OtjnddlmZddlmZddlmZddlmZddlm Z ddl m Z ddl m Z mZmZmZdd lmZd Zd Zd Zd ZdS)) defaultdictAdd)Mul)S)construct_domain)PolyNonlinearError)SDM sdm_irrefsdm_particular_from_rrefsdm_nullspace_from_rref) filldedentct|}t||\}}t|||}|j}|js|jr>|d}t|\}}} |r|d|krdSt||dz|} t||j ||| \} } tt} | D]9\}}| ||||:t%| | D]^\}}||}|D]<\}}| |||||z=_d| D} t&j}t+|t+| z D]}|| |<| S)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 c(i|]\}}|t|Sr).0stermss [/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/polys/matrices/linsolve.py z_linsolve..ms" 6 6 6ha1c5k 6 6 6)len_linear_eq_to_dictsympy_dict_to_dmdomain is_RealFieldis_ComplexFieldto_ddmrrefto_sdmr r ronerlistitemsappendto_sympyziprZeroset)eqssymsnsymseqsdictconstAaugKArrefpivotsnzcolsPV nonpivotssolivnpiVisymzerors r _linsolver?0s0 IIE(T22NGU GUD 1 1D A  ~0*0{{}}!!##A&--//&dOOE66&*%%t !a88A+5!%OOLAy d  C ++1 DG AJJqMM****y!$$55R3iHHJJ 5 5DAq QL  ajjmm 3 4 4 4 4 5 7 6#))++ 6 6 6C 6D YYS !A JrcB t|jd|D}t|dd\}}tt || t |}t |}tt |t |g}t ||D]K\} } fd| D} | r |  | |<| r|| Ltt|||dzf|} | S)z?Convert a system of dict equations to a sparse augmented matrixc3>K|]}|VdS)N)values)res r z#sympy_dict_to_dm..zs* @ @ @ @ @ @ @ @rT)field extensionc4i|]\}}||Srr)rrcelem_map sym2indexs rrz$sympy_dict_to_dm..s'CCC1)A, CCCrr ) r*unionrdictr(rranger%r&r enumerate) eqs_coeffseqs_rhsr,elemsr1elems_Kneqsr-r.eqrhseqdictsdm_augrIrJs @@rrrxs) CLL  @ @Z @ @ @ AE!%ttDDDJAwCw''((H z??D IIESuU||,,--IGz7++##CCCCCC CCC  +%c]NF5M  # NN6 " " ")G$$tUQY&7;;G Nrcg}g}t|}|D]}|jrt|j|\}}t|j|\}} ||z}| D] \} } | |vr|| xx| zcc<| || <!d|D}||} } nt||\} } || || ||fS)amConvert 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]) ci|] \}}||| Srr)rkr:s rrz&_linear_eq_to_dict..s#999daq9Q999r)r* is_Equality _lin_eq2dictlhsrUr%r&)r+r,coeffsindsymsetrCcoeffrcRtRrZr:rHds rrrsF C YYF  = +'v66LE5!!%00FB RKE  " "1::!HHHMHHHH !rE!HH99ekkmm999E%qAA6**DAq a 1 3;rcD ||vrtj|tjifS|jrt t }g}|jD]_}t||\}}||| D] \}}|||!`t| d| D} | fS|j rdx} } g}|jD]R}t||\}}|s||-| |} |} 4ttd|ztj| |  ifS fd| D} | z| fS||s|ifStd|z)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}) c(i|]\}}|t|Srr)rr=r^s rrz _lin_eq2dict..s"III{sFc6lIIIrNz- nonlinear cross-term: %sc"i|] \}}||z Srr)rr=rHras rrz _lin_eq2dict..s#@@@QS%!)@@@rznonlinear term: %s)rr)Oneis_Addrr$argsr\r&r%ris_Mulr rr _from_args has_xfree) ar` terms_list coeff_listaicitimijcijr terms_coeffras @rr\r\s  F{{v15z!! #; &&  & , ,B!"f--FB   b ! ! !HHJJ , ,S3&&s++++ ,Z IIj6F6F6H6HIIIe| ;""  & 6 6B!"f--FB 6!!"%%%% )502354*5*5666z** ="9 @@@@%++--@@@EK'. . [[ ;"u  !5!9:::rN) collectionsrsympy.core.addrsympy.core.mulrsympy.core.singletonrsympy.polys.constructorrsympy.polys.solversr sdmr r r rsympy.utilities.miscrr?rrr\rrrrs:$#####""""""444444222222,+++++EEEP&###L5;5;5;5;5;r