This paper is devoted to a study of multivariate nonhomogeneous refinement equations of the form ϕ ( x ) = g ( x ) + ∑ α ∈ Z s a ( α ) ϕ ( M x − α ) , x ∈ R s , \begin{equation*} \phi (x) = g(x) + \sum _{\alpha \in \mathbb {Z}^s} a(\alpha ) \phi (Mx-\alpha ), \qquad x \in \mathbb {R}^s, \end{equation*} where ϕ = ( ϕ 1 , … , ϕ r ) T \phi = (\phi _1,\ldots ,\phi _r)^T is the unknown, g = ( g 1 , … , g r ) T g = (g_1,\ldots ,g_r)^T is a given vector of functions on R s \mathbb {R}^s , M M is an s × s s \times s dilation matrix, and a a is a finitely supported refinement mask such that each a ( α ) a(\alpha ) is an r × r r \times r (complex) matrix. Let ϕ 0 \phi _0 be an initial vector in ( L 2 ( R s ) ) r (L_2(\mathbb {R}^s))^r . The corresponding cascade algorithm is given by ϕ k := g + ∑ α ∈ Z s a ( α ) ϕ k − 1 ( M ⋅ − α ) , k = 1 , 2 , … . \begin{equation*} \phi _k := g + \sum _{\alpha \in \mathbb {Z}^s} a(\alpha ) \phi _{k-1}({M \cdot } - \alpha ), \qquad k=1,2,\ldots . \end{equation*} In this paper we give a complete characterization for the L 2 L_2 -convergence of the cascade algorithm in terms of the refinement mask a a , the nonhomogeneous term g g , and the initial vector of functions ϕ 0 \phi _0 .