def mpc_besseljn(n, z, prec, rounding=round_fast): negate = n < 0 and n & 1 n = abs(n) origprec = prec prec += 20 + bitcount(abs(n)) zre, zim = z zre = to_fixed(zre, prec) zim = to_fixed(zim, prec) z2re = (zre**2 - zim**2) >> prec z2im = (zre * zim) >> (prec - 1) if not n: sre = tre = MP_ONE << prec sim = tim = MP_ZERO else: re, im = complex_int_pow(zre, zim, n) sre = tre = (re // int_fac(n)) >> ((n - 1) * prec + n) sim = tim = (im // int_fac(n)) >> ((n - 1) * prec + n) k = 1 while abs(tre) + abs(tim) > 3: p = -4 * k * (k + n) tre, tim = tre * z2re - tim * z2im, tim * z2re + tre * z2im tre = (tre // p) >> prec tim = (tim // p) >> prec sre += tre sim += tim k += 1 if negate: sre = -sre sim = -sim re = from_man_exp(sre, -prec, origprec, rounding) im = from_man_exp(sim, -prec, origprec, rounding) return (re, im)
def mpc_besseljn(n, z, prec, rounding=round_fast): negate = n < 0 and n & 1 n = abs(n) origprec = prec zre, zim = z mag = max(zre[2]+zre[3], zim[2]+zim[3]) prec += 20 + n*bitcount(n) + abs(mag) if mag < 0: prec -= n * mag zre = to_fixed(zre, prec) zim = to_fixed(zim, prec) z2re = (zre**2 - zim**2) >> prec z2im = (zre*zim) >> (prec-1) if not n: sre = tre = MP_ONE << prec sim = tim = MP_ZERO else: re, im = complex_int_pow(zre, zim, n) sre = tre = (re // int_fac(n)) >> ((n-1)*prec + n) sim = tim = (im // int_fac(n)) >> ((n-1)*prec + n) k = 1 while abs(tre) + abs(tim) > 3: p = -4*k*(k+n) tre, tim = tre*z2re - tim*z2im, tim*z2re + tre*z2im tre = (tre // p) >> prec tim = (tim // p) >> prec sre += tre sim += tim k += 1 if negate: sre = -sre sim = -sim re = from_man_exp(sre, -prec, origprec, rounding) im = from_man_exp(sim, -prec, origprec, rounding) return (re, im)
def mpf_besseljn(n, x, prec, rounding=round_fast): negate = n < 0 and n & 1 n = abs(n) origprec = prec prec += 20 + bitcount(abs(n)) x = to_fixed(x, prec) x2 = (x**2) >> prec if not n: s = t = MP_ONE << prec else: s = t = (x**n // int_fac(n)) >> ((n - 1) * prec + n) k = 1 while t: t = ((t * x2) // (-4 * k * (k + n))) >> prec s += t k += 1 if negate: s = -s return from_man_exp(s, -prec, origprec, rounding)
def mpf_besseljn(n, x, prec): negate = n < 0 and n & 1 n = abs(n) origprec = prec prec += 20 + bitcount(abs(n)) x = to_fixed(x, prec) x2 = (x**2) >> prec if not n: s = t = MP_ONE << prec else: s = t = (x**n // int_fac(n)) >> ((n-1)*prec + n) k = 1 while t: t = ((t * x2) // (-4*k*(k+n))) >> prec s += t k += 1 if negate: s = -s return from_man_exp(s, -prec, origprec, round_nearest)
def mpf_besseljn(n, x, prec, rounding=round_fast): prec += 50 negate = n < 0 and n & 1 mag = x[2] + x[3] n = abs(n) wp = prec + 20 + n * bitcount(n) if mag < 0: wp -= n * mag x = to_fixed(x, wp) x2 = (x**2) >> wp if not n: s = t = MP_ONE << wp else: s = t = (x**n // int_fac(n)) >> ((n - 1) * wp + n) k = 1 while t: t = ((t * x2) // (-4 * k * (k + n))) >> wp s += t k += 1 if negate: s = -s return from_man_exp(s, -wp, prec, rounding)
