Piers W. Lawrence

Stability of rootfinding for barycentric Lagrange interpolants

Computing the roots of a univariate polynomial can be reduced to computing the eigenvalues of an associated companion matrix. For the monomial basis, these computations have been shown to be numerically stable under certain conditions. However, for certain applications, polynomials are more naturally expressed in other bases, such as the Lagrange basis or orthogonal polynomial bases. For the Lagrange basis, the equivalent stability results have not been published. We show that computing the roots of a polynomial expressed in barycentric form via the eigenvalues of an associated companion matrix pair is numerically stable, and give a bound for the backward error. Numerical experiments show that the error bound is approximately an order of magnitude larger than the backward error. We also discuss the matter of scaling and balancing the companion matrix to bring it closer to a normal pair. With balancing, we are able to produce roots with small backward error.

Algorithm 917: Complex Double-Precision Evaluation of the Wright ω Function

This article describes an efficient and robust algorithm and implementation for the evaluation of the Wright ω function in IEEE double precision arithmetic over the complex plane.

Constructing Strong Linearizations of Matrix Polynomials Expressed in Chebyshev Bases

The need to solve polynomial eigenvalue problems for matrix polynomials expressed in nonmonomial bases has become very important. Among the most important bases in numerical applications are the Chebyshev polynomials of the first and second kind. In this work, we introduce a new approach for constructing strong linearizations for matrix polynomials expressed in Chebyshev bases, generalizing the classical colleague pencil, and expanding the arena in which to look for linearizations of matrix polynomials expressed in Chebyshev bases. We show that any of these linearizations is a strong linearization regardless of whether the matrix polynomial is regular or singular. In addition, we show how to recover eigenvectors, minimal indices, and minimal bases of the polynomial from those of any of the new linearizations. As an example, we also construct strong linearizations for matrix polynomials of odd degree that are symmetric (resp., Hermitian) whenever the matrix polynomials are symmetric (resp., Hermitian).

Block Kronecker linearizations of matrix polynomials and their backward errors

We introduce a new family of strong linearizations of matrix polynomials—which we call “block Kronecker pencils”—and perform a backward stability analysis of complete polynomial eigenproblems. These problems are solved by applying any backward stable algorithm to a block Kronecker pencil, such as the staircase algorithm for singular pencils or the QZ algorithm for regular pencils. This stability analysis allows us to identify those block Kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial. The global backward error analysis in this work presents for the first time the following key properties: it is a rigorous analysis valid for finite perturbations (i.e., it is not a first order analysis), it provides precise bounds, it is valid simultaneously for a large class of linearizations, and it establishes a framework that may be generalized to other classes of linearizations. These features are related to the fact that block Kronecker pencils are a particular case of the new family of “strong block minimal bases pencils”, which are robust under certain perturbations and, so, include certain perturbations of block Kronecker pencils.

Exploring the Security Vulnerabilities of LoRa

Internet-of-Things (IoT) deployments increasingly incorporate long range communication technologies. To support this transition, wide area IoT deployments are employing LoRa as their communication technology of choice due to its low power consumption and long range. The security of LoRa networks and devices is currently being put to the test in the wild, and has already become a major challenge. New features and characteristics of LoRa technology also intorduce new vulnerabilities against security attacks. In this paper, we investigate potential security vulnerabilities in LoRa. In particular, we analyze the LoRa network stack and discuss the possible susceptibility of LoRa devices to different types of attacks using commercial-off-the-shelf hardware. Our analysis shows that the long range transmissions of LoRa are vulnerable to multiple security attacks.

Numerical stability of barycentric Hermite root-finding

Computing the roots of a polynomial expressed in the Lagrange basis or a Hermite interpolational basis can be reduced to computing the eigenvalues of the corresponding companion matrix [2]. The result we present here is that roots of a polynomial computed via this method are exactly the roots of a polynomial with slightly perturbed coefficients.

Backward Error Analysis of Polynomial Eigenvalue Problems Solved by Linearization

We perform a backward error analysis of polynomial eigenvalue problems solved via linearization. Through the use of dual minimal bases, we unify the construction of strong linearizations for many different polynomial bases. By inspecting the prototypical linearizations for polynomials expressed in a number of classical bases, we are able to identify a small number of driving factors involved in the growth of the backward error. One of the primary factors is found to be the norm of the block vector of coefficients of the polynomial, which is consistent with the current literature. We derive upper bounds for the backward errors for specific linearizations, and these are shown to be reasonable estimates for the computed backward errors.

Fast Reduction of Generalized Companion Matrix Pairs for Barycentric Lagrange Interpolants

For a barycentric Lagrange interpolant

μPnP-WAN: Experiences with LoRa and its deployment in DR Congo

p(z)

BACKWARD ERROR OF POLYNOMIAL EIGENVALUE PROBLEMS SOLVED BY LINEARIZATION OF LAGRANGE INTERPOLANTS

, the roots of