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Returns (r,err) where r is the estimated location of the root and err is a positive number estimating the error of the asymptotic expansion. rjrrr r¢éüÿÿÿrŒéiàÿÿÿéSiÖiÃr r˜iÀÿÿÿi%iÿXiO2iuÈ_éiéRr|iiß iÑ i,†i il"qi»QYgr )rr*Úlenr-)rÚkindÚprimer6r?r[rRÚs1Ús2Zs3Zs4Zs5r¶raÚerrÚirrr Úmcmahons>     (4@,8D rÆcs€|dkrtdƒ‚|d}g‰g‰xZˆ |||¡‰‡‡fdd„ˆDƒ‰‡‡fdd„t|dƒDƒ}t|ƒ|krp|S|d}q"WdS)zî Given f known to have exactly n simple roots within [a,b], return a list of n intervals isolating the roots and having opposite signs at the endpoints. TODO: this can be optimized, e.g. by reusing evaluation points. rzn cannot be less than 1csg|]}ˆ ˆ|ƒ¡‘qSr)Úsign)rr)rrwrr ú Jsz)generalized_bisection..cs8g|]0}ˆ|ˆ|ddkrˆ|ˆ|df‘qS)rr r)rrÅ)ÚpointsÚsignsrr rÈKsrN)r7Zlinspacer*r¿)rrwrQrRrÚNZ ok_intervalsr)rrwrÉrÊr Úgeneralized_bisection;s rÌcCs|j||dddS)NZillinoisF)ZsolverÚverify)r£)rrwÚabrrr Úfind_in_intervalQsrÏg{®Gáz„?cs˜ˆj}t|ˆ ˆ¡ˆ |¡ƒd}zf|ˆ_ˆ ˆ¡‰t|ƒ}t|ƒ}ˆdkrVtdƒ‚|dkrftdƒ‚|dkrvtdƒ‚|dkr |r’‡‡fdd „} n‡‡fd d „} |d krÊ|r¼‡‡fd d „} n‡‡fd d „} |dkr2|r2|dkr2ˆdkròˆjSˆdkr2d ˆ ˆdˆˆd ¡} tˆ| | dd | fƒS||ˆ|f|kr\tˆ| |||ˆ|fƒSt ˆ||ˆ|ƒ\} } | |kr’tˆ| | || |fƒS|dkr¦|s¦d} |dkrº|rºd} |d krÎ|sÎd} |d krâ|râd} |d} xœt ˆ||ˆ| ƒ\}} | |krzt ˆ||ˆ| dƒ\}}t ˆ| | d||| ƒ}x*t |ƒD]\}}||||ˆ|df<qDWtˆ| ||dƒS| d } qìWWd|ˆ_XdS)Nrrzv cannot be negativerzm cannot be less than 1)rrz prime should lie between 0 and 1csˆjˆ|ddS)Nr)r3)r)r)rr6rr r¡cr zbessel_zero..cs ˆ ˆ|¡S)N)r)r)rr6rr r¡dr rcsˆjˆ|ddS)Nr)r3)rB)r)rr6rr r¡fr cs ˆ ˆ|¡S)N)rB)r)rr6rr r¡gr g333333@gÍÌÌÌÌÌü?gš™™™™™é?g@gà?) rr!r(r+r$r7ÚzerorUrÏrÆrÌÚ enumerate)rrÀrÁr6r?ZisoltolZ_interval_cacherZworkprecrwrrÄÚlowrÚr1Úr2Zerr2Z intervalsrrÎr)rr6r Ú bessel_zeroTsf    rÕcCst|d|||ƒ S)aÿ For a real order `\nu \ge 0` and a positive integer `m`, returns `j_{\nu,m}`, the `m`-th positive zero of the Bessel function of the first kind `J_{\nu}(z)` (see :func:`~mpmath.besselj`). Alternatively, with *derivative=1*, gives the first nonnegative simple zero `j'_{\nu,m}` of `J'_{\nu}(z)`. The indexing convention is that used by Abramowitz & Stegun and the DLMF. Note the special case `j'_{0,1} = 0`, while all other zeros are positive. In effect, only simple zeros are counted (all zeros of Bessel functions are simple except possibly `z = 0`) and `j_{\nu,m}` becomes a monotonic function of both `\nu` and `m`. The zeros are interlaced according to the inequalities .. math :: j'_{\nu,k} < j_{\nu,k} < j'_{\nu,k+1} j_{\nu,1} < j_{\nu+1,2} < j_{\nu,2} < j_{\nu+1,2} < j_{\nu,3} < \cdots **Examples** Initial zeros of the Bessel functions `J_0(z), J_1(z), J_2(z)`:: >>> from mpmath import * >>> mp.dps = 25; mp.pretty = True >>> besseljzero(0,1); besseljzero(0,2); besseljzero(0,3) 2.404825557695772768621632 5.520078110286310649596604 8.653727912911012216954199 >>> besseljzero(1,1); besseljzero(1,2); besseljzero(1,3) 3.831705970207512315614436 7.01558666981561875353705 10.17346813506272207718571 >>> besseljzero(2,1); besseljzero(2,2); besseljzero(2,3) 5.135622301840682556301402 8.417244140399864857783614 11.61984117214905942709415 Initial zeros of `J'_0(z), J'_1(z), J'_2(z)`:: 0.0 3.831705970207512315614436 7.01558666981561875353705 >>> besseljzero(1,1,1); besseljzero(1,2,1); besseljzero(1,3,1) 1.84118378134065930264363 5.331442773525032636884016 8.536316366346285834358961 >>> besseljzero(2,1,1); besseljzero(2,2,1); besseljzero(2,3,1) 3.054236928227140322755932 6.706133194158459146634394 9.969467823087595793179143 Zeros with large index:: >>> besseljzero(0,100); besseljzero(0,1000); besseljzero(0,10000) 313.3742660775278447196902 3140.807295225078628895545 31415.14114171350798533666 >>> besseljzero(5,100); besseljzero(5,1000); besseljzero(5,10000) 321.1893195676003157339222 3148.657306813047523500494 31422.9947255486291798943 >>> besseljzero(0,100,1); besseljzero(0,1000,1); besseljzero(0,10000,1) 