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Starting from an GMRES error estimate proposed by Elman in terms of the ratio of the smallest eigenvalue of the hermitian part and the norm of some non-symmetric matrix, we propose some asymptotically tighter bound in terms of the same ratio. Here we make use of a recent deep result of Crouzeix et al. on the norm of functions of matrices.
We consider Tucker-like approximations with an r × r × r core tensor for three-dimensional n × n × n arrays in the case of r n and possibly very large n (up to 10 4 − 10 6). As the approximation contains only O(rn + r 3) parameters, it is natural to ask if it can be computed using only a small amount of entries of the given array. A similar question for(More)
If a matrix has a small rank then it can be multiplied by a v ector with many s a vings in memory and arithmetic. As was recently shown by the author, the same applies to the matrices which might be of full classical rank but have a small mosaic rank. The mosaic-skeleton approximations seem to have imposing applications to the solution of large dense(More)
The goal of this work is the presentation of some new formats which are useful for the approximation of (large and dense) matrices related to certain classes of functions and nonlocal (integral, integro-differential) operators, especially for high-dimensional problems. These new formats elaborate on a sum of few terms of Kronecker products of smaller-sized(More)
Multilevel Toeplitz matrices generated by QTT tensor-structured vectors and convolution with logarithmic complexity Abstract We consider two operations in the QTT format: composition of a multilevel Toeplitz matrix generated by a given multidimensional vector and convolution of two given multidimensional vectors. We show that low-rank QTT structure of the(More)
DESCRIPTION. Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other(More)