Related papers: A numerical method for computing the Jordan Canoni…
We give an explanation of the $\mathrm{d}\log$-form of the coefficient matrix of canonical differential equations using the projection of ($n$+1)-$\mathrm{d}\log$ forms onto $n$-$\mathrm{d}\log$ forms. This projection is done using the…
The multiplication of matrices is an important arithmetic operation in computational mathematics. In the context of hierarchical matrices, this operation can be realized by the multiplication of structured block-wise low-rank matrices,…
We comment on two randomized algorithms for constructing low-rank matrix decompositions. Both algorithms employ the Subsampled Randomized Hadamard Transform [14]. The first algorithm appeared recently in [9]; here, we provide a novel…
We study numerical conformal mappings of planar Jordan domains with boundaries consisting of finitely many circular arcs and compute the moduli of quadrilaterals for these domains. Experimental error estimates are provided and, when…
This article studies how to form CUR decompositions of low-rank matrices via primarily random sampling, though deterministic methods due to previous works are illustrated as well. The primary problem is to determine when a column submatrix…
A Cylindrical Algebraic Decomposition (CAD) is a decomposition of R^n into a finite collection of semialgebraic cells. A CAD satisfies the "frontier condition" if, for every cell C, there is a collection of cells of the decomposition whose…
There is growing body of learning problems for which it is natural to organize the parameters into matrix, so as to appropriately regularize the parameters under some matrix norm (in order to impose some more sophisticated prior knowledge).…
This paper presents a robotic system (\textit{Chitrakar}) which autonomously converts any image of a human face to a recognizable non-self-intersecting loop (Jordan Curve) and draws it on any planar surface. The image is processed using…
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…
In this paper, we study the equality constrained nonlinear least squares problem, where the Jacobian matrices of the objective function and constraints are unavailable or expensive to compute. We approximate the Jacobian matrices via…
This paper presents a randomized algorithm for computing the near-optimal low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging techniques to compute low-rank matrix approximations at a fraction of the cost of…
We use convex relaxation techniques to provide a sequence of solutions to the matrix completion problem. Using the nuclear norm as a regularizer, we provide simple and very efficient algorithms for minimizing the reconstruction error…
The eigenvalue problem plays a central role in linear algebra and its applications in control and optimization methods. In particular, many matrix decompositions rely upon computation of eigenvalue-eigenvector pairs, such as diagonal or…
The problem of polynomial regression in which the usual monomial basis is replaced by the Bernstein basis is considered. The coefficient matrix A of the overdetermined system to be solved in the least squares sense is then a rectangular…
This paper presents a canonical duality approach for solving a general topology optimization problem of nonlinear elastic structures. By using finite element method, this most challenging problem can be formulated as a mixed integer…
We first propose a concise singular value decomposition of dual matrices. Then, the randomized version of the decomposition is presented. It can significantly reduce the computational cost while maintaining the similar accuracy. We analyze…
Given a family of nearly commuting symmetric matrices, we consider the task of computing an orthogonal matrix that nearly diagonalizes every matrix in the family. In this paper, we propose and analyze randomized joint diagonalization (RJD)…
Let $J_r$ denote a full $r \times r$ Jordan block matrix with eigenvalue $1$ over a field $F$ of characteristic $p$. For positive integers $r$ and $s$ with $r \leq s$, the Jordan canonical form of the $r s \times r s$ matrix $J_{r} \otimes…
A new method for solving systems of linear algebraic equations of a special type arising in solving problems of image reconstruction has been proposed. This method, due to a certain symmetry of the matrix and the choice of the voxel…
We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the…