9\c@ sdZddlmZmZddlmZmZmZddlm Z ddl m Z ddl m Z mZmZmZmZddlmZddlmZmZmZmZdd lmZdd lmZdd lmZdd lm Z dd l!m"Z"ddl#m$Z$m%Z%m&Z&ddl'm(Z(ddl)m*Z*ddl+m,Z,dZ-dZ.dZ/e0dZ1dZ2dZ3e0ej4e5dZ6dZ7e5dZ8dZ9gdZ:dS(s<Tools for solving inequalities and systems of inequalities. i(tprint_functiontdivision(tSymboltDummytsympify(titerable(t factor_terms(t RelationaltEqtGetLttNe(tInterval(t FiniteSettUniontEmptySett Intersection(tImageSet(tS(t expand_mul(tAbs(tAnd(tPolytPolynomialErrortparallel_poly_from_expr(t_nsort(tsift(t filldedentcC s t|tstdn|jrt|jd|}|tjkrXtjgS|tj krqtj gSt d|n|j dt g}}|dkrx]|D](\}}t||}|j|qWn*|dkrFtj}x|tjdfgD]4\} }t|| tt}|j|| }q Wn|jdkrbd } nd} dt } } |d krd } n^|d krd} nI|d krd t} } n,|d krdt} } ntd |tjt} } xt|D]\}}|drd| | krK|jdt|| | | n| || } } } q| | kr| r|jdt|| t| |t} } q| | kr| r|jdt||qqW| | kr |jdttj| t| n|S(s@Solve a polynomial inequality with rational coefficients. Examples ======== >>> from sympy import Poly >>> from sympy.abc import x >>> from sympy.solvers.inequalities import solve_poly_inequality >>> solve_poly_inequality(Poly(x, x, domain='ZZ'), '==') [{0}] >>> solve_poly_inequality(Poly(x**2 - 1, x, domain='ZZ'), '!=') [Interval.open(-oo, -1), Interval.open(-1, 1), Interval.open(1, oo)] >>> solve_poly_inequality(Poly(x**2 - 1, x, domain='ZZ'), '==') [{-1}, {1}] See Also ======== solve_poly_inequalities s8For efficiency reasons, `poly` should be a Poly instanceis%could not determine truth value of %stmultiples==s!=iit>t=s<=s'%s' is not a valid relationiN(t isinstanceRt ValueErrort is_numberRtas_exprRttruetRealstfalseRtNotImplementedErrort real_rootstFalseR tappendtNegativeInfinitytInfinitytTruetLCtNonetreversedtinsert(tpolytreltttrealst intervalstroott_tintervaltlefttrighttsignteq_signtequalt right_opent multiplicity((s9lib/python2.7/site-packages/sympy/solvers/inequalities.pytsolve_poly_inequalitysh       #              cC s@ddlm}|g|D]}t|D] }|^q*qS(sSolve polynomial inequalities with rational coefficients. Examples ======== >>> from sympy.solvers.inequalities import solve_poly_inequalities >>> from sympy.polys import Poly >>> from sympy.abc import x >>> solve_poly_inequalities((( ... Poly(x**2 - 3), ">"), ( ... Poly(-x**2 + 1), ">"))) Union(Interval.open(-oo, -sqrt(3)), Interval.open(-1, 1), Interval.open(sqrt(3), oo)) i(R(tsympyRR@(tpolysRtpts((s9lib/python2.7/site-packages/sympy/solvers/inequalities.pytsolve_poly_inequalitiesrscC s^tj}xN|D]F}|s"qnttjtjg}x|D]\\}}}t|||}t|d}g} xM|D]E} x<|D]4} | j| } | tjk r| j| qqWqW| }g} xH|D]@} x|D]} | | 8} qW| tjk r| j| qqW| }|sAPqAqAWx|D]} |j| }q=WqW|S(saSolve a system of rational inequalities with rational coefficients. Examples ======== >>> from sympy.abc import x >>> from sympy import Poly >>> from sympy.solvers.inequalities import solve_rational_inequalities >>> solve_rational_inequalities([[ ... ((Poly(-x + 1), Poly(1, x)), '>='), ... ((Poly(-x + 1), Poly(1, x)), '<=')]]) {1} >>> solve_rational_inequalities([[ ... ((Poly(x), Poly(1, x)), '!