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

Liu, Ching-Sung; Guo, Chen; Lin, Wen‐Wei

doi:10.1007/s00211-017-0869-7

Cited by 18 publications

(11 citation statements)

References 21 publications

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“…We assume that Jf (x * , λ * ) in (3.1) is nonsingular, but we do not assume that λ * I − (m − 1)T (x * ) is nonsingular. When m = 2, the Z-eigenvalue problem here is the same as the H-eigenvalue problem studied in [13] for all m ≥ 2, and it is shown there that λ * I − T (x * ) is always singular and Jf (x * , λ * ) is always nonsingular. For m ≥ 3, however, the difference of these two assumptions is not that big, but the assumption that λ * I − (m − 1)T (x * ) is nonsingular is still the stronger assumption.…”

Section: Local Quadratic Convergence Of Mnimentioning

confidence: 99%

“…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.…”

Section: )mentioning

confidence: 99%

“…After some preliminaries in Section 2, we propose in Section 3 a modified Newton iteration for finding a nonnegative Z-eigenpair of a nonnegative tensor, in the spirit of [13] for the H-eigenvalue problem. If we compare our algorithm here to that in [13] (although they are for two different problems), we no longer try to select parameters θ k to guarantee the monotonic convergence of the sequence approximating a nonnegative Z-eigenvalue and we effectively use θ k = 1 all the time here. When the tensor has more than one nonnegative eigenpairs, we expect to find some of them by using different initial vectors in our algorithm.…”

Section: )mentioning

confidence: 99%

“…When the tensor has more than one nonnegative eigenpairs, we expect to find some of them by using different initial vectors in our algorithm. Unlike the algorithm in [13] for the H-eigenvalue problem, our algorithm here does not naturally preserve nonnegativity in approximate Z-eigenpairs. Instead, the nonnegativity is preserved through some intervention when needed.…”

Section: )mentioning

confidence: 99%

“…We now prove the local quadratic convergence of Algorithm 3.1. We assume m ≥ 3 since the result holds for m = 2 by [13].…”

Section: Since the Fréchet Derivative Of Ax [M−1]mentioning

confidence: 99%

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

2018

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We propose a modified Newton iteration for finding some nonnegative Z-eigenpairs of a nonnegative tensor. When the tensor is irreducible, all nonnegative eigenpairs are known to be positive. We prove local quadratic convergence of the new iteration to any positive eigenpair of a nonnegative tensor, under the usual assumption guaranteeing the local quadratic convergence of the original Newton iteration. A big advantage of the modified Newton iteration is that it seems capable of finding a nonnegative eigenpair starting with any positive unit vector. Special attention is paid to transition probability tensors.x is an H-eigenvector, then cx is also an H-eigenvector for any c = 0. The same is not true in general for Z-eigenvectors. That is why we require x = 1 in (1.3) with

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Section: Local Quadratic Convergence Of Mnimentioning

confidence: 99%