\hnSSKJr SSKJr SSKJr SSKJr SSKJ r SSK J r SSK J r JrJrJr SS KJr S rS rS rS rg)) defaultdict)Add)Mul)S)construct_domain)PolyNonlinearError)SDM sdm_irrefsdm_particular_from_rrefsdm_nullspace_from_rref) filldedentc[U5n[X5up4[X4U5nURnUR(dUR (a/UR 5R5SR5n[U5upxn U(a USU:Xag[XrS-U5n [XvRX(U 5up[[5n U R5H*upXUR!UR#U55 M, [%X5HLunnUUnUR5H-upXUR!UUR#U5-5 M/ MN U R5VVs0sHunnU['U6_M n nn[(R*n[-U5[-U 5- HnUU U'M U $s snnf)aSolve a linear system of equations. Examples ======== Solve a linear system with a unique solution: >>> from sympy import symbols, Eq >>> from sympy.polys.matrices.linsolve import _linsolve >>> x, y = symbols('x, y') >>> eqs = [Eq(x + y, 1), Eq(x - y, 2)] >>> _linsolve(eqs, [x, y]) {x: 3/2, y: -1/2} In the case of underdetermined systems the solution will be expressed in terms of the unknown symbols that are unconstrained: >>> _linsolve([Eq(x + y, 0)], [x, y]) {x: -y, y: y} rNr )len_linear_eq_to_dictsympy_dict_to_dmdomain is_RealFieldis_ComplexFieldto_ddmrrefto_sdmr r r onerlistitemsappendto_sympyziprrZeroset)eqssymsnsymseqsdictconstAaugKArrefpivotsnzcolsPV nonpivotssolivnpiVisymstermszeros U/var/www/auris/envauris/lib/python3.13/site-packages/sympy/polys/matrices/linsolve.py _linsolver90s0 IE(2NG GD 1D A  ~~**{{}!!#A&--/&dOE6&*% !a8A+5%%OLA d C  G AJJqM*y$R3iHHJDA QL  ajjm 3 4% +.))+ 6+ha1c5k>+C 6 66D YS !A" J 7sGc[U5R"SU56n[USSS9upE[[ X555n[ U5n[ U5n[[ U[ U555n /n [ X5HVupU R5V Vs0sH upXXn_M nn nU (aXl*X'U(dMEU RU5 MX [[U 5XxS-4U5nU$s snn f)z?Convert a system of dict equations to a sparse augmented matrixc3@# UHoR5v M g7f)N)values).0es r8 #sympy_dict_to_dm..zs @ZZsT)field extensionr ) r!unionrdictrrrangerrr enumerate) eqs_coeffseqs_rhsr#elemsr(elems_Kelem_mapneqsr$ sym2indexr%eqrhsr5ceqdictsdm_augs r8rrxs L   @Z @ AE!%ttDJAC'(H z?D IESuU|,-IGz+8: C ), + C %]NFM 6 NN6 " , )G$tQY&7;G N DsC8c/n/n[U5nUHnUR(a[URU5upg[URU5upXh-nU R 5HupX;aXz==U -ss'MU *Xz'M UR 5V V s0sHupU (dMX_M nn n XgpO [XT5upUR U 5 UR U 5 M X#4$s sn n f)aIConvert a system Expr/Eq equations into dict form, returning the coefficient dictionaries and a list of syms-independent terms from each expression in ``eqs```. Examples ======== >>> from sympy.polys.matrices.linsolve import _linear_eq_to_dict >>> from sympy.abc import x >>> _linear_eq_to_dict([2*x + 3], {x}) ([{x: 2}], [3]) )r! is_Equality _lin_eq2dictlhsrOrr)r"r#coeffsindsymsetr>coeffr6cRtRkr1rPds r8rrsF C YF  =='v6LE!!%%0FB KE :HMH !rEH # ',kkm9mdaqTQTmE9q*DA a 1 %& ; :s  C1/C1cX;a"[RU[R04$UR(a[ [ 5n/nUR HMn[XA5upVURU5 UR5HupxX'RU5 M MO [U6n UR5V V s0sHupU [U 6_M n n n X4$UR(aS=p/nUR HHn[XA5upVU(dURU5 M*U cUn Un M3[[SU-55e [R"U5n U cU 04$U R5V Vs0sH upXU-_M n n nX-U 4$UR!U5(dU04$[SU-5es sn n fs snn f)areturn (c, d) where c is the sym-independent part of ``a`` and ``d`` is an efficiently calculated dictionary mapping symbols to their coefficients. A PolyNonlinearError is raised if non-linearity is detected. The values in the dictionary will be non-zero. Examples ======== >>> from sympy.polys.matrices.linsolve import _lin_eq2dict >>> from sympy.abc import x, y >>> _lin_eq2dict(x + 2*y + 3, {x, y}) (3, {x: 1, y: 2}) Nz- nonlinear cross-term: %sznonlinear term: %s)rr Oneis_AddrrargsrUrrris_Mulrrr _from_args has_xfree)arY terms_list coeff_listaicitimijcijrZr4rWr6 terms_coeffrPs r8rUrUs  {vv155z!!  &  &&B!"-FB   b !HHJ&&s+' Z 6@6F6F6HI6H{sc6l"6HI| "" &&B!"-FB!!"% )502354*566z* ="9 27++-@-S!)^-E@'. . [[ "u  !5!9::5J*As F=;GN) collectionsrsympy.core.addrsympy.core.mulrsympy.core.singletonrsympy.polys.constructorrsympy.polys.solversrsdmr r r r sympy.utilities.miscrr9rrrUr8rys?:$"42,EP&#L5;rx