U Zʹjn@sddlmZddlmZddlmZddlmZddlm Z ddl m Z ddl m Z mZmZmZdd lmZd d Zd d ZddZddZdS)) defaultdictAdd)Mul)S)construct_domain)PolyNonlinearError)SDM sdm_irrefsdm_particular_from_rrefsdm_nullspace_from_rref) filldedentcCsHt|}t||\}}t|||}|j}|js4|jrH|d}t |\}}} |rj|d|krjdSt ||d|} t ||j ||| \} } t t} | D] \}}| ||||qt| | D]>\}}||}|D]$\}}| |||||qqdd| D} tj}t|t| D]}|| |<q4| S)aSolve a linear system of equations. Examples ======== Solve a linear system with a unique solution: >>> from sympy import symbols, Eq >>> from sympy.polys.matrices.linsolve import _linsolve >>> x, y = symbols('x, y') >>> eqs = [Eq(x + y, 1), Eq(x - y, 2)] >>> _linsolve(eqs, [x, y]) {x: 3/2, y: -1/2} In the case of underdetermined systems the solution will be expressed in terms of the unknown symbols that are unconstrained: >>> _linsolve([Eq(x + y, 0)], [x, y]) {x: -y, y: y} rNr cSsi|]\}}|t|qSr).0stermsrr\/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/polys/matrices/linsolve.py msz_linsolve..)len_linear_eq_to_dictsympy_dict_to_dmdomainZ is_RealFieldZis_ComplexFieldZto_ddmZrrefZto_sdmr r r onerlistitemsappendZto_sympyziprZeroset)eqssymsnsymseqsdictconstZAaugKZArrefZpivotsZnzcolsPVZ nonpivotsZsolivZnpiZVisymzerorrrr _linsolve0s.    r-c st|jdd|D}t|ddd\}}tt||t|}t|}tt|t|g}t||D]@\} } fdd| D} | r|  | |<| rh|| qht t |||df|} | S)z?Convert a system of dict equations to a sparse augmented matrixcss|]}|VqdS)N)values)rerrr zsz#sympy_dict_to_dm..T)field extensioncsi|]\}}||qSrr)rrcZelem_mapZ sym2indexrrrsz$sympy_dict_to_dm..r ) r unionrdictrrrangerrr enumerate) Z eqs_coeffsZeqs_rhsr"Zelemsr&Zelems_KZneqsr#r$eqrhsZeqdictZsdm_augrr4rrxs rcCsg}g}t|}|D]}|jrt|j|\}}t|j|\}} ||8}| D],\} } | |krp|| | 8<qN| || <qNdd|D}||} } nt||\} } || || q||fS)amConvert a system Expr/Eq equations into dict form, returning the coefficient dictionaries and a list of syms-independent terms from each expression in ``eqs```. Examples ======== >>> from sympy.polys.matrices.linsolve import _linear_eq_to_dict >>> from sympy.abc import x >>> _linear_eq_to_dict([2*x + 3], {x}) ([{x: 2}], [3]) cSsi|]\}}|r||qSrr)rkr*rrrrsz&_linear_eq_to_dict..)r Z is_Equality _lin_eq2dictlhsr:rr)r!r"coeffsindsymsetr/coeffrZcRZtRr;r*r3drrrrs$     rc sT||krtj|tjifS|jrtt}g}|jD]<}t||\}}||| D]\}}|||qTq0t |dd| D} | fS|j r0d} } g}|jD]D}t||\}}|s||q| dkr|} |} qt t d|qt|| dkr ifSfdd| D} | | fSn ||sD|ifSt d|dS)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}) cSsi|]\}}|t|qSrr)rr+r>rrrrsz _lin_eq2dict..Nz- nonlinear cross-term: %scsi|]\}}||qSrr)rr+r3rArrrsznonlinear term: %s)rrZOneZis_Addrrargsr<rrrZis_MulrrrZ _from_argsZ has_xfree) ar@Z terms_listZ coeff_listZaicitiZmijZcijrZ terms_coeffrrCrr<sD        r<N) collectionsrZsympy.core.addrZsympy.core.mulrZsympy.core.singletonrZsympy.polys.constructorrZsympy.polys.solversrZsdmr r r r Zsympy.utilities.miscrr-rrr<rrrrs       H&