Luca GemignaniUniversità di Pisa  UNIPI · Department of Computer Science
Luca Gemignani
Prof.
About
119
Publications
10,090
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the fulltext. Learn more
1,358
Citations
Introduction
Luca Gemignani currently works at the Department of Computer Science, Università di Pisa. Luca does research in numerical linear algebra and applied mathematics.
Publications
Publications (119)
We provide a new approach to obtain solutions of linear differential problems set in a Banach space and equipped with nonlocal boundary conditions. From this approach we derive a family of numerical schemes for the approximation of the solutions. We show by numerical tests that these schemes are numerically robust and computationally efficient.
This paper presents the results of a preliminary experimental investigation of the performance of a stationary iterative method based on a block staircase splitting for solving singular systems of linear equations arising in Markov chain modelling. From the experiments presented, we can deduce that the method is well suited for solving block banded...
This paper presents the results of a preliminary experimental investigation of the performance of a stationary iterative method based on a block staircase splitting for solving singular systems of linear equations arising in Markov chain modelling. From the experiments presented, we can deduce that the method is well suited for solving block banded...
The paper is concerned with efficient numerical methods for solving a linear system $\phi(A) x= b$, where $\phi(z)$ is a $\phi$function and $A\in \mathbb R^{N\times N}$. In particular in this work we are interested in the computation of ${\phi(A)}^{1} b$ for the case where $\phi(z)=\phi_1(z)=\displaystyle\frac{e^z1}{z}, \quad \phi(z)=\phi_2(z)=\...
We present some accelerated variants of fixed point iterations for computing the minimal nonnegative solution of the unilateral matrix equation associated with an M/G/1type Markov chain. These variants derive from certain staircase regular splittings of the block Hessenberg Mmatrix associated with the Markov chain. By exploiting the staircase pr...
We present some accelerated variants of fixed point iterations for computing the minimal nonnegative solution of the unilateral matrix equation associated with an M/G/1type Markov chain. These schemes derive from certain staircase regular splittings of the block Hessenberg Mmatrix associated with the Markov chain. By exploiting the staircase pro...
We present a class of fast subspace algorithms based on orthogonal iterations for structured matrices/pencils that can be expressed as small rank perturbations of unitary matrices. The representation of the matrix by means of a new datasparse factorization—named LFR factorization—using orthogonal Hessenberg matrices is at the core of these algorit...
In this paper, we introduce a family of rational approximations of the reciprocal of a ϕfunction involved in the explicit solutions of certain linear differential equations, as well as in integration schemes evolving on manifolds. The derivation and properties of this family of approximations applied to scalar and matrix arguments are presented. M...
The algebraic characterization of dual univariate interpolating subdivision schemes is investigated. Specifically, we provide a constructive approach for finding dual univariate interpolating subdivision schemes based on the solutions of certain associated polynomial equations. The proposed approach also makes it possible to identify conditions for...
Some variants of the (block) Gauss–Seidel iteration for the solution of linear systems with Mmatrices in (block) Hessenberg form are discussed. Comparison results for the asymptotic convergence rate of some regular splittings are derived: in particular, we prove that for a lowerHessenberg Mmatrix \(\rho (P_{GS})\ge \rho (P_S)\ge \rho (P_{AGS})\)...
Some variants of the (block) GaussSeidel iteration for the solution of linear systems with $M$matrices in (block) Hessenberg form are discussed. Comparison results for the asymptotic convergence rate of some regular splittings are derived: in particular, we prove that for a lowerHessenberg Mmatrix $\rho(P_{GS})\geq \rho(P_S)\geq \rho(P_{AGS})$,...
We present a class of fast subspace tracking algorithms based on orthogonal iterations for structured matrices/pencils that can be represented as small rank perturbations of unitary matrices. The algorithms rely upon an updated data sparse factorization  named LFR factorization  using orthogonal Hessenberg matrices. These new subspace trackers...
The algebraic characterization of dual univariate interpolating subdivision schemes is investigated. Specifically, we provide a constructive approach for finding dual univariate interpolating subdivision schemes based on the solutions of certain associated polynomial equations. The proposed approach also makes possible to identify conditions for th...
