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We develop a general method for analyzing the renormalization group equations (RGE's) to one-loop and two-loop order in grand unified theories (GUT's) which can have an arbitrary number of symmetry-breaking scales and/or mass scales. The method is more efficient, convenient, and accurate than the current method for determining the ranges of symmetry-breaking scales and/or mass scales, as well as the ranges of unified gauge couplings. It does not involve the usual numerical integration of the RGE's at all, but in fact solves the RGE's analytically by a perturbative approach. The results are presented graphically and analyzed using simple geometrical considerations. We apply the method to two-step symmetry-breaking chains of SO(10) GUT's and compare the results with those obtained by numerical integration. We discover that the numerical method itself often introduces errors which are comparable with two-loop corrections. Our method avoids these. We also apply our method to two-step symmetry-breaking chains of supersymmetric SO(10) GUT's.
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