='), ... ((Poly(-x + 1), Poly(1, x)), '>=')]]) Union(Interval.open(-oo, 0), Interval.Lopen(0, 1)) See Also ======== solve_poly_inequality s==( RRR R*R+R@t intersectR)tunion(teqstresultt_eqstglobal_intervalstnumertdenomR2tnumer_intervalstdenom_intervalsR5tnumer_intervaltglobal_intervalR8tdenom_interval((s9lib/python2.7/site-packages/sympy/solvers/inequalities.pytsolve_rational_inequalitiess6       cC st}g}|rtjntj}x|D]}g}x|D]}t|trb|\}} n3|jr|j|j|j }} n |d}} |tj krtj tj d} } } nD|tj krtj tj d} } } n|jj\} } y%t| | f|\\} } } Wn#tk rOttdnX| jjs| j| jt} } }n| jj} | jp| js| | }t|d| }|t||dtM}q>|j| | f| fq>W|r+|j|q+q+W|r|t|M}tg|D]C}|D]6\\}}}|j|r9||jfdf^q9q/g}||8}n| r|r|j }n|r|j!|}n|S(s$Reduce a system of rational inequalities with rational coefficients. Examples ======== >>> from sympy import Poly, Symbol >>> from sympy.solvers.inequalities import reduce_rational_inequalities >>> x = Symbol('x', real=True) >>> reduce_rational_inequalities([[x**2 <= 0]], x) Eq(x, 0) >>> reduce_rational_inequalities([[x + 2 > 0]], x) (-2 < x) & (x < oo) >>> reduce_rational_inequalities([[(x + 2, ">")]], x) (-2 < x) & (x < oo) >>> reduce_rational_inequalities([[x + 2]], x) Eq(x, -2) s==s only polynomials and rational functions are supported in this context. it relational("R,RR$RRttuplet is_Relationaltlhstrhstrel_opR#tZerotOneR%ttogethertas_numer_denomRRRtdomaintis_Exacttto_exactR(t get_exacttis_ZZtis_QQRtsolve_univariate_inequalityR)RSthastonetevalft as_relational(texprstgenRTtexactRHtsolutiont_exprsRJtexprR2RLRMtoptR^titntdR7texclude((s9lib/python2.7/site-packages/sympy/solvers/inequalities.pytreduce_rational_inequalitiessT    "  # S  c s|jtkr$ttdnfd|}idd6dd6}g}xf|D]^\}}||jkrt|d|}nt| d||}|j|g|q`Wt||S(sReduce an inequality with nested absolute values. Examples ======== >>> from sympy import Abs, Symbol >>> from sympy.solvers.inequalities import reduce_abs_inequality >>> x = Symbol('x', real=True) >>> reduce_abs_inequality(Abs(x - 5) - 3, '<', x) (2 < x) & (x < 8) >>> reduce_abs_inequality(Abs(x + 2)*3 - 13, '<', x) (-19/3 < x) & (x < 7/3) See Also ======== reduce_abs_inequalities sr can't solve inequalities with absolute values containing non-real variables. c  sg}|js|jr|j}x|jD]{}|}|sL|}q+g}xK|D]C\}}x4|D],\}}|j|||||fqlWqYW|}q+Wn|jr|j} | jstdn|j }x|D]#\}}|j|| |fqWnt |t r|jd}xm|D]S\}}|j||t |dgf|j| |t |dgfq@Wn|gfg}|S(Ns'Only Integer Powers are allowed on Abs.i(tis_Addtis_MultfunctargsR)tis_Powtexpt is_IntegerR tbaseRRR R ( RnRitoptargRmRxtcondst_exprt_condsRq(t_bottom_up_scan(s9lib/python2.7/site-packages/sympy/solvers/inequalities.pyR4s4   (    #+RRs>=s<=i(tis_realR(t TypeErrorRtkeysRR)Rt(RnR2RjRitmappingt inequalitiesR((Rs9lib/python2.7/site-packages/sympy/solvers/inequalities.pytreduce_abs_inequalitys' cC s/tg|D]\}}t|||^q S(sUReduce a system of inequalities with nested absolute values. Examples ======== >>> from sympy import Abs, Symbol >>> from sympy.abc import x >>> from sympy.solvers.inequalities import reduce_abs_inequalities >>> x = Symbol('x', real=True) >>> reduce_abs_inequalities([(Abs(3*x - 5) - 7, '<'), ... (Abs(x + 25) - 13, '>')], x) (-2/3 < x) & (x < 4) & (((-oo < x) & (x < -38)) | ((-12 < x) & (x < oo))) >>> reduce_abs_inequalities([(Abs(x - 4) + Abs(3*x - 5) - 7, '<')], x) (1/2 < x) & (x < 4) See Also ======== reduce_abs_inequality (RR(RiRjRnR2((s9lib/python2.7/site-packages/sympy/solvers/inequalities.pytreduce_abs_inequalitiesksc1 sa ddlm}ddlm}m}m}ddlm} ddlm } m } m } ddlm } }|}j tkrtj}|s|S|j|Sj dkrtddtyji|6Wqtk rttd qXnd}tjkr%|}n%tjkr@tj}n jj}||}|tjkrt|}j|d }|tjkr|}q|tjkrtj}qn|dk r|||}j}|d ks|d krCj|j d r|}qj|j!d stj}qnZ|d ks[|dkrj|j!d ry|}qj|j d stj}qn|j!