arXiv subjects

Leili Rafiee Sevyeri

Publications and source records attributed to Leili Rafiee Sevyeri.

8 recordsLinked to original sources

arXiv2026

On Parametric Linear System Solving

Parametric linear systems are linear systems of equations in which some symbolic parameters, that is, symbols that are not considered to be candidates for elimination or solution in the course of analyzing the problem, appear in the coefficients of the system. In this paper we assume that the symbolic parameters appear polynomially in the coefficients and that the only variables to be solved for are those of the linear system. The consistency of the system and expression of the solutions may vary depending on the values of the parameters. It is well-known that it is possible to specify a covering set of regimes, each of which is a Zariski-constructible condition on the parameters together with a solution description valid under that condition. We provide a method of solution that requires time polynomial in the matrix dimension and the degrees of the polynomials when there are up to three parameters. We also discuss examples suggesting how the method may be useful beyond the formal three-parameter setting. In previous methods the number of regimes needed is exponential in the system dimension and polynomial degree of the parameters. Our approach exploits the Hermite and Smith normal forms that may be computed when the system coefficient domain is mapped to the univariate polynomial domain over suitably constructed fields. Our method identifies {intrinsic singularities} and {ramification points} where the algebraic and geometric structure of the matrix changes. Parametric eigenvalue problems are addressed as well.

math.RA↗

arXiv2022

Bohemian Matrix Geometry

A Bohemian matrix family is a set of matrices all of whose entries are drawn from a fixed, usually discrete and hence bounded, subset of a field of characteristic zero. Originally these were integers -- hence the name, from the acronym BOunded HEight Matrix of Integers (BOHEMI) -- but other kinds of entries are also interesting. Some kinds of questions about Bohemian matrices can be answered by numerical computation, but sometimes exact computation is better. In this paper we explore some Bohemian families (symmetric, upper Hessenberg, or Toeplitz) computationally, and answer some open questions posed about the distributions of eigenvalue densities.

cs.SC↗

arXiv2021

Generalized Standard Triples for Algebraic Linearizations of Matrix Polynomials

We define \emph{generalized standard triples} $\mathbf{X}$, $\mathbf{Y}$, and $L(z) = z\mathbf{C}_{1} - \mathbf{C}_{0}$, where $L(z)$ is a linearization of a regular matrix polynomial $\mathbf{P}(z) \in \mathbb{C}^{n \times n}[z]$, in order to use the representation $\mathbf{X}(z \mathbf{C}_{1}~-~\mathbf{C}_{0})^{-1}\mathbf{Y}~=~\mathbf{P}^{-1}(z)$ which holds except when $z$ is an eigenvalue of $\mathbf{P}$. This representation can be used in constructing so-called \emph{algebraic linearizations} for matrix polynomials of the form $\mathbf{H}(z) = z \mathbf{A}(z)\mathbf{B}(z) + \mathbf{C} \in \mathbb{C}^{n \times n}[z]$ from generalized standard triples of $\mathbf{A}(z)$ and $\mathbf{B}(z)$. This can be done even if $\mathbf{A}(z)$ and $\mathbf{B}(z)$ are expressed in differing polynomial bases. Our main theorem is that $\mathbf{X}$ can be expressed using the coefficients of the expression $1 = \sum_{k=0}^\ell e_k ϕ_k(z)$ in terms of the relevant polynomial basis. For convenience, we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases. We account for the possibility of common similarity transformations.

math.NA↗

arXiv2021

Approximate GCD in Lagrange bases

For a pair of polynomials with real or complex coefficients, given in any particular basis, the problem of finding their GCD is known to be ill-posed. An answer is still desired for many applications, however. Hence, looking for a GCD of so-called approximate polynomials where this term explicitly denotes small uncertainties in the coefficients has received significant attention in the field of hybrid symbolic-numeric computation. In this paper we give an algorithm, based on one of Victor Ya. Pan, to find an approximate GCD for a pair of approximate polynomials given in a Lagrange basis. More precisely, we suppose that these polynomials are given by their approximate values at distinct known points. We first find each of their roots by using a Lagrange basis companion matrix for each polynomial, cluster the roots of each polynomial to identify multiple roots, and then "marry" the two polynomials to find their GCD. At no point do we change to the monomial basis, thus preserving the good conditioning properties of the original Lagrange basis. We discuss advantages and drawbacks of this method. The computational cost is dominated by the rootfinding step; unless special-purpose eigenvalue algorithms are used, the cost is cubic in the degrees of the polynomials. In principle, this cost could be reduced but we do not do so here.