It is shown that the problem of balancing a nonnegative matrix by positive diagonal matrices can be recast as a nonlinear eigenvalue problem with eigenvector nonlinearity. Based on this equivalent formulation some adaptations of the power method and Arnoldi process are proposed for computing the dominant eigenvector which defines the structure of t...
Some fast algorithms for computing the eigenvalues of a (block) companion matrix have recently appeared in the literature. In this paper we generalize the approach to encompass unitary plus low rank matrices of the form \(A=U + XY^H\) where U is a general unitary matrix. Three important cases for applications are U unitary diagonal, U unitary block...
Efficient eigenvalue solvers for unitary plus lowrank matrices exploit the structural properties of the Hessenberg reduction performed in the preprocessing phase. Recently some twostage fast algorithms have been proposed for computing the Hessenberg reduction of a matrix $A = D + UV^H$, where $D$ is a unitary $n \times n$ diagonal matrix and $U,...
It is shown that the problem of balancing a nonnegative matrix by positive diagonal matrices can be recast as a constrained nonlinear multiparameter eigenvalue problem. Based on this equivalent formulation some adaptations of the power method and Arnoldi process are proposed for computing the dominant eigenvector which defines the structure of the...
Some fast algorithms for computing the eigenvalues of a block companion matrix $A = U + XY^H$, where $U\in \mathbb C^{n\times n}$ is unitary block circulant and $X, Y \in\mathbb{C}^{n \times k}$, have recently appeared in the literature. Most of these algorithms rely on the decomposition of $A$ as product of scalar companion matrices which turns in...
In this paper we introduce a family of rational approximations of the inverse of a $\phi$ function involved in the explicit solutions of certain linear differential equations as well as in integration schemes evolving on manifolds. For symmetric banded matrices these novel approximations provide a computable reconstruction of the associated matrix...
We develop two fast algorithms for Hessenberg reduction of a structured matrix $A = D + UV^H$ where $D$ is a real or unitary $n \times n$ diagonal matrix and $U, V \in\mathbb{C}^{n \times k}$. The proposed algorithm for the real case exploits a twostage approach by first reducing the matrix to a generalized Hessenberg form and then completing the...
In this paper we focus on the solution of shifted quasiseparable systems and of more general parameter dependent matrix equations with quasiseparable representations. We propose an efficient algorithm exploiting the invariance of the quasiseparable structure under diagonal shifting and inversion. This algorithm is applied to compute various functio...
We design a fast implicit real QZ algorithm for eigenvalue computation of structured companion pencils arising from linearizations of polynomial rootfinding problems. The modified QZ algorithm computes the generalized eigenvalues of an N×N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssy...
In this paper we consider the application of polynomial rootfinding methods to the solution of the tridiagonal matrix eigenproblem. All considered solvers are based on evaluating the Newton correction. We show that the use of scaled threeterm recurrence relations complemented with error free transformations yields some compensated schemes which s...
Pseudosplines are a rich family of functions that allows the user to meet various demands for balancing polynomial reproduction (i.e., approximation power), regularity and support size. Such a family includes, as special members, Bspline functions, universally known for their usefulness in different fields of application. When replacing polynomia...
In this paper we propose a variation of the Ehrlich–Aberth method for the simultaneous refinement of the zeros of Hpalindromic polynomials.
In this paper we present a novel matrix method for polynomial rootfinding. We approximate the roots by computing the eigenvalues of a permuted version of the companion matrix associated with the polynomial. This form, referred to as a lower staircase form of the companion matrix in the literature, has a block upper Hessenberg shape with possibly no...
The klevel subdivision symbol of a binary exponential pseudospline is
defined by a suitable polynomial "correction" of the klevel subdivision symbol
of an exponential Bspline. The polynomial correction is such that the
resulting subdivision symbol is the one of minimal support that fulfills the
conditions for reproduction of the space of expone...