def mpf_besseljn(n, x, prec, rounding=round_fast): prec += 50 negate = n < 0 and n & 1 mag = x[2]+x[3] n = abs(n) wp = prec + 20 + n*bitcount(n) if mag < 0: wp -= n * mag x = to_fixed(x, wp) x2 = (x**2) >> wp if not n: s = t = MP_ONE << wp else: s = t = (x**n // int_fac(n)) >> ((n-1)*wp + n) k = 1 while t: t = ((t * x2) // (-4*k*(k+n))) >> wp s += t k += 1 if negate: s = -s return from_man_exp(s, -wp, prec, rounding)
def mpf_expint(n, x, prec, rnd=round_fast, gamma=False): """ 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 """ sign, man, exp, bc = x if not man: if gamma: if x == fzero: # Actually gamma function pole if n <= 0: return finf return mpf_gamma_int(n, prec, rnd) if x == finf: return fzero, None # TODO: could return finite imaginary value at -inf return fnan, fnan else: if x == fzero: if n > 1: return from_rational(1, n - 1, prec, rnd), None else: return finf, None if x == finf: return fzero, None return fnan, fnan n_orig = n if gamma: n = 1 - n wp = prec + 20 xmag = exp + bc # Beware of near-poles if xmag < -10: raise NotImplementedError nmag = bitcount(abs(n)) have_imag = n > 0 and sign negx = mpf_neg(x) # Skip series if direct convergence if n == 0 or 2 * nmag - xmag < -wp: if gamma: v = mpf_exp(negx, wp) re = mpf_mul(v, mpf_pow_int(x, n_orig - 1, wp), prec, rnd) else: v = mpf_exp(negx, wp) re = mpf_div(v, x, prec, rnd) else: # Finite number of terms, or... can_use_asymptotic_series = -3 * wp < n <= 0 # ...large enough? if not can_use_asymptotic_series: xi = abs(to_int(x)) m = min(max(1, xi - n), 2 * wp) siz = -n * nmag + (m + n) * bitcount(abs(m + n)) - m * xmag - ( 144 * m // 100) tol = -wp - 10 can_use_asymptotic_series = siz < tol if can_use_asymptotic_series: r = ((-MP_ONE) << (wp + wp)) // to_fixed(x, wp) m = n t = r * m s = MP_ONE << wp while m and t: s += t m += 1 t = (m * r * t) >> wp v = mpf_exp(negx, wp) if gamma: # ~ exp(-x) * x^(n-1) * (1 + ...) v = mpf_mul(v, mpf_pow_int(x, n_orig - 1, wp), wp) else: # ~ exp(-x)/x * (1 + ...) v = mpf_div(v, x, wp) re = mpf_mul(v, from_man_exp(s, -wp), prec, rnd) elif n == 1: re = mpf_neg(mpf_ei(negx, prec, rnd)) elif n > 0 and n < 3 * wp: T1 = mpf_neg(mpf_ei(negx, wp)) if gamma: if n_orig & 1: T1 = mpf_neg(T1) else: T1 = mpf_mul(T1, mpf_pow_int(negx, n - 1, wp), wp) r = t = to_fixed(x, wp) facs = [1] * (n - 1) for k in range(1, n - 1): facs[k] = facs[k - 1] * k facs = facs[::-1] s = facs[0] << wp for k in range(1, n - 1): if k & 1: s -= facs[k] * t else: s += facs[k] * t t = (t * r) >> wp T2 = from_man_exp(s, -wp, wp) T2 = mpf_mul(T2, mpf_exp(negx, wp)) if gamma: T2 = mpf_mul(T2, mpf_pow_int(x, n_orig, wp), wp) R = mpf_add(T1, T2) re = mpf_div(R, from_int(int_fac(n - 1)), prec, rnd) else: raise NotImplementedError if have_imag: M = from_int(-int_fac(n - 1)) if gamma: im = mpf_div(mpf_pi(wp), M, prec, rnd) else: im = mpf_div( mpf_mul(mpf_pi(wp), mpf_pow_int(negx, n_orig - 1, wp), wp), M, prec, rnd) return re, im else: return re, None
