Jesús Abderramán Marrero

Universidad Politécnica de Madrid | UPM · Departamento de Matemática aplicada a las Tecnologías de la Información

After a short overview, improvements (based on the Kronecker product) are proposed for the eigenvalues of (N × N) block-Toeplitz tridiagonal (block-TT) matrices with (K × K) matrix-entries, common in applications. Some extensions of the spectral properties of the Toeplitz-tridiagonal matrices are pointed-out. The eigenvalues of diagonalizable symme...

SHARE LINK before October 19, 2019 https://authors.elsevier.com/a/1Zeo41LkiPMMeu ABSTRACT: A linear-time solver with total flop count $10n-11$ and $4n+2$ of memory space is introduced for solving determined comrade linear systems of degree~$n$. This algorithm is based on a suitable factorization satisfied for every comrade matrix coefficient. It...

ABSTRACT: A fast and reliable numerical solver is proposed to handle determined opposite-bordered tridiagonal linear systems Ax=b. A precise sequence of Givens rotations on the nonsingular matrix coefficient A allows us to work with a simpler linear system. This equivalent system, which admits always a trivial LU factorization, can be solved in li...

SHARE LINK: https://www.tandfonline.com/eprint/RyiHkpFibVkCJQrHruKX/full Abstract: The nice inversion properties of Toeplitz–Hessenberg matrices from a Hessenbergian representation of Catalan's numbers encourage us to provide explicit inversion formulas (in terms of basic arithmetical operations involving entries from the original matrix) for the...

In this paper, we use the arrowhead matrices as a tool to study graph theory. More precisely, we deal with an interesting class of directed multigraphs, the hub-directed multigraphs. We associate the arrowhead matrices with the adjacency matrices of a class of directed multigraph, and we obtain new properties of the second objects by using properti...

To overcome several limitations of symbolic algorithms introduced recently for matrices of large order, a fast numerical solver is proposed for the matrix linear equation , where the coefficient matrix is a general nonsingular bordered tridiagonal matrix. Its sparse structure is preserved through partial Givens reduction. In particular, the matrix...

To improve on the shortcomings observed in symbolic algorithms introduced recently for related matrices, a reliable numerical solver is proposed for computing the solution of the matrix linear equation . The matrix coefficient is a nonsingular bordered -tridiagonal matrix. The particular structure of is exploited through an incomplete or full Given...

The arrowhead matrices define a class of one-term Sylvester matrix (OTSM) operators on a finite-dimensional Hilbert space through an elementary UDL factorization. It enables us to consider the infinite invertible arrowhead matrices UDL factored properly for introducing, under suitable conditions, the arrowhead operators and their associated class o...

A comprehensive treatment on compact representations for the solutions of linear difference equations with variable coefficients, of both -th and unbounded order, is presented. The equivalence between their celebrated combinatorial and determinantal representations is considered. A corresponding representation is proposed using determined nested su...

After a concise survey, the expanded Ikebe algorithm for inverting the lower half plus the superdiagonal of an n × n unreduced upper Hessenberg matrix H is extended to general nonsingular upper Hessenberg matrices by computing, in the reduced case, a block diagonal form of the factor matrix HL in the inverse factorization . This factorization enabl...

SHARE LINK: http://www.tandfonline.com/eprint/IrPRXbtb93aHQB8ys8yX/full ABSTRACT Some procedures are proposed for well known as well as new factorizations of the inverses of nonsingular generalized Hessenberg matrices . Moreover, we take advantage in such decompositions of low-rank structures of Hessenberg-like matrices . This suite of decompo...

A block matrix analysis is proposed to justify, and modify, a known algorithm for computing in O(n)O(n) time the determinant of a nonsingular n×nn×n pentadiagonal matrix (n≥6)(n≥6) having nonzero entries on its second subdiagonal. Also, we describe a procedure for computing the inverse matrix with acceptable accuracy in O(n2)O(n2) time. In the gene...

Here a known result on the structure of finite Hessenberg matrices is extended to infinite Hessenberg matrices. Its consequences for the example of infinite Hessenberg–Toeplitz matrices are described. The results are applied also to the inversion of infinite tridiagonal matrices via recurrence relations. Moreover, since there are available free par...

We present some explicit formulas for calculating the principal pth root of a square matrix. The main tools are based on various polynomial decompositions of the principal matrix pth root and well-known properties of the linear recursive sequences.

A method, based on recurrence relations, is proposed for evaluating classical inverses of infinite (unreduced as well as reduced) tridiagonal matrices using a known result on inversion of finite Hessenberg matrices, applicable also on tridiagonal matrices. The recurrence relations for the inverse way are also provided.

A modification of the Ikebe algorithm for computing the lower half of the inverse of an (unreduced) upper Hessenberg matrix, extended to compute the entries of the superdiagonal, is considered in this paper. It enables us to compute the inverse of a quasiseparable Hessenberg matrix in O(n2)O(n2) times. A new factorization expressing the inverse of...

A representation for the entries of the inverse of general tridiagonal matrices is based on the determinants of their principal submatrices. It enables us to introduce, through the linear recurrence relations satisfied by such determinants, a simple algorithm for the entries of the inverse of any tridiagonal nonsingular matrix, reduced as well as u...

