^MhD2dZddlmZmZddlZddlmZddlm Z m Z ddl m Z  ddl Z ddl m Z mZmZmZmZmZmZmZmZmZmZddlmZn #e$rYnwxYwd Zd Zd Zd Zd ZdZ dZ!e"dkr e!dSdS)zPrecompute coefficients of several series expansions of Wright's generalized Bessel function Phi(a, b, x). See https://dlmf.nist.gov/10.46.E1 with rho=a, beta=b, z=x. )ArgumentParserRawTextHelpFormatterN)quad)minimize_scalar curve_fit)time) EulerGammaRationalSSum factorialgamma gammasimppi polygammasymbolszeta)hornerc8d}td\}}}}g}g}g}t||zt|z t||z|zz |dtjf}t|t j|z |z}td|dzD]} | ||  |d } | td|dtd} | d| zz} ||| zt| z |t!| |t!| | z d} | dz } t#gd |||gD]F\} }tt%|D]$}| d | d |d t'||zz } %G| S) zATylor series expansion of Phi(a, b, x) in a=0 up to order 5. a b x krcdSNrargss g/var/www/html/test/jupyter/venv/lib/python3.11/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||3aAJ5GHHJq%)A,,&3J1eAg   3 3q!$$))!Q//88::??AA))IaOOQ//79oo66  2' Aill"###     f..00112222HA MMAAq 2211as1vv 1 1A $d$$Q$$$s1Q4yy0 0AA 1 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_seriesrH7s\  A a4*  a$q'')A!H4q!QqSkBB CCr!cPtjt|||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!ct dtd\}}}td\}}}t|t|td|i}g}g}g} gt t j|z t||zt|z t ||zzz |dtj fz} tddzD]| | |d} | t!ddt d} | dzz} | | z t z } dkr| t fd } | ddzz  t!d dd tdz} ||ztz |t+| | | t j| d |tt3D]/}|t|z|<0d }|dz }|dz }|dz }|dz }|dz }|dz }t5ddg||gD]H\}}tt3|D]&}|d|d|dz }|t7||z }'Itt3D]}|d|dz }|t7|z }|d|dz }|t7| |t|t|tdidz }|dz }|dz }|dz }t;fdtdz D}|| d  |z }|d |tdkz }|d!z }t;fd"td z D}|| d |z }|d |tdkz }|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_Z3rrcdSrrrs rrz(series_small_a_small_b..er r!r"c2t||dzzS)Nr)rJ)r:r9r<r6s rrz(series_small_a_small_b..ms9Q57193M3Mr!)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] + ..cRg|]#}|zt|z |dzz$S)rr .0r:Cr8s r z*series_small_a_small_b..5CCCq1Yq\\!AacF*CCCr!z test successful = z- Test B[3] == C[2] + b*C[3] + b^2/2*C[4] + ..cRg|]#}|zt|z |dzz$S)rErRrSs rrVz*series_small_a_small_b..rWr!)rrr rrr*r+r r r r)r,r-r.r/r0rr1seriesremoveOr2rPolycoeffsreverser4r3r5evalfsum)r7r9r:M_PIM_EGM_Z3c_subsr#r$r%r;r=r>pg_partrBr?r@rAtestrUr8r<r6s @@@@rseries_small_a_small_brfFs- E##JAq!Q/00D$$ D$q''4 8F A A A A q%)A,,& AqD1 eAaCEll *Q1:,>??@J1eAg  q!$$))!Q//88::??AA))IaOOQ//79oo66  2'v+eAhh& 66ooi&M&M&M&M&MOOG~~aeAgai~88 Yq!__baj99    Aill"###       1Q499V$$a((//11AIIKKK 3q66]]00!y||#--//!NA FFAA 22AA A ASzAq6**as1vv  A $d$$Q$$$ $A QqTNAA 3q66]] a    S1YY ^A^^^ S1D*dBd1ggFGG%))  <.gzHelper function g according to Wright (1935) g(n, rho, v) = (1 + (rho+2)/3 * v + (rho+2)*(rho+3)/(2*3) * v^2 + ...) Note: Wright (1935) uses square root of above definition. rLc6|dkstd|dkrdS|dz ||tt|dz|zt|dzz ttd|ztdz z ||zzzS)Nrzmust have n >= 0rrErL) ValueErrorrr)clsr<rhovgs revalz!asymptotic_series..g.evals66 !3444aqq1c1~~c!eAguSU|| ;<<ac 588 344556T:::r!N__name__ __module__ __qualname____doc__nargs classmethodrqrpsrrprjsI    : : : :  : : :r!rpc2eZdZdZdZefdZdS)!asymptotic_series..coef_CzCalculate coefficients C_m for integer m. C_m is the coefficient of v^(2*m) in the Taylor expansion in v=0 of Gamma(m+1/2)/(2*pi) * (2/(rho+1))^(m+1/2) * (1-v)^(-b) * g(rho, v)^(-m-1/2) rLc|dkstdtd}d|z | zd|z||| tddz zz}||d|z|dt d|zz }|t |tddzdtzz d|dzz |tddzzzz}|S)Nrzmust have m >= 0rorrE)rlrr r-r.r rr)rmmrnbetaror;resrps rrqz&asymptotic_series..coef_C.evals66 !3444 AA#$!!AaCa..A2hq!nn;L*MMJ//!QqS))..q!44y1~~ECq8Aq>>122ad;s1uIXa^^);<=>CJr!Nrrrysrcoef_Cr{sI           r!rz 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]) rc6g|]}|Sr) denominatorrTr9s rrVz%asymptotic_series..s EEEa!