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

Abstract

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