Linear algebra for tensor problems
Computing
Approximation of $2^d\times2^d$ Matrices Using Tensor Decomposition
SIAM Journal on Matrix Analysis and Applications
Decay in Functions of Multiband Matrices
SIAM Journal on Matrix Analysis and Applications
Numerical Solution of the Hartree-Fock Equation in Multilevel Tensor-Structured Format
SIAM Journal on Scientific Computing
Superfast solution of linear convolutional Volterra equations using QTT approximation
Journal of Computational and Applied Mathematics
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Important matrix-valued functions f (A) are, e.g., the inverse A −1, the square root $${\sqrt{A}}$$ and the sign function. Their evaluation for large matrices arising from pdes is not an easy task and needs techniques exploiting appropriate structures of the matrices A and f (A) (often f (A) possesses this structure only approximately). However, intermediate matrices arising during the evaluation may lose the structure of the initial matrix. This would make the computations inefficient and even infeasible. However, the main result of this paper is that an iterative fixed-point like process for the evaluation of f (A) can be transformed, under certain general assumptions, into another process which preserves the convergence rate and benefits from the underlying structure. It is shown how this result applies to matrices in a tensor format with a bounded tensor rank and to the structure of the hierarchical matrix technique. We demonstrate our results by verifying all requirements in the case of the iterative computation of A −1 and $${\sqrt{A}}$$.