B ed @sVdZddlmZmZmZddlmZmZmZddl m Z ddl m Z m Z mZmZddlmZmZmZmZddlmZddlmZdd lmZmZdd lmZdd lmZm Z m!Z!dd l"m#Z#dd l$m%Z%m&Z&ddl'm(Z(m)Z)ddl*m+Z+ddZ,ddZ-ddZ.d)ddZ/ddZ0ddZ1dej2dfddZ3d d!Z4d*d"d#Z5d$d%Z6gfd&d'Z7d(S)+ztjgS|tj krPtj gSt d||j ddg}}|dkrx$|D]\}}t ||}||q|Wn|dkrtj}x8|tjdfgD]$\} }t || d d }||| }qWnP|dkrd} nd } d \} } |d krd} nD|d kr,d } n4|dkr@d\} } n |dkrTd\} } n td|tjd } } xt|D]\}}|dr| | kr|dt || | | | || } } } nT| | kr| s|dt || d | |d } } n"| | krv| rv|dt ||qvW| | kr<|dt tj| d | |S)aSolve a polynomial inequality with rational coefficients. Examples ======== >>> from sympy import solve_poly_inequality, Poly >>> from sympy.abc import x >>> 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 z8For efficiency reasons, `poly` should be a Poly instancerz%could not determine truth value of %sF)Zmultiplez==z!=T)NF>=)rTz<=)r Tz'%s' is not a valid relation) isinstancer ValueErroras_expr is_numberr rtrueRealsfalseEmptySetNotImplementedErrorZ real_rootsr appendNegativeInfinityInfinityZLCreversedinsert)Zpolyreltreals intervalsroot_intervalleftrightsignZeq_signequalZ right_openZ multiplicityr=g/work/yifan.wang/ringdown/master-ringdown-env/lib/python3.7/site-packages/sympy/solvers/inequalities.pysolve_poly_inequalitysh                    r?cCstdd|DS)aSolve polynomial inequalities with rational coefficients. Examples ======== >>> from sympy import Poly >>> from sympy.solvers.inequalities import solve_poly_inequalities >>> 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)) cSsg|]}t|D]}|qqSr=)r?).0psr=r=r> }sz+solve_poly_inequalities..)r)Zpolysr=r=r>solve_poly_inequalitiesosrDcCstj}x|D]}|sq ttjtjg}x|D]\\}}}t|||}t|d}g} x8|D]0} x*|D]"} | | } | tjk rd| | qdWqZW| }g} x6|D].} x|D] } | | 8} qW| tjk r| | qW| }|s,Pq,Wx|D]} || }qWq W|S)a3Solve a system of rational inequalities with rational coefficients. Examples ======== >>> from sympy.abc import x >>> from sympy import solve_rational_inequalities, Poly >>> 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 z==) rr+r r.r/r? intersectr-union)eqsresult_eqsZglobal_intervalsnumerdenomr2Znumer_intervalsZdenom_intervalsr5Znumer_intervalZglobal_intervalr8Zdenom_intervalr=r=r>solve_rational_inequalitiess6           rLTc sd}g}|rtjntj}xp|D]f}g}xL|D]B}t|trL|\}} n&|jrh|j|j|j}} n |d}} |tj krtj tj d} } } n0|tj krtj tj d} } } n| \} } yt| | f\\} } } Wn"tk rttdYnX| jjs"| | d} } }| j} | jsd| jsd| | }t|d| }|t|ddM}q2|| | f| fq2W|r ||q W|r|t|M}tfdd|Dg}||8}|s|r|}|r|}|S) a8Reduce a system of rational inequalities with rational coefficients. Examples ======== >>> from sympy import 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 >>> reduce_rational_inequalities([[(x + 2, ">")]], x) -2 < x >>> reduce_rational_inequalities([[x + 2]], x) Eq(x, -2) This function find the non-infinite solution set so if the unknown symbol is declared as extended real rather than real then the result may include finiteness conditions: >>> y = Symbol('y', extended_real=True) >>> reduce_rational_inequalities([[y + 2 > 0]], y) (-2 < y) & (y < oo) Tz==z only polynomials and rational functions are supported in this context. Fr) relationalcs6g|].}|D]$\\}}}|r ||jfdfq qS)z==)hasone)r@indr7)genr=r>rCsz0reduce_rational_inequalities..)rr)r+r$tupleZ is_Relationallhsrhsrel_opr(ZeroOner*Ztogetheras_numer_denomrrrdomainZis_ExactZto_exactZ get_exactZis_ZZZis_QQr solve_univariate_inequalityr-rLZevalf as_relational)exprsrSrMexactrGZsolution_exprsrIexprr2rJrKoptr[excluder=)rSr>reduce_rational_inequalitiessT             rdcs|jdkrttdfdd|}ddd}g}xL|D]D\}}||kr`t|d|}nt| d||}||g|q>Wt||S) aReduce an inequality with nested absolute values. Examples ======== >>> from sympy import reduce_abs_inequality, Abs, Symbol >>> 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 Fzs Cannot solve inequalities with absolute values containing non-real variables. c s:g}|js|jr~|j}xd|jD]Z}|}|s4|}qg}x:|D]2\}}x(|D] \}}||||||fqLWq>W|}qWn|jr|j} | jstd|j }x|D]\}}||| |fqWnnt |t r,|jd}xR|D]>\}}|||t |dgf|| |t |dgfqWn |gfg}|S)Nz'Only Integer Powers are allowed on Abs.r)Zis_AddZis_Mulfuncargsr-is_Powexp is_Integerr%baser$rr r ) rar^opargr`rfcondsZ_exprZ_condsrQ)_bottom_up_scanr=r>rn6s4      " z.reduce_abs_inequality.._bottom_up_scanr!z>=)r"z<=r)is_extended_real TypeErrorrkeysr r-rd)rar2rSr^mapping inequalitiesrmr=)rnr>reduce_abs_inequalitys  '  rtcstfdd|DS)aReduce a system of inequalities with nested absolute values. Examples ======== >>> from sympy import reduce_abs_inequalities, Abs, Symbol >>> x = Symbol('x', extended_real=True) >>> reduce_abs_inequalities([(Abs(3*x - 5) - 7, '<'), ... (Abs(x + 25) - 13, '>')], x) (-2/3 < x) & (x < 4) & (((-oo < x) & (x < -38)) | ((-12 < x) & (x < oo))) >>> reduce_abs_inequalities([(Abs(x - 4) + Abs(3*x - 5) - 7, '<')], x) (1/2 < x) & (x < 4) See Also ======== reduce_abs_inequality csg|]\}}t||qSr=)rt)r@rar2)rSr=r>rCsz+reduce_abs_inequalities..)r)r^rSr=)rSr>reduce_abs_inequalitiesms ruFc( sddlm}|tjdkr*ttdn2|tjk r\td|d|}|rX| }|S}|}j dkrtj }|s||S| |Sj dkrt ddd y |iWn tk rttd YnXd}tjkr|}ntjkrtj }n jj} t| } | tjkrVt| } | d} | tjkrB|}n| tjkrtj }n| dk rt| |} j} | d kr| jdr|}n| jdstj }n6| d kr| jdr|}n| jdstj }|j|j}}||tjkrtd| dd|}|}|dkr| \}}y>|jkrNt | jd krNt!t"| |}|dkrht!Wn6t!tfk rttd#t$dYnXt| fdd}g}x&|D]}|%t"||qW|st&|}djko jdk}yt'|j(t)|j|j}t)||t*|t|j|j|j|k|j|k}t+dd|Drt,|ddd}n^t-|dd}|drty&|d}t |d krt*t.|}Wntk rtYnXWntk rtdYnXtj}t/tjkrd}t)}y@t0t/|}t1|ts|x6|D].