Abstract. We prove that the inverse of a positive-definite matrix can be approximated by a weighted-sum of a small number of matrix exponentials. Combining this with a previous result [6], we establish an equivalence between matrix inversion and exponentiation up to polylogarithmic factors. In particular, this connection justifies the use of Laplacian solvers for designing fast semi-definite pr...

We describe the relationship between certain reinforcement learning (RL) methods based on dynamic programming (DP) and a class of unorthodox Monte Carlo methods for solving systems of linear equations proposed in the 1950's. These methods recast the solution of the linear system as the expected value of a statistic suitably defined over sample paths of a Markov chain. The significance of our ob...

The inversion of the Vandermonde matrix has received much attention for its role in the solution of some problems of numerical analysis and control theory. This work deals with the problem of getting an explicit formula for the generic element of the inverse. We derive two algorithms in O(n) and O(n) and compare them with the Parker–Traub and the Björck–Pereyra algorithms. 2005 Elsevier Inc. Al...

A novel kind of recurrent implicit dynamics together with its electronic realization is proposed and exploited for real-time matrix inversion. Compared to conventional explicit neural dynamics, our proposed model in the form of implicit dynamics has the following advantages: (a) can coincide better with systems in practice; and (b) has higher abilities in representing dynamic systems. More impo...

In this paper we present two different variants of method for symmetric matrix inversion, based on modified Gaussian elimination. Both methods avoid computation of square roots and have a reduced machine time’s spending. Further, both of them can be used efficiently not only for positive (semi-) definite, but for any non-singular symmetric matrix inversion. We use simulation to verify results, ...

We consider the questions of inversion modulo a regular chain in dimension zero and of matrix inversion modulo such a regular chain. We show that a well-known idea, Leverrier’s algorithm, yields new results for these questions.

A direct and exact method for calculating the density of states for systems with localized potentials is presented. The method is based on explicit inversion of the operator E−H. The operator is written in the discrete variable representation of the Hamiltonian, and the Toeplitz property of the asymptotic part of the obtained infinite matrix is used. Thus, the problem is reduced to the inversio...

A triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Conversely a square matrix is called upper triangular if all the entries below the main diagonal are zero. [6] For a square matrix M , M−1 is the inverse matrix where M ×M−1 = I and I denotes the n× n identity matrix. We say that the problem siz...

In this paper we present two different variants of method for symmetric matrix inversion, based on modified Gaussian elimination. Both methods avoid computation of square roots and have a reduced machine time’s spending. Further, both of them can be used efficiently not only for positive (semi-) definite, but for any non-singular symmetric matrix inversion. We use simulation to verify results, ...

The inversion of matrix is widely used in many scientific applications and engineering calculations and this operation requires a large number of computational efforts and storage space. In order to reduce the consumed time and to increase efficiency, a parallel algorithm for matrix inversion based on classic Gauss-Jordan elimination with pivoting is proposed. Upon the multi-core DSPs platform,...