r0 i_indices j_indicesMrrlolr s r$_extract DFM._extract`sc HH5>?Y+A!$+Y?YY0~~c$++66,?s A+A& A+&A+cURup4/n/nUHKnUS:aXs-nOUnSUs=::aU:dO [SUSUR35eURU5 MM UHKn U S:aX-n OU n SU s=::aU:dO [SU SUR35eURU 5 MM URXV5$)zExtract a submatrix.rzInvalid row index z for Matrix of shape zInvalid column index )r rappendr) r0rcolslistrOrnew_rowsnew_colsri_posrj_poss r$extract DFM.extracths zzA1u>> #5aS8Mdjj\!Z[[ OOE "A1u>> #8;PQUQ[Q[P\!]^^ OOE "}}X00r.cxURup4[U5Un[U5UnURXV5$)z Slice a DFM.)r rr)r0rowslicecolslicerOrrrs r$ extract_sliceDFM.extract_slices:zz!HX& !HX& }}Y22r.c:URUR*5$zNegate a DFM matrix.r1r#rXs r$negDFM.negs}}dhhY''r.cRURURUR-5$)zAdd two DFM matrices.rrfs r$addDFM.add}}TXX 122r.cRURURUR- 5$)zSubtract two DFM matrices.rrfs r$subDFM.subrr.c>URURU-5$)z1Multiply a DFM matrix from the right by a scalar.rrfs r$mulDFM.muls}}TXX-..r.c<URXR-5$)z0Multiply a DFM matrix from the left by a scalar.rrfs r$rmulDFM.rmuls}}UXX-..r.cxUR5RUR55R5$)z/Elementwise multiplication of two DFM matrices.)r\mul_elementwiser~rfs r$rDFM.mul_elementwises*{{},,U\\^<CCEEr.cURUR4nURURUR-X R5$)zMultiply two DFM matrices.)r*r+rr#r!)r0rgr s r$matmul DFM.matmuls6EJJ'yyEII-ukkBBr.c"UR5$r)rrXs r$__neg__ DFM.__neg__sxxzr.cNURU5nURU"U6X5$)zReturn a zero DFM matrix.)rr)rr r!rs r$zeros DFM.zeross)""6*xxe e44r.cJ[R"X5R5$)zReturn a one DFM matrix.)rvonesr~)rr r!s r$rDFM.onessxx&--//r.cJ[R"X5R5$)z%Return the identity matrix of size n.)rveyer~)rrr!s r$rDFM.eyeswwq!((**r.cJ[R"X5R5$)zReturn a diagonal matrix.)rvdiagr~)rrr!s r$rDFM.diagsxx)0022r.c\UR5RX5R5$)z/Apply a function to each entry of a DFM matrix.)r\ applyfuncr~)r0rr!s r$r DFM.applyfuncs"{{}&&t4;;==r.cURURR5URUR4UR 5$)zTranspose a DFM matrix.)rr# transposer+r*r!rXs r$r DFM.transposes3yy++- 499/Et{{SSr.cUR5R"UVs/sHo"R5PM sn6R5$s snf)zHorizontally stack matrices.)r\hstackr~r0othersos r$r DFM.hstack8{{}##&%A&Qhhj&%ABIIKK%AA cUR5R"UVs/sHo"R5PM sn6R5$s snf)zVertically stack matrices.)r\vstackr~rs r$r DFM.vstackr r cURnURup#[[X#55Vs/sHoAXD4PM sn$s snf)z$Return the diagonal of a DFM matrix.)r#r rmin)r0rrOrrs r$diagonal DFM.diagonals= HHzz!&s1y!12!1A!$!1222sAcURn[UR5H6n[[X R55HnXU4(dM g M8 g)z2Return ``True`` if the matrix is upper triangular.FT)r#rr*rr+r0rrrs r$is_upper DFM.is_uppersI HHtyy!A3q)),-T77 ."r.cURn[UR5H1n[US-UR5HnXU4(dM g M3 g)z2Return ``True`` if the matrix is lower triangular.rFTr#rr*r+rs r$is_lower DFM.is_lowersJ HHtyy!A1q5$)),T77 -"r.cPUR5=(a UR5$)z*Return ``True`` if the matrix is diagonal.)rrrXs r$ is_diagonalDFM.is_diagonals}}24==?2r.cURn[UR5H-n[UR5HnXU4(dM g M/ g)z1Return ``True`` if the matrix is the zero matrix.FTrrs r$is_zero_matrixDFM.is_zero_matrixsD HHtyy!A499%T77 &"r.c>UR5R5$)z5Return the number of non-zero elements in the matrix.)r\nnzrXs r$r$DFM.nnz{{}  ""r.c>UR5R5$)z7Return the strongly connected components of the matrix.)r\sccrXs r$r(DFM.sccr&r.rrc6URR5$)a Compute the determinant of the matrix