A fast implicit QR algorithm for eigenvalue computation of low rank corrections of Hermitian matrices is adjusted to work with matrix pencils arising from zerofinding problems for polynomials expressed in Chebyshevlike bases. The modified QZ algorithm computes the generalized eigenvalues of certain [TEX equation: N\times N] rank structured matrix...
A fast implicit QR algorithm for eigenvalue computation of low rank corrections of unitary matrices is adjusted to work with matrix pencils arising from polynomial zerofinding problems. The modified QZ algorithm computes the generalized eigenvalues of certain $ N\times N$ rank structured matrix pencils using $ O(N^2)$ flops and $ O(N)$ memory stor...
This paper is concerned with the reduction of a unitary matrix U to CMVlike
shape. A Lanczostype algorithm is presented which carries out the reduction
by computing the block tridiagonal form of the Hermitian part of U, i.e., of
the matrix U+U^H. By elaborating on the Lanczos approach we also propose an
alternative algorithm using elementary mat...
It is well known that if a matrix $A\in\mathbb C^{n\times n}$ solves the
matrix equation $f(A,A^H)=0$, where $f(x, y)$ is a linear bivariate polynomial,
then $A$ is normal; $A$ and $A^H$ can be simultaneously reduced in a finite
number of operations to tridiagonal form by a unitary congruence and, moreover,
the spectrum of $A$ is located on a strai...
In this paper, we elaborate on the implicit shifted QR eigenvalue algorithm given in [D.A. Bini, P. Boito, Y. Eidelman, L. Gemignani, I. Gohberg, A fast implicit QR eigenvalue algorithm for companion matrices, Linear Algebra Appl. 432 (2010), 20062031]. The algorithm is substantially simplified and speeded up while preserving its numerical robustn...
The paper describes some modifications of Newton’s method for refining the zeros of evengrade f(x)twined (f(x)egt) polynomials, defined as polynomials whose roots appear in pairs {x
i
,f(x
i
)}. Particular attention is given to evengrade palindromic (egp) polynomials. The algorithms are derived from certain symmetric division processes for comp...
This paper describes an algebraic construction of bivariate interpolatory subdivision masks induced by threedirectional box spline subdivision schemes. Specifically, given a threedirectional box spline, we address the problem of defining a corresponding interpolatory subdivision scheme by constructing an appropriate correction mask to convolve wi...
An algorithm based on the EhrlichAberth rootfinding method is presented for
the computation of the eigenvalues of a Tpalindromic matrix polynomial. A
structured linearization of the polynomial represented in the Dickson basis is
introduced in order to exploit the symmetry of the roots by halving the total
number of the required approximations. T...
In this paper we describe a general, computationally feasible strategy to deduce a family of interpolatory nonstationary subdivision schemes from a symmetric nonstationary, noninterpolatory one satisfying quite mild assumptions. To achieve this result we extend our previous work [C.Conti, L.Gemignani, L.Romani, Linear Algebra Appl. 431 (2009), n...
Diagonal plus semiseparable matrices are constructed, the eigenvalues of which are algebraic numbers expressed by simple closed trigonometric formulas.
Subdivision schemes are nowadays customary in curve and surface modeling. In this paper the problem of designing interpolatory subdivision schemes is considered. The idea is to modify a given approximating subdivision scheme just enough to satisfy the interpolation requirement. From an algebraic point of view this leads to the solution of a general...
SUMMARY An implicit version of the shifted QR eigenvalue algorithm given in
[D.~A.~Bini, Y.~Eidelman, I.~Gohberg, L.~Gemignani,
SIAM J. Matrix Anal. Appl. 29 (2007), no. 2, 566585]
is presented for computing the eigenvalues of an $n\times n$ companion
matrix
using $O(n^2)$ flops and $O(n)$ memory storage.
Numerical experiments and comparisons con...
A general scheme is proposed for computing the QRfactorization of certain displacement structured matrices, including Cauchylike, Vandermonde like, Toeplitzlike and Hankellike matrices, hereby extending some earlier work for the QRfactorization of the Cauchy matrix. The algorithm employs a chasing scheme for the recursive construction of a di...