}||krH||rH|j rH|t)|7}qHWn|j|j}} xt,|t)| D]}||}!|| kr<||}"t2||}#|#|kr<|#j r<||#r<|!r|"r|t||7}n@|!r|t3||7}n(|"r,|t4||7}n|t5||7}|}qWx|D]}$|t)|$8}qLWWn tk rtj}d}YnX|tj krt!td#||f||}tj g}%|j}||kr||r|j6r|%7t)|x|D]~}&|&} |t2|| r"|%7t|| dd|&|kr8|8|&n6|&|krV|8|&||&}'n|}'|'rn|%7t)|&| }qW|j} | |kr|| r| j6r|%7t)| |t2|| r|%7t5|| t/tjkr|r||}nt9t:|%||#|}|s|S| |S)aSSolves 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 :func:`sympy.solvers.solveset.solvify`. Notes ===== Currently, we cannot solve all the inequalities due to limitations in :func:`sympy.solvers.solveset.solvify`. Also, the solution returned for trigonometric inequalities are restricted in its periodic interval. See Also ======== sympy.solvers.solveset.solvify: solver returning solveset solutions with solve's output API Examples ======== >>> from sympy import solve_univariate_inequality, Symbol, sin, Interval, S >>> x = Symbol('x') >>> solve_univariate_inequality(x**2 >= 4, x) ((2 <= x) & (x < oo)) | ((-oo < x) & (x <= -2)) >>> 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) r)denomsFz| Inequalities in the complex domain are not supported. Try the real domain by setting domain=S.Reals)rM continuousNrST) extended_realz When gen is real, the relational has a complex part which leads to an invalid comparison like I < 0. )r"z<=)r!z>=rz The inequality, %s, cannot be solved using solve_univariate_inequality. xcst|}y|d}Wntk r:tj}YnX|tjtjfkrP|S|jdkr`tjS|d}|j r||dSt d|dS)NrFr#z!relationship did not evaluate: %s) subsrrerprr*r(rorQZ is_comparabler,)ryvr) expanded_erarSr=r>valids     z*solve_univariate_inequality..valid=z!=css|] }|jVqdS)N)r')r@r|r=r=r> Gsz.solve_univariate_inequality..) separatedcSs|jS)N)ro)ryr=r=r>Jz-solve_univariate_inequality..z'sorting of these roots is not supportedz %s contains imaginary parts which cannot be made 0 for any value of %s satisfying the inequality, leading to relations like I < 0. );sympy.solvers.solversrvZ is_subsetrr)r,rr\ intersectionr]ror+rxreplacerpr(r*rUrVrrXrrerrWsupinfr/r rErZ free_symbolslenr%rrzrextendrsetboundaryrlistallrrsortedrrr$_ptZRopenZLopenopen is_finiter-removerr)(rarSrMr[rwrvrvZ_genZ_domaineZperiodconstZfranger2rrrQrRZsolnsr~Z singularitiesZ include_xZdiscontinuitiesZcritical_pointsr4ZsiftedZ make_realcheckZim_solazstartendZ valid_startZvalid_zptrBZsol_setsry_validr=)r}rarSr>r\s29                                               r\cCs|js|js||d}n|jr.|jr.tj}n|jr>|jdksN|jrV|jdkrVtd|jrb|jsn|jrx|jrx||}}|jr|jr|d}q|jr|tj}q|d}n0|jr|jr|tj}n|jr|d}n|d}|S)z$Return a point between start and endr#Nz,cannot proceed with unsigned infinite valuesr) is_infiniterrXZis_extended_positiver%Zis_extended_negativeZHalf)rrrr=r=r>rs.          rc Csddlm}||jkr|S|j|kr*|j}|j|krD||jjkrD|Sdd}d}tj}|j|j}yBt||}| dkr| | d}n|s| dkrt Wn0t t fk r|syt|gg|}Wnt k rt||}YnX||||} | tjkr.