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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…
This paper introduces an efficient algorithm for computing the general oscillatory matrix functions. These computations are crucial for solving second-order semi-linear initial value problems. The method is exploited using the scaling and…
The Schur decomposition of a square matrix $A$ is an important intermediate step of state-of-the-art numerical algorithms for addressing eigenvalue problems, matrix functions, and matrix equations. This work is concerned with the following…
Triangular factorizations are an important tool for solving integral equations and partial differential equations with hierarchical matrices ($\mathcal{H}$-matrices). Experiments show that using an $\mathcal{H}$-matrix LR factorization to…
The computation of matrix functions is a well-studied problem. Of special importance are the exponential and the logarithm of a matrix, where the latter also raises existence and uniqueness questions. This is particularly relevant in the…
Matrix functions with potential applications have a major role in science and engineering. One of the fundamental matrix functions, which is particularly important due to its connections with certain matrix differential equations and other…
We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\em weakened} notion of a polynomial GCD modulo a regular chain, which…
The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Pad\'e approximation, sometimes accompanied by the Schur decomposition. The main computational…
An algorithm for computing an analytic function of a matrix $A$ is described. The algorithm is intended for the case where $A$ has some close eigenvalues, and clusters (subsets) of close eigenvalues are separated from each other. This…
In this paper, we introduce novel fast matrix inversion algorithms that leverage triangular decomposition and recurrent formalism, incorporating Strassen's fast matrix multiplication. Our research places particular emphasis on triangular…
We propose an efficient algorithm for computing a common eigenvector of a finite set of square matrices. As an immediate consequence we obtain an algorithm for determining whether the matrices admit a simultaneous triangulation, and, if so,…
Let $T$ be a square matrix with a real spectrum, and let $f$ be an analytic function. The problem of the approximate calculation of $f(T)$ is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that $T$…
Transformers have revolutionized natural language processing, but their use for numerical computation has received less attention. We study the approximation of matrix functions, which map scalar functions to matrices, using neural networks…
The computation of triangular decompositions are based on two fundamental operations: polynomial GCDs modulo regular chains and regularity test modulo saturated ideals. We propose new algorithms for these core operations relying on modular…
Deterministic recursive algorithms for the computation of matrix triangular decompositions with permutations like LU and Bruhat decomposition are presented for the case of commutative domains. This decomposition can be considered as a…
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…
Algorithms and implementations for computing the sign function of a triangular matrix are fundamental building blocks in algorithms for computing the sign of arbitrary square real or complex matrices. We present novel recursive and cache…
This paper introduces a novel general-purpose algorithm for Pauli decomposition that employs matrix slicing and addition rather than expensive matrix multiplication, significantly accelerating the decomposition of multi-qubit matrices. In a…
In a recent paper, a new method was proposed to find the common invariant subspaces of a set of matrices. This paper invstigates the more general problem of putting a set of matrices into block triangular or block-diagonal form…
Finding the inverse of a matrix is an open problem especially when it comes to engineering problems due to their complexity and running time (cost) of matrix inversion algorithms. An optimum strategy to invert a matrix is, first, to reduce…