In this paper we present a general strategy to deduce a family of interpolatory masks from a symmetric Hurwitz noninterpolatory one. This brings back to a polynomial equation involving the symbol of the noninterpolatory scheme we start with. The solution of the polynomial equation here proposed, tailored for symmetric Hurwitz subdivision symbols,...
Two fast algorithms that use orthogonal similarity transformations to convert a symmetric rationally generated Toeplitz matrix to tridiagonal form are developed, as a means of finding the eigenvalues of the matrix efficiently. The reduction algorithms achieve cost efficiency by exploiting the rank structure of the input Toeplitz matrix. The propose...
In this paper we address the problem of efficiently computing all the eigenvalues of a large N×N Hermitian matrix modified by a possibly non Hermitian perturbation of low rank. Previously proposed fast adaptations of the
QR algorithm are considerably simplified by performing a preliminary transformation of the matrix by similarity into an upper
Hes...
In this paper it is shown that Neville elimination is suited to exploit the rank structure of an orderr quasiseparable matrix A∈Cn×n by providing a condensed decomposition of A as product of unit bidiagonal matrices, all together specified by O(nr) parameters, at the cost of O(nr3) flops. An application of this result for eigenvalue computation of...
In this paper, we present two fast numerical methods for computing the QR factorization of an n×n Cauchylike matrix C, C=QR, with data points lying on the real axis or on the unit circle in the complex plane. It is shown that the rows of the Qfactor of C are the eigenvectors of a rank structured matrix partially determined by some prescribed spec...
We consider the problem of completion of a matrix with a specified lower triangular part to a unitary matrix. In this paper we obtain the necessary and sufficient conditions of existence of a unitary completion without any additional constraints and give a general formula for this completion. The paper is mainly focused on matrices with the specifi...
The connection between Gauss quadrature rules and the algebraic eigenvalue problem for a Jacobi matrix was first exploited
in the now classical paper by Golub and Welsch (Math. Comput. 23(106), 221–230, 1969). From then on many computational problems arising in the construction of (polynomial) Gauss quadrature formulas have been
reduced to solving...
In this paper we discuss the use of structured matrix meth ods for the numerical approximation of the zeros of a univari ate polynomial. In particular, it is shown that rootfinding algorithms based on floatingpoint eigenvalue computation can benefit from the structure of the matrix problem to re duce their complexity and memory requirements by...
Let ℋn ∪ ℂn×n be the class of n × n Hessenberg matrices A which are rankone modifications of a unitary matrix, that is, A = H + uwH, where H is unitary and u, w ∈ ℂn. The class ℋn includes three wellknown subclasses: unitary Hessenberg matrices, companion (Frobenius) matrices, and fellow matrices. The paper presents some novel fast adaptations of...
In this paper we design a fast new algorithm for reducing an N × N quasiseparable matrix to upper Hessenberg form via a sequence of N − 2 unitary transformations. The new reduction is especially useful when it is followed by the QR algorithm to obtain a complete set of eigenvalues of the original matrix. In particular, it is shown that in a number...
In this paper, we present a novel method for solving the unitary Hessenberg eigenvalue problem. In the first phase, an algorithm is designed to transform the unitary matrix into a diagonalplussemiseparable form. Then we rely on our earlier adaptation of the QR algorithm to solve the dpss eigenvalue problem in a fast and robust way. Exploiting the...
Matrix methods based on the QR eigenvalue algorithm applied to a companion matrix are customary for polynomial rootfinding.
These methods take advantage of recent results showing that the quasiseparable structure of the input matrix is maintained
under the iterative process. The property enables the QRiteration for a companion matrix to be perfor...
We devise a fast fractionfree algorithm for the computation of the triangular factorization of Bernstein–Bezoutian matrices with entries over an integral domain. Our approach uses the Bareiss fractionfree variant of Gaussian elimination, suitably modified to take into account the structural properties of Bernstein–Bezoutian matrices. The algorith...
