o 8Vab @sdZddlmZmZmZmZmZmZmZm Z m Z m Z m Z m Z mZmZmZmZmZmZmZmZddlmZmZmZmZmZddlmZmZmZm Z m!Z!m"Z"m#Z#m$Z$m%Z%ddl&m'Z'ddl(m)Z)ddl*m+Z+ddl,m-Z-dd l.m/Z/m0Z0dd l1m2Z2dd l3Z3ed \ Z4Z5Z6Z7Z8Z9Z:Z;ZddZ?ddZ@ddZAddZBddZCddZDddZEddZFdd ZGd!d"ZHd#d$ZId%d&ZJd'd(ZKd)d*ZLd+d,ZMd-d.ZNd/d0ZOd1d2ZPd3d4ZQd5d6ZRd7d8ZSd9d:ZTd;d<ZUe0d=d>ZVd?d@ZWdAdBZXdCdDZYd S)EzBTests for algorithms for computing symbolic roots of polynomials. )SsymbolsSymbolWildRationalsqrtpowsimpsincospiIIntervalreimexpZZ Piecewiseacosroot conjugate)Polycyclotomic_poly intervalsnrootsrootof) root_factors roots_linearroots_quadratic roots_cubic roots_quarticroots_cyclotomicroots_binomialpreprocess_rootsroots) legendre_poly)PolynomialError)_nsort)cartes)raisesslow)verify_numericallyNza,b,c,d,e,q,t,x,y,zcCstdd|D}tt|d|t|d|ksJt|t|dD]}||d||ks@||dt||ks@dSq%dS)NcSsg|] }|jr dndqS)ris_real.0ir1B/usr/lib/python3/dist-packages/sympy/polys/tests/test_polyroots.py sz_check..r+FT)sumlistsortedrangelenr)r#ZnrealZixr1r1r2_checks(r:cCs*ttdtdttddgksJdS)Nr4r+)rrxrr1r1r1r2test_roots_linear(s*r=cCsttdtdtddgksJttdtddtttdddgks)Jttdtddtt tddttddgksIJttdtddtdtdttdddttddgkspJttdtddtdttddtt dt t tt tt t tt dt t dtt }tt|tt tt tt ttt t t tt t t dtt t dtt t tt tt ttt t t tt t t dtt t dtt gks Jtt tddtdt t}t|tdt dd t dt tdt ddt dt gksFJttdt d dtt ddt}t|dt ddgkskJttdtddt}t|}|t|ksJtd tdd td }d d |jddDdd |jddDksJtdddD]!\}}}t|td|t|}t|}|t|ksJqdS)Nr4rr;r+iicSg|]}|dqSr4nr/wr1r1r2r3Fz(test_roots_quadratic..T)ZradicalscSrCrDrErGr1r1r2r3GrIF)r4)rr;)rrr<rr rr: all_rootsaecdbyr&r')frZ_aZ_bZ_cr#r1r1r2test_roots_quadratic,s>"0@N&T TT "@ ,  rTc Csttdttdtt}t|ttdtdtddttd tdtddttdtdtd dttd tdtd diksSJdS)NrAr4r+)rr<r r#reqr1r1r2test_issue_7724Ns rXcstdtddgttttt}tddttddfdd|D}t|tttttks;Jt fdd |DsIJdS) Nr+rJr?r4csg|]}|tqSr1)subsrQr/rS)zr1r2r3[sz#test_issue_8438..c3s0|]}tt|ijddddkVqdS)r4TZchoprN)rZrQr<rFr.pr\r1r2 _s.z"test_issue_8438..) rrQr<Zas_exprrrr setrZall)r#Zpostr1r^r2test_issue_8438Ws rccstdtddtttdd}t|sJttddtddtfddtdD}ttddtddt}||ksLJt|sRJtdtdd}t|sdJttdtd dd ksvJdS) NrAr+r4rUr@cg|]}t|qSr1rr.rRr1r2r3frIz#test_issue_8285.. )rr<rKr:r8r9)r#ror1rgr2test_issue_8285bs* "   (rjcCsttddttdd}t|sJttddtddt}t|s/Jttdtd}t|sAJttddtddt}t|sXJdS)Nr4rAr@r>r+)rr<rKr:r#r1r1r2test_issue_8289qs$ "  "rlcsttdddtdddtdddtdtdtdtdtdtgks1JtddtddfddtdD}t|ksRJt|dS)Nr+r4rArhcrer1rfr.r_r1r2r3rIz$test_issue_14291..)rr<rKr rr8r:)ansr1rmr2test_issue_14291}s*0 rocCs>ttdtttttdd}t|}t|dksJdS)Nr>EXdomain)rrQrr<r#r9)rWZroots_dr1r1r2test_issue_13340s"rscsxttdtdddttdddttdd td d ttt}tfd d |Ds:JdS)NrAr> r4iiiii ic3s|] }|dkVqdSrNr1r[rVr1r2r`z#test_issue_14522..)rr<r r#rb)roots_eqr1rVr2test_issue_14522sRrycCs@tttddtdttdt}|dtsJdS)NrAr@r4r>r)rrtr<Zhas)solr1r1r2test_issue_15076s*r|cCsbttddtdtddtddtdtddtt}t|}d|vs/JdS)NrArdr4r>@ir)rr<rr#)rWrxr1r1r2test_issue_16589sJr~c CslttdtdtgdksJtttddtddtdtgdks,Jttd}tttdtt||tj tdtd|tj tdtdgks[JtttddtdtjttddtjttddgksJttdtddtddtdtdtjdt ddddt dddksJtd dtddtd}t |ddd tt|tdd kr1t dddtd t t d td d dddtd t t d td d  dtddt dddtd t t d td d  dtddt ddgks4JJdS)Nr4r>)rrrr+r+r+r+r;rT)trigmultiple)r rdrJr@)rrr<rrQrHalfrr rr#r rr r )rSrWr1r1r2test_roots_cubics,"6  ,0("  4>>rc sbtttdtgdksJtttdtdtgdgdgdgdfvs,Jtttdtdtgdgd gd gd fvsIJtttdtt}tjttdd tjttdd tjtj g}t |t d t |t d ks}Jt gdD]e\}\}}}}|d kr| |d td|td }n|dkr|||d t dd|td|td}td|td|td |t|ttt}t fdd|DsJqttttdtdtdd td t ddtt}t fdd|DsJtdddtdd tddtd tddttt} t fdd| DsMJtttdtdttt}t dd|DsnJtt ddt dddtt ddt dd dtt ddd df} | D]ttt}t fd!dt||DsJqdS)"NrArrrrr>)r;rrr)rr;rr)rrr;r)rrrr;)r+rrr)rr+rr)rrr+r)rrrr+r4key))r+r4r>r)r>i r+r4r>rAr)rr?r>)r?rUr)r@ir?rdrtc3s(|]}t|jdddkVqdS)Tr]rN)rZr<rF)r/ZairVr1r2r`s&z%test_roots_quartic..r+c3s"|] }tt|dVqdSrvr*rZr<r.rVr1r2r`s r\T)negative rUcs(g|]}tt|fdffdqS)r;rrr.)rWr\r1r2r3s(z&test_roots_quartic..css|] }t|tkVqdSN)typerr.r1r1r2r`rwr;)rQr\rJcs$g|]\}}t||dqS)r)r*rZ)r/rHs)repr1r2r3s$)rrr<rrr rZeroOner7hash enumeraterrbqrrQdictrZzip) ZlhsZrhsr0rLrPrNrOrnr{ZzansZrepsr1)rWrr\r2test_roots_quarticsT  6$2,< 0"&rc CsXtddtttdtdddtdtdtddttDr*JdS) Ncss|]}t|tVqdSr) isinstancerr.r1r1r2r`sz#test_issue_21287..rAr4r>rUr;r+)anyrrr<r r1r1r1r2test_issue_21287s <rc CsttdtdddgksJttdtdddgksJttdtddtddttddtddttddgksAJttdtddt tgksQJttdtddtjttddtjttddgksrJttd tddtt d  tt t d tt d  tt t d tt tdd  tt t tdd tt tdd  tt t tdd tt tdd tt t tdd tt tdd tt t tdd gksJttd tddtd dttddtd