In previous work 0305-4470/29/19/031 (Skala L and Cizek J 1996 J. Phys. A: Math. Gen. 29 L129, 1996 J. Phys. A: Math. Gen. 29 6467), a new method of calculating perturbation energies for one-dimensional problems based on the linear dependence of the perturbation wavefunctions on the perturbation energies has been suggested. It is shown in this letter that this method can be extended to multi-dimensional problems and the linearity can be used not only at a boundary point but also at an arbitrary point inside the integration region. Degenerate eigenvalues are also discussed. The resulting perturbation theory is very simple and can be used at large orders.