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We present a quantum algorithm for estimating the matrix determinant based on quantum spectral sampling. The algorithm estimates the logarithm of the determinant of an $n \times n$ positive sparse matrix to an accuracy $\epsilon$ in time…
There have been several algorithms designed to optimise matrix multiplication. From schoolbook method with complexity $O(n^3)$ to advanced tensor-based tools with time complexity $O(n^{2.3728639})$ (lowest possible bound achieved), a lot of…
Efficient matrix determinant calculations have been studied since the 19th century. Computers expand the range of determinants that are practically calculable to include matrices with symbolic entries. However, the fastest determinant…
Two quadrature-based algorithms for computing the matrix fractional power $A^\alpha$ are presented in this paper. These algorithms are based on the double exponential (DE) formula, which is well-known for its effectiveness in computing…
The fastest known algorithms for dealing with structured matrices, in the sense of the displacement rank measure, are randomized. For handling classical displacement structures, they achieve the complexity bounds…
Matrices with low-rank structure are ubiquitous in scientific computing. Choosing an appropriate rank is a key step in many computational algorithms that exploit low-rank structure. However, estimating the rank has been done largely in an…
Fast algorithms for matrix multiplication, namely those that perform asymptotically fewer scalar operations than the classical algorithm, have been considered primarily of theoretical interest. Apart from Strassen's original algorithm, few…
The stories told in this paper are dealing with the solution of finite, infinite, and biinfinite Toeplitz-type systems. A crucial role plays the off-diagonal decay behavior of Toeplitz matrices and their inverses. Classical results of…
We demonstrate that a modification of the classical index calculus algorithm can be used to factor integers. More generally, we reduce the factoring problem to finding an overdetermined system of multiplicative relations in any factor base…
A factorization of the inverse of a Hermetian positive definite matrix based on a diagonal by diagonal recurrence formulae permits the inversion of Toeplitz Block Toeplitz matrices using minimized matrix-vector products, with a complexity…
In the work we propose an algorithm for a Wiener -- Hopf factorization of scalar polynomials based on notions of indices and essential polynomials. The algorithm uses computations with finite Toeplitz matrices and permits to obtain…
In this paper, we develop fast procedures for solving linear systems arising from discretization of ordinary and partial differential equations with Caputo fractional derivative w.r.t time variable. First, we consider a finite difference…
We study the spectrum of the product of two Toeplitz operators. Assume that the symbols of these operators are continuous and real-valued and that one of them is non-negative. We prove that the spectrum of the product of finite section…
Matrix multiplication is a fundamental task in almost all computational fields, including machine learning and optimization, computer graphics, signal processing, and graph algorithms (static and dynamic). Twin-width is a natural complexity…
We investigate the joint convergence of independent random Toeplitz matrices with complex input entries that have a pair-correlation structure, along with deterministic Toeplitz matrices and the backward identity permutation matrix.…
As was initially shown by Brent, exponentials of truncated power series can be computed using a constant number of polynomial multiplications. This note gives a relatively simple algorithm with a low constant factor.
Given a nonsingular $n \times n$ matrix of univariate polynomials over a field $\mathbb{K}$, we give fast and deterministic algorithms to compute its determinant and its Hermite normal form. Our algorithms use…
Multipoint polynomial evaluation and interpolation are fundamental for modern symbolic and numerical computing. The known algorithms solve both problems over any field of constants in nearly linear arithmetic time, but the cost grows to…
The theory of Toeplitz quantization presented in our previous paper is extended and further developed to include diverse and interesting non-commutative realizations of the classical Euclidean plane. This is done using Hilbert spaces of…
Several applied problems are characterized by the need to numerically solve equations with an operator function (matrix function). In particular, in the last decade, mathematical models with a fractional power of an elliptic operator and…