Krylov subspace projection method for Sylvester tensor equation with low rank right-hand side

Extended Krylov subspace methods for solving Sylvester and Stein tensor equations

“…For both sets, the right hand sides are constructed randomly. We point out that this example shows that the methods presented in the current paper are more efficient comparing to the ones presented in [4]. Figures 1 and 2 show the efficiency of the proposed methods in term of convergence speed, table 3 shows that the Sylvester EGA process is more efficient in terms of precision of the approximate solution.…”

“…Next, we give an upper bound for the norm of the residual tensor, which will be used as a stopping criterion in the Sylvester EGA algorithm without computing the whole residual tensor at each iteration. We first recall the following lemma [4] to be used later Lemma 4.3. Let W mi be the basis generated by the EGA algorithm applied to the pairs (A (i) , B (i) ) and X ∈ R J1ו••×J N with J i = 2Rm i and Z ∈ R K1ו••×K N with K i = 2m i .…”

“…We run an example where the rank R = 5, where the right hand side is constructed randomly. [4] (example 2 and 3). The coefficient matrices A (i) , i = 1, 2, 3, are generated using the following Matlab command…”

“…Some well-known Krylov subspace methods have been studied in their tensor format by Beik et al in [2]. Arnoldi-based methods to solve (1) with low rank right hand sides have been proposed in [4]. Many theoretical results such as the expressions of residuals have been generalized to the tensor case.…”

The LSQR method for solving tensor least-squares problems

“…where A = A (N ) ⊗ • • • ⊗ A (2) ⊗ A (1) and ⊗ denotes the Kronecker product. The operator "vec" stacks the columns of a matrix or a tensor to form a vector.…”