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We consider the numerical solution of a c-stable linear equation in the tensor product space R n 1 ×· · ·×n d , arising from a discretized elliptic partial differential equation in R d . Utilizing the stability, we produce an equivalent d-stable generalized Stein-like equation, which can be solved iteratively. For large-scale problems defined by sparse and structured matrices, the methods can be modified for further efficiency, producing algorithms of O( ∑ i n i ) + O(n s ) computational complexity, under appropriate assumptions (with n s being the flop count for solving a linear system associated with A i − I n i ). Illustrative numerical examples will be presented. KEYWORDS Cayley transform, elliptic partial differential equation, Kronecker product, large-scale problem, linear equation, Stein equation, Sylvester equationfor a constant c ∈ R d . In many similar applications, A i can be formulated to be c-stable (with eigenvalues in the open left-half plane). We make use of the stability of A i and transform the corresponding LET_d (also described as c-stable) to an equivalent d-stable Stein-like equation with tensor product structure (SET), which are defined by matrices with spectra inside the unit Numer Linear Algebra Appl. 2017;24:e2106.wileyonlinelibrary.com/journal/nla

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MSC: 15A06 15A24 15A30 15A69 15A72 65F10 65N22 Keywords: Column pivoting Gram-Schmidt orthogonalization Kronecker product Large-scale problem Linear equation Low-rank representation Multilinear algebra QR decompositionTensor computation is intensive and difficult. Invariably, a vital component is the truncation of tensors, so as to control the memory and associated computational requirements. Various tensor toolboxes have been designed for such a purpose, in addition to transforming tensors between different formats. In this paper, we propose a simple Q-less QR truncation technique for tensors {x (i) } with x (i) ∈ R n1×···×nd in the simple and natural Kronecker product form. It generalizes the QR decomposition with column pivoting, adapting the well-known Gram-Schmidt orthogonalization process. The main difficulty lies in the fact that linear combinations of tensors cannot be computed or stored explicitly. All computations have to be performed on the coefficients α i in an arbitrary tensor v = i α i x (i) . The orthonormal Q factor in the QR decomposition X ≡ [x (1) , · · · , x (p) ] = QR cannot be computed but expressed as XR −1 when required. The resulting algorithm has an O(p 2 dn) computational complexity, with n = max n i . Some illustrative examples in the numerical solution of tensor linear equations are presented.

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