Related papers: Efficient Computation of a Canonical Form for a Ge…
We study from a theoretical viewpoint the fundamental problem of efficiently computing the stationary distribution of general classes of structured Markov processes. In strong contrast with previous work, we consider this fundamental…
We describe a family of new algorithms for finding the canonical image of a set of points under the action of a permutation group. This family of algorithms makes use of the orbit structure of the group, and a chain of subgroups of the…
Motivated by the work of Boston, Jones and Goksel, we propose a Markov model for the factorisation of post-critically finite (PCF) cubic polynomials f. Using the information encoded in the critical orbits, we define a Markov model for PCF…
We present a class of algorithms based on rational Krylov methods to compute the action of a generalized matrix function on a vector. These algorithms incorporate existing methods based on the Golub-Kahan bidiagonalization as a special…
It is known that computing the permanent of the matrix $1+A$, where $A$ is a finite-rank matrix, requires a number of operations polynomial in the matrix size. Motivated by the boson-sampling proposal of restricted quantum computation, I…
This is a book about computational aspects of modular forms and the Galois representations attached to them. The main result is the following: Galois representations over finite fields attached to modular forms of level one can, in almost…
Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…
The theme of symbolic computation in algebraic categories has become of utmost importance in the last decade since it enables the automatic modeling of modern algebra theories. On this theoretical background, the present paper reveals the…
Canonical matrices are given for (a) bilinear forms over an algebraically closed or real closed field; (b) sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution; and (c) sesquilinear…
Given a square, nonsingular matrix of univariate polynomials $\mathbf{F}\in\mathbb{K}[x]^{n\times n}$ over a field $\mathbb{K}$, we give a deterministic algorithm for finding the determinant of $\mathbf{F}$. The complexity of the algorithm…
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…
Based on the column pivoted QR decomposition, we propose some randomized algorithms including pass-efficient ones for the generalized CUR decompositions of matrix pair and matrix triplet. Detailed error analyses of these algorithms are…
In a previous paper, we have shown that any Boolean formula can be encoded as a linear programming problem in the framework of Bayesian probability theory. When applied to NP-complete algorithms, this leads to the fundamental conclusion…
This work introduces a canonical structure for a broad class of unconstrained first-order algorithms that admit a Lur'e representation, including systems with relative degree greater than one, e.g., systems with delayed gradient feedback.…
We study the computability of the operator norm of a matrix with respect to norms induced by linear operators. Our findings reveal that this problem can be solved exactly in polynomial time in certain situations, and we discuss how it can…
Computing the volume of a polytope in high dimensions is computationally challenging but has wide applications. Current state-of-the-art algorithms to compute such volumes rely on efficient sampling of a Gaussian distribution restricted to…
This material is a rewriting and expansion of notes for beginning graduate students in seminars in combinatorics (Department of Mathematics, University of California San Diego). Solid skills in linear and multilinear algebra were required…
Canonical forms for congruence and *congruence of square complex matrices were given by Horn and Sergeichuk in [Linear Algebra Appl. 389 (2004) 347-353], based on Sergeichuk's paper [Math. USSR, Izvestiya 31 (3) (1988) 481-501], which…
This paper is concerned with the problem of finding a quadratic common Lyapunov function for a family of stable linear systems. We present gradient iteration algorithms which give deterministic convergence for finite system families and…
We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer…