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In this paper, we present an efficient algorithm to simulate the response of a multicomponent induction well-logging (MCIL) tool in a cylindrically layered anisotropic formation (transversely isotropic media). The tool consists of three pairs of orthogonally arranged transmitter-receiver coils for measuring the magnetic field around a borehole for oil and gas exploration. First, we will derive the EM field components to present the coupled transverse magnetic and transverse electric modes in the spectral domain for an arbitrary observation layer from the Maxwell equation and obtain the spectral-domain Green's function. The new local transmission and both the local and generalized reflections for the media are given. After that, we use a cubic spline interpolation to realize the inverse Fourier transformation and obtain the spatial-domain Green's function for the investigation of the response characteristics of the MCIL tool. Finally, we give some numerical results to explore the response characteristics of the tool in several different environments.