We introduce a class
of n×n structured matrices which includes three wellknown classes of generalized companion matrices: tridiagonal plus rankone matrices (comrade matrices), diagonal plus rankone matrices and arrowhead matrices. Relying on the structure properties of
, we show that if A ∈
then A
′=RQ ∈
, where A=QR is the QR decomposition of A...
A new class of functional iterations is introduced for the numerical solution of the matrix equation C+AX+BX2=0 where A,B,C are given m × m matrices and X is the unknown matrix. Each iteration in this class is globally convergent, selfcorrecting, and the local convergence is linear with an arbitrarily large speed. The new iterations, which rely on...
The linear space of all proper rational functions with prescribed poles is considered. Given a set of points z i in the complex plane and the weights w i we define the discrete inner product 〈ϕ,ψ〉:=∑ i=0 n w i  2 ϕ(z i ) ¯ψ(z i )· We derive a method to compute the coefficients of a recurrence relation generating a set of orthonormal rational basi...
An algorithm based on the EhrlichAberth iteration is presented for the computation of the zeros of $p(\lambda)=\det(T\lambda I)$, where $T$ is a real irreducible nonsymmetric tridiagonal matrix. The algorithm requires the evaluation of $p(\lambda)/p'(\lambda)=1/\mathrm{trace}(T\lambda I)^{1}$, which is done by exploiting the QR factorization...
The problem of solving large Mmatrix linear systems with sparse coefficient matrix in block Hessenberg form is here addressed. In previous work of the authors a divideandconquer strategy was proposed and a backward error analysis of the resulting algorithm was presented showing its effectiveness for the solution of computational problems of queu...
An algorithm based on the EhrlichAberth iteration is presented for the computation of the zeros of p(#) = det(T  #I), where T is an irreducible nonsymmetric tridiagonal matrix. The algorithm requires the evaluation of p(#)/p (#) = 1/trace(T  #I) , which is done by exploiting the QR factorization of T  #I and the semiseparable structure of (T ...
Several computational and structural properties of Bezoutian matrices expressed with respect to the Bernstein polynomial basis are shown. The exploitation of such properties allows the design of fast algorithms for the solution of BernsteinBezoutian linear systems without never making use of potentially illconditioned reductions to the monomial b...
Univariate polynomial rootfinding is the oldest classical problem of mathematics and computational mathematics, and is still an important research topic, due to its impact on computational algebra and geometry. The Weierstrass (DurandKerner) approach and its variations as well as matrix methods based on the QR algorithm are among the most popular...
We approximate polynomial roots numerically as the eigenvalues of a unitary diagonal plus rankone matrix. We rely on our earlier adaptation of the algorithm, which exploits the semiseparable matrix structure to approximate the eigenvalues in a fast and robust way, but we substantially improve the performance of the resulting algorithm at the initi...
We show that the shifted QR iteration applied to a companion matrix F maintains the weakly semiseparable structure of F. More precisely, if Ai iI = QiRi, Ai+1 := RiQi + iI, i = 0; 1; : : :, where A0 = F , then we prove that Qi, Ri and Ai are semiseparable matrices having semiseparability rank at most 1, 4 and 3, respectively. This structural proper...
Let p(z) be a polynomial of degree n having zeros 1
m

m+1
n
. This paper is concerned with the problem of efficiently computing the coefficients of the factors u(z)=
i=1
m
(z–
i
) and l(z)=
i=m+1
n
(z–
i
) of p(z) such that a(z)=z
–m
p(z)=(z
–m
u(z))l(z) is the spectral factorization of a(z). To perform this task the following twostage...
In this paper we study both direct and inverse eigenvalue problems for diagonalplussemiseparable (dpss) matrices. In particular, we show that the computation of the eigenvalues of a symmetric dpss matrix can be reduced by a congruence transformation to solving a generalized symmetric definite tridiagonal eigenproblem. Using this reduction, we dev...
