Explicit inverse of a tridiagonal k−Toeplitz matrix

Fonseca, Carlos M. da; Petronilho, J.

doi:10.1007/s00211-005-0596-3

Cited by 68 publications

(51 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this case, the Schrödinger operator corresponds to a second order linear difference equation with constant coefficients, so its solution can be expressed in terms of Chebyshev polynomials. The expression obtained coincides with that published by Fonseca and Petronilho in [6, Corollary 4.1] and [7,Equation 4.26].…”

Section: The Inverse Of a Tridiagonal (P R)-toeplitz Matrixsupporting

confidence: 90%

“…If r = 1, the Jacobi (p, 1)-Toeplitz matrices are the ones so-called tridiagonal pToeplitz matrices, see for instance [7], whose coefficients are periodic with period p. When p = 1 too, the Jacobi (1, 1)-Toeplitz matrices are the matrices referenced at the beginning of this section, the tridiagonal and Toeplitz matrices. Note also that Jacobi (1, r)-Toeplitz matrices are those whose diagonals are geometrical sequences with ratio r.…”

Section: The Inverse Of a Tridiagonal (P R)-toeplitz Matrixmentioning

confidence: 98%

See 1 more Smart Citation

Explicit inverse of a tridiagonal ( p , r )-Toeplitz matrix

Encinas¹,

Jiménez²

2018

Linear Algebra and its Applications

View full text Add to dashboard Cite

Tridiagonal matrices appears in many contexts in pure and applied mathematics, so the study of the inverse of these matrices becomes of specific interest. In recent years the invertibility of nonsingular tridiagonal matrices has been quite investigated in different fields, not only from the theoretical point of view (either in the framework of linear algebra or in the ambit of numerical analysis), but also due to applications, for instance in the study of sound propagation problems or certain quantum oscillators. However, explicit inverses are known only in a few cases, in particular when the tridiagonal matrix has constant diagonals or the coefficients of these diagonals are subjected to some restrictions like the tridiagonal k-Toeplitz matrices, such that their three diagonals are formed by k-periodic sequences.The recent formulae for the inversion of tridiagonal k-Toeplitz matrices are based, more o less directly, on the solution of difference equations with periodic coefficients, though all of them use complex formulation that in fact don't take into account the periodicity of the coefficients.This contribution presents the explicit inverse of a tridiagonal matrix (p, r)-Toeplitz, which diagonal coefficients are in a more general class of sequences than periodic ones, that we have called quasi-periodic sequences. A tridiagonal matrix A = (a ij ) of order n + 2 is (p, r)-Toeplitz if there exists m ∈ N \ {0} such that n + 2 = mp andwe develop a procedure that reduces any linear second order difference equation with periodic coefficients to a difference equation of the same kind but with constant coefficients. Therefore, the solutions of the former equations can be expressed in terms of Chebyshev polynomials This fact explain why Chebyshev polynomials are ubiquitous in the above mentioned papers In addition, we show that this results are true when the coefficients are in a more general class of sequences, that we have called quasi-periodic sequences As a by-product, the inversion of these class of tridiagonal matrices could be explicitly obtained through the resolution of boundary value problems on a path Making use of the theory of orthogonal polynomials, we will give the explicit inverse of tridiagonal 2-Toeplitz and 3-Toeplitz matrices, based on recent results from which enables us to state some conditions for the existence of A − 1. We obtain explicit formulas for the entries of the inverse of a based on results from the theory of orthogonal polynomials and it is shown that the entries of the inverse of such a matrix are given in terms of Chebyshev polynomials of the second kind.como corolario, constante k-top, geometricas It is well-known that any second order linear difference equation with constant coefficients is equivalent to a Chebyshev equation. The aim of this work is to extend this result to any second order linear difference equation with quasi-periodic coefficients.We say that a sequence z, z : Z −→ C, is quasi-periodic with period p ∈ N \ {0} if there exists r ∈ C \ {0} such thatGiven a ...

show abstract

Section: The Inverse Of a Tridiagonal (P R)-toeplitz Matrixsupporting

confidence: 90%

Section: The Inverse Of a Tridiagonal (P R)-toeplitz Matrixmentioning

confidence: 98%

Explicit inverse of a tridiagonal ( p , r )-Toeplitz matrix

Encinas¹,

Jiménez²

2018

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…For more general spectral results on tridiagonal r-Toeplitz matrices, the reader is referred to [2,3].…”

Section: Eigenvalues Of a Tridiagonal 2-toeplitz Matrixmentioning

confidence: 99%

Eigenvalues of a certain class of tridiagonal matrices: A brief comment

Fonseca

2015

Applied Mathematics and Computation

View full text Add to dashboard Cite

“…This kind of polynomial mappings arise in problems from Quantum Mechanics and Physics (see, e.g., [2,5,4,13] and the references therein) as well as in connection with the so-called sieved OPs (see, e.g., [1,6,8,13,17] and references therein). Also, similar transformation laws for OPs were used to solve some algebraic problems in matrix theory [20,10]. In this paper we explore this kind of transformations in another direction, more precisely to solve the following inverse problem (P) in below concerning OPUC.…”

Section: Introductionmentioning

confidence: 98%

Orthogonal polynomials on the unit circle via a polynomial mapping on the real line

Petronilho¹

2008

Journal of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

Let { n } n 0 be a sequence of monic orthogonal polynomials on the unit circle (OPUC) with respect to a symmetric and finite positive Borel measure d on [0, 2 ] and let −1, 0 , 1 , 2 , . . . be the associated sequence of Verblunsky coefficients. In this paper we study the sequence { n } n 0 of monic OPUC whose sequence of Verblunsky coefficients is where b 1 , b 2 , . . . , b N−1 are N − 1 fixed real numbers such that b j ∈ (−1, 1) for all j = 1, 2, . . . , N − 1, so that { n } n 0 is also orthogonal with respect to a symmetric and finite positive Borel measure d on the unit circle. We show that the sequences of monic orthogonal polynomials on the real line (OPRL) corresponding to { n } n 0 and { n } n 0 (by Szegö's transformation) are related by some polynomial mapping, giving rise to a one-to-one correspondence between the monic OPUC { n } n 0 on the unit circle and a pair of monic OPRL on (a subset of) the interval [−1, 1]. In particular we prove thatsupported on (a subset of) the union of 2N intervals contained in [0, 2 ] such that any two of these intervals have at most one common point, and where, up to an affine change in the variable, N−1 and cos ϑ N are algebraic polynomials in cos of degrees N − 1 and N (respectively) defined only in terms of 0 , b 1 , . . . , b N−1 . This measure induces a measure on the unit circle supported on the union of 2N arcs, pairwise symmetric with respect to the real axis. The restriction to symmetric measures (or real Verblunsky coefficients) is needed in order that Szegö's transformation may be applicable.

show abstract

Explicit inverse of a tridiagonal k−Toeplitz matrix

Cited by 68 publications

References 24 publications

Explicit inverse of a tridiagonal ( p , r )-Toeplitz matrix

Explicit inverse of a tridiagonal ( p , r )-Toeplitz matrix

Eigenvalues of a certain class of tridiagonal matrices: A brief comment

Orthogonal polynomials on the unit circle via a polynomial mapping on the real line

Contact Info

Product

Resources

About