| import torch
|
| import math
|
| from typing import Optional
|
|
|
|
|
|
|
| def _phi(j, neg_h):
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| remainder = torch.zeros_like(neg_h)
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|
|
| for k in range(j):
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| remainder += (neg_h)**k / math.factorial(k)
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| phi_j_h = ((neg_h).exp() - remainder) / (neg_h)**j
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|
|
| return phi_j_h
|
|
|
| def calculate_gamma(c2, c3):
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| return (3*(c3**3) - 2*c3) / (c2*(2 - 3*c2))
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|
|
|
|
| def _gamma(n: int,) -> int:
|
| """
|
| https://en.wikipedia.org/wiki/Gamma_function
|
| for every positive integer n,
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| Γ(n) = (n-1)!
|
| """
|
| return math.factorial(n-1)
|
|
|
| def _incomplete_gamma(s: int, x: float, gamma_s: Optional[int] = None) -> float:
|
| """
|
| https://en.wikipedia.org/wiki/Incomplete_gamma_function#Special_values
|
| if s is a positive integer,
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| Γ(s, x) = (s-1)!*∑{k=0..s-1}(x^k/k!)
|
| """
|
| if gamma_s is None:
|
| gamma_s = _gamma(s)
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|
|
| sum_: float = 0
|
|
|
| for k in range(s):
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| numerator: float = x**k
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| denom: int = math.factorial(k)
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| quotient: float = numerator/denom
|
| sum_ += quotient
|
| incomplete_gamma_: float = sum_ * math.exp(-x) * gamma_s
|
| return incomplete_gamma_
|
|
|
| def phi(j: int, neg_h: float, ):
|
| """
|
| For j={1,2,3}: you could alternatively use Kat's phi_1, phi_2, phi_3 which perform fewer steps
|
|
|
| Lemma 1
|
| https://arxiv.org/abs/2308.02157
|
| ϕj(-h) = 1/h^j*∫{0..h}(e^(τ-h)*(τ^(j-1))/((j-1)!)dτ)
|
|
|
| https://www.wolframalpha.com/input?i=integrate+e%5E%28%CF%84-h%29*%28%CF%84%5E%28j-1%29%2F%28j-1%29%21%29d%CF%84
|
| = 1/h^j*[(e^(-h)*(-τ)^(-j)*τ(j))/((j-1)!)]{0..h}
|
| https://www.wolframalpha.com/input?i=integrate+e%5E%28%CF%84-h%29*%28%CF%84%5E%28j-1%29%2F%28j-1%29%21%29d%CF%84+between+0+and+h
|
| = 1/h^j*((e^(-h)*(-h)^(-j)*h^j*(Γ(j)-Γ(j,-h)))/(j-1)!)
|
| = (e^(-h)*(-h)^(-j)*h^j*(Γ(j)-Γ(j,-h))/((j-1)!*h^j)
|
| = (e^(-h)*(-h)^(-j)*(Γ(j)-Γ(j,-h))/(j-1)!
|
| = (e^(-h)*(-h)^(-j)*(Γ(j)-Γ(j,-h))/Γ(j)
|
| = (e^(-h)*(-h)^(-j)*(1-Γ(j,-h)/Γ(j))
|
|
|
| requires j>0
|
| """
|
| assert j > 0
|
| gamma_: float = _gamma(j)
|
| incomp_gamma_: float = _incomplete_gamma(j, neg_h, gamma_s=gamma_)
|
| phi_: float = math.exp(neg_h) * neg_h**-j * (1-incomp_gamma_/gamma_)
|
| return phi_
|
|
|
|
|
|
|
| from mpmath import mp, mpf, factorial, exp
|
|
|
|
|
| mp.dps = 80
|
|
|
| def phi_mpmath_series(j: int, neg_h: float) -> float:
|
| """
|
| Arbitrary‐precision phi_j(-h) via the remainder‐series definition,
|
| using mpmath’s mpf and factorial.
|
| """
|
| j = int(j)
|
| z = mpf(float(neg_h))
|
| S = mp.mpf('0')
|
| for k in range(j):
|
| S += (z**k) / factorial(k)
|
| phi_val = (exp(z) - S) / (z**j)
|
| return float(phi_val)
|
|
|
|
|
|
|
| class Phi:
|
| def __init__(self, h, c, analytic_solution=False):
|
| self.h = h
|
| self.c = c
|
| self.cache = {}
|
| if analytic_solution:
|
|
|
| self.phi_f = phi_mpmath_series
|
| self.h = mpf(float(h))
|
| self.c = [mpf(c_val) for c_val in c]
|
|
|
|
|
| else:
|
| self.phi_f = phi
|
|
|
|
|
| def __call__(self, j, i=-1):
|
| if (j, i) in self.cache:
|
| return self.cache[(j, i)]
|
|
|
| if i < 0:
|
| c = 1
|
| else:
|
| c = self.c[i - 1]
|
| if c == 0:
|
| self.cache[(j, i)] = 0
|
| return 0
|
|
|
| if j == 0 and type(c) in {float, torch.Tensor}:
|
| result = math.exp(float(-self.h * c))
|
| else:
|
| result = self.phi_f(j, -self.h * c)
|
|
|
| self.cache[(j, i)] = result
|
|
|
| return result
|
|
|
|
|
|
|
| from mpmath import mp, mpf, gamma, gammainc
|
|
|
| def superphi(j: int, neg_h: float, ):
|
| gamma_: float = gamma(j)
|
| incomp_gamma_: float = gamma_ - gammainc(j, 0, float(neg_h))
|
| phi_: float = float(math.exp(float(neg_h)) * neg_h**-j) * (1-incomp_gamma_/gamma_)
|
| return float(phi_)
|
|
|
|
|
