a Z>m?Z?m@Z@mAZAmBZBmCZCmDZDmEZEmFZFmGZGmHZHmIZImJZJmKZKmLZLmMZMmNZNmOZOmPZPmQZQmRZRmSZSdd lTmUZUmVZVmWZWmXZXmYZYmZZZm[Z[m\Z\m]Z]m^Z^m_Z_m`Z`maZambZbmcZcmdZdmeZemfZfmgZgmhZhmiZimjZjmkZkmlZlmmZmmnZne>d d Zoe>d d Zpe>ddZqe>ddZre>ddZse>ddZte>ddZue?esZve?erZwe?epZxe?eqZye?eoZze?etZ{e?euZ|dZ}iZ~e!dZe!dZddZee}ZdkddZdldd Zd!d"Zd#d$Zd%d&Zefd'd(Zefd)d*Zefd+d,Zefd-d.ZiZd/d0Zd1ZiZefd2d3Zedfd4d5Zedd6fd7d8Zefd9d:Zefd;d<ZdZd=d>Zgagagad?d@ZdAdBZdCZdDdEZdFZedGksJdHZdIZiZiZdJdKeedDZdLdMZdNdOZdPdQZdRdSZdTdUZdVdWZdmdYdZZdnd[d\Zdod]d^Zdpd_d`ZdqdadbZdrdcddZdsdedfZdtdgdhZefdidjZdS)uao ----------------------------------------------------------------------- This module implements gamma- and zeta-related functions: * Bernoulli numbers * Factorials * The gamma function * Polygamma functions * Harmonic numbers * The Riemann zeta function * Constants related to these functions ----------------------------------------------------------------------- N)xrange)MPZMPZ_ZEROMPZ_ONE MPZ_THREEgmpy) list_primesifacifac2moebius)- round_floor round_ceiling round_downround_up round_nearest round_fastlshift sqrt_fixed isqrt_fastfzerofonefnonefhalfftwofinffninffnanfrom_intto_intto_fixed from_man_exp from_rationalmpf_posmpf_negmpf_absmpf_addmpf_submpf_mul mpf_mul_intmpf_divmpf_sqrt mpf_pow_int mpf_rdiv_int mpf_perturbmpf_lempf_ltmpf_gt mpf_shift negative_rndreciprocal_rndbitcountto_float mpf_floormpf_sign ComplexResult) constant_memodef_mpf_constantmpf_pipi_fixed ln2_fixed log_int_fixedmpf_ln2mpf_expmpf_logmpf_powmpf_cosh mpf_cos_sin mpf_cosh_sinhmpf_cos_sin_pi mpf_cos_pi mpf_sin_piln_sqrt2pi_fixedmpf_ln_sqrt2pi sqrtpi_fixed mpf_sqrtpi cos_sin_fixed exp_fixed)mpc_zerompc_onempc_halfmpc_twompc_abs mpc_shiftmpc_posmpc_negmpc_addmpc_submpc_mulmpc_div mpc_add_mpf mpc_mul_mpf mpc_div_mpf mpc_mpf_div mpc_mul_int mpc_pow_intmpc_logmpc_expmpc_pow mpc_cos_pi mpc_sin_pimpc_reciprocal mpc_square mpc_sub_mpfcCs|d}t|>}}d\}}}|r|d|dd|d9}|dd|d|dd}|d|dd |dd |d|dd|d}||7}|d7}q|d ?S) N)rrr r(r)precaonestnrzf/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/mpmath/libmp/gammazeta.py catalan_fixedHs   < r|c Cs*t||dd}t}td}t|>}}t|}tt|||d}}d} ttd| |} t| ||} t | ||} t | |} | ||| |?} | dkrq|| 7}||d| d|d| 7}| d7} t |d| d| d|}t|||}qN||>t |}t t|| |} t | |} | S)N?rmrd)intrrrr<r2r(r% mpf_bernoullir*r r)r>rAr!) rtwprwfacrxONEpiZpipowZtwopi2ryZzeta2ntermKrzrzr{khinchin_fixedms.      rcCs|d}td|d}t|>}t}td|D]}|t|||d7}q.t||}||||7}|||dd7}|d}d}d} d} td} d}||>| ||} t| | } td||} t| | |}t || |}t ||}t |dkrqz||8}| || | | ||| df\}} }} | || | | ||| df\}} }} |d7}t | d|d|d|} qt |}|d 9}||>|d|?}|t|7}|t ttd|| ||7}|d }tt|| |}t ||S) NgQ?rmrlrr )rrrranger?rr!rr(r*r absr)r= euler_fixedrBrA)rtrNrrwkZlogNZpNrubjrDBrrArzrzr{glaisher_fixedsH      **  rcCs|d7}t|>}td|>}d}t}|r||7}||d9}|d|ddd|d}d|d|dd |d|}|d7}q$|d ?S) NrjMr rmrrorr)rrr)rtdrryrwrzrzr{ apery_fixeds   $ rc Csd}||7}tt|dtddd}d|}| t|}}t|>}}d}||d|d}||d|||}||7}||7}tt|t|dkrq|d7}qX|||>|S)Nrrrmrr)rmathlogr>rmaxr) rtextrapryrUrVrrzrzr{rs"  rcCsp|d}d}t|}t||}|tkr(qft||}t|t||}t|t||}t||}|d7}qt ||S)Nrjrmr) mpf_euler mpf_zeta_intrrBr)r r*rr&r )rtrmrwrxrzrzr{ mertens_fixed+s    rcsdd}d|dt}fdddD}fdd|D}d}t|}td D]2}t|tt||}t||||||<qRt||| }t||d d tkrqt||}|d 7}q@t|td }t|td}t ||S)Ncs$tfddtddDS)Nc3s&|]}|st||>VqdS)N)r .0rryrzr{ >z-twinprime_fixed..I..r)sumrrrzrr{I=sztwinprime_fixed..Irmrcsg|]}td|qS)r)r"rrrrzr{ Arz#twinprime_fixed..)rmrlrcsg|]}t||qSrz)r(rrrzr{rBrrrryri'i ) rrrr(r'r,r#rr*r )rtrresprimesZppowersryruirzrr{twinprime_fixed;s$     ri rlrcCs(t|d}tdd|||dS)z5Accurately estimate the size of B_n (even n > 2 only)rmgS㥛@r}gK7A`@)rrr)ryZlgnrzrzr{bernoulli_sizes rcCsV|dkr4|dkrtd|dkr$tS|dkr4ttS|d@r@tS|tkrz|t|ddkrzt|\}}t||||pvt S|t krt |||S|d}|d|d @7}t |}|r|\}}||vr|s||St||||S|\} } } || d krJt |||SnB|d krt |||Sdti}dtd tg\} } } }||ft |<| |krN| d } t| } d}td| |}| d krt}n| }td| d dD]}|| d |\}}}}}|r| }|t||||7}d |}|| d || d || d|| d|| d|| |9}|d |d |d |d|d|d|}q| dkrt| dt|}| dkrt| dt|}| d krt| dt|}t|||}tt|||t| |}||| <| d7} | | d| d| | d} | d kr8| d| d| | d| d } | | | g|dd<qJ||S)z.Computation of Bernoulli numbers (numerically)rmrz)Bernoulli numbers only defined for n >= 0rg?rrkrrrrrlr N) ValueErrorrr$rrBERNOULLI_PREC_CUTOFFrbernfracr"r MAX_BERNOULLI_CACHEmpf_bernoulli_hugebernoulli_cachegetr#rrrrrrr-f3f6r!r*r'r)ryrtrndrqrcachedZnumbersstaterbinZbin1ZcaseZszbmrwZsexprurZusignZumanZuexpZubcuZj6rrzrzr{rsv         H8   $rcCs|d}|tt|d}t|d|}t|t|||}t|tt|| |}t|d|}|d@srt |}t |||p~t S)Nrrmrrl) rrr mpf_gamma_intr(rr,r<r2r$r#r)ryrtrrZpiprecvrzrzr{rsrcCst|}|dkrgd|S|d@r(dSd}t|dD]}||ds8||9}q8t|tt|dd}t||}t|t|}t|t }||fS)a Returns a tuple of integers `(p, q)` such that `p/q = B_n` exactly, where `B_n` denotes the `n`-th Bernoulli number. The fraction is always reduced to lowest terms. Note that for `n > 1` and `n` odd, `B_n = 0`, and `(0, 1)` is returned. **Examples** The first few Bernoulli numbers are exactly:: >>> from mpmath import * >>> for n in range(15): ... p, q = bernfrac(n) ... print("%s %s/%s" % (n, p, q)) ... 0 1/1 1 -1/2 2 1/6 3 0/1 4 -1/30 5 0/1 6 1/42 7 0/1 8 -1/30 9 0/1 10 5/66 11 0/1 12 -691/2730 13 0/1 14 7/6 This function works for arbitrarily large `n`:: >>> p, q = bernfrac(10**4) >>> print(q) 2338224387510 >>> print(len(str(p))) 27692 >>> mp.dps = 15 >>> print(mpf(p) / q) -9.04942396360948e+27677 >>> print(bernoulli(10**4)) -9.04942396360948e+27677 .. note :: :func:`~mpmath.bernoulli` computes a floating-point approximation directly, without computing the exact fraction first. This is much faster for large `n`. **Algorithm** :func:`~mpmath.bernfrac` works by computing the value of `B_n` numerically and then using the von Staudt-Clausen theorem [1] to reconstruct the exact fraction. For large `n`, this is significantly faster than computing `B_1, B_2, \ldots, B_2` recursively with exact arithmetic. The implementation has been tested for `n = 10^m` up to `m = 6`. In practice, :func:`~mpmath.bernfrac` appears to be about three times slower than the specialized program calcbn.exe [2] **References** 1. MathWorld, von Staudt-Clausen Theorem: http://mathworld.wolfram.com/vonStaudt-ClausenTheorem.html 2. 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