+ in^RIHt^RIHt^RIHt^RIHt^RIH t ^RI H t ^RI H t HtHtHt^RIHtR tR tR tR tR #)) defaultdict)Add)Mul)S)construct_domain)PolyNonlinearError)SDM sdm_irrefsdm_particular_from_rrefsdm_nullspace_from_rref) filldedentc  \V4p\W4wr4\W4V4pVPpVP'gVP 'd4VP 4P4^,P4p\V4wrxp V'dVR,V8XdR#\Wr^,V4p \WvPW(V 4wr\\4p V P4F2wrWV,,P!VP#V44K4 \%W4F\wppVV,pVP4F9wrWV,,P!VVP#V4,4K; K^ V P4UUu/uFwppV\'V!bK p pp\(P*p\-V4\-V 4, FpVV V&K V #uuppi)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} N)len_linear_eq_to_dictsympy_dict_to_dmdomain is_RealFieldis_ComplexFieldto_ddmrrefto_sdmr r r onerlistitemsappendto_sympyziprrZeroset)eqssymsnsymseqsdictconstAaugKArrefpivotsnzcolsPV nonpivotssolivnpiVisymstermszeros&& }/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/polys/matrices/linsolve.py _linsolver80s0 IE(2NG GD 1D A  ~~~***{{}!!#A&--/&dOE6&*% !a8A+5%%OLA d C  G AJJqM*y$R3iHHJDA QL  ajjm 3 4% +.))+ 6+ha1c5k>+C 6 66D YS ! !A" J 7s-G?c&\V4P!RV4!p\VRRR7wrE\\ W544p\ V4p\ V4p\\ V\ V444p .p \ W4FcwrV P4U Uu/uFwrW,Wn,bK pp pV 'd Wl,)W&V'gKRV PV4Ke \\V 4Wx^,3V4pV#uupp i)z?Convert a system of dict equations to a sparse augmented matrixc3@"TFqP4xK R#5i)N)values).0es& r7 #sympy_dict_to_dm..zs @ZZsT)field extension) r unionrdictrrrangerrr enumerate) eqs_coeffseqs_rhsr"elemsr'elems_Kelem_mapneqsr# sym2indexr$eqrhsr4ceqdictsdm_augs&&& r7rrxs L   @Z @ AE!%ttDJAC'(H z?D IESuU|,-IGz+8: C ), + C %]NFM 6 NN6 " , )G$tQY&7;G N DsD c .p.p\V4pVFpVP'd\VPV4wrg\VPV4wrWh,pV P 4F&wrW9dWz;;,V ,uu&K!V )Wz&K( VP 4U U u/uFwrV 'gKWbK pp p YgrM \WT4wrVP V 4VP V 4K W#3#uup p i)aIConvert a system Expr/Eq equations into dict form, returning the coefficient dictionaries and a list of syms-independent terms from each expression in ``eqs```. Examples ======== >>> from sympy.polys.matrices.linsolve import _linear_eq_to_dict >>> from sympy.abc import x >>> _linear_eq_to_dict([2*x + 3], {x}) ([{x: 2}], [3]) )r is_Equality _lin_eq2dictlhsrNrr)r!r"coeffsindsymsetr=coeffr5cRtRkr0rOds&& r7rrsF C YF  ==='v6LE!!%%0FB KE :HMH !rEH # ',kkm9mdaqTQTmE9q*DA a 1 %& ; :s , C?=C?c W9d#\PV\P/3#VP'd\ \ 4p.pVP FQp\WA4wrVVPV4VP4FwrxW',PV4K KS \V!p VP4U U u/uFwrV \V !bK p p p W3#VP'dR;r.pVP FNp\WA4wrVV'gVPV4K+V fTp Tp K5\\RV,44h \P!V4p V fV /3#V P4U Uu/uFwrWV,bK p p pW,V 3#VP!V4'gV/3#\RV,4huup p iuupp i)areturn (c, d) where c is the sym-independent part of ``a`` and ``d`` is an efficiently calculated dictionary mapping symbols to their coefficients. A PolyNonlinearError is raised if non-linearity is detected. The values in the dictionary will be non-zero. Examples ======== >>> from sympy.polys.matrices.linsolve import _lin_eq2dict >>> from sympy.abc import x, y >>> _lin_eq2dict(x + 2*y + 3, {x, y}) (3, {x: 1, y: 2}) Nz- nonlinear cross-term: %sznonlinear term: %s)rrOneis_AddrrargsrTrrris_Mulrr r _from_args has_xfree)arX terms_list coeff_listaicitimijcijrYr3rVr5 terms_coeffrOs&& r7rTrTs  {vv155z!!  &  &&B!"-FB   b !HHJ&&s+' Z 6@6F6F6HI6H{sc6l"6HI| "" &&B!"-FB!!"% )502354*566z* ="9 27++-@-S!)^-E@'. . [[ "u  !5!9::5J*As G GN) collectionsrsympy.core.addrsympy.core.mulrsympy.core.singletonrsympy.polys.constructorrsympy.polys.solversrsdmr r r r sympy.utilities.miscr r8rrrTr7rxs?:$"42,EP&#L5;rw