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Various numerical linear algebra problems can be formulated as evaluating bivariate function of matrices. The most notable examples are the Fr\'echet derivative along a direction, the evaluation of (univariate) functions of…
Superregular matrices have the property that all of their submatrices, which can be full rank are so. Lower triangular superregular matrices are useful for e.g., maximum distance separable convolutional codes as well as for (sequential)…
In this paper, a numerical method is proposed for canonical polyadic (CP) decomposition of small size tensors. The focus is primarily on decomposition of tensors that correspond to small matrix multiplications. Here, rank of the tensors is…
The Hadamard decomposition is a powerful technique for data analysis and matrix compression, which decomposes a given matrix into the element-wise product of two or more low-rank matrices. In this paper, we develop an efficient algorithm to…
Basing on a modification of the "Dichotomy Algorithm" (Terekhov, 2010), we propose a parallel procedure for solving tridiagonal systems of equations with Toeplitz matrices. Taking the structure of the Toeplitz matrices, we may substantially…
We show that the maximum rank of block lower triangular Toeplitz block matrices equals their term rank if the blocks fulfill a structural condition, i.e., only the locations but not the values of their nonzeros are fixed.
Block majorization-minimization (BMM) is a simple iterative algorithm for nonconvex optimization that sequentially minimizes a majorizing surrogate of the objective function in each block coordinate while the other block coordinates are…
Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which…
Existing block Kaczmarz methods face challenges in balancing computational efficiency and convergence for large sparse linear systems with scattered nonzero patterns, due to costly partitioning strategies and non-orthogonal projections. In…
This article introduces randomized block Gram-Schmidt process (RBGS) for QR decomposition. RBGS extends the single-vector randomized Gram-Schmidt (RGS) algorithm and inherits its key characteristics such as being more efficient and having…
In this paper, we consider the problem of Robust Matrix Completion (RMC) where the goal is to recover a low-rank matrix by observing a small number of its entries out of which a few can be arbitrarily corrupted. We propose a simple…
The cost of data input can dominate the run-time of quantum algorithms. Here, we consider data input of arithmetically structured matrices via block encoding circuits, the input model for the quantum singular value transform and related…
Although QR iterations dominate in eigenvalue computations, there are several important cases when alternative LR-type algorithms may be preferable. In particular, in the symmetric tridiagonal case where differential qd algorithm with…
We establish a new iterative method for solving a class of large and sparse linear systems of equations with three-by-three block coefficient matrices having saddle point structure. Convergence properties of the proposed method are studied…
This paper introduces an algorithm designed to approximate quantum transformation matrix with a restricted number of gates by using the block decomposition technique. Addressing challenges posed by numerous gates in handling large qubit…
In this paper we show how to recover a spectral approximations to broad classes of structured matrices using only a polylogarithmic number of adaptive linear measurements to either the matrix or its inverse. Leveraging this result we obtain…
This article presents a fast solver for the dense "frontal" matrices that arise from the multifrontal sparse elimination process of 3D elliptic PDEs. The solver relies on the fact that these matrices can be efficiently represented as a…
A scalable algorithm for solving compact banded linear systems on distributed memory architectures is presented. The proposed method factorizes the original system into two levels of memory hierarchies, and solves it using parallel cyclic…
Optimization problems with constraints in the form of a partial differential equation arise frequently in the process of engineering design. The discretization of PDE-constrained optimization problems results in large-scale linear systems…
In this paper, two accelerated divide-and-conquer algorithms are proposed for the symmetric tridiagonal eigenvalue problem, which cost $O(N^2r)$ {flops} in the worst case, where $N$ is the dimension of the matrix and $r$ is a modest number…