using FLINT. Examples ======== >>> from sympy import Matrix >>> M = Matrix([[1, 2], [3, 4]]) >>> dfm = M.to_DM().to_dfm() >>> dfm [[1, 2], [3, 4]] >>> dfm.det() -2 Notes ===== Calls the ``.det()`` method of the underlying FLINT matrix. For :ref:`ZZ` or :ref:`QQ` this calls ``fmpz_mat_det`` or ``fmpq_mat_det`` respectively. At the time of writing the implementation of ``fmpz_mat_det`` uses one of several algorithms depending on the size of the matrix and bit size of the entries. The algorithms used are: - Cofactor for very small (up to 4x4) matrices. - Bareiss for small (up to 25x25) matrices. - Modular algorithms for larger matrices (up to 60x60) or for larger matrices with large bit sizes. - Modular "accelerated" for larger matrices (60x60 upwards) if the bit size is smaller than the dimensions of the matrix. The implementation of ``fmpq_mat_det`` clears denominators from each row (not the whole matrix) and then calls ``fmpz_mat_det`` and divides by the product of the denominators. See Also ======== sympy.polys.matrices.domainmatrix.DomainMatrix.det Higher level interface to compute the determinant of a matrix. )r#detrXs r$r+DFM.det sbxx||~r.c^URR5R5SSS2$)a; Compute the characteristic polynomial of the matrix using FLINT. Examples ======== >>> from sympy import Matrix >>> M = Matrix([[1, 2], [3, 4]]) >>> dfm = M.to_DM().to_dfm() # need ground types = 'flint' >>> dfm [[1, 2], [3, 4]] >>> dfm.charpoly() [1, -5, -2] Notes ===== Calls the ``.charpoly()`` method of the underlying FLINT matrix. For :ref:`ZZ` or :ref:`QQ` this calls ``fmpz_mat_charpoly`` or ``fmpq_mat_charpoly`` respectively. At the time of writing the implementation of ``fmpq_mat_charpoly`` clears a denominator from the whole matrix and then calls ``fmpz_mat_charpoly``. The coefficients of the characteristic polynomial are then multiplied by powers of the denominator. The ``fmpz_mat_charpoly`` method uses a modular algorithm with CRT reconstruction. The modular algorithm uses ``nmod_mat_charpoly`` which uses Berkowitz for small matrices and non-prime moduli or otherwise the Danilevsky method. See Also ======== sympy.polys.matrices.domainmatrix.DomainMatrix.charpoly Higher level interface to compute the characteristic polynomial of a matrix. N)r#charpolycoeffsrXs r$r/ DFM.charpoly?s*Txx  "))+DbD11r.c^URnURup#X#:wa [S5eU[:Xa[ SU-5eU[ :XdUR (a*URURR55$[SU-5e![a [S5ef=f)a Compute the inverse of a matrix using FLINT. Examples ======== >>> from sympy import Matrix, QQ >>> M = Matrix([[1, 2], [3, 4]]) >>> dfm = M.to_DM().to_dfm().convert_to(QQ) >>> dfm [[1, 2], [3, 4]] >>> dfm.inv() [[-2, 1], [3/2, -1/2]] >>> dfm.matmul(dfm.inv()) [[1, 0], [0, 1]] Notes ===== Calls the ``.inv()`` method of the underlying FLINT matrix. For now this will raise an error if the domain is :ref:`ZZ` but will use the FLINT method for :ref:`QQ`. The FLINT methods for :ref:`ZZ` and :ref:`QQ` are ``fmpz_mat_inv`` and ``fmpq_mat_inv`` respectively. The ``fmpz_mat_inv`` method computes an inverse with denominator. This is implemented by calling ``fmpz_mat_solve`` (see notes in :meth:`lu_solve` about the algorithm). The ``fmpq_mat_inv`` method clears denominators from each row and then multiplies those into the rhs identity matrix before calling ``fmpz_mat_solve``. See Also ======== sympy.polys.matrices.domainmatrix.DomainMatrix.inv Higher level method for computing the inverse of a matrix. z!cannot invert a non-square matrixzfield expected, got %szmatrix is not invertiblez#DFM.inv() is not implemented for %s) r!r r rr rr;r1r#invZeroDivisionErrorr r>)r0KrOrs r$r3DFM.invksn KKzz 6()LM M 7 81 <= = "W M}}TXX\\^44 &&Ka&OP P % M01KLL Ms (BB,cUR5R5upnUR5UR5U4$)z*Return the LU decomposition of the matrix.)r\lur~)r0LUswapss r$r8DFM.lus3kkm&&( exxz188:u,,r.cUR5R5upUR5UR54$)z*Return the QR decomposition of the matrix.)r\qrr~)r0QRs r$r>DFM.qrs/{{}!xxz188:%%r.c URUR:Xd'[SUR<SUR<35eURR(d[SUR-5eURup#URupEX$:wa[ SU<SU<SU<SU<35eX54nX#:wa;UR 5R UR 55R5$URRUR5nURXvUR5$![a [S5ef=f)a Solve a matrix equation using FLINT. Examples ======== >>> from sympy import Matrix, QQ >>> M = Matrix([[1, 2], [3, 4]]) >>> dfm = M.to_DM().to_dfm().convert_to(QQ) >>> dfm [[1, 2], [3, 4]] >>> rhs = Matrix([1, 2]).to_DM().to_dfm().convert_to(QQ) >>> dfm.lu_solve(rhs) [[0], [1/2]] Notes ===== Calls the ``.solve()`` method of the underlying FLINT matrix. For now this will raise an error if the domain is :ref:`ZZ` but will use the FLINT method for :ref:`QQ`. The FLINT methods for :ref:`ZZ` and :ref:`QQ` are ``fmpz_mat_solve`` and ``fmpq_mat_solve`` respectively. The ``fmpq_mat_solve`` method uses one of two algorithms: - For small matrices (<25 rows) it clears denominators between the matrix and rhs and uses ``fmpz_mat_solve``. - For larger matrices it uses ``fmpq_mat_solve_dixon`` which is a modular approach with CRT reconstruction over :ref:`QQ`. The ``fmpz_mat_solve`` method uses one of four algorithms: - For very small (<= 3x3) matrices it uses a Cramer's rule. - For small (<= 15x15) matrices it uses a fraction-free LU solve. - Otherwise it uses either Dixon or another multimodular approach. See Also ======== sympy.polys.matrices.domainmatrix.DomainMatrix.lu_solve Higher level interface to solve a matrix equation. zDomains must match: z != zField expected, got %szMatrix size mismatch: z * z vs z Matrix det == 0; not invertible.) r!r is_Fieldr rr\lu_solver~r#solver4r r)r0rhsrOrrk sol_shapesols r$rD DFM.lu_solves \{{cjj($++szz Z[ [ {{## 84;; FG Gzzyy 6QPQSTVWXY YF 6;;=))#**,7>>@ @ Q((..)Cyy55! Q,-OP P Qs 5%D66E czUR[:Xa[URSS5nUbURR 5up#pEUR upgUR X&U4UR5UR X6U4UR5UR XFU4UR5UR XPR UR54$UR5R 5uppUR5nU R5nU R5nU R5nX#XE4$)z Fraction-free LU decomposition of DFM. Explanation =========== Uses `python-flint` if possible for a matrix of integers