math.NA↗

arXiv2021

Equivalences for Linearizations of Matrix Polynomials

One useful standard method to compute eigenvalues of matrix polynomials ${\bf P}(z) \in \mathbb{C}^{n\times n}[z]$ of degree at most $\ell$ in $z$ (denoted of grade $\ell$, for short) is to first transform ${\bf P}(z)$ to an equivalent linear matrix polynomial ${\bf L}(z)=z{\bf B}-{\bf A}$, called a companion pencil, where ${\bf A}$ and ${\bf B}$ are usually of larger dimension than ${\bf P}(z)$ but ${\bf L}(z)$ is now only of grade $1$ in $z$. The eigenvalues and eigenvectors of ${\bf L}(z)$ can be computed numerically by, for instance, the QZ algorithm. The eigenvectors of ${\bf P}(z)$, including those for infinite eigenvalues, can also be recovered from eigenvectors of ${\bf L}(z)$ if ${\bf L}(z)$ is what is called a "strong linearization" of ${\bf P}(z)$. In this paper we show how to use algorithms for computing the Hermite Normal Form of a companion matrix for a scalar polynomial to direct the discovery of unimodular matrix polynomial cofactors ${\bf E}(z)$ and ${\bf F}(z)$ which, via the equation ${\bf E}(z){\bf L}(z){\bf F}(z) = \mathrm{diag}( {\bf P}(z), {\bf I}_n, \ldots, {\bf I}_n)$, explicitly show the equivalence of ${\bf P}(z)$ and ${\bf L}(z)$. By this method we give new explicit constructions for several linearizations using different polynomial bases. We contrast these new unimodular pairs with those constructed by strict equivalence, some of which are also new to this paper. We discuss the limitations of this experimental, computational discovery method of finding unimodular cofactors.

math.NA↗

arXiv2019

Approximate GCD in a Bernstein basis

We adapt Victor Y. Pan's root-based algorithm for finding approximate GCD to the case where the polynomials are expressed in Bernstein bases. We use the numerically stable companion pencil of Gudbjörn F. Jónsson to compute the roots, and the Hopcroft-Karp bipartite matching method to find the degree of the approximate GCD. We offer some refinements to improve the process.

math.NA↗

arXiv2019

Stirling's Original Asymptotic Series from a Formula like one of Binet's and its Evaluation by Sequence Acceleration

We give an apparently new proof of Stirling's original asymptotic formula for the behavior of $\ln z!$ for large $z$. Stirling's original formula is not the formula widely known as "Stirling's formula", which was actually due to De Moivre. We also show by experiment that this old formula is quite effective for numerical evaluation of $\ln z!$ over $\mathbb{C}$, when coupled with the sequence acceleration method known as Levin's $u$-transform. As an homage to Stirling, who apparently used inverse symbolic computation to identify the constant term in his formula, we do the same in our proof.

math.HO↗

arXiv2018

The Runge Example for Interpolation and Wilkinson's Examples for Rootfinding

We look at two classical examples in the theory of numerical analysis, namely the Runge example for interpolation and Wilkinson's example (actually two examples) for rootfinding. We use the modern theory of backward error analysis and conditioning, as instigated and popularized by Wilkinson, but refined by Farouki and Rajan. By this means, we arrive at a satisfactory explanation of the puzzling phenomena encountered by students when they try to fit polynomials to numerical data, or when they try to use numerical rootfinding to find polynomial zeros. Computer algebra, with its controlled, arbitrary precision, plays an important didactic role.

math.NA↗