Darboux transformation and perturbation of linear functionals

28 pages, no figures.-- MSC2000 codes: 42C05; 15A23.MR#: MR2055354 (2005b:15027)Zbl#: Zbl 1055.42016Let L be a quasi-definite linear functional defined on the linear space of polynomials with real coefficients. In the literature, three canonical transformations of this functional are studied: $\bold{xL}$, $\bold L+\bold C\delta (\bold x)$ and $\frac {1}{\bold x}\bold L +\bold C\delta(\bold x)$ where $\delta(x)$ denotes the linear functional $(\delta(x))(x )=\delta_{k,0}$, and $\delta_{k,0}$ is the Kronecker symbol. Let us consider the sequence of monic polynomials orthogonal with respect to L. This sequence satisfies a three-term recurrence relation whose coefficients are the entries of the so-called monic Jacobi matrix. In this paper we show how to find the monic Jacobi matrix associated with the three canonical perturbations in terms of the monic Jacobi matrix associated with L. The main tools are Darboux transformations. In the case that the LU factorization of the monic Jacobi matrix associated with xL does not exist and Darboux transformation does not work, we show how to obtain the monic Jacobi matrix associated with $\bold x \bold L$ as a limit case. We also study perturbations of the functional L that are obtained by combining the canonical cases. Finally, we present explicit algebraic relations between the polynomials orthogonal with respect to L and orthogonal with respect to the perturbed functionals.The work of the authors has been partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant BFM 2003-06335-C03-02 and NATO collaborative grant PST.CLG.979738.Publicad

Eigenvectors and minimal bases for some families of Fiedler-like linearizations

In this paper, we obtain formulas for the left and right eigenvectors and minimal bases of some families of Fiedler-like linearizations of square matrix polynomials. In particular, for the families of Fiedler pencils, generalized Fiedler pencils and Fiedler pencils with repetition. These formulas allow us to relate the eigenvectors and minimal bases of the linearizations with the ones of the polynomial. Since the eigenvectors appear in the standard formula of the condition number of eigenvalues of matrix polynomials, our results may be used to compare the condition numbers of eigenvalues of the linearizations within these families and the corresponding condition number of the polynomial eigenvalue problem.Publicad

Polynomial perturbations of bilinear functionals and Hessenberg matrices

20 pages, no figures.-- MSC2000 codes: 42C05; 15A23.MR#: MR2209234 (2008c:42024)Zbl#: Zbl 1134.42015This paper deals with symmetric and non-symmetric polynomial perturbations of symmetric quasi-definite bilinear functionals. We establish a relation between the Hessenberg matrices associated with the initial and the perturbed functionals using LU and QR factorizations. Moreover we give an explicit algebraic relation between the sequences of orthogonal polynomials associated with both functionals.The work of the authors has been partially supported by Dirección General de Investigación (Ministerio de Educacion y Ciencia) of Spain under Grant BFM 2003-06335-C03-02 and INTAS Research Network NeCCA INTAS 03-51-6637.Publicad

Continuous symmetric Sobolev inner products

24 pages, no figures.-- MSC2000 codes: 42C05.MR#: MR1953647 (2003j:42029)Zbl#: Zbl pre05368623In this paper we consider the sequence of monic polynomials (Qn) orthogonal with respect to a symmetric Sobolev inner product. If Q_2n(x)=Pn(x^2) and Q_2n+1(x)=xRn(x^2), then we deduce the integral representation of the inner products such that (Pn) and (Rn) are, respectively, the corresponding sequences of monic orthogonal polynomials. In the semiclassical case, algebraic relations between such sequences are deduced. Finally, an application of the above results to Freud-Sobolev polynomials is given.The work of the second author was partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant BFM 2000-0206-C04-01 and INTAS project INTAS 2000-272.Publicad

