o ֭jD2@sdZddlmZmZddlZddlmZddlm Z m Z ddl m Z z&ddl Z ddl m Z mZmZmZmZmZmZmZmZmZmZddlmZWn eyQYnwd d Zd d Zd dZddZddZddZ ddZ!e"dkrwe!dSdS)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)hornercCsZd}td\}}}}g}g}g}t||t|t||||dtjf}t|t||}td|dD]B} | ||  |d } | t d|dt dd} | d| 9} ||| t| |t| |t| |  q9d} | d 7} tgd |||gD]\} }tt|D]}| d | d |d t||7} qq| S)zATylor series expansion of Phi(a, b, x) in a=0 up to order 5. a b x krcWdSNrargsrrd/var/www/html/CropPilot/venv/lib/python3.10/site-packages/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)orderabxkr r!r" expressionntermx_partsnamecirrrseries_small_as..  $r@cCsBtd}d|ttd|t|||d|d|dfS)zSymbolic expansion of digamma(z) in z=0 to order n. See https://dlmf.nist.gov/5.7.E4 and with https://dlmf.nist.gov/5.5.E2 r7rr)rrr' summationr)zr9r7rrr dg_series7s ,rDcCstt|||||S)z8Symbolic expansion of polygamma(k, z) in z=0 to order n.)r'r*rD)r7rCr9rrr pg_seriesAsrEc svdtd\}}}td\}}}t|t|td|i}g}g}g} gtt|t||t|t|||dt j f} t ddD]n| | |d} | tddtdd} | d 9} | | t} dkr| tfd d} | jddd  td dd td} ||t|t| | | qJt| d |t tD]}|t||<qd}|d7}|d7}|d7}|d7}|d7}|d7}tddg||gD]"\}}t t|D]}|d|d|d7}|t||7}q qt tD]1}|d|d7}|t|7}|d|d7}|t| |t|t|tdid7}q+|d7}|d7}|d7}tfd d!t dD}|| d  |}|d"|t dk7}|d#7}tfd$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_Z3rrcWrrrrrrrrerz(series_small_a_small_b..rcst||dS)Nr)rE)r7r6)r9r3rrrm)r9rAzDTylor 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)r r!r#r$r%z # C[z C[z/ Test if B[i] does have the assumed structure.z" C[i] are derived from B[1] alone.z: Test B[2] == C[1] + b*C[2] + b^2/2*C[3] + b^3/6*C[4] + ..c(g|]}|t||dqS)rr .0r7Cr5rr (z*series_small_a_small_b..z test successful = z- Test B[3] == C[2] + b*C[3] + b^2/2*C[4] + ..crJ)rArKrLrNrrrPrQ)rrrrr r'r(r r r r&r)r*r+r,r-rr.seriesremoveOr/rPolycoeffsreverser1r0r2evalfsum)r4r6r7M_PIM_EGM_Z3c_subsr r!r"r8r:r;pg_partr?r<r=r>testr)rOr5r9r3rseries_small_a_small_bFs ,        r_c sd}GfdddtjGfdddtj}td\}}}|d||}d}|d 7}|d 7}|d 7}|d 7}|d 7}|d7}|d7}td|dD]R}|||||d||}ddt|D} t| } || |tj }| |d|i}|d|d| d|d7}|d|dt |d7}qJddl } | d} | d|}| d} | d|}|dd}|d d!}| d"} | d#|}|S)$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 c$eZdZdZdZefddZdS)zasymptotic_series..gzHelper function g according to Wright (1935) g(n, rho, v) = (1 + (rho+2)/3 * v + (rho+2)*(rho+3)/(2*3) * v^2 + ...) Note: Wright (1935) uses square root of above definition. rFcsl|dkstd|dkrdS|d||tt|d|t|dttd|td||S)Nrzmust have n >= 0rrArF) ValueErrorrr )clsr9rhovgrrevalsz!asymptotic_series..g.evalN__name__ __module__ __qualname____doc__nargs classmethodrhrrfrrrg rgcra)z!asymptotic_series..coef_CzCalculate coefficients C_m for integer m. C_m is the coefficient of v^(2*m) in the Taylor expansion in v=0 of Gamma(m+1/2)/(2*pi) * (2/(rho+1))^(m+1/2) * (1-v)^(-b) * g(rho, v)^(-m-1/2) rFcs|dkstdtd}d|| d|||| tdd}||d||dtd|}|t|tdddtd|d|tdd}|S)Nrzmust have m >= 0rerrA)rbrr r*r+r r r)rcmrdbetarer8resrfrrrhs.$z&asymptotic_series..coef_C.evalNrirrfrrcoef_Crprtz 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) denominatorrMr6rrrrPrGz%asymptotic_series..zC[z ] = C[0] / (z * Ap1[z]) z] *= z Nz xa\*\*(\d+)zA[\1]z b\*\*(\d+)zB[\1]xap1zAp1[1]xar4z (\d{10,})z\1.)r'Functionrr)r,rTrUlcmcollectfactorxreplacer2recompilesubr.)r3rtrxr5rwC0r<r?exprr|r~re_are_b re_digitsrrfrasymptotic_seriess>            rc sddd!fdd gdgdgd t\g}tjD]|tfd d d d ddidjq6t|}|d}dd}t t |||dddd}d}|d7}|d7}|d7}|d7}|d dd|D7}|S)"aFit 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)epsr4r5r6phieps_arrrfps0z$optimal_epsilon_integral..fp{Gz?dcs(tfdddtj|dddS)zCompute Arc length of f. Note that the arc length of a function f from t0 to t1 is given by int_t0^t1 sqrt(1 + f'(t)^2) dt c std|dS)NrrA)rsqrt)r)r4r5rrr6rrrsz=optimal_epsilon_integral..arclength..rr)epsrellimit)rrr)rr4r5r6rr)r)r4r5rr6r arclength s z+optimal_epsilon_integral..arclength) MbP?g?g?g?rrArr`)rrr ) rg?rArr2rig@@g@g@cs|SNr)r)rdata_adata_bdata_xr?rrrsz*optimal_epsilon_integral..)riBoundedxatolr)boundsmethodoptions)r4r5r6rc Ssx|d}|d}|d} ||td|t|dd|t| |t| ||dt||S)z#Compute parametric function to fit.r4r5r6gr)rr(log) dataA0A1A2A3A4A5r4r5r6rrrfunc*s0z&optimal_epsilon_integral..funcrtrf)rrz7Fit 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|]}|dqS)z.5grrvrrrrP:rGz,optimal_epsilon_integral..N)rr) rmeshgridflattenr)sizer/rr6arraylistrjoin)best_epsdfr func_paramsr<r)rrrrrr?roptimal_epsilon_integralsB      rcCst}tttd}|jdtgddd|}ddddd dd dd}||jd dt d t|d dddS)N) descriptionformatter_classaction)rrArFrzchose 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.)typechoiceshelpcS ttSr)printr@rrrrrL zmain..cSrr)rr_rrrrrMrcSrr)rrrrrrrNrcSrr)rrrrrrrOrcSstdS)NzInvalid input.)rrrrrrQsr#<z.1fz minutes elapsed. ) rrrmr add_argumentint parse_argsgetrr)t0parserrswitchrrrmain>s r__main__)#rmargparserrnumpyrscipy.integraterscipy.optimizerrrr'rr r r r r rrrrrsympy.polys.polyfuncsr ImportErrorr@rDrEr_rrrrjrrrrs.  4 " YYF