Nonsparse companion Hessenberg matrices

In recent years, there has been a growing interest in companion matrices. Sparse companion matrices are well known: every sparse companion matrix is equivalent to a Hessenberg matrix of a particular simple type. Recently, Deaett et al. [Electron. J. Linear Algebra, 35:223--247, 2019] started the systematic study of nonsparse companion matrices. They proved that every nonsparse companion matrix is nonderogatory, although not necessarily equivalent to a Hessenberg matrix. In this paper, the nonsparse companion matrices which are unit Hessenberg are described. In a companion matrix, the variables are the coordinates of the characteristic polynomial with respect to the monomial basis. A PB-companion matrix is a generalization, in the sense that the variables are the coordinates of the characteristic polynomial with respect to a general polynomial basis. The literature provides examples with Newton basis, Chebyshev basis, and other general orthogonal bases. Here, the PB-companion matrices which are unit Hessenberg are also described.

Fast and accurate approximation of the angle-averaged redistribution function for polarized radiation

Context. Modeling spectral line profiles taking frequency redistribution effects into account is a notoriously challenging problem from the computational point of view, especially when polarization phenomena (atomic polarization and polarized radiation) are taken into account. Frequency redistribution effects are conveniently described through the redistribution function formalism, and the angle-averaged approximation is often introduced to simplify the problem. Even in this case, the evaluation of the emission coefficient for polarized radiation remains computationally costly, especially when magnetic fields are present or complex atomic models are considered. Aims. We aim to develop an efficient algorithm to numerically evaluate the angle-averaged redistribution function for polarized radiation. Methods. The proposed approach is based on a low-rank approximation via trivariate polynomials whose univariate components are represented in the Chebyshev basis. Results. The resulting algorithm is significantly faster than standard quadrature-based schemes for any target accuracy in the range [10−6, 10−2].

Numerical Analysis of the Chebyshev Collocation Method for Functional Volterra Integral Equations

The collocation method based on Chebyshev basis functions, coupled Picard iterative process, is proposed to solve a functional Volterra integral equation of the second kind. Using the Banach Fixed Point Theorem, we prove theorems on the existence and uniqueness solutions in the L2-norm. We also provide the convergence and stability analysis of the proposed method, which indicates that the numerical errors in the L2-norm decay exponentially, provided that the kernel function is sufficiently smooth. Numerical results are presented and they confirm the theoretical prediction of the exponential rate of convergence.

Application of the Orthogonal Polynomial Fitting Method in Estimating PM2.5 Concentrations in Central and Southern Regions of China

Sufficient and accurate air pollutant data are essential to analyze and control air contamination problems. An orthogonal polynomial fitting (OPF) method using Chebyshev basis functions is introduced to produce spatial distributions of fine particle (PM2.5) concentrations in central and southern regions of China. Idealized twin experiments (IE1 and IE2) are designed to validate the feasibility of the OPF method. IE1 is designed in accordance with the most common distribution of PM2.5 concentrations in China, whereas IE2 represents a common distribution in spring and autumn. In both idealized experiments, prescribed distributions are successfully estimated by the OPF method with smaller errors than kriging or Cressman interpolations. In practical experiments, cross-validation is employed to assess the interpolation results. Distributions of PM2.5 concentrations are well improved when OPF is applied. This suggests that errors decrease when the fitting order increases and arrives at the minimum when both orders reach 6. Results calculated by the OPF method are more accurate than kriging and Cressman interpolations if appropriate fitting orders are selected in practical experiments.

Analysis and comparison of numerical algorithms for finding the GCD of certain types of polynomials in the Chebyshev basis

This research investigates on the numerical methods for computing the greatest common divisors (GCD) of two polynomials in the orthogonal basis without having to convert to the power series form. Previous implementations were conducted using the Gauss Elimination with partial pivoting (GEPP) and QR Householder algorithms, respectively. This work proceeds to seek for a better approximate solution by comparing the results of the implementations with the QR with column pivoting (QRCP) algorithm. The results reveal that QRCP is as competent as the GEPP algorithm, up to a certain degree, giving a reasonably good approximate solution. It is also found that normalizing the columns of the associated coefficient matrix slightly reduces the condition number of the matrix but has no significant effect on the GCD solutions when applying the GEPP and QR Householder algorithms. However equilibration of the columns by computing its ∞-norm is capable to improve the solution when QRCP is applied. Comparing the three algorithms on some test problems, QR Householder outperforms the rest and is able to give a good approximate solution in the worst case condition when the smallest element of the matrix is 1, the entries ranging up to 15 digits integers.