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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
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