Newton---Noda iteration for finding the Perron pair of a weakly irreducible nonnegative tensor

We present a Newton---Noda iteration (NNI) for computing the Perron pair of a weakly irreducible nonnegative mth-order tensor $${\mathscr {A}}$$A, by combining the idea of Newton's method with the idea of the Noda iteration. The method requires the selection of a positive parameter $$\theta _k$$źk in the kth iteration, and produces a scalar sequence approximating the spectral radius of $$\mathscr {A}$$A and a positive vector sequence approximating the Perron vector. We propose a halving procedure to determine the parameters $$\theta _k$$źk, starting with $$\theta _k=1$$źk=1 for each k, such that the scalar sequence is monotonically decreasing. Convergence of this sequence to the spectral radius of $${\mathscr {A}}$$A (and convergence of the vector sequence to the Perron vector) is guaranteed for any initial positive unit vector, as long as the sequence $$\{\theta _k\}$${źk} so chosen is bounded below by a positive constant. In this case, we always have $$\theta _k=1$$źk=1 near convergence and the convergence is quadratic. Very often, the halving procedure will return $$\theta _k=1$$źk=1 (i.e., no halving is actually used) for each k. If the tensor is semisymmetric, $$m\ge 4$$mź4, and $$\theta _k=1$$źk=1, then the computational work in the kth iteration of NNI is roughly the same as that for one iteration of the Ng---Qi---Zhou algorithm, which is linearly convergent for the smaller class of weakly primitive tensors.