We develop a superfast method for the solution of (n+m)×(n+m) Sylvester’s resultant linear systems associated with two real polynomials a(z) and c(z) of degree n and m, respectively, where a(z) is a stable polynomial, i.e., all its roots lie inside the unit circle, whereas c(z) is an antistable polynomial, i.e, zmc(z−1) is stable. The proposed sch...
The problem of solving large linear systems whose coefficient matrix is a sparse Mmatrix in block Hessenberg form has recently received much attention, especially for applications in Markov chains and queueing theory. Stewart proposed a recursive algorithm which is shown to be backward stable. Although the theoretical derivation of such an algorit...
We present a fast algorithm for computing the QR factorization of Cauchylike matrices with real nodes. The algorithm is based on the existence of certain generalized recurrences among the columns of Q and R T , does not require squaring the matrix, and fully exploits the displacement structure of Cauchylike matrices. Moreover, we describe a class...
The space of all proper rational functions with prescribed real poles is considered. Given a set of points zi on the real line and the weights wi, we define the discrete inner product (formula in paper). In this paper we derive an efficient method to compute the coefficients of a recurrence relation generating a set of orthonormal rational basis fu...
In this paper we propose a superfast implementation of Wilson's method for the spectral factorization of Laurent polynomials based on a preconditioned conjugate gradient algorithm. The new computational scheme follows by exploiting several recently established connections between the considered factorization problem and the solution of certain disc...
We describe a stable algorithm, having linear complexity, for the solution of bandedplussemiseparable linear systems. The
algorithm exploits the structural properties of the inverse of a semiseparable matrix. Stability is achieved by combining
these properties with partial pivoting techniques. Several numerical experiments are shown to confirm th...
We present a fast algorithm for computing the QR factorization of Cauchy matrices with real nodes. The algorithm works for almost any input matrix, does not require squaring the matrix, and fully exploits the displacement structure of Cauchy matrices. We prove that, if the determinant of a certain semiseparable matrix is nonzero, a three term recu...
We relate polynomial computations with operations involving infinite band Toeplitz matrices and show applications to the numerical solution of Markov chains, of nonlinear matrix equations, to spectral factorizations and to the solution of finite Toeplitz systems. In particular two matrix versions of Graeffe's iteration are introduced and their conv...
A classical result of structured numerical linear algebra states that the inverse of a nonsingular semiseparable matrix is a tridiagonal matrix. Such a property of a semiseparable matrix has been proved to be useful for devising linear complexity solvers, for establishing recurrence relations among its columns or rows and, moreover, for efficiently...
We show that some classes of matrix equations can be reduced to solving a quadratic matrix equation of the kind X2A + XB + C = 0 where A,B,C,X are m × m matrices or semiinfinite matrices. The problem of computing the minimal solution, if it exists, of the latter equation is reduced to computing the matrix coefficients Ho and H1 of the Laurent mat...
In this paper we present a suitable generalization of the classical Graeffe's iteration by showing that it provides effective algorithms for the fast evaluation of polynomials and residues of rational and meromorphic functions. In particular, if g(z) = q(z)/p(z) is a rational function with n finite poles &xgr;1,…,&xgr;n and &xgr; is an initial appr...
By providing a matrix version of Koenig's theorem we reduce the problem of evaluating the coefficients of a monic factor
r(z) of degree h of a power series f(z) to that of approximating the first h entries in the first column of the inverse of an Toeplitz matrix in block Hessenberg form for sufficiently large values of n. This matrix is reduced to...
Consider the homogeneous linear difference equation of order n, y(n+m)+(a 1 +p 1 (m))y(n+m1)+⋯+(a n +p n (m))y(m)=0, where a n ≠0, the associated polynomial q(t)=t n +∑ i=1 n a i t ni has zeros ξ i , 1≤i≤n, with distinct moduli, and lim m→∞ p j (m)=0,1≤j≤n· Then, Poincaré’s theorem asserts that every nontrivial solution {y(m)} is such that lim m→...
A polynomial is called a Hurwitz polynomial (sometimes, when the coefficients are real, a stable polynomial) if all its roots have real part strictly less than zero. In this paper we present a numerical method for computing the coefficients of the Hurwitz factor f(z) of a polynomial p(z). It is based on a polynomial description of the classical LR...
