2016
DOI: 10.1002/nla.2033
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On the Krylov subspace methods based on tensor format for positive definite Sylvester tensor equations

Abstract: This paper deals with studying some of well-known iterative methods in their tensor forms to solve a Sylvester tensor equation. More precisely, the tensor form of the Arnoldi process and full orthogonalization method are derived by using a product between two tensors. Then tensor forms of the conjugate gradient and nested conjugate gradient algorithms are also presented. Rough estimation of the required number of operations for the tensor form of the Arnoldi process is obtained, which reveals the advantage of … Show more

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Cited by 68 publications
(8 citation statements)
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“…For simplicity, we consider that  is a 3-order n-dimensional tensor, that is,  ∈ R n×n×n . By Huyer and Neumaier, 25 it is easy to know that the required number of operations for computing (1) i, * )n z (A (2) j, * )n z (A (3) k, * ), where the notation n z (A i, * ) refers to the number of nonzero entries of the i-th row of the matrix A. So the main costs for computing () are given as follows.…”
Section: Algorithm 4 Bicgstab_btf Algorithmmentioning
confidence: 99%
“…For simplicity, we consider that  is a 3-order n-dimensional tensor, that is,  ∈ R n×n×n . By Huyer and Neumaier, 25 it is easy to know that the required number of operations for computing (1) i, * )n z (A (2) j, * )n z (A (3) k, * ), where the notation n z (A i, * ) refers to the number of nonzero entries of the i-th row of the matrix A. So the main costs for computing () are given as follows.…”
Section: Algorithm 4 Bicgstab_btf Algorithmmentioning
confidence: 99%
“…In this section, we present four numerical examples to show the effectiveness of our methods for solving (1) and (2). The low dimensional tensor equations ( 12), ( 13), ( 16) and (19) will be solved by the recursive blocked algorithms presented in [5].…”
Section: Numerical Examplesmentioning
confidence: 99%
“…Solving those linear systems can be a real challenge, since the associated matrices are too large. For N=2, (1) and (2) are respectively reduced to…”
mentioning
confidence: 99%
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