a Z>m?Z?m@Z@mAZAmBZBmCZCmDZDmEZEmFZFmGZGmHZHmIZImJZJmKZKmLZLmMZMmNZNdd l mOZOdd lPmQZQmRZRmSZSGd d d eTZUd dZVedkrddZVe fddZWddZXe fddZYddZZddZ[ddZ\ddZ]e dfd d!Z^e dfd"d#Z_e fd$d%Z`e fd&d'Zae dfd(d)ZbdMd*d+ZcdNd,d-Zde d.fd/d0Zee fd1d2Zfe fd3d4Zge fd5d6Zhe fd7d8Zie fd9d:Zje fd;d<Zke fd=d>Zle fd?d@Zme fdAdBZne fdCdDZoe fdEdFZpe fdGdHZqe fdIdJZre fdKdLZsdS)Oz This module implements computation of hypergeometric and related functions. In particular, it provides code for generic summation of hypergeometric series. Optimized versions for various special cases are also provided. N)MPZ_ZEROMPZ_ONEBACKENDxrangeexec_)gcd)% ComplexResult round_fast round_nearest negative_rndbitcountto_fixed from_man_expfrom_intto_int from_rationalfzerofonefnoneftwofinffninffnanmpf_signmpf_addmpf_absmpf_posmpf_cmpmpf_ltmpf_lempf_gt mpf_min_max mpf_perturbmpf_neg mpf_shiftmpf_submpf_mulmpf_div sqrt_fixedmpf_sqrt mpf_rdiv_int mpf_pow_int to_rational) mpf_pimpf_expmpf_logpi_fixed mpf_cos_sinmpf_cosmpf_sinr* agm_fixed)mpc_onempc_sub mpc_mul_mpfmpc_mulmpc_negcomplex_int_powmpc_div mpc_add_mpf mpc_sub_mpfmpc_logmpc_addmpc_pos mpc_shift mpc_is_infnanmpc_zerompc_sqrtmpc_abs mpc_mpf_div mpc_squarempc_exp)ifac) mpf_gamma_int mpf_euler euler_fixedc@s eZdZdS) NoConvergenceN)__name__ __module__ __qualname__rRrRe/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/mpmath/libmp/libhyper.pyrN+srNc Cs|\}}}}d|}d|||d|||d|f}d|v}|dk}|pN|} g} | j} g} g} g}g}g}g}g}g}| d| d| d| d| r| d |r| d | d | d | d n| d| d | d| d| d| d| d|r2| d| d| d| d| dt|D]\}}ddg||k}|dkr| |g||k|| d|||fq:|dkr| |g||k|| d|||||fq:|dkr2||g||k|| d|| d | d| d| d||f| d| d ||fn|dkr||g||k|| d!|| d"| d | d#| d | d| d| d$||f| d| d%||f| d| d| d&||f| d| d'||fntq:t|}t|}t||}||d}||d}| d(| d)| d*| rP| d+| d,d-d.d/| Dd0d/| Dd1d/|D}d-d2d/|Dd3d/|Dd4d/| Dd5g}|r| d6|| d7|| d8|r| d9| d:| d;| rt|D]}| d<||||fq|D]}| d=d>t|q4|D]}| d?d>t|qTt|D]}| d@||||fqx|D]}| dAd>t|q|D]}| dBd>t|q|r|r| dC| dD| dEn| dF| dGn0|r0| dH| dD| dEn| dI| dJ|D]*}| dKd>t|| dD| dEqD|D]j}| dLd>t|| dMd>t|| dNd>t|| dOd>t|| dPd>t|qtnt|D]}| d<||||fq|D]}| d=d>t|q|D]}| d?d>t|q.|rZ| dQn| dI| r| dR| dS| dT| dUn| dR| dV| dU| dW| dX| D]}| dYd>t|q|D]}| dZd>t|q| D]}| d[d>t|q|D]}| d\d>t|q|D]}| d]d>t|q6|D]}| d^d>t|qV|D]}| d_d>t|qv|D]}| d`d>t|q| r| da| db| dc| dd| de| df| dg| dh| di| d| dj| dkn0| da| dc| dl| d| dj| dmdndodp| D} dq|| } i}t | t || ||fS)rz Returns a function that sums a generalized hypergeometric series, for given parameter types (integer, rational, real, complex). zhypsum_%i_%i_%s_%s_%sNCz$MAX = kwargs.get('maxterms', wp*100)zHIGH = MPZ_ONE<= 0:z ZRE = xm << offsetzelse:z ZRE = xm >> (-offset)zoffset = ye + wpz ZIM = ym << offsetz ZIM = ym >> (-offset)ABZz%sINT_%i = coeffs[%i]Qz!