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A factorization of the inverse of a Hermetian positive definite matrix based on a diagonal by diagonal recurrence formulae permits the inversion of Block Toeplitz matrices, using only matrix-vector products, and with a complexity of…
The Cayley-Hamilton problem of expressing functions of matrices in terms of only their eigenvalues is well-known to simplify to finding the inverse of the confluent Vandermonde matrix. Here, we give a highly compact formula for the inverse…
Can one recover a matrix efficiently from only matrix-vector products? If so, how many are needed? This paper describes algorithms to recover matrices with known structures, such as tridiagonal, Toeplitz, Toeplitz-like, and hierarchical…
Unbounded (and bounded) Toeplitz operators (TO) with rational symbols are analysed in detail showing that they are densely defined closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains,…
We derive novel explicit formulas for the inverses of truncated block Toeplitz matrices that correspond to a multivariate minimal stationary process. The main ingredients of the formulas are the Fourier coefficients of the phase function…
Classic cache-oblivious parallel matrix multiplication algorithms achieve optimality either in time or space, but not both, which promotes lots of research on the best possible balance or tradeoff of such algorithms. We study modern…
This paper deals with simultaneously fast and in-place algorithms for formulae where the result has to be linearly accumulated: some of the output variables are also input variables, linked by a linear dependency. Fundamental examples…
In recent years, randomized algorithms have established themselves as fundamental tools in computational linear algebra, with applications in scientific computing, machine learning, and quantum information science. Many randomized matrix…
Matrix data sets are common nowadays like in biomedical imaging where the Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) modality produces data sets of 3D symmetric positive definite matrices anchored at voxel positions capturing the…
This paper discusses the explicit inverse of a class of seven-diagonal (near) Toeplitz matrices, which arises in the numerical solutions of nonlinear fourth-order differential equation with a finite difference method. A non-recurrence…
An alternative to the matrix inverse procedure is presented. Given a bit register which is arbitrarily large, the matrix inverse to an arbitrarily large matrix can be peformed in ${\cal O}(N^2)$ operations, and to matrix multiplication on a…
We present a new algorithm for fast matrix multiplication using tensor decompositions which have special features. Thanks to these features we obtain exponents lower than what the rank of the tensor decomposition suggests. In particular for…
We present a novel class of methods to compute functions of matrices or their action on vectors that are suitable for parallel programming. Solving appropriate simple linear systems of equations in parallel (or computing the inverse of…
Matrices are typically considered over fields or rings. Motivated by applications in parametric differential equations and data-driven modeling, we suggest to study matrices with entries from a Hilbert space and present an elementary theory…
This paper studies matrix-valued truncated Toeplitz operators, which are a vectorial generalisation of truncated Toeplitz operators. It is demonstrated that, although there exist matrix-valued truncated Toeplitz operators without a matrix…
We discuss quantum algorithms that calculate numerical integrals and descriptive statistics of stochastic processes. With either of two distinct approaches, one obtains an exponential speed increase in comparison to the fastest known…
The problem of approximating a sampled function using sums of a fixed number of complex exponentials is considered. We use alternating projections between fixed rank matrices and Hankel matrices to obtain such an approximation. Convergence,…
In order to find the outcome probabilities of quantum mechanical systems like the optical networks underlying Boson sampling, it is necessary to be able to compute the permanents of unitary matrices, a computationally hard task. Here we…
Motivated mainly by applications to partial differential equations with random coefficients, we introduce a new class of Monte Carlo estimators, called Toeplitz Monte Carlo (TMC) estimator for approximating the integral of a multivariate…
For some kernel matrices, low-rank approximations can be quickly obtained via analytic techniques. One important class of analytic methods that has received attention in recent years is based on the use of proxy points. Accuracy analysis…