--//EEEr!zC[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/r[r\lcmcollectfactorxreplacer5recompilesubr1)r6rrr8rC0r?rBexprrrre_are_b re_digitsrps @rasymptotic_seriesrs E:::::::EN:::(,+&&KB4 2q  B,A ::A @@A ))A A A ##A //A 1eAg  **q"a  B"qyL1;;==EE5:d+;+;+B+B+D+DEEE6""v ''))11!U\BB}}bdD\** 7! 7 7 7 7 7 7 77 )! ) )#d)) ) ) )) III ::n % %D 1A ::m $ $D 1A &(##A $A <((I fa  A Hr!c  d d fd gdgdgdtj\cg}tjD]6 |t  fdd d d d i j7tj|}|d}d}tt|||ddd}d}|dz }|dz }|dz }|dz }|d d|Dz }|S)aFit optimal choice of epsilon for integral representation. The integrand of int_0^pi P(eps, a, b, x, phi) * dphi can exhibit oscillatory behaviour. It stems from the cosine of P and can be minimized by minimizing the arc length of the argument f(phi) = eps * sin(phi) - x * eps^(-a) * sin(a * phi) + (1 - b) * phi of cos(f(phi)). We minimize the arc length in eps for a grid of values (a, b, x) and fit a parametric function to it. ctjd|z| }|tj|z||z|ztj||zzz dz|z S)zDerivative of f w.r.t. phi.g?r)nppowercos)epsr7r8r9phieps_as rfpz$optimal_epsilon_integral..fpsTcA2&&RVC[[ 1q55=26!c'??#BBQFJJr!{Gz?dcbtfddtj|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 Rtjd|dzzS)NrrE)rsqrt)rr7r8rrr9s rrz=optimal_epsilon_integral..arclength..s-BBsAq!S,A,A1,D(D E Er!rr)epsrellimit)rrr)rr7r8r9rrrs```` r arclengthz+optimal_epsilon_integral..arclength sL EEEEEEEEru!..../1 1r!) MbP?g?g?g?rrErrh)rrr ) rg?rErr2rig@@g@g@cD|SNr)rrdata_adata_bdata_xrBs rrz*optimal_epsilon_integral..s( #vay&)28))=)=r!)riBoundedxatolr)boundsmethodoptions)r7r8r9rc F|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)))&a1q5kBF1II55RVRC!G__8LLRVBF^^!345666 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, cg|]}|dS)z.5grrs rrVz,optimal_epsilon_integral..:s4441qJJ444r!)rr) rmeshgridflattenr,sizer2rr9arraylistrjoin) best_epsdfr func_paramsr?rrrrrrBs @@@@@@roptimal_epsilon_integralrsKKK 1111115 4 4F   F E E EF[@@FFF$nn..0@0@$nn..FFFH 6;     ========#/#,wo G G GHI     x!!H  B 777yr2e9UCCCAFGGKBA &&A NNA JJA ,,A44 444 5 55A Hr!cjt}ttt}|dt gdd|}dddd d}||jd td t|z d z d ddS)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.)typechoiceshelpc8ttSr)printrCrr!rrzmain..Ls~//00r!c8ttSr)rrfrr!rrzmain..Ms57788r!c8ttSr)rrrr!rrzmain..Ns02233r!c8ttSr)rrrr!rrzmain..Os799::r!c tdS)NzInvalid input.)rrr!rrzmain..QsE*:$;$;r!r&<z.1fz minutes elapsed. ) rrrvr add_argumentint parse_argsgetrr)t0parserrswitchs rmainr>s B ,@BBBF sLLLP     D008833::F =FJJt{;;<<>>> 8 R 8 8 8 899999r!__main__)#rvargparserrnumpyrscipy.integraterscipy.optimizerrrr*r r r r r rrrrrrsympy.polys.polyfuncsr ImportErrorrCrHrJrfrrrrsrr!rrs :9999999 55555555 LLLBBBBBBBBBBBBBBBBBBBBBBBBBB,,,,,,,   D    DCCC/// V V V rV V V rC C C L:::. zDFFFFFs$A AA