%sP_%i, %sQ_%i = coeffs[%i]._mpq_Rz%xsign, xm, xe, xbc = coeffs[%i]._mpf_z %sREAL_%i = xm << offsetz %sREAL_%i = xm >> (-offset)z__re, __im = coeffs[%i]._mpc_zxsign, xm, xe, xbc = __rezysign, ym, ye, ybc = __imz %sCRE_%i = xm << offsetz %sCRE_%i = xm >> (-offset)z %sCIM_%i = ym << offsetz %sCIM_%i = ym >> (-offset)zfor n in xrange(1,10**8):z if n in magnitude_check:z" p_mag = bitcount(abs(PRE))z. p_mag = max(p_mag, bitcount(abs(PIM)))z% magnitude_check[n] = wp-p_magz * cSsg|]}ddt|qS)zAINT_##replacestr.0irRrRrS z%make_hyp_summator..cSsg|]}ddt|qS)zAP_#r[r\r_rRrRrSrbrccSsg|]}ddt|qS)zBQ_#r[r\r_rRrRrSrbrccSsg|]}ddt|qS)zBINT_#r[r\r_rRrRrSrbrccSsg|]}ddt|qS)zBP_#r[r\r_rRrRrSrbrccSsg|]}ddt|qS)zAQ_#r[r\r_rRrRrSrbrcnz mul = z div = z if not div:z if not mul:z breakz raise ZeroDivisionErrorz$ PRE = PRE * AREAL_%i // BREAL_%iz PRE = (PRE * AREAL_#) >> wpr[z PRE = (PRE << wp) // BREAL_#z$ PIM = PIM * AREAL_%i // BREAL_%iz PIM = (PIM * AREAL_#) >> wpz PIM = (PIM << wp) // BREAL_#zI PRE, PIM = (mul*(PRE*ZRE-PIM*ZIM))//div, (mul*(PIM*ZRE+PRE*ZIM))//divz PRE >>= wpz PIM >>= wpz* PRE = ((mul * PRE * ZRE) >> wp) // divz* PIM = ((mul * PIM * ZRE) >> wp) // divz= PRE, PIM = (PRE*ZRE-PIM*ZIM)//div, (PIM*ZRE+PRE*ZIM)//divz$ PRE = ((PRE * ZRE) >> wp) // divz$ PIM = ((PIM * ZRE) >> wp) // divz; PRE, PIM = PRE*ACRE_#-PIM*ACIM_#, PIM*ACRE_#+PRE*ACIM_#z% mag = BCRE_#*BCRE_#+BCIM_#*BCIM_#z re = PRE*BCRE_# + PIM*BCIM_#z im = PIM*BCRE_# - PRE*BCIM_#z PRE = (re << wp) // magz PIM = (im << wp) // magz* PRE = ((PRE * mul * ZRE) >> wp) // divz SRE += PREz SIM += PIMz1 if (HIGH > PRE > LOW) and (HIGH > PIM > LOW):z breakz if HIGH > PRE > LOW:z if n > MAX:zc raise NoConvergence('Hypergeometric series converges too slowly. Try increasing maxterms.')z AINT_# += 1z BINT_# += 1z AP_# += AQ_#z BP_# += BQ_#z AREAL_# += onez BREAL_# += onez ACRE_# += onez BCRE_# += onez%a = from_man_exp(SRE, -wp, prec, 'n')z%b = from_man_exp(SIM, -wp, prec, 'n')zif SRE:z if SIM:z( magn = max(a[2]+a[3], b[2]+b[3])z else:z magn = a[2]+a[3]z elif SIM:z magn = b[2]+b[3]z magn = -wp+1zreturn (a, b), True, magnz magn = a[2]+a[3]zreturn a, False, magn css|]}d|VqdS)z NrR)r`linerRrRrS +rcz$make_hyp_summator..zBdef %s(coeffs, z, prec, wp, epsshift, magnitude_check, **kwargs): ) joinappend enumerate ValueErrorlenminranger]r^rglobals) keypq param_typesztypeZpstringfnameZhave_complex_paramZhave_complex_argZ have_complexsourceaddZaintZaratZbintZbratZarealZbrealZacomplexZbcomplexraflagWZl_arealZl_brealZcancellable_realZnoncancellable_real_numZnoncancellable_real_den multiplierZdivisork namespacerRrRrSmake_hyp_summator@sp  "                (  (      (              r}Zsagecs4ddlm|\fdd}d|fS)z Returns a function that sums a generalized hypergeometric series, for given parameter types (integer, rational, real, complex). r)hypsum_internalc s||||||| SNrR)ZcoeffszprecwpZepsshiftZmagnitude_checkkwargsr~rqrsrrrtrRrS_hypsum?s z"make_hyp_summator.._hypsumz(none))Zsage.libs.mpmath.ext_mainr~)rprrRrrSr}8s  cCs||\}}}}|s8|tkrtS|tkr(tS|tkr4tStS||}tj}|dkrd|dd||dkr|rzttd||Sttd||S|| krt |d}t t |d|d} t || ||S|t |d} t t|| } | | | ?