U Hv fR2@sdZddlmZmZddlZddlmZddlm Z m Z ddl m Z zLddl Z ddl m Z mZmZmZmZmZmZmZmZmZmZddlmZWnek rYnXd d Zd d Zd dZddZddZddZ ddZ!e"dkre!dS)zPrecompute coefficients of several series expansions of Wright's generalized Bessel function Phi(a, b, x). See https://dlmf.nist.gov/10.46.E1 with rho=a, beta=b, z=x. )ArgumentParserRawTextHelpFormatterN)quad)minimize_scalar curve_fit)time) EulerGammaRationalSSum factorialgamma gammasimppi polygammasymbolszeta)hornercCs`d}td\}}}}g}g}g}t||t|t||||dtjf}t|t||}td|dD]} | ||  |d } | t d|dt dd} | d| 9} ||| t| |t| |t| |  qrd} | d 7} td d d g|||gD]@\} }tt|D](}| d | d|dt||7} q.q| S)zATylor series expansion of Phi(a, b, x) in a=0 up to order 5. a b x krcWsdSNrargsrrK/tmp/pip-unpacked-wheel-96ln3f52/scipy/special/_precompute/wright_bessel.py&z series_small_a..z=Tylor series expansion of Phi(a, b, x) in a=0 up to order 5. zAPhi(a, b, x) = exp(x)/gamma(b) * sum(A[i] * X[i] * B[i], i=0..5) AXB [] = )rr r r r Infinitysympyexprangediffsubssimplifydoitrreplaceappendrziplenstr)orderabxkrr r! expressionntermx_partsnamecirrrseries_small_as.. *r?cCsBtd}d|ttd|t|||d|d|dfS)zSymbolic expansion of digamma(z) in z=0 to order n. See https://dlmf.nist.gov/5.7.E4 and with https://dlmf.nist.gov/5.5.E2 r6rr)rrr&Z summationr)zr8r6rrr dg_series7s ,rBcCstt|||||S)z8Symbolic expansion of polygamma(k, z) in z=0 to order n.)r&r)rB)r6rAr8rrr pg_seriesAsrCc szdtd\}}}td\}}}t|t|td|i}g}g}g} gtt|t||t|t|||dt j f} t ddD]މ| | |d} | tddtdd} | d 9} | | t} dkrD| tfd d} | jddd  td dd td} ||t|t| | | qt| d |t tD]}|t||<qd}|d7}|d7}|d7}|d7}|d7}|d7}tddg||gD]D\}}t t|D],}|d|d|d7}|t||7}qqt tD]b}|d|d7}|t|7}|d|d7}|t| |t|t|tdid7}qZ|d7}|d7}|d7}tfd d!t dD}|| d  |}|d"|t dk7}|d#7}tfd$d!t d D}|| d |}|d"|t dk7}|S)%aTylor series expansion of Phi(a, b, x) in a=0 and b=0 up to order 5. Be aware of cancellation of poles in b=0 of digamma(b)/Gamma(b) and polygamma functions. digamma(b)/Gamma(b) = -1 - 2*M_EG*b + O(b^2) digamma(b)^2/Gamma(b) = 1/b + 3*M_EG + b*(-5/12*PI^2+7/2*M_EG^2) + O(b^2) polygamma(1, b)/Gamma(b) = 1/b + M_EG + b*(1/12*PI^2 + 1/2*M_EG^2) + O(b^2) and so on. rrzM_PI M_EG M_Z3rrcWsdSrrrrrrrerz(series_small_a_small_b..rcst||dS)Nr)rC)r6r5)r8r2rrrmr)r8r@zDTylor series expansion of Phi(a, b, x) in a=0 and b=0 up to order 5.z9 Phi(a, b, x) = exp(x) * sum(A[i] * X[i] * B[i], i=0..5) z B[0] = 1 z&B[i] = sum(C[k+i-1] * b**k/k!, k=0..) z M_PI = piz M_EG = EulerGammaz M_Z3 = zeta(3)rr r"r#r$z # C[z C[z/ Test if