|j }}||tj"krt#d |tt}qn|dkrJ |j$\}}yX|j%kr,t&|j%dkr,t'n| ||}|dkrSt'nWn?t't(fk rt(tdj)t*dnXt|fd}g}x0| D]}|j+| ||qW|s ||}ndjko&jdk}yt,|j-t.|j!|j }t.||t/|j0t#|j!|j |j!|k|j |k}t1d|Drt2|dtd } ntt3|d}!|!drt(ny5|!t} t&| dkr t/t4| } nWntk r:t(nXWnt(k r[t(dnXtj5}"|tjkrt}#t.}$y| ||}%t6|%t#sxp|%D]:}&|&|kr||&r|&j r|$t.|&7}$qqWn+|%j!|%j }'}(xt2|t.|(D]}&||'})|'|(kr||&}*t7|'|&}+|+|kr|+j r||+r|)r|*r|$t#|'|&7}$q|)r|$t#j8|'|&7}$q|*r|$t#j9|'|&7}$q|$t#j:|'|&7}$qn|&}'q%Wx|D]},|$t.|,8}$q WWn tk rItj5}$t}#nXt6|$trt'tdj)||fn|"j;|$}"ntjg}-}.|j!}'||'r|'j<r|.j=t.|'nx| D]}/|/}(|t7|'|(r!|.j=t#|'|(ttn|/|kr=|j>|/nJ|/|kre|j>|/||/}0n|}0|0r|.j=t.|/n|(}'qW|j }(||(r|(j<r|.j=t.|(n|t7|'|(r|.j=t#j:|'|(n|tjkr# |#r# |"j;|}qJ t?t@|.|"|j)|}n|sT |S|j|S(s0Solves a real univariate inequality. Parameters ========== expr : Relational The target inequality gen : Symbol The variable for which the inequality is solved relational : bool A Relational type output is expected or not domain : Set The domain over which the equation is solved continuous: bool True if expr is known to be continuous over the given domain (and so continuous_domain() doesn't need to be called on it) Raises ====== NotImplementedError The solution of the inequality cannot be determined due to limitation in `solvify`. Notes ===== Currently, we cannot solve all the inequalities due to limitations in `solvify`. Also, the solution returned for trigonometric inequalities are restricted in its periodic interval. See Also ======== solvify: solver returning solveset solutions with solve's output API Examples ======== >>> from sympy.solvers.inequalities import solve_univariate_inequality >>> from sympy import Symbol, sin, Interval, S >>> x = Symbol('x') >>> solve_univariate_inequality(x**2 >= 4, x) ((2 <= x) & (x < oo)) | ((x <= -2) & (-oo < x)) >>> solve_univariate_inequality(x**2 >= 4, x, relational=False) Union(Interval(-oo, -2), Interval(2, oo)) >>> domain = Interval(0, S.Infinity) >>> solve_univariate_inequality(x**2 >= 4, x, False, domain) Interval(2, oo) >>> solve_univariate_inequality(sin(x) > 0, x, relational=False) Interval.open(0, pi) i(tim(tcontinuous_domaint periodicitytfunction_range(tdenoms(t solveset_realtsolvifytsolveset(tsolveRjtreals When gen is real, the relational has a complex part which leads to an invalid comparison like I < 0. iRs<=Rs>=is The inequality, %s, cannot be solved using solve_univariate_inequality. txc sjt|}yj|d}Wntk rJtj}nX|tjtjfkrg|S|jtkr}tjS|j d}|j rj|dSt d|dS(Niis!relationship did not evaluate: %s( tsubsRRwRRR%R#RR(Rqt is_comparableR&(Rtvtr(t expanded_eRnRj(s9lib/python2.7/site-packages/sympy/solvers/inequalities.pytvalids    t=s!=cs s|]}|jVqdS(N(R!