dttddtddttddtddttddgks-Jttd tddtd dtdtd dtdtddtdtddtdgksaJttdtdddd dgksrJttdtdddd dgksJttdtdddd t dd dt ddgksJttdtdddd t tgksJttd tdddd t dd  t dd d t dd ddt dd dt dd t dd dgksJttdtdddd dt ddt ddgksJdS)Nr+TZpolysr4r;r>rAr@rYrd )ZfactorrU) r rr<rr rrrr r r rr1r1r1r2test_roots_cyclotomicsn2 *((&&        N rc Cs:ttdttdgksJttdtdtgdksJttdtdttddgks2Jdtddd}ttdtddt| |t| |t||t||tgksbJttttd dtd d d }td d d }tt|td|t}tt|td|t}t|dt|dksJt|dt|dksJt ddddD]'\}}}}||kr|dkrqt|t|||} t| } | t | ksJqt tdtddt dtdt tddt dtdddt ddt tddt dtddiksJdS)NrUrrArr4rJrhr>rda1T)Z nonnegativeb1r+)r+r4)r;r+)r4r>rArUrtr;)r!rr<rr r:rrrr'r&r#rQr) ArrZr0r1rLrPrrFr_rnr1r1r2test_roots_binomial#s2"&.   ""rcCs`tttdtt}tt|t\}}|dksJ|ttttdtttks.Jtdtdtdtdttt}tt|t\}}|dtksUJ|ttdtdtttkshJtdtdtdtdt}tt|t\}}|dtksJ|ttdtdttksJtdtdttt}tt|t\}}|dtksJ|ttdtttksJtdtdt}tt|t\}}|dtksJ|ttdttksJtd\}}}}d|d|d|dd|t|d |d |d d |d |d|d td|d d|d|d|dtd|dd|d|d|dtd|dd|d|d|dtd|d d|d|d |dtd d|dd|||d td d|dd|dtdd}tt|t\}}|d||||dksJ|dtddtd dtd dtddtddtddtddtd ksJttd tdtttt tttfd!}ttd tttt ttd!}t|t|fks.JdS)"Nr4r+r>E,F,J,L[r|)s2^rdrt83ll8rY]$=,yr@r/@:rUrhD!^rAq Pyqidi Bi߰lul/1g&l glFW&krq) rLrQr<rPr"rrNrrr)rRZcoeffZpolyEFJLgr1r1r2test_roots_preprocessingDsh &,&$""****.& "<   0$rcCs tdtiks JttttjdiksJttdttjdiks#Jttdtdtdttd dtddtddiksEJtdtdttdddiksWJtdtddttdddikskJtdtddttdddiksJtdtddttdddiksJttddttdtjdtj dt diksJttdddttdtjdtj dt diksJtdtddttdddiksJtdtddttd ddiksJtdtddttdddiks Jtdtddttd ddiks!Jtdtddttdddiks8Jtdtddttd ddiksOJtttt dtt tdikseJtttt dtt tdiks|Jttdt dttttdtjdiksJttddtddttjdtj diksJttd dtddtdtdttjddt ddtjddt ddiksJttd dtt ddtt dddt ddtt dddt d dtt dddt d dtt dddtjdtj dtdt diks2Jd tdd tddtddtddtd dtddtd dtd}t|tjdtd dtddtdddtd ddtdddtdddiksJttt t ttt t t ttt t t tt t diksJttdtdtdtddiksJttdtdtdtddtd dtddtddiksJttdtdtdtdtddtd dtddtddtddiksJttddtddtd ttd ddtddtdiks>Jttddtddtd tdddtddtdtd dikseJttdttddtddtd ttjdtjdtd ddtddtdiksJtjtdd}}dt dd|}d|d}|d}t dtd}||}| |}ttdtdtdtdd| ||d| ||||d| ||||diksJtddtdtdtddttdtd}dddD\} } t d} t|t| | t dd t| d| | t dd t| d| | t dd t| d| | t dd t| diksdJtdtdtdtd}ddd D\} } }t|t| | d|t|| d||d| | d|t|| d||d| | d|t|| d||d| | d|t|| d||diksJtddttdddtdtdttdtd}t|ttjdtjtjttjt