We have established an iterative scheme to calculate with 15-digit accuracy the numerical values of Ambartsumian-Chandrasekhar's H-functions for anisotropic scattering characterized by the four-term phase function: the method incorporates some advantageous features of the iterative procedure of Kawabata (2015, Astrophys. Space Sci. 358:32) and the doubleexponential integration formula (DE-formula) of Takahashi and Mori (1974, Publ. RIMS, Kyoto Univ. 9, 721), which proved highly effective in Kawabata (2016, Astrophys. Space Sci. 361:373). Actual calculations of the H-functions have been carried out employing 27 selected cases of the phase function, 56 values of the single scattering albedo ̟ 0 , and 36 values of an angular variable µ(= cos θ), with θ being the zenith angle specifying the direction of incidence and/or emergence of radiation. Partial results obtained for conservative isotropic scattering, Rayleigh scattering, and anisotropic scattering due to a full four-term phase function are presented. They indicate that it is important to simultaneously verify accuracy of the numerical values of the H-functions for µ < 0.05, the domain often neglected in tabulation. As a sample application of the isotropic scattering H-function, an attempt is made in Appendix to simulate by iteratively solving the Ambartsumian equation the values of the plane and spherical albedos of a semi-infinite, homogeneous atmosphere calculated by Rogovtsov and Borovik (2016, J. Quant. Spectr. Radiat. Transf. 183, 128), who employed their analytical representations for these quantities and the singleterm and two-term Henyey-Greenstein phase functions of appreciably high degrees of anisotropy. While our results are in satisfactory agreement with theirs, our Kiyoshi Kawabata Department of Physics, Tokyo University of Science, Shinjukuku, Tokyo 162-8601, Japan E-mail: kawabata@rs.kagu.tus.ac.jp procedure is in need of a faster algorithm to routinely deal with problems involving highly anisotropic phase functions giving rise to near-conservative scattering.