A method of exact analysis for the cylindrical fiber with an arbitrary but azimuthally symmetric refractive-index distribution is given. The index distribution is simulated by a multilayer structure and the analysis is formulated exactly by using iterative multiplication of four by four matrices. Then the four by four matrices are approximately separated into two systems of two by two matrices with a good accuracy for the fiber with small index difference. In this way, the computing time is reduced almost 1/6 times compared with the former method. As an example of the application, the propagation modes of a focusing fiber are identified which previously were not exactly known. It is revealed that the group-velocity differences between the modes which are degenerate in the scalar analysis can be larger than those of the non-degenerate modes.