(t.0R((s9lib/python2.7/site-packages/sympy/solvers/inequalities.pys >st separatedcS s|jS(N(R(R((s9lib/python2.7/site-packages/sympy/solvers/inequalities.pytAts'sorting of these roots is not supporteds %s contains imaginary parts which cannot be made 0 for any value of %s satisfying the inequality, leading to relations like I < 0. N(ARARtsympy.calculus.utilRRRtsympy.solvers.solversRtsympy.solvers.solvesetRRRRRR(RRRhR.RR,txreplaceRRR#R%RWRXRZRRwRYtsuptinfR+R R]t free_symbolstlenR R&RRtextendtsettboundaryR tlistt intersectiontallRRtsortedR$Rt_pttRopentLopentopenRFt is_finiteR)tremoveRR(1RnRjRTR^t continuousRRRRRRRRRt_gent_domaintrvtetperiodtconsttfrangeR2RRRqRrtsolnsRt singularitiest include_xtdiscontinuitiestcritical_pointsR4tsiftedt make_realtchecktim_soltatztstarttendt valid_starttvalid_ztptRDtemptytsol_setsRt_valid((RRnRjs9lib/python2.7/site-packages/sympy/solvers/inequalities.pyRds&:           $   &    !        !   !      %       $cC sD|j r%|j r%||d}n|jrC|jrCtj}n|jr[|jdkss|jr|jdkrtdn|jr|js|jr|jr||}}n|jr|jr|d}q@|jr|tj}q@|d}nE|jr@|jr|tj}q@|jr3|d}q@|d}n|S(s$Return a point between start and endis,cannot proceed with unsigned infinite valuesiN(t is_infiniteRRZt is_positiveR.R t is_negativetHalf(RRR((s9lib/python2.7/site-packages/sympy/solvers/inequalities.pyRs.           cC sddlm}||jkr#|S|j|kr>|j}n|j|krc||jjkrc|Sd}d }tj}|j|j}ybt ||}|j dkr|j |j d}n"| r|j dkrt nWntt fk r|ssyt|gg|}Wn tk rDt||}nX||||} | tjkr||||tjkr|j||kt}n|||| } | tjkr|||| tjkr|j| |kt}|j|| kt}n|tjkr| tjkr9||kn ||k}| tjk rpt| |k|}qpqqt |}nXg} |d kr|j } d} | j|dt\}}| |8} | |8} t| }|j|dt\}} |jtks=|j|jko)d knrO|jd krO|} tj}n| |} |jrw|j | | }n|jj | | }||j||jB}||}x||D]t}tt|d|d |}t |tr|j|kr||||jtjkr3| j!|q3qqWx|| |fD]j}||||tjkrE||||tjk rE| j!||kr||kn ||kqEqEWn| j!|t| S( sReturn the inequality with s isolated on the left, if possible. If the relationship is non-linear, a solution involving And or Or may be returned. False or True are returned if the relationship is never True or always True, respectively. If `linear` is True (default is False) an `s`-dependent expression will be isolated on the left, if possible but it will not be solved for `s` unless the expression is linear in `s`. Furthermore, only "safe" operations which don't change the sense of the relationship are applied: no division by an unsigned value is attempted unless the relationship involves Eq or Ne and no division by a value not known to be nonzero is ever attempted. Examples ======== >>> from sympy import Eq, Symbol >>> from sympy.solvers.inequalities import _solve_inequality as f >>> from sympy.abc import x, y For linear expressions, the symbol can be isolated: >>> f(x - 2 < 0, x) x < 2 >>> f(-x - 6 < x, x) x > -3 Sometimes nonlinear relationships will be False >>> f(x**2 + 4 < 0, x) False Or they may involve more than one region of values: >>> f(x**2 - 4 < 0, x) (-2 < x) & (x < 2) To restrict the solution to a relational, set linear=True and only the x-dependent portion will be isolated on the left: >>> f(x**2 - 4 < 0, x, linear=True) x**2 < 4 Division of only nonzero quantities is allowed, so x cannot be isolated by dividing by y: >>> y.is_nonzero is None # it is unknown whether it is 0 or not True >>> f(x*y < 1, x) x*y < 1 And while an equality (or inequality) still holds after dividing by a