Related papers: Average conditioning of underdetermined polynomial…
Condition numbers of random polynomial systems have been widely studied in the literature under certain coefficient ensembles of invariant type. In this note we introduce a method that allows us to study these numbers for a broad family of…
We study the average condition number for polynomial eigenvalues of collections of matrices drawn from various random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with Gaussian entries…
Let $M$ be an arbitrary $n$ by $n$ matrix. We study the condition number a random perturbation $M+N_n$ of $M$, where $N_n$ is a random matrix. It is shown that, under very general conditions on $M$ and $M_n$, the condition number of $M+N_n$…
We consider the sensitivity of real zeros of structured polynomial systems to perturbations of their coefficients. In particular, we provide explicit estimates for condition numbers of structured random real polynomial systems, and extend…
In this paper, we consider the different eigenvalue condition numbers for matrix polynomials used in the literature and we compare them. One of these condition numbers is a generalization of the Wilkinson condition number for the standard…
Polynomial preconditioning is an important tool in solving large linear systems and eigenvalue problems. A polynomial from GMRES can be used to precondition restarted GMRES and restarted Arnoldi. Here we give methods for indefinite matrices…
We compute the exact value of the squared condition number for the polynomial eigenvalue problem, when the input matrices have entries coming from the standard complex Gaussian distribution, showing that in general this problem is quite…
This article investigates parameter estimation of affine term structure models by means of the generalized method of moments. Exact moments of the affine latent process as well as of the yields are obtained by using results derived for…
This paper is devoted to parameter estimation for partially observed polynomial state space models. This class includes discretely observed affine or more generally polynomial Markov processes. The polynomial structure allows for the…
In this paper, the normwise condition number of a linear function of the equality constrained linear least squares solution called the partial condition number is considered. Its expression and closed formulae are first presented when the…
We consider the sensitivity of real roots of polynomial systems with respect to perturbations of the coefficients. In particular - for a version of the condition number defined by Cucker, Krick, Malajovich, and Wschebor - we establish new…
Polynomial preconditioning can improve the convergence of the Arnoldi method for computing eigenvalues. Such preconditioning significantly reduces the cost of orthogonalization; for difficult problems, it can also reduce the number of…
Theorem 1 is a formula expressing the mean number of real roots of a random multihomogeneous system of polynomial equations as a multiple of the mean absolute value of the determinant of a random matrix. Theorem 2 derives closed form…
We propose a general error analysis related to the low-rank approximation of a given real matrix in both the spectral and Frobenius norms. First, we derive deterministic error bounds that hold with some minimal assumptions. Second, we…
Many numerical problems with input $x$ and output $y$ can be formulated as a system of equations $F(x, y) = 0$ where the goal is to solve for $y$. The condition number measures the change of $y$ for small perturbations to $x$. From this…
We show a Condition Number Theorem for the condition number of zero counting for real polynomial systems. That is, we show that this condition number equals the inverse of the normalized distance to the set of ill-posed systems (i.e., those…
The tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with…
We study the probability distribution of the number of common zeros of a system of $m$ random $n$-variate polynomials over a finite commutative ring $R$. We compute the expected number of common zeros of a system of polynomials over $R$.…
Estimating the condition numbers of random structured matrices is a well known challenge, linked to the design of efficient randomized matrix algorithms. We deduce such estimates for Gaussian random Toeplitz and circulant matrices. The…
In optimal prediction methods one estimates the future behavior of underresolved systems by solving reduced systems of equations for expectations conditioned by partial data; renormalization group methods reduce the number of variables in…