B[i] does have the assumed structure.z# C[i] are derived from B[1] allone.z: Test B[2] == C[1] + b*C[2] + b^2/2*C[3] + b^3/6*C[4] + ..cs(g|] }|t||dqS)rr .0r6Cr4rr sz*series_small_a_small_b..z test successful = z- Test B[3] == C[2] + b*C[3] + b^2/2*C[4] + ..cs(g|] }|t||dqS)r@rGrHrJrrrLs)rrrrr r&r'r r r r%r(r)r*r+r,rr-ZseriesZremoveOr.rPolycoeffsreverser0r/r1Zevalfsum)r3r5r6ZM_PIZM_EGZM_Z3Zc_subsrr r!r7r9r:Zpg_partr>r;r<r=testr)rKr4r8r2rseries_small_a_small_bFs~ ,     "   rRc sd}GfdddtjGfdddtj}td\}}}|d||}d}|d 7}|d 7}|d 7}|d 7}|d 7}|d7}|d7}td|dD]}|||||d||}ddt|D} t| } || |tj }| |d|i}|d|d| d|d7}|d|dt |d7}qddl } | d} | d|}| d} | d|}|dd}|d d!}| d"} | d#|}|S)$aAsymptotic expansion for large x. Phi(a, b, x) ~ Z^(1/2-b) * exp((1+a)/a * Z) * sum_k (-1)^k * C_k / Z^k Z = (a*x)^(1/(1+a)) Wright (1935) lists the coefficients C_0 and C_1 (he calls them a_0 and a_1). With slightly different notation, Paris (2017) lists coefficients c_k up to order k=3. Paris (2017) uses ZP = (1+a)/a * Z (ZP = Z of Paris) and C_k = C_0 * (-a/(1+a))^k * c_k cs$eZdZdZdZefddZdS)zasymptotic_series..gzHelper function g according to Wright (1935) g(n, rho, v) = (1 + (rho+2)/3 * v + (rho+2)*(rho+3)/(2*3) * v^2 + ...) Note: Wright (1935) uses square root of above definition. rDcsr|dkstdn\|dkrdS|d||tt|d|t|dttd|td||SdS)Nrzmust have n >= 0rr@rD) ValueErrorrr )clsr8rhovgrrevals z!asymptotic_series..g.evalN__name__ __module__ __qualname____doc__nargs classmethodrZrrXrrrYsrYcs$eZdZdZdZefddZdS)z!asymptotic_series..coef_CzCalculate coefficients C_m for integer m. C_m is the coefficient of v^(2*m) in the Taylor expansion in v=0 of Gamma(m+1/2)/(2*pi) * (2/(rho+1))^(m+1/2) * (1-v)^(-b) * g(rho, v)^(-m-1/2) rDcs|dkstdtd}d|| d|||| tdd}||d||dtd|}|t|tdddtd|d|tdd}|S)Nrzmust have m >= 0rWrr@)rTrr r)r*r r r)rUmrVbetarWr7resrXrrrZs.$z&asymptotic_series..coef_C.evalNr[rrXrrcoef_Csrez xa b xap1rz!Asymptotic expansion for large x z.Phi(a, b, x) = Z**(1/2-b) * exp((1+a)/a * Z) z3 * sum((-1)**k * C[k]/Z**k, k=0..6) zZ = pow(a * x, 1/(1+a)) zA[k] = pow(a, k) zB[k] = pow(b, k) zAp1[k] = pow(1+a, k) z#C[0] = 1./sqrt(2. * M_PI * Ap1[1]) rcSsg|] }|qSr) denominatorrIr5rrrrLsz%asymptotic_series..zC[z ] = C[0] / (z * Ap1[z]) z] *= z Nz xa\*\*(\d+)zA[\1]z b\*\*(\d+)zB[\1]xap1zAp1[1]xar3z (\d{10,})z\1.)r&ZFunctionrr(r+rMrNZlcmZcollectfactorZxreplacer1recompilesubr-)r2rerir4rhZC0r;r>exprrjrkZre_aZre_bZ re_digitsrrXrasymptotic_seriess>            roc sFddd0fdd