311.8018681873704508125112 3139.236339643802482833973 31413.57032947022399485808 Zeros of functions with large order:: >>> besseljzero(50,1) 57.11689916011917411936228 >>> besseljzero(50,2) 62.80769876483536093435393 >>> besseljzero(50,100) 388.6936600656058834640981 >>> besseljzero(50,1,1) 52.99764038731665010944037 >>> besseljzero(50,2,1) 60.02631933279942589882363 >>> besseljzero(50,100,1) 387.1083151608726181086283 Zeros of functions with fractional order:: >>> besseljzero(0.5,1); besseljzero(1.5,1); besseljzero(2.25,4) 3.141592653589793238462643 4.493409457909064175307881 15.15657692957458622921634 Both `J_{\nu}(z)` and `J'_{\nu}(z)` can be expressed as infinite products over their zeros:: >>> v,z = 2, mpf(1) >>> (z/2)**v/gamma(v+1) * \ ... nprod(lambda k: 1-(z/besseljzero(v,k))**2, [1,inf]) ... 0.1149034849319004804696469 >>> besselj(v,z) 0.1149034849319004804696469 >>> (z/2)**(v-1)/2/gamma(v) * \ ... nprod(lambda k: 1-(z/besseljzero(v,k,1))**2, [1,inf]) ... 0.2102436158811325550203884 >>> besselj(v,z,1) 0.2102436158811325550203884 r)rÕ)rr6r?r3rrr Ú besseljzero‰sprÖcCst|d|||ƒ S)aÆ For a real order `\nu \ge 0` and a positive integer `m`, returns `y_{\nu,m}`, the `m`-th positive zero of the Bessel function of the second kind `Y_{\nu}(z)` (see :func:`~mpmath.bessely`). Alternatively, with *derivative=1*, gives the first positive zero `y'_{\nu,m}` of `Y'_{\nu}(z)`. The zeros are interlaced according to the inequalities .. math :: y_{\nu,k} < y'_{\nu,k} < y_{\nu,k+1} y_{\nu,1} < y_{\nu+1,2} < y_{\nu,2} < y_{\nu+1,2} < y_{\nu,3} < \cdots **Examples** Initial zeros of the Bessel functions `Y_0(z), Y_1(z), Y_2(z)`:: >>> from mpmath import * >>> mp.dps = 25; mp.pretty = True >>> besselyzero(0,1); besselyzero(0,2); besselyzero(0,3) 0.8935769662791675215848871 3.957678419314857868375677 7.086051060301772697623625 >>> besselyzero(1,1); besselyzero(1,2); besselyzero(1,3) 2.197141326031017035149034 5.429681040794135132772005 8.596005868331168926429606 >>> besselyzero(2,1); besselyzero(2,2); besselyzero(2,3) 3.384241767149593472701426 6.793807513268267538291167 10.02347797936003797850539 Initial zeros of `Y'_0(z), Y'_1(z), Y'_2(z)`:: >>> besselyzero(0,1,1); besselyzero(0,2,1); besselyzero(0,3,1) 2.197141326031017035149034 5.429681040794135132772005 8.596005868331168926429606 >>> besselyzero(1,1,1); besselyzero(1,2,1); besselyzero(1,3,1) 3.683022856585177699898967 6.941499953654175655751944 10.12340465543661307978775 >>> besselyzero(2,1,1); besselyzero(2,2,1); besselyzero(2,3,1) 5.002582931446063945200176 8.350724701413079526349714 11.57419546521764654624265 Zeros with large index:: >>> besselyzero(0,100); besselyzero(0,1000); besselyzero(0,10000) 311.8034717601871549333419 3139.236498918198006794026 31413.57034538691205229188 >>> besselyzero(5,100); besselyzero(5,1000); besselyzero(5,10000) 319.6183338562782156235062 3147.086508524556404473186 31421.42392920214673402828 >>> besselyzero(0,100,1); besselyzero(0,1000,1); besselyzero(0,10000,1) 313.3726705426359345050449 3140.807136030340213610065 31415.14112579761578220175 Zeros of functions with large order:: >>> besselyzero(50,1) 53.50285882040036394680237 >>> besselyzero(50,2) 60.11244442774058114686022 >>> besselyzero(50,100) 387.1096509824943957706835 >>> besselyzero(50,1,1) 56.96290427516751320063605 >>> besselyzero(50,2,1) 62.74888166945933944036623 >>> besselyzero(50,100,1) 388.6923300548309258355475 Zeros of functions with fractional order:: >>> besselyzero(0.5,1); besselyzero(1.5,1); besselyzero(2.25,4) 1.570796326794896619231322 2.798386045783887136720249 13.56721208770735123376018 r)rÕ)rr6r?r3rrr Ú besselyzeroûsYr×N)r)r)r)r)r)F)r)rF)rT)rT)r)r)2Z functionsrrr r rr9rBrFrHrIrKrNrLrVrWrdrerfrgrirkrlrrrsr{rr€rƒr„r†rŠr›rœr¥r¦r§rªr«r¬r±r´r·r¹rÆrÌrÏrÕrÖr×rrrr Úsx   B # #                  [ K   +     "'5 q