Discrete-continuous symmetrized Sobolev inner products

23 pages, no figures.-- MSC2000 code: 42C05.MR#: MR2069523 (2005f:42053)Zbl#: Zbl 1048.42023^aThis paper deals with the bilinear symmetrization problem associated with Sobolev inner products. Let $\{Q_n\}_{n=0}^{\infty}$ be the sequence of monic polynomials orthogonal with respect to a Sobolev inner product of order 1 when one of the measures is discrete and the other one is a nondiscrete positive Borel measure. Furthermore, assume that the supports of such measures are symmetric with respect to the origin so that the corresponding odd moments vanish. We consider the orthogonality properties of the sequences of monic polynomials $\{P_n\}_{n=0}^{\infty}$ and $\{R_n\}_{n=0}^{\infty}$ such that $Q_{2n}(x)=P_n(x^2)$, $Q_{2n+1}(x)=xR_n(x^2)$. Moreover, recurrence relations for $\{P_n\}_{n=0}^{\infty}$ and $\{R_n\}_{n=0}^{\infty}$ are obtained as well as explicit algebraic relations between them.The work of the second author has been partially supported by KRF-2002-070-C00004. The work of the third author has been partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant BFM 2000-0206-C04-01 and INTAS project INTAS 2000-272.Publicad

Large vector spaces of block-symmetric strong linearizations of matrix polynomials

Given a matrix polynomial P(lambda) = Sigma(k)(i=0) lambda(i) A(i) of degree k, where A(i) are n x n matrices with entries in a field F, the development of linearizations of P(lambda) that preserve whatever structure P(lambda) might posses has been a very active area of research in the last decade. Most of the structure-preserving linearizations of P(lambda) discovered so far are based on certain modifications of block-symmetric linearizations. The block-symmetric linearizations of P(lambda) available in the literature fall essentially into two classes: linearizations based on the so-called Fiedler pencils with repetition, which form a finite family, and a vector space of dimension k of block-symmetric pencils, called DL(P), such that most of its pencils are linearizations. One drawback of the pencils in DL(P) is that none of them is a linearization when P(lambda) is singular. In this paper we introduce new vector spaces of block,symmetric pencils, most of which are strong linearizations of P(lambda). The dimensions of these spaces are O(n(2)), which, for n >= root k, are much larger than the dimension of DL(P). When k is odd, many of these vector spaces contain linearizations also when P(lambda) is singular. The coefficients of the block-symmetric pencils in these new spaces can be easily constructed as k x k block-matrices whose n x n blocks are of the form 0, +/-alpha I-n, +/-alpha A(i), or arbitrary n x n matrices, where a is an arbitrary nonzero scalar.The research of F. M. Dopico was partially supported by the Ministerio de Economía y Competitividad of Spain through grant MTM-2012-3254

Conditioning and backward errors of eigenvalues of homogeneous matrix polynomials under Möbius transformations

We present the first general study on the effect of Möbius transformations on the eigenvalue condition numbers and backward errors of approximate eigenpairs of polynomial eigenvalue problems (PEPs). By usingthe homogeneous formulation of PEPs, we are able to obtain two clear andsimple results. First, we show that if the matrix inducing the Möbius transformation is well-conditioned, then such transformation approximately preservesthe eigenvalue condition numbers and backward errors when they are definedwith respect to perturbations of the matrix polynomial which are small relativeto the norm of the whole polynomial. However, if the perturbations in eachcoefficient of the matrix polynomial are small relative to the norm of that coefficient, then the corresponding eigenvalue condition numbers and backwarderrors are preserved approximately by the Möbius transformations induced bywell-conditioned matrices only if a penalty factor, depending on the norms ofthose matrix coefficients, is moderate. It is important to note that these simple results are no longer true if a non-homogeneous formulation of the PEP isused

A comparison of eigenvalue condition numbers for matrix polynomials

In this paper, we consider the different condition numbers for simple eigenvalues of 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 eigenvalue problem. This number has the disadvantage of only being defined for finite eigenvalues. In order to give a unified approach to all the eigenvalues of a matrix polynomial, both finite and infinite, two (homogeneous) condition numbers have been defined in the literature. In their definition, very different approaches are used. One of the main goals of this note is to show that, when the matrix polynomial has a moderate degree, both homogeneous condition numbers are essentially the same and one of them provides a geometric interpretation of the other. We also show how the homogeneous condition numbers compare with the "Wilkinson-like" eigenvalue condition number and how they extend this condition number to zero and infinite eigenvalues.