SHARE LINK: http://www.tandfonline.com/eprint/NwUsJppIFNuZu8ccKVky/full ABSTRACT: In this note, we propose an explicit representation with the nested sums for the entries of the inverses of general tridiagonal nonsingular matrices. Its equivalence with other particular representations, based on the combinatorial expressions or the continued frac...

We describe a method for evaluating both the Fibonacci-Hörner and the polynomial decomposition of the principal matrix logarithm, with a view to solve the lifting problem of its explicit computation. The Binet formula for linear recursive sequences serves as a triggering factor for giving the exact formula. We supply some illustrative examples.

Adequate conditions, using a known result on a class of finite Hessenberg matrices, are here proposed to make available finite, and infinite, matrices as a rank one perturbation of strictly upper triangular matrices UV+T. Some characterizations on such matrices for generating Orthogonal Polynomials sequences are also considered.

SHARE LINK: http://www.tandfonline.com/eprint/zyJ8tqbTqbWu6MJbnBZr/full ABSTRACT: The nested sums applied to a general three-term recurrence relation permits us to give compact representations of orthogonal polynomial sequences, which satisfy such kind of linear recurrence. We illustrate this model on particular examples of classical as well as n...

A change to the Szegő matrix recurrence relation, satisfied by orthonormal polynomials on the unit circle, gives rise to a linear map by the action of matrices belonging to the group SU (1,1). The companion factorization of such matrices, via 2nd-order linear homogeneous difference equations, provides a compact representation of the orthogonal poly...

Ikebe algorithm for computing the lower half of the inverse of any (unreduced) upper Hessenberg matrix is extended here to compute the entries of the superdiagonal. It gives rise to an algorithm of inversion based on the factorization H^-1=H_L.U^-1. The lower Hessenberg matrix H_L is a quasiseparable one and U^-1 is upper triangular, with diagonal...

The general representation for the elements of the inverse of any Hessenberg matrix of finite order is here extended to the reduced case with a new proof. Those entries are given with proper Hessenbergians from the original matrix. It justifies both the use of linear recurrences of unbounded order for such computations on matrices of intermediate o...

SHARE LINK: http://www.tandfonline.com/eprint/Fg7bcjuc7VXs2E4m2pCY/full ABSTRACT: For a large class of discrete matrix difference equations many qualitative problems remain unsolved. The companion matrix factorization is applied here to the shift matrices associated to linear non-autonomous area-preserving maps. It permits us to introduce second...

After a brief introduction and justification of the method, a simple numerical algorithm for the inversion of tridiagonal nonsingular matrices (unreduced as well as reduced) is introduced. We maintain here the numerical approach, without invoking the symbolic computation, wich has produced recent advances in the inversion of such matrices.

The companion factorization for nonsingular matrices belonging to the general linear group GL(n;C)GL(n;C) is studied here. The entries in the last row of the companion matrices are explicitly represented in terms of determinants of proper sub-matrices of the matrix being factorized. The computation of the inverse of a nonsingular matrix is given as...

A new proof of the general representation for the entries of the inverse of any unreduced Hessenberg matrix of finite order is found. Also this formulation is extended to the inverses of reduced Hessenberg matrices. Those matrices are given with proper Hessenbergians from the original matrix. It justifies both the use of linear recurrences for such...

An explicit representation for the determinant of any upper Hessenberg matrix in C^nxn, with non null diagonal terms, is achieved by means of a quasitriangular matrix with the same determinant value. It gives rise to a representation with nested functions on matrix elements h_i,j from de original Hessenberg matrix. Like an interesting application,...

The component functions {Ψn (ε)} (n 2 Z+) from difference Schrödinger operators, can be formulated in a second order linear difference equation. Then the Harper equation, associated to almost-Mathieu operator, is a prototypical example. Its spectral behavior is amazing. Here, due the cosine coefficient in Harper equation, the component functions ar...

A new procedure to solve some fluid problems formulated in elliptical partial differential equations is presented. A Genetic Algorithm with a dynamical encoding and a partial grid sampling is proposed for it as the advantages of solving the problem without using all grid nodes at the same time, and of adjusting step grid, without increasing the com...

A practical dynamical model of an efficient Simple Genetic Algorithm is presented, introducing in the matrix of the Nix and Vose Markov model a practical postulate related to the schema theorem, that induces deterministic correction factors in the matrix, through Heavisides unitary step function. This alteration permits SGA to evolve by efficient d...

In this paper it is attempted to improve the Genetic-Statistic Algorithm by studying the relations between several search parameters (mutation "μ", chromosomés length "L", population " n " and number of genertions by iteration "t"). It has been made a study over twenty-four objective functions in eight different dimensions. When the convergence is...

Palabras clave: Ecuación en diferencias lineal general, Función anidada, Hessenbergiano. Resumen Representaciones explícitas de las soluciones de la ecuacíón en diferencias lineal son obtenidas, en términos de los coeficientes variables de la recurrencia, tanto en el caso general como en el problema de valor inicial. Estas representaciones se basan...

A constructive theory for the general solution of kth-order difference equation is given as in a forthcoming paper of the author. As complement of the analytical theory [George D. Birkhoff, General theory of linear difference equations, Transactions of the American Mathematical Society, volume 12, number 2, pages 243–284, 1911], this constructive a...