def mpf_expint(n, x, prec, rnd=round_fast, gamma=False): """ 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 """ sign, man, exp, bc = x if not man: if gamma: if x == fzero: # Actually gamma function pole if n <= 0: return finf return mpf_gamma_int(n, prec, rnd) if x == finf: return fzero, None # TODO: could return finite imaginary value at -inf return fnan, fnan else: if x == fzero: if n > 1: return from_rational(1, n-1, prec, rnd), None else: return finf, None if x == finf: return fzero, None return fnan, fnan n_orig = n if gamma: n = 1-n wp = prec + 20 xmag = exp + bc # Beware of near-poles if xmag < -10: raise NotImplementedError nmag = bitcount(abs(n)) have_imag = n > 0 and sign negx = mpf_neg(x) # Skip series if direct convergence if n == 0 or 2*nmag - xmag < -wp: if gamma: v = mpf_exp(negx, wp) re = mpf_mul(v, mpf_pow_int(x, n_orig-1, wp), prec, rnd) else: v = mpf_exp(negx, wp) re = mpf_div(v, x, prec, rnd) else: # Finite number of terms, or... can_use_asymptotic_series = -3*wp < n <= 0 # ...large enough? if not can_use_asymptotic_series: xi = abs(to_int(x)) m = min(max(1, xi-n), 2*wp) siz = -n*nmag + (m+n)*bitcount(abs(m+n)) - m*xmag - (144*m//100) tol = -wp-10 can_use_asymptotic_series = siz < tol if can_use_asymptotic_series: r = ((-MP_ONE) << (wp+wp)) // to_fixed(x, wp) m = n t = r*m s = MP_ONE << wp while m and t: s += t m += 1 t = (m*r*t) >> wp v = mpf_exp(negx, wp) if gamma: # ~ exp(-x) * x^(n-1) * (1 + ...) v = mpf_mul(v, mpf_pow_int(x, n_orig-1, wp), wp) else: # ~ exp(-x)/x * (1 + ...) v = mpf_div(v, x, wp) re = mpf_mul(v, from_man_exp(s, -wp), prec, rnd) elif n == 1: re = mpf_neg(mpf_ei(negx, prec, rnd)) elif n > 0 and n < 3*wp: T1 = mpf_neg(mpf_ei(negx, wp)) if gamma: if n_orig & 1: T1 = mpf_neg(T1) else: T1 = mpf_mul(T1, mpf_pow_int(negx, n-1, wp), wp) r = t = to_fixed(x, wp) facs = [1] * (n-1) for k in range(1,n-1): facs[k] = facs[k-1] * k facs = facs[::-1] s = facs[0] << wp for k in range(1, n-1): if k & 1: s -= facs[k] * t else: s += facs[k] * t t = (t*r) >> wp T2 = from_man_exp(s, -wp, wp) T2 = mpf_mul(T2, mpf_exp(negx, wp)) if gamma: T2 = mpf_mul(T2, mpf_pow_int(x, n_orig, wp), wp) R = mpf_add(T1, T2) re = mpf_div(R, from_int(int_fac(n-1)), prec, rnd) else: raise NotImplementedError if have_imag: M = from_int(-int_fac(n-1)) if gamma: im = mpf_div(mpf_pi(wp), M, prec, rnd) else: im = mpf_div(mpf_mul(mpf_pi(wp), mpf_pow_int(negx, n_orig-1, wp), wp), M, prec, rnd) return re, im else: return re, None