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We have established an iterative scheme to calculate with 15-digit accuracythe numerical values of Ambartsumian-Chandrasekhar's H-functions foranisotropic scattering characterized by the four-term phase function: themethod incorporates some advantageous features of the iterative procedure ofKawabata (2015) and the double-exponential integration formula~(DE-formula) ofTakahashi and Mori (1974), which proved highly effective in Kawabata (2016).Actual calculations of the H-functions have been carried out employing 27selected cases of the phase function, 56 values of the single scattering albedo$\varpi_0$, and 36 values of an angular variable $\mu(=\cos \theta)$, with$\theta$ being the zenith angle specifying the direction of incidence and/oremergence of radiation.  Partial results obtained for conservative isotropic scattering, Rayleighscattering, and anisotropic scattering due to a full four-term phase functionare presented. As a sample application of the isotropic scattering H-function,an attempt is made in Appendix to simulate by iteratively solving theAmbartsumian equation the values of the plane and spherical albedos of asemi-infinite, homogeneous atmosphere calculated by Rogovtsov and Borovik(2016), who employed their analytical representations for these quantities andthe single-term and two-term Henyey-Greenstein phase functions of appreciablyhigh degrees of anisotropy, to find that our results are in satisfactoryagreement with theirs.

15-Digit Accuracy Calculations of Ambartsumian-Chandrasekhar's $H$-Functions for Four-Term Phase Functions with the Double-Exponential Formula