tKgD2dZddlmZmZddlZddlmZddlm Z m Z ddl m Z  ddl Z ddl m Z mZmZmZmZmZmZmZmZmZmZddlmZd Zd Zd Zd Zd ZdZ dZ!e"dk(re!yy#e$rY*wxYw)zPrecompute coefficients of several series expansions of Wright's generalized Bessel function Phi(a, b, x). See https://dlmf.nist.gov/10.46.E1 with rho=a, beta=b, z=x. )ArgumentParserRawTextHelpFormatterN)quad)minimize_scalar curve_fit)time) EulerGammaRationalSSum factorialgamma gammasimppi polygammasymbolszeta)hornercd}td\}}}}g}g}g}t||zt|z t||z|zz |dtj f}t|t j|z |z}td|dzD]} |j|| j|djj} | jtd|djtd} | d| zz} |j|| zt| z |jt!| |jt!| | z jd} | dz } t#gd |||gD]9\} }tt%|D]}| d | d |d t'||zz } ;| S) zATylor series expansion of Phi(a, b, x) in a=0 up to order 5. a b x krcyNrargss k/home/alanp/www/video.onchill/myenv/lib/python3.12/site-packages/scipy/special/_precompute/wright_bessel.pyz series_small_a..&Az=Tylor series expansion of Phi(a, b, x) in a=0 up to order 5. zAPhi(a, b, x) = exp(x)/gamma(b) * sum(A[i] * X[i] * B[i], i=0..5) )AXB [] = )rr r rr Infinitysympyexprangediffsubssimplifydoitrreplaceappendrziplenstr)orderabxkr#r$r% expressionntermx_partsnamecis rseries_small_arCs E#JAq!Q A A AQT)A,&uQqSU|3aAJJ5GHJq%))A,&3J1eAg q!$))!Q/88:??A))IaOQ/79o6  2' Ail"#   f..012 IA MMAAq 2as1vA 2dV1QCt$s1Q4y0 0A3 Hr!ctd}d|z tz tjd|zt |z||dz zz|d|dzfzS)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 r:r"r)rr r* summationr)zr<r:s r dg_seriesrH7sW  A a4*  a$q')A!H4q!QqSkB CCr!cJtjt|||z||S)z8Symbolic expansion of polygamma(k, z) in z=0 to order n.)r*r-rH)r:rGr<s r pg_seriesrJAs ::i1Q3'A ..r!c dtd\}}}}td\}}}t|t|td|i}g}g} g} g} t |t j |z t||zt|z t ||z|zz |dtjfz} tddzD]Q| j|j|djj} | jt!d|dj#t d}|dzz}| |z t |z }dk\rv|j#t fd }|j%|ddzz j'jt!d dd tdzj}|j)|ztz | j)t+|| j)|Tt j,| dj||j/} | j1tt3| D]$}| |t|zj| |<&d }|dz }|dz }|dz }|dz }|dz }|dz }t5ddg|| gD];\}}tt3|D]}|d|d|dz }|t7||z }!=tt3| D]k}|d|dz }|t7| |z }|d|dz }|t7| |j|t|t|tdij9dz }m|dz }|dz }|dz }t;tdz Dcgc]}||zt|z | |dzzc}}|| d j|z j}|d|tdk(z }|d z }t;td z Dcgc]}||zt|z | |d zzc}}|| dj|z j}|d|tdk(z }|Scc}wcc}w)!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_Z3rrcyrrrs rrz(series_small_a_small_b..er r!r"c*t||dzzS)Nr)rJ)r:r9r<r6s rrz(series_small_a_small_b..ms9Q57193Mr!)r<rEzDTylor 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$r&r'r(z # C[z C[z/ Test if B[i] does have the assumed structure.z" C[i] are derived from B[1] alone.z: Test B[2] == C[1] + b*C[2] + b^2/2*C[3] + b^3/6*C[4] + ..z test successful = z- Test B[3] == C[2] + b*C[3] + b^2/2*C[4] + ..)rrr rrr*r+r r r r)r,r-r.r/r0rr1seriesremoveOr2rPolycoeffsreverser4r3r5evalfsum)r7r8r9r:M_PIM_EGM_Z3c_subsr#r$r%Cr;r=r>pg_partrBr?r@rAtestr<r6s @@rseries_small_a_small_br_Fs E#JAq!Q/0D$$ D$q'4 8F A A A A q%))A,& AqD1 eAaCEl *Q1::,>?@J1eAg q!$))!Q/88:??A))IaOQ/79o6  2'v+eAh& 6ooi&MOG~~aeAgai~8 Yq!_baj9    Ail"#   +0  1Q499V$a(//1AIIK 3q6]!y|#--/! OA FFAA 22AA A ASzAq6*as1vA 2dV1QCt$ $A QqTNA+ 3q6] vaS  S1Y tA3d^ S1D*dBd1gFG%)   <.