ddttdd tddtjtjttjt ddttdd tddiksEJttt t tddtddtd tddikseJttt t tddtddtd tddiksJttddtd!d"tjdtj diksJttddtd#d"tdt diksJttdtdttjdtj diksJttdtdtd$d%d&tjdiksJttddtd!dd'tj tjgksJttddtd#dd'tt gks Jtd(dd)\}}td||ddt||d}t|td*d"d||diks7Jttdtdd+tjtjtjgksKJtd,tdd+gksWJtd tdtdtdtdtd}t|t ttdttddt tttddtttdddt tttddtttdddtttdttddttttddtttdddttttddtttdddiksJtdd|d}t|t ttdttddt tttddtttdddt tttddtttdddtttdttddttttddtttdddttttddtttdddt dtdikspJttdd-td}d.d|Dd/}d0d t d1}|dt|ddd2t|dksJtddt dtdd3dt dtdt d dtdtd4d5}t|dt ddddt ddt d ddiksJtd6tdd7t dtdd8tdd0t dtdd9tdd:t dtdd;tddtd4d5}t|t d ddddt dddt ddd?dt ddd dt ddiksiJttddtdd t dtdd t dtd@tdAdAt dtd4d5}t|dt ddddt ddddt dddiksJtttt dddtd4d5t d tddtj t dtddt d tddtj t dtddt d tdddik sJdS)BNr+rr4r>rAr;rUrhr?r@rdi iii irYir}rF)ZcubicsrJT!cSg|]}t|qSr1rr[r1r1r2r3s ztest_roots0..))rr)r+rcSrr1rr[r1r1r2r3))r+rA)r+rd)rUrdZfilterr cSs|jSr)Z is_positive)rSr1r1r2sztest_roots0..)Z predicate)rrza, b)realRri(cSsg|]}|jr|qSr1r,)r/Zrxr1r1r2r3rIrli2rrrprq)}-rr)r#r<rrexpandrr rrLrPZ NegativeOnerrNrOrrZr\rQrr r r rr)rRZr1_2Zr1_3Zx0Zx1Zx2Zx3Zx4Zx5Zr13_20Zr1_20s2Zr1_4Zr1_8Zr5_8arbrr_rrSZ real_rootZcrrWr1r1r2 test_roots0sXD$(((26,,....,.44,&  :   ( L, ,$ J( 0(     <  **** D:: @@.*. ,(($(0**((  **((   *P<@  0  D B ,,rcstd\}}}}d||||||}d|||||||||||||||}tt|ddgksNJtt|ddgks]Jtd\}}}} } } } || || || | | | | | | } || | | | || | | | || }tt| || gdksJddddtt}tdd fd d |DDrJd S) z6Just test that calculating these roots does not hang. z a,b,c,d,xr4r+zzz,yy,xx,zy,zx,yx,krr>rdcss|]}|VqdSrr1r.r1r1r2r`@sz"test_roots_slow..cs g|] }|jddqS)Tr])rZrF)r/ZrirRr<r1r2r3@s z#test_roots_slow..N)rr6r#valueskeysr)rLrPrNrOf1f2ZzzZyyZxxZzyZzxZyxkZe1Ze2rr1rr2test_roots_slow,s$H00"*rcCsttdtdtdd}ttdtdtdd}t||D]\}}t||dks-Jqtddtd td d d tdtdd dtdd}t|dd}d}t||D]\}}t||dksmJq_dS)Nr4r+Trg?g-q=rAg@g@r>gS@g8@gb@g"@)g؋~)g)c g3! @g @绽|=)r#r<rabsr)R1R2rr2rRr1r1r2test_roots_inexactCs2 rc Cstd\}}}}d|d|d|dd|t|d|d|dd|d |d |d td |d d |d |d|d td|dd|d|d|dtd|dd|d|d |dtd |d d|d |d |d td d |dd|||dtdd|d d|dtdd}t|tiksJt|tdd}gd}td}|||||d }t|t|ksJt||D]\} } | |} | durt| || dksJqdS)NrrrdrtrrYrrr@r4rrrUr>rhrrArrrrrrTr)g?