||||tjkr.|||kd}|||| } | tjkr|||| tjkr|| |kd}||| kd}|tjkr| tjkr||kn||k}| tjk rt| |k|}nt|}YnXg} |dkrv| } d} | j|dd\}}| |8} | |8} t| }|j|d d\}} |jd ksd|j|jkrTdkrnnn|jd krn|} tj}| |} |jr| | | }n|j | | }||j||jB}||}x`||D]T}tt|d||d }t|tr|j|kr||||jtjkr| |qWx\| |fD]N}||||tjkr$||||tjk r$| ||krf||kn||kq$W| |t| S) aReturn 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 do not 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) r)rvcSsFy*|||}|tjkr|S|dkr(dS|Stk r@tjSXdS)N)TF)rzrNaNrp)ierBrPr{r=r=r>classify s  z#_solve_inequality..classifyNrT)Zas_AddF)z!=z==)linear)rrvrrVr0rUrr/rZdegreerer&r,rrdr\r(r*rzrZas_independentris_zeroZ is_negativeZ is_positiverWrY_solve_inequalityr r$r-)rrBrrvrrZoorarAZokooZoknoormrrVbZaxZefrZbeginning_denomsZcurrent_denomsrRcrPr=r=r>rsJ               & rcstii}}g}x|D]}|j|j}}|t}t|dkrF|nF|j|@} t| dkr| |tt |d|qn t t d| r| g||fq|fdd} | rtdd| Dr| g||fq|tt |d|qWg} g} x(|D]\} | t| gqWx&|D]\} | t| qFWt| | |S)NrrzZ inequality has more than one symbol of interest. cs |o|jp|jo|jj S)N)rNZ is_Functionrgrhri)u)rSr=r>rs z&_reduce_inequalities..css|]}t|tVqdS)N)r$r)r@rPr=r=r>rsz'_reduce_inequalities..)rUrWZatomsrrpoprr-rr r,rZ is_polynomial setdefaultfindritemsrdrur)rssymbolsZ poly_partZabs_partotherZ inequalityrar2genscommon componentsZ poly_reducedZ abs_reducedr^r=)rSr>_reduce_inequalities{s6        rcsPt|s|g}dd|D}tjdd|D}t|s@|g}t|pJ||@}tdd|Drnttddd|Dfd d|D}fd d |D}g}x|D]z}t|tr||j |j d }n|d krt |d }|dkrqn|dkrt jS|j jrtd|||qW|}~t||}|ddDS)aEReduce a system of inequalities with rational coefficients. Examples ======== >>> from sympy.abc import x, y >>> from sympy 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) cSsg|] }t|qSr=)r)r@rPr=r=r>rCsz'reduce_inequalities..cSsg|] }|jqSr=)r)r@rPr=r=r>rCscss|]}|jdkVqdS)FN)ro)r@rPr=r=r>rsz&reduce_inequalities..zP inequalities cannot contain symbols that are not real. cSs&i|]}|jdkrt|jdd|qS)NT)rx)rorname)r@rPr=r=r> sz'reduce_inequalities..csg|]}|qSr=)r)r@rP)recastr=r>rCscsh|]}|qSr=)r)r@rP)rr=r> sz&reduce_inequalities..r)TFTFz%could not determine truth value of %scSsi|]\}}||qSr=r=)r@kr{r=r=r>rs)rrrFanyrprr$r rerUr&rVr rr*r'r,r-rrr)rsrrZkeeprPrr=)rr>reduce_inequalitiess@       rN)T)F)8__doc__Zsympy.calculus.utilrrrZ sympy.corerrrZsympy.core.exprtoolsrZsympy.core.relationalr r r r Zsympy.sets.setsr rrrZsympy.core.singletonrZsympy.core.functionrZ$sympy.functions.elementary.complexesrrZ sympy.logicrZ sympy.polysrrrZsympy.polys.polyutilsrZsympy.solvers.solvesetrrZsympy.utilities.iterablesrrZsympy.utilities.miscrr?rDrLrdrtrur)r\rrrrr=r=r=r>s8      [B ZQ)! .3