Related papers: An Exact Algorithm for Computing the Structure of …
A precision matrix is the inverse of a covariance matrix. In this paper, we study the problem of estimating the precision matrix with a known graphical structure under high-dimensional settings. We propose a simple estimator of the…
An efficient algorithm for computing eigenvectors of a matrix of integers by exact computation is proposed. The components of calculated eigenvectors are expressed as polynomials in the eigenvalue to which the eigenvector is associated, as…
We present a new algorithm for computing the straight skeleton of a polygon. For a polygon with $n$ vertices, among which $r$ are reflex vertices, we give a deterministic algorithm that reduces the straight skeleton computation to a…
We present a combination of two algorithms that accurately calculate multiple roots of general polynomials. Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and…
In this pedagogical article, we present a simple direct matrix method for analytically computing the Jacobian of nonlinear algebraic equations that arise from the discretization of nonlinear integro-differential equations. The method is…
Using exact computer arithmetic, it is possible to determine the (exact) solution of a numerical model without rounding error. For such purposes, a corresponding system of equations should be exactly defined, either directly or by…
A new robust algorithm for the numerical computation of biarcs, i.e. $G^1$ curves composed of two arcs of circle, is presented. Many algorithms exist but are based on geometric constructions, which must consider many geometrical…
In this short article I introduce the "jordan" package which provides functionality for working with different types of Jordan algebra. I give some numerical verification of the Jordan identity for the five types of Jordan algebras. The…
Obtaining the inverse of a large symmetric positive definite matrix $\mathcal{A}\in\mathbb{R}^{p\times p}$ is a continual challenge across many mathematical disciplines. The computational complexity associated with direct methods can be…
We present a numerical method for computing the logarithmic capacity of compact subsets of $\mathbb{C}$, which are bounded by Jordan curves and have finitely connected complement. The subsets may have several components and need not have…
In this paper, we present a determinist Jordan normal form algorithms based on the Fadeev formula: \[(\lambda \cdot I-A) \cdot B(\lambda)=P(\lambda) \cdot I\] where $B(\lambda)$ is $(\lambda \cdot I-A)$'s comatrix and $P(\lambda)$ is $A$'s…
In this work, new closed-form formulas for the matrix exponential are provided. Our method is direct and elementary, it gives tractable and manageable formulas not current in the extensive literature on this essential subject. Moreover,…
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 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 introduce efficient algorithm for the multiplication of split-octonions. The direct multiplication of two split-octonions requires 64 real multiplications and 56 real additions. More effective solutions still do not exist.…
Many standard linear algebra problems can be solved on a quantum computer by using recently developed quantum linear algebra algorithms that make use of block encodings and quantum eigenvalue/singular value transformations. A block encoding…
Our goal is to find accurate and efficient algorithms, when they exist, for evaluating rational expressions containing floating point numbers, and for computing matrix factorizations (like LU and the SVD) of matrices with rational…
This paper introduces an efficient algorithm for computing the best approximation of a given matrix onto the intersection of linear equalities, inequalities and the doubly nonnegative cone (the cone of all positive semidefinite matrices…
This paper presents a study on Jordan maps over matrix rings with some functional equations related to additive maps on these rings. We first show that every Jordan left (right) centralizer over a matrix ring is a left (right) centralizer.…
We introduce a new set of algorithms to compute Jacobi matrices associated with measures generated by infinite systems of iterated functions. We demonstrate their relevance in the study of theoretical problems, such as the continuity of…