non-zero quantity >>> nz = Symbol('nz', nonzero=True) >>> f(Eq(x*nz, 1), x) Eq(x, 1/nz) the sign must be known for other inequalities involving > or <: >>> f(x*nz <= 1, x) nz*x <= 1 >>> p = Symbol('p', positive=True) >>> f(x*p <= 1, x) x <= 1/p When there are denominators in the original expression that are removed by expansion, conditions for them will be returned as part of the result: >>> f(x < x*(2/x - 1), x) (x < 1) & Ne(x, 0) i(RcS s_yC|j||}|tjkr(|S|ttfkr>dS|SWntk rZtjSXdS(N(RRtNaNR,R(R(tieRDRpR((s9lib/python2.7/site-packages/sympy/solvers/inequalities.pytclassifys iitas_Adds!=s==tlinearN(s!=s==("RRRRXR/RWR.RR+RtdegreeRwR"R&RRtRdR#R%RR,Rtas_independentRR(tis_zeroRRRYR[t_solve_inequalityRRR)(RRDRRRRtooRnRCtokootoknooRRRXtbtaxtefRtbeginning_denomstcurrent_denomsRrtcRp((s9lib/python2.7/site-packages/sympy/solvers/inequalities.pyRsJ !    *'         5 c sii}}g}x_|D]W}|j|j}}|jt}t|dkrc|jne|j|@} t| dkr| j|jtt |d|qnt t d|j r|j gj||fq|jfd} | rOtd| DrO|j gj||fq|jtt |d|qWg} g} x3|jD]%\} | jt| gqWx0|jD]"\} | jt| qWt| | |S(NiisZ inequality has more than one symbol of interest. c s,|jo+|jp+|jo+|jj S(N(Ret is_FunctionRyRzR{(tu(Rj(s9lib/python2.7/site-packages/sympy/solvers/inequalities.pyRscs s|]}t|tVqdS(N(RR(RRp((s9lib/python2.7/site-packages/sympy/solvers/inequalities.pys s(RWRYtatomsRRtpopRR)RRR&Rt is_polynomialt setdefaulttfindRtitemsRtRR(Rtsymbolst poly_parttabs_parttothert inequalityRnR2tgenstcommont componentst poly_reducedt abs_reducedRi((Rjs9lib/python2.7/site-packages/sympy/solvers/inequalities.pyt_reduce_inequalitiesqs6    " ""&c st|s|g}ng|D]}t|^q}tjg|D]}|j^qG}t|sw|g}nt|p||@}td|Drttdnd|Dg|D]}|j^q}fd|D}g}x|D]}t |t rM|j |j j |jj d}n$|ttfkrqt|d}n|tkrq n|tkrtjS|j jrtd|n|j|q W|}~t||}|jdjDS(sReduce a system of inequalities with rational coefficients. Examples ======== >>> from sympy import sympify as S, Symbol >>> from sympy.abc import x, y >>> from sympy.solvers.inequalities import reduce_inequalities >>> reduce_inequalities(0 <= x + 3, []) (-3 <= x) & (x < oo) >>> reduce_inequalities(0 <= x + y*2 - 1, [x]) (x < oo) & (x >= 1 - 2*y) cs s|]}|jtkVqdS(N(RR((RRp((s9lib/python2.7/site-packages/sympy/solvers/inequalities.pys ssP inequalities cannot contain symbols that are not real. cS s7i|]-}|jdkrt|jdt|qS(RN(RR.RtnameR,(RRp((s9lib/python2.7/site-packages/sympy/solvers/inequalities.pys s c sh|]}|jqS((R(RRp(trecast(s9lib/python2.7/site-packages/sympy/solvers/inequalities.pys s is%could not determine truth value of %scS si|]\}}||qS(((RtkR((s9lib/python2.7/site-packages/sympy/solvers/inequalities.pys s (RRRRGRtanyRRRRRRwRWR"RXR,R(RRR%R!R&R)RR(RRRpRtkeepR((Rs9lib/python2.7/site-packages/sympy/solvers/inequalities.pytreduce_inequalitiess@  (   " +   N(;t__doc__t __future__RRt sympy.coreRRRtsympy.core.compatibilityRtsympy.core.exprtoolsRtsympy.core.relationalRRR R R t sympy.setsR tsympy.sets.setsR RRRtsympy.sets.fancysetsRtsympy.core.singletonRtsympy.core.functionRtsympy.functionsRt sympy.logicRt sympy.polysRRRtsympy.polys.polyutilsRtsympy.utilities.iterablesRtsympy.utilities.miscRR@RERSR,RtRRR$R(RdRRRR(((s9lib/python2.7/site-packages/sympy/solvers/inequalities.pyts8(" \  C R R  ! 3