A Positivity Preserving Inverse Iteration for Finding the Perron Pair of an Irreducible Nonnegative Third Order Tensor

Abstract

We propose an inverse iterative method for computing the Perron pair of an irreducible nonnegative third order tensor. The method involves the selection of a parameter θ k in the kth iteration. For every positive starting vector, the method converges quadratically and is positivity preserving in the sense that the vectors approximating the Perron vector are strictly positive in each iteration. It is also shown that θ k = 1 near convergence. The computational work for each iteration of the proposed method is less than four times (three times if the tensor is symmetric in modes two and three, and twice if we also take the parameter to be 1 directly) that for each iteration of the Ng-Qi-Zhou algorithm, which is linearly convergent for essentially positive tensors.

Computing the dominant eigenpair of an essentially nonnegative tensor via a homotopy method

“…Their algorithm is proved to be locally quadratically convergent when the nonnegative tensor is irreducible. Liu, Guo, and Lin [12] recently proposed an algorithm that combines Newton's and Noda's iterations for third order nonnegative tensors. This algorithm preserves positivity and is shown to be quadratically convergent to the Perron pair for irreducible nonnegative tensors.…”

A modified Newton iteration for finding nonnegative Z-eigenpairs of a nonnegative tensor

“…The positive H-eigenpair (x * , λ * ) may be found by the NQZ algorithm [14], whose (linear) convergence is guaranteed for the smaller class of weakly primitive tensors [7]. In [12,13], we present a modified Newton iteration, called the Newton-Noda iteration, for finding the unique positive H-eigenpair. The method requires the selection of a positive parameter θ k in the kth iteration, and naturally keeps the positivity in the approximate eigenpairs.…”

“…The method requires the selection of a positive parameter θ k in the kth iteration, and naturally keeps the positivity in the approximate eigenpairs. For m = 3, a practical procedure for choosing θ k is given in [12], which guarantees the global convergence of the method. For a general m, a different practical procedure for choosing θ k is given in [13], and the global convergence of the method is almost certain.…”

Continuation methods for computing Z-/H-eigenpairs of nonnegative tensors

“…In addition, if A ∈ R The convergence of NQZ appears to be linear for weakly primitive tensors. -NNI [22,23] is guaranteed to compute the largest H-eigenvalue of a weakly irreducible nonnegative tensor. The convergence rate is quadratic when it is near convergence.…”

“…Some modeled versions of the power-type method have been proposed in [24,34,35]. Recently, Liu, Guo and Lin [22,23] proposed a Newton-Noda iteration (NNI) for finding the largest H-eigenvalue of weakly irreducible nonnegative tensors.…”