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. rLc|dk\s td|dk(ry|dz ||tt|dz|zt|dzz ttd|ztdz z ||zzzS)Nrzmust have n >= 0rrErL) ValueErrorrr)clsr<rhovgs revalz!asymptotic_series..g.evals6 !344a1c1~c!eAguSU| ;<ac 58 34556T:::r!N__name__ __module__ __qualname____doc__nargs classmethodrjrisrrircs!   :  :r!ric*eZdZdZdZefdZy)!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) rLcr|dk\s tdtd}d|z | zd|z||| tddz zz}|j|d|zj |dt d|zz }|t |tddzdtzz d|dzz |tddzzzz}|S)Nrzmust have m >= 0rhrrE)rerr r-r.r rr)rfmrgbetarhr;resris rrjz&asymptotic_series..coef_C.evals6 !344 AA#$!AaCa.A2hq!n;L*MMJ//!QqS)..q!4y1~ECq8Aq>12ad;s1uIXa^);<=>CJr!Nrkrrsrcoef_Crts!     r!ryz 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]) rzC[z ] = C[0] / (z * Ap1[z]) z] *= z Nz xa\*\*(\d+)zA[\1]z b\*\*(\d+)zB[\1]xap1zAp1[1]xar7z (\d{10,})z\1.)r*Functionrr,r/rSrT denominatorlcmcollectfactorxreplacer5recompilesubr1)r6ryr{r8rzC0r?rBexprr9rrre_are_b re_digitsris @rasymptotic_seriesrs E:ENN:(,+&KB4 2q B,A ::A @@A ))A A A ##A //A 1eAg q"a B"qyL1;;=+0::d+;+B+B+DE+Da!--/+DE6"v '')11!U\\B}}bdD\* r!Ls$ 77 r!E#d)D )) ::n %D 1A ::m $D 1A &(#A $A <(I fa A H#FsHc f d d fd gdgdgd tj \ jj jc g}t jD]3 |j t  fdddd d i j5tj|} |d }d }tt|||ddd}d}|dz }|dz }|dz }|dz }|dj|Dcgc]}|dc}z }|Scc}w)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. ctjd|z| }|tj|z||z|ztj||zzz dz|z S)zDerivative of f w.r.t. phi.g?r)nppowercos)epsr7r8r9phieps_as rfpz$optimal_epsilon_integral..fpsScA2&RVVC[ 1q55=266!c'?#BBQFJJr!c\tfddtj|ddS)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 Ntjd|dzzS)NrrE)rsqrt)rr7r8rrr9s rrz=optimal_epsilon_integral..arclength..s%BsAq!S,A1,D(D Er!rd)epsrellimit)rrr)rr7r8r9rrrs```` r arclengthz+optimal_epsilon_integral..arclength s+ Eruu!../1 1r!) MbP?g?g?g?rrErra)rrr ) rg?rErr2rig@@g@g@c,|SNr)rrdata_adata_bdata_xrBs rrz*optimal_epsilon_integral..s #vay&)28))=r!)riBoundedxatolr)boundsmethodoptions)r7r8r9rc >|d}|d}|d} ||ztjd|zztj|dd|zz tj| zz|tj| |zzz |dtj||zzz zzS)z#Compute parametric function to fit.r7r8r9gr)rr+log) dataA0A1A2A3A4A5r7r8r9s rfuncz&optimal_epsilon_integral..func*s I I IQq))&&a1q5kBFF1I55RVVRC!G_8LLRVVBF^!34566 7r!rtrf)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, z.5g)g{Gz?r) rmeshgridflattenr,sizer2rr9arraylistrjoin) best_epsdfr func_paramsr?r9rrrrrrBs @@@@@@roptimal_epsilon_integralrsRK 15F F EF[[@FFF$nn.0@$nn.FFFH 6;;  =#/#,wo GHIq   xx!H B 7yr2e9UCAFGKBA &&A NNA JJA ,,A 4 1qgJ 4 55A H5s D. c.t}ttt}|j dt gdd|j }dddd d}|j|jd td t|z d z d dy)N) descriptionformatter_classaction)rrErLrzchose 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.)typechoiceshelpc(ttSr)printrCrr!rrzmain..Ls ~/0r!c(ttSr)rr_rr!rrzmain..Ms 578r!c(ttSr)rrrr!rrzmain..Ns 023r!c(ttSr)rrrr!rrzmain..Os 79:r!ctdS)NzInvalid input.)rrr!rrzmain..Qs E*:$;r!r&<z.1fz minutes elapsed. ) rrror add_argumentint parse_argsgetrr)t0parserrswitchs rmainr>s B ,@BF sLP    D083:F =FJJt{{;<> B R$$7 89r!__main__)#roargparserrnumpyrscipy.integraterscipy.optimizerrrr*r r r r r rrrrrrsympy.polys.polyfuncsr ImportErrorrCrHrJr_rrrrlrr!rrs : 5 BBBB,  DC/ V rV rC L:. zFI   s$A++A32A3