ddd d d d d dddg dd d ddgd dd d dddddddddg t\g}tjD]0|tfddddd did!jqt|}|d"}d#d$}t t |||d%d&d'd}d(}|d)7}|d*7}|d+7}|d,7}|d- d.d/|D7}|S)1aFit optimal choice of epsilon for integral representation. The integrand of int_0^pi P(eps, a, b, x, phi) * dphi can exhibit oscillatory behaviour. It stems from the cosine of P and can be minimized by minimizing the arc length of the argument f(phi) = eps * sin(phi) - x * eps^(-a) * sin(a * phi) + (1 - b) * phi of cos(f(phi)). We minimize the arc length in eps for a grid of values (a, b, x) and fit a parametric function to it. cSsBtd|| }|t||||t||d|S)zDerivative of f w.r.t. phi.g?r)nppowercos)epsr3r4r5phiZeps_arrrfpsz$optimal_epsilon_integral..fp{Gz?dcs(tfdddtj|dddS)zCompute Arc length of f. Note that the arg length of a function f fro t0 to t1 is given by int_t0^t1 sqrt(1 + f'(t)^2) dt c std|dS)Nrr@)rpsqrt)rt)r3r4rsrur5rrrrz=optimal_epsilon_integral..arclength..rrw)epsrellimit)rrpr)rsr3r4r5ryrz)ru)r3r4rsr5r arclength sz+optimal_epsilon_integral..arclengthMbP?g?g?g?rr@rrSr g?2ig@@g@g@cs|SNr)rs)r{data_adata_bdata_xr>rrrsz*optimal_epsilon_integral..)r|iZBoundedZxatol)Zboundsmethodoptions)r3r4r5rsc Ssx|d}|d}|d} ||td|t|dd|t| |t| ||dt||S)z#Compute parametric function to fit.r3r4r5gr)rpr'log) dataZA0A1A2ZA3ZA4ZA5r3r4r5rrrfunc*s0z&optimal_epsilon_integral..funcrsZtrf)rz7Fit optimal eps for integrand P via minimal arc length zwith parametric function: zBoptimal_eps = (A0 * b * exp(-a/2) + exp(A1 + 1 / (1 + a) * log(x) z= - A2 * exp(-A3 * a) + A4 / (1 + exp(A5 * a))) z Fitted parameters A0 to A5 are: z, cSsg|]}d|qS)z{:.5g})formatrgrrrrL:sz,optimal_epsilon_integral..)rvrw) rpZmeshgridflattenr(sizer.rr5arraylistrjoin)Zbest_epsZdfrZ func_paramsr;r)r{rrrrur>roptimal_epsilon_integralsB      rcCst}tttd}|jdtddddgdd|}d d d d d d d d d}||jdd t d t|ddS)N) descriptionformatter_classactionrr@rDr}zchose what expansion to precompute 1 : Series for small a 2 : Series for small a and small b 3 : Asymptotic series for large x This may take some time (>4h). 4 : Fit optimal eps for integral representation.)typechoiceshelpcSs ttSr)printr?rrrrrLrzmain..cSs ttSr)rrRrrrrrMrcSs ttSr)rrorrrrrNrcSs ttSr)rrrrrrrOr)rr@rDr}cSstdS)NzInvalid input.)rrrrrrQrz {:.1f} minutes elapsed. <) rrr_r add_argumentint parse_argsgetrrr)t0parserrswitchrrrmain>sr__main__)#r_argparserrZnumpyrpZscipy.integraterZscipy.optimizerrrr&rr r r r r rrrrrZsympy.polys.polyfuncsr ImportErrorr?rBrCrRrorrr\rrrrs(  4" YYF