otherwise uses the DDM method. See Also ======== sympy.polys.matrices.ddm.DDM.fflu ffluN) r!rgetattrr#rLr rr\r~) r0rLPr9Dr:rOrddm_pddm_lddm_dddm_us r$rLDFM.fflus ;;" 488VT2D!XX]]_ azzIIaQ5IIaQ5IIaQ5IIaT[[9  &*[[]%7%7%9"e LLN LLN LLN LLNQzr.cfUR5R5upUR5U4$)/Return a basis for the nullspace of the matrix.)r\ nullspacer~)r0r nonpivotss r$rW DFM.nullspace5s+&002zz|Y&&r.NcdUR5RUS9up#UR5U4$)rV)pivots)r{nullspace_from_rrefr~)r0r[sdmrXs r$r\DFM.nullspace_from_rrefKs0::&:Izz|Y&&r.cZUR5R5R5$)z+Return a particular solution to the system.)r\ particularr~rXs r$r`DFM.particularQs {{}'')0022r.cSnU"U5nU"U5nSUs=:aS:d O [S5eURupxURR5U:wa [ S5eURR XX4US9$)zACall the fmpz_mat.lll() method but check rank to avoid segfaults.c[R"U5(a+[UR5[UR5- $[U5$rN)rof_typefloat numerator denominator)xs r$to_floatDFM._lll..to_float_s5zz!}}Q[[)E!--,@@@Qxr.g?rz delta must be between 0.25 and 1z-Matrix must have full row rank for Flint LLL.) transformdeltaetar#gram)rr r#rankr lll) r0rkrlrmr#rnrirOrs r$_lllDFM._lllUs{ smeaAB Bzz 88==?a MN Nxx||i#UY|ZZr.cUR[:wa[SUR-5eURUR:a [ S5eUR US9nURU5$)aCompute LLL-reduced basis using FLINT. See :meth:`lll_transform` for more information. Examples ======== >>> from sympy import Matrix >>> M = Matrix([[1, 2, 3], [4, 5, 6]]) >>> M.to_DM().to_dfm().lll() [[2, 1, 0], [-1, 1, 3]] See Also ======== sympy.polys.matrices.domainmatrix.DomainMatrix.lll Higher level interface to compute LLL-reduced basis. lll_transform Compute LLL-reduced basis and transform matrix. ZZ expected, got %s,Matrix must not have more rows than columns.)rl)r!rr r*r+rrqr1)r0rlr#s r$rpDFM.lllss`, ;;"  5 CD D YY "MN Niiei$}}S!!r.cTUR[:wa[SUR-5eURUR:a [ S5eUR SUS9up#URU5nURX0RUR4UR5nXE4$)aCompute LLL-reduced basis and transform using FLINT. Examples ======== >>> from sympy import Matrix >>> M = Matrix([[1, 2, 3], [4, 5, 6]]).to_DM().to_dfm() >>> M_lll, T = M.lll_transform() >>> M_lll [[2, 1, 0], [-1, 1, 3]] >>> T [[-2, 1], [3, -1]] >>> T.matmul(M) == M_lll True See Also ======== sympy.polys.matrices.domainmatrix.DomainMatrix.lll Higher level interface to compute LLL-reduced basis. lll Compute LLL-reduced basis without transform matrix. rtruT)rkrl) r!rr r*r+rrqr1r)r0rlr#TbasisT_dfms r$ lll_transformDFM.lll_transforms2 ;;"  5 CD D YY "MN NT7 c" !ii3T[[A|r.rkrN)FgGz?gRQ?zbasisapprox)g?)R__name__ __module__ __qualname____firstlineno____doc__fmtis_DFMis_DDMr% classmethodrr1r(rCrpropertyrKr]rbrhrlrprsr\r{r~rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr!r$r(rr+r/r3r8r>rDrLrWr\r`rqrpr{__static_attributes__rkr.r$rrEs D C F F ,7 M MGGMM,11"/E,,!3FF CC ' '"*AA&99&99$* M ] ]71<3(33//F C5500 ++ 33>TLL3 3##W-0.0dW-)2.)2VW-FQ.FQP- & W-H6.H6TB',' 3[<W-".":W- . r.)rv)rzN)sympy.external.gmpyrsympy.external.importtoolsrsympy.utilities.decoratorrsympy.polys.domainsrr exceptionsr r r r r rr__doctest_skip__r__all__rsympy.polys.matrices.ddmrvrzrkr.r$rsvT-48&7u g ''+l l ,l `)(r.