Roots of bivariate polynomial systems via determinantal representations

We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two-parameter eigenvalue problem. The first determinantal representation is suitable for polynomials with scalar or matrix coefficients, and consists of matrices with asymptotic order $n^2/4$, where $n$ is the degree of the polynomial. The second representation is useful for scalar polynomials and has asymptotic order $n^2/6$. The resulting method to compute the roots of a system of two bivariate polynomials is competitive with some existing methods for polynomials up to degree 10, as well as for polynomials with a small number of terms.Comment: 22 pages, 9 figure

NiceGraph Program and its Applications in Chemistry

Recently, the problem of drawing graphs has become a hot subject in mathematical and computer sciences. In the present paper, two of the graph drawing algorithms, namely those of Kamada-Kawai and Fruchterman- Reingold, are for the first time applied to chemistry in their original two dimensional (2D) versions as well as in their generalized three dimensional (3D)version developed by us. In addition, the algorithm based on the adjacency matrix eigenvectors has been also tested. All three algorithms in their 2D and 3D versions have been tested on a series of molecules, especially on fullerenes and toroidal pure carbon cages, the so-called torusenes. The conforrnations obtained offer a rather good guess of starting geometries for more sophisticated methods. The drawings obtained by the Fruchterman-Reingold algorithm are superior to those generated by the Kamada-Kawai algorithm. In addition, all molecular graphs studied have also been represented by the so-called Schlegel diagrams for whose generation a novel algorithm was developed. Schlegel diagrams are important for identifying and analyzing the topological properties of large spatial carbon clusters

Jacobi-Davidson methods for polynomial two-parameter eigenvalue problems

We propose Jacobi-Davidson type methods for polynomial two-parameter eigenvalue problems (PMEP). Such problems can be linearized as singular two-parameter eigenvalue problems, whose matrices are of dimension k(k+1)n/2, where k is the degree of the polynomial and n is the size of the matrix coefficients in the PMEP. When k^2n is relatively small, the problem can be solved numerically by computing the common regular part of the related pair of singular pencils. For large k^2n, computing all solutions is not feasible and iterative methods are required. When k is large, we propose to linearize the problem first and then apply Jacobi-Davidson to the obtained singular two-parameter eigenvalue problem. The resulting method may for instance be used for computing zeros of a system of scalar bivariate polynomials close to a given target. On the other hand, when k is small, we can apply a Jacobi-Davidson type approach directly to the original matrices. The original matrices are projected onto a low-dimensional subspace, and the projected polynomial two-parameter eigenvalue problems are solved by a linearization. Keywords: Polynomial two-parameter eigenvalue problem (PMEP), quadratic two-parameter eigenvalue problem (QMEP), Jacobi-Davidson, correction equation, singular generalized eigenvalue problem, bivariate polynomial equations, determinantal representation, delay differential equations (DDEs), critical delays

Solving singular generalized eigenvalue problems. Part II: projection and augmentation

Generalized eigenvalue problems involving a singular pencil may be very challenging to solve, both with respect to accuracy and efficiency. While Part I presented a rank-completing addition to a singular pencil, we now develop two alternative methods. The first technique is based on a projection onto subspaces with dimension equal to the normal rank of the pencil while the second approach exploits an augmented matrix pencil. The projection approach seems to be the most attractive version for generic singular pencils because of its efficiency, while the augmented pencil approach may be suitable for applications where a linear system with the augmented pencil can be solved efficiently

Numerical methods for rectangular multiparameter eigenvalue problems, with applications to finding optimal ARMA and LTI models

Standard multiparameter eigenvalue problems (MEPs) are systems of $k\ge 2$ linear $k$-parameter square matrix pencils. Recently, a new form of multiparameter eigenvalue problems has emerged: a rectangular MEP (RMEP) with only one multivariate rectangular matrix pencil, where we are looking for combinations of the parameters for which the rank of the pencil is not full. Applications include finding the optimal least squares autoregressive moving average (ARMA) model and the optimal least squares realization of autonomous linear time-invariant (LTI) dynamical system. For linear and polynomial RMEPs, we give the number of solutions and show how these problems can be solved numerically by a transformation into a standard MEP. For the transformation we provide new linearizations for quadratic multivariate matrix polynomials with a specific structure of monomials and consider mixed systems of rectangular and square multivariate matrix polynomials. This numerical approach seems computationally considerably more attractive than the block Macaulay method, the only other currently available numerical method for polynomial RMEPs.Comment: 26 page

Uniform determinantal representations

The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak-Hochstenbach, and we investigate the performance of new representations in their root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research directions.Comment: 23 pages, 3 figures, 4 table

Solving singular generalized eigenvalue problems by a rank-completing perturbation

Generalized eigenvalue problems involving a singular pencil are very challenging to solve, both with respect to accuracy and efficiency. The existing package Guptri is very elegant but may sometimes be time-demanding, even for small and medium-sized matrices. We propose a simple method to compute the eigenvalues of singular pencils, based on one perturbation of the original problem of a certain specific rank. For many problems, the method is both fast and robust. This approach may be seen as a welcome alternative to staircase methods