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Computation of a general integral of Fermi-Dirac distribution by McDougall-Stoner method. (English) Zbl 1334.65060
Summary: We extended the method of J. McDougall and E. C. Stoner [Philos. Trans. R. Soc. Lond., Ser. A 237, 67–104 (1938; Zbl 0018.16201)] to the computation of a general integral of the Fermi-Dirac distribution, $$F(\eta)$$. When $$\eta > 0$$, the new method splits $$F(\eta)$$ into a sum of three parts, $$A(\eta), B(\eta)$$, and $$C(\eta)$$, and integrates them exactly and/or numerically by means of the double-exponential quadrature rules. As $$\eta$$ increases, $$B(\eta) / F(\eta)$$ damps algebraically while $$C(\eta) / F(\eta)$$ decays exponentially. Thus, they can be ignored when $$\eta$$ exceeds certain threshold values depending on the input error tolerance. When $$A(\eta)$$ is exactly computable and $$\eta$$ is sufficiently large such that $$B(\eta)$$ and $$C(\eta)$$ are negligible, the new method was 60–1000 times faster than the numerical integration of $$F(\eta)$$. Even if $$B(\eta)$$ and/or $$C(\eta)$$ are not negligible, their smallness and rapid decay significantly accelerate their numerical quadrature so that the speed-up factor becomes 3–20 unless $$\eta$$ is less than 2.0–4.5. On the other hand, when $$A(\eta)$$ is numerically integrated, the speed-up factor diminishes to 2–5. Still, the superiority of the new method to the numerical integration of $$F(\eta)$$ is unchanged.

##### MSC:
 65D30 Numerical integration 60E05 Probability distributions: general theory
##### Software:
Algorithm 745; DLMF; Fermi-Dirac; Mathematica; QUADPACK
Full Text:
##### References:
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