} | dd} }}|rF| | | ?|} | d|d}|d@r4| |8} n| |7} |d7}q| | d>tt| | } |rl| } t| | ||S) NrgW?ri90)rrrrrrmathlogr#r%r*r.r(absrr)r1r)xrrndsignmanexpbcsizeZlgcrtt2stermr{rRrRrSmpf_erfOs>    "      rcCs t|}|dd|krdSdS)Nr ףp= ?TF)r)rrrdrRrRrSerfc_check_series|srcCs|\}}}}|s8|tkrtS|tkr(tS|tkr4tStS|d}||}|tdd|7}|pd|dk} | stt||s| rttt ||dt |||St |d} ttt ||t | ddd||St |>} } d} dt||d|?}d}| d|d|>|} |dkr | | ksT| s*qT|d@r>| | 8} n| | 7} | } |d7}q| |>tt||} t| | |} ttt|||||}tt|| ||||}|S)Nrrr rr)rrrrrrmaxrr&rr rintrrr)r1rr/r$r'r()rrrrrrrrmagZ regular_erfrdrrZ term_prevrr{ryrRrRrSmpf_erfcs@      (    rcCs:|}}d}|r6|||?|}|||7}|d7}q |SNrrrR)rrrrr{rRrRrS ei_taylors  rc Cst}|}}|}}d}||||dkr||||||?||||||?}}|||7}|||7}|d7}q||fS)Nrr)r) zrezimr_abssretresimtimr{rRrRrScomplex_ei_taylors2   rcCsNt|>}||>|}}||}d}|rJ||||?}||7}|d7}q$|Sr)r)rronerrr{rRrRrS ei_asymptotics rc Cst}t|>}|||||?}||>|}}| |>|}} ||} |} d} |||| dkr||| || |?||| || |?}} | |7} | | 7} | d7} | |krRtqR| | fS)Nrir)rrrN) rrrrrMZxrerZximrrrr{rRrRrScomplex_ei_asymptotics 2rFcCsz|r t|}|\}}}}|r4|s4|tkr,tStd|r4d|||f}||} |d} | | k} | s|dkrt||>} n || ?} | t| ddk} | r| | krt} nttt|| | | } t | t || | } t | |||} nT| dtt |7} t|| }t || t| } t| | }t|| }t||||} n4|tkrDt} n$|tkrTt} n|tkrdt} nt} |rvt| } | S)NzE1(x) for x < 0rrV-?rr)r$rrr rrrrrr'r/r(rrrMr0rrr)rrre1rrrrxabsxmagr can_use_asympZxabsintvut1rrRrRrSmpf_eisH       rcCs|r t|}|\}}|\}}}} |\} } } } |tkrx|rhtt|||}|s\tt||}nt}||fSt|||tfS|tkr|r| sttfS|d}|| }| | }t||}||k}|stt|tt|}|t |ddk}z|r||krt tf}nt|| }|}||}t| >}|r>|r>||7}|d7}|||| ?}qt || }|rht |t || d| | }n t ||| }t |t|| ||}n|dkrtt|||}nh|dkr|d | krtt|| }|r| d@rt|}nt |t ||d| | }t|| }}dg|d}td|dD]}||d|||<q.|ddd }|d| >}td|dD]>}|d@r||||8}n||||7}||| ?}qrt|| | }t |t || }|rt |t || | | }t||}t |tt|d||}nt| rtt|d }|rXt t| |||}| d@r~t|}n&t t t| t || d| | |||}||fS|dfSdS) z E_n(x), n an integer, x real With gamma=True, computes Gamma(n,x) (upper incomplete gamma function) Returns (real, None) if real, otherwise (real, imag) The imaginary part is an optional branch cut term rNrrirdrr)rrrKrrNotImplementedErrorr rr$r/r'r,r(rrmrrrrrrnrrrJr.) rdrrrgammarrrrZn_origrrZnmagZ have_imagZnegxrreZcan_use_asymptotic_seriesximZsizZtolrrrZT1Zfacsr{ZT2rZrimrRrRrS mpf_expintUs      6            &rcCst||}|| |?}|dkr6dt|>d}}}n||d}}}|rx||||d|?}|||7}|d7}qFt|| S)z2 0 - 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