. The problem of solving large linear systems whose coecient matrix is a sparse Mmatrix in block Hessenberg form has recently received a great interest especially for applications in Markov chains and queueing theory. Stewart proposed a recursive algorithm which is shown to be backward stable provided that the systems to be solved at the intermedi...
This paper is concerned with the ecient solution of (block) Hessenberg linear systems whose coecient matrix is a Toeplitz matrix in (block) Hessenberg form plus a band matrix. Such problems arise, for instance, when we apply a computational scheme based on the use of dierence equations for the computation of many signicant special functions and qua...
. This paper is concerned with the solution of linear systems with coefficient matrices which are Vandermondelike matrices modified by adding lowrank corrections. Hereafter we refer to these matrices as to modified Vandermondelike matrices. The solution of modified Vandermondelike linear systems arises in the approximation theory both when we us...
Given two polynomials with coefficients over Z[k], the associated Bezout matrix B(k) with entries over Z[k] defines a parametric family of Bezout matrices with entries over Z. It is intended in this paper to propose a hybrid approach for determining the inertia of B(k) for any value of k in some real interval. This yields an efficient solution to c...
An algorithm for the computation of the LU factorization over the integers of an n Theta n Bezoutian B is presented. The algorithm requires O(n 2 ) arithmetic operations and involves integers having at most O(n log nc) bits, where c is an upper bound of the moduli of the integer entries of B. As an application, by using the correlations between Bez...
In this paper we deal with the numerical approximation of a factor of a polynomial. Our approach is based on the relations between matrix transforms and functional iterations. We show that a generalized LR algorithm applied to an $n\times n$ Hessenberg matrix A may be viewed in a polynomial setting as an iterative method for the computation of a si...
Consider an n × n lower triangular matrix L whose (i + 1)st row is defined by the coefficients of the real polynomial pi(x) of degree i such that {pi(x)} is a set of orthogonal polynomials satisfying a standard threeterm recurrence relation. If H is an n × n real Hankel matrix with nonsingular leading principal submatrices, then will be referred t...
The Lanczos method and its variants can be used to solve efficiently the rational interpolation problem. In this paper we
present a suitable fast modification of a general lookahead version of the Lanczos process in order to deal with polynomials
expressed in the Chebyshev orthogonal basis. The proposed approach is particularly suited for approxim...
this article we propose a generalization of these efficient computational schemes in order to deal with generalized polynomials
this paper we present a suitable fast modification of a general lookahed version of the Lanczos process in order to deal with polynomials expressed in the Chebyshev orthogonal basis. The proposed approach is particularly suited for rational interpolation at Chebyshev points, that is, at the zeros of Chebyshev polynomials. In fact, in this case it...
We derive a closed inversion formula for an np × np square block Hankel matrix Hn − 1 = (Wi − j) with entries Wi from the ring of the p × p matrices over a field. The representation of H−1n − 1 relies upon a strong structurepreserving property of the Schur complements of the nonsingular leading principal submatrices of a certain generalized Bezout...
We present an iterative numerical method for solving two classical stability problems for a polynomial p(x) of degree n: the RouthHurwitz and the SchurCohn problems. This new method relies on the construction of a polynomial sequence fp (k) (x)g k2N , p (0) (x) = p(x) , such that p (k) (x) quadratically converges to (x Gamma 1) p (x + 1) nGammap w...
. By representing the LR algorithm of Rutishauser and its variants in a polynomial setting, we derive numerical methods for approximating either all of the roots or a number k of the roots of minimum modulus of a given polynomial p(t) of degree n. These methods share the convergence properties of the LR matrix iteration but, unlike it, they can be...
. Fast orthogonalization schemes for m Theta n Vandermonde matrices V = (z j i ), introduced by Demeure, are extended to compute both a QR factorization and an inverse QR factorization of Vandermondelike matrices P = (p j (z i )) where the polynomials p j (z) satisfy a threeterm recurrence relation. In this way we are able to solve least squares...