It is knowyn [5, Theorem 2.7] that if A > 0 (every element of A is nonnegative), then A has a inonnegative real eigenvalue equal to its spectral radius. If in addition A is irreducible, then this eigeuivalue is positive and simple. If a nonnegative irreducible matrix has k eigenvalues of modulus p(A), then A is said to be k-cyclic if k > I and primitive if k = 1. In a recent paper [8], Yamamoto iintroduced an iterative method for finding nested upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. Although the method applies to both cyclic and primitive matrices, it involves matrix squaring, so its utility may be offset by a substantial increase in computing costs over the power method [11, [2], [5]. These increased costs might be justified for cyclic matrices (for which the power method fails to converge) or for matrices with dominance ratio close to unity, a condition which slows the convergence of the power method. However, such justification would depend on comparisons of the convergence properties of the competing methods. In an attempt to make theoretical comparisons of Yamamoto's method with the classical power method, a third "hybrid" method was developed. This new method also applies to cyclic matrices, and, like Yamamoto's method, involves matrix squaring. This paper establishes the convergence properties of the new method and compares it with the other two. The hybrid method is shown to give as good (and in most cases considerably better) bounds as Yamamoto's method or the power method. A practical comparison supports the theoretical findings, which indicate that the