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This paper is concerned with nonhomogeneous refinement equations of the form ϕ(x)=∑lower limit α∈Zs a(α)ϕ(Mx−α)+g(x), x∈Rs, where the vector of functions ϕ=(ϕ1,…,ϕr)T is unknown, g is a given vector of compactly supported functions on Rs, a is a finitely supported sequence of r×r matrices called the refinement mask, and M is an s×s integer matrix such that limn→∞M−n=0. Our approach will be to consider the convergence rates of the cascade algorithms associated with nonhomogeneous refinement equations mentioned above. The cascade algorithms associated with mask a, nonhomogeneous term g, and dilation matrix M generates a sequence ϕn, n=1,2,…, by the iterative process ϕn(x)=∑lower limit α∈Zs a(α)ϕn−1(Mx−α)+g(x), x∈Rs, from a starting vector of function ϕ0 in (Lp(Rs))r (0The aim of this paper is to give a characterization of the convergence rates of the cascade algorithms associated with a,g,ϕ0 and dilation matrix M in (Lp(Rs))r (0 in terms of the p-norm joint spectral radius of a finite collection of some linear operators determined by the sequence a and the set E restricted to a certain invariant subspace, where the set E is a complete set of representatives of the distinct cosets of the quotient group Zs/MZs containing 0. Some examples are provided to illustrate the method. [Copyright &y& Elsevier]