|YcgOXLzGX@g'I^g@g |۰n@gdZw@g5`e@gS$Ȫ@gJm@rHr) rr<r#evalfrr9rmatchr) rrrrrRrrrHr_rrrr1r1r2test_roots_preprocessedUs8"****.&  &rc Csddtdtddtddtd}t|d d \}}t|}t|d d }d d |D}dd |D}||}dd |D}t|ddd}t|ddd}||fD]"}t||D]\}}|jrl||vskJq^t|t|f|vsxJq^qWdS)Nipiir4i r>1rAT)rbrcSsg|] \\}}}t||qSr1)r r/rLrP_r1r1r2r3ysz$test_roots_mixed..cSs8g|]\\}}}tt|t|tt|t|qSr1)r rrrr1r1r2r3zs(cSsg|]}|qSr1)rr[r1r1r2r3~rcS|SrZsort_keyr<r1r1r2rz"test_roots_mixed..rcSrrrrr1r1r2rr) r<rrr#r7rr-rr) rR_reZ_imZ_nrootsZ_srootsZ _intervalsZ_rootsr0rSr1r1r2test_roots_mixedrs&0  rc CsttdttdtgksJtttttttgksJttddttdtdgks1JttdttttttttgksHJttdddtdtdtdtdttttttttgksnJtttddtddttdtttdtttddtgksJtdtddtddtdtdtd dtttddtddtddgksJdS) Nr+r4rArrrdrr@Q)rrr<rQrr r1r1r1r2test_root_factorss&.2*4( rcsRd}t|tddttjjfddjdd}dd |Dgd ks'JdS) Nr}TrcsjdddS)Nr>rU)rFZmaxsteps)rr1rmr1r2rsztest_nroots1..r>rEcSg|]}t|qSr1strr[r1r1r2r3rz test_nroots1..)@z-0.999z-0.996z-0.991z-0.983z-0.973z-0.961z-0.946z-0.930z-0.911z-0.889z-0.866z-0.841z-0.813z-0.784z-0.753z-0.720z-0.685z-0.649z-0.611z-0.572z-0.531z-0.489z-0.446z-0.402z-0.357z-0.311z-0.265z-0.217z-0.170z-0.121z-0.0730z-0.0243z0.0243z0.0730z0.121z0.170z0.217z0.265z0.311z0.357z0.402z0.446z0.489z0.531z0.572z0.611z0.649z0.685z0.720z0.753z0.784z0.813z0.841z0.866z0.889z0.911z0.930z0.946z0.961z0.973z0.983z0.991z0.996z0.999)r$r<r(mpmathZmpZ NoConvergencer)rFr#r1rmr2 test_nroots1s   rcCsjttddtdt}|jdd}dd|Dgdks J|jdd}dd|Dgd ks3JdS) NrUr>r+rEcSrr1rr[r1r1r2r3rz test_nroots2..)z-0.332z-0.839 - 0.944*Iz-0.839 + 0.944*Iz1.01 - 0.937*Iz1.01 + 0.937*IcSrr1rr[r1r1r2r3r)z-0.33199z-0.83907 - 0.94385*Iz-0.83907 + 0.94385*Iz1.0051 - 0.93726*Iz1.0051 + 0.93726*I)rr<r)r_r#r1r1r2 test_nroots2s     rcCs>ttttdtdtttttdddksJdS)Nr>r4T)Z composite)r9r#rrQrr<r1r1r1r2test_roots_composites>rcs.ttdttdttfdddS)Nr>r+cstSrrkr1rVr1r2rrz"test_issue_19113..)r r<r(r%r1r1rVr2test_issue_19113sr)Z__doc__Zsympyrrrrrrrr r r r r rrrrrrrrZ sympy.polysrrrrrZsympy.polys.polyrootsrrrrrr r!r"r#Zsympy.polys.orthopolysr$Zsympy.polys.polyerrorsr%Zsympy.polys.polyutilsr&Zsympy.utilities.iterablesr'Zsympy.testing.pytestr(r)Zsympy.testing.randtestr*rrLrPrNrOrMrrzr<rQr\r:r=rTrXrcrjrlrorsryr|r~rrrrrrrrrrrrrrrrr1r1r1r2sTX,     "    :0!<-