Centrosymmetric Matrices in the Sinc Collocation Method for Sturm-Liouville Problems

Recently, we used the Sinc collocation method with the double exponential transformation to compute eigenvalues for singular Sturm-Liouville problems. In this work, we show that the computation complexity of the eigenvalues of such a differential eigenvalue problem can be considerably reduced when its operator commutes with the parity operator. In this case, the matrices resulting from the Sinc collocation method are centrosymmetric. Utilizing well known properties of centrosymmetric matrices, we transform the problem of solving one large eigensystem into solving two smaller eigensystems. We show that only 1/(N+1) of all components need to be computed and stored in order to obtain all eigenvalues, where (2N+1) corresponds to the dimension of the eigensystem. We applied our result to the Schr\"odinger equation with the anharmonic potential and the numerical results section clearly illustrates the substantial gain in efficiency and accuracy when using the proposed algorithm.Comment: 11 pages, 4 figure

Pseudo-centrosymmetric matrices, with applications to counting perfect matchings

We consider square matrices A that commute with a fixed square matrix K, both with entries in a field F not of characteristic 2. When K^2=I, Tao and Yasuda defined A to be generalized centrosymmetric with respect to K. When K^2=-I, we define A to be pseudo-centrosymmetric with respect to K; we show that the determinant of every even-order pseudo-centrosymmetric matrix is the sum of two squares over F, as long as -1 is not a square in F. When a pseudo-centrosymmetric matrix A contains only integral entries and is pseudo-centrosymmetric with respect to a matrix with rational entries, the determinant of A is the sum of two integral squares. This result, when specialized to when K is the even-order alternating exchange matrix, applies to enumerative combinatorics. Using solely matrix-based methods, we reprove a weak form of Jockusch's theorem for enumerating perfect matchings of 2-even symmetric graphs. As a corollary, we reprove that the number of domino tilings of regions known as Aztec diamonds and Aztec pillows is a sum of two integral squares.Comment: v1: Preprint; 11 pages, 7 figures. v2: Preprint; 15 pages, 7 figures. Reworked so that linear algebraic results are over a field not of characteristic 2, not over the real numbers. Accepted, Linear Algebra and its Application

On Discrete Gauss-Hermite Functions and Eigenvectors of the Discrete Fourier Transform

The problem of furnishing an orthogonal basis of eigenvectors for the discrete Fourier transform (DFT) is fundamental to signal processing. Recent developments in the area of discrete fractional Fourier analysis also rely upon the ability to furnish a basis of eigenvectors for the DFT or its centralized version. However, none of the existing approaches are able to furnish a commuting matrix where both the eigenvalue spectrum and the eigenvectors are a close match to corresponding properties of the continuous differential Gauss-Hermite operator. Furthermore, any linear combination of commuting matrices produced by existing approaches also commutes with the DFT, thereby bringing up the question of uniqueness. In this paper, inspired by concepts from quantum mechanics in finite dimensions, we present an approach that furnishes a basis of orthogonal eigenvectors for both versions of the DFT. This approach also furnishes a commuting matrix whose eigenvalue spectrum is a very close approximation to that of the Gauss--Hermite differential operator and consequently a framework for a unique definition of the discrete Gauss--Hermite operato

Inverse eigenproblem for centrosymmetric and centroskew matrices and their approximation

In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors fxigm i=1 in Cn and a set of complex numbers f¸igm i=1, find a centrosymmetric or centroskew matrix C in Rn£n such that fxigm i=1 and f¸igm i=1 are the eigenvectors and eigenvalues of C respectively. We then consider the best approximation problem for the IEPs that are solvable. More precisely, given an arbitrary matrix B in Rn£n, we find the matrix C which is the solution to the IEP and is closest to B in the Frobenius norm.We show that the best approximation is unique and derive an expression for it

Block Decompositions and Applications of Generalized Reflexive Matrices

Generalize reflexive matrices are a special class of matrices that have the relation where and are some generalized reflection matrices. The nontrivial cases ( or ) of this class of matrices occur very often in many scientific and engineering applications. They are also a generalization of centrosymmetric matrices and reflexive matrices. The main purpose of this paper is to present block decomposition schemes for generalized reflexive matrices of various types and to obtain their decomposed explicit block-diagonal structures. The decompositions make use of unitary equivalence transformations and, therefore, preserve the singular values of the matrices. They lead to more efficient sequential computations and at the same time induce large-grain parallelism as a by-product, making themselves computationally attractive for large-scale applications. A numerical example is employed to show the usefulness of the developed explicit decompositions for decoupling linear least-square problems whose coefficient matrices are of this class into smaller and independent subproblems

A Thesis Submitted in Partial Fulfilmen of the Requirements for the Degree of Doctor of Philosophy in Mathematics

The Cayley transform method is a Newton-like method for solving in- verse eigenvalue problems. If the problem is large, one can solve the Ja- cobian equation by iterative methods. However, iterative methods usually oversolve the problem in the sense that they require far more (inner) it- erations than is required for the convergence of the Newton (outer) itera- tions. In this paper, we develop an inexact version of the Cayley transform method. Our method can reduce the oversolving problem and improves the e±ciency with respect to the exact version. We show that the convergence rate of our method is superlinear and that a good tradeo® between the required inner and outer iterations can be obtained

Estudio de la clase de matrices {K,s+1}-potentes

En esta tesis doctoral se han introducido y analizado de manera exhaustiva una nueva clase de matrices denominada matrices {K,s+1}-potentes. Estas matrices contienen como casos particulares las matrices {s+1}-potentes, periódicas, centrosimétricas, mirrorsimétricas, circulantes, etc. Estos últimos tipos de matrices son de gran utilidad en diferentes áreas tales como transmisión de líneas multiconductor, antenas, ondas, sistemas eléctricos y mecánicos, y teoría de la comunicación, entre otros. En el capítulo 1 se han presentado algunos resultados básicos. En el capítulo 2 se han obtenido diferentes propiedades de las matrices {K,s+1}-potentes relacionadas con la suma, el producto, la inversa, la adjunta, la semejanza y la suma directa. Posteriormente, se han encontrado caracterizaciones de las matrices {K,s+1}-potentes desde distintos puntos de vista: usando teoría espectral, mediante potencias de matrices, a partir de inversas generalizadas, y mediante una representación por bloques de una matriz de índice 1. Luego, en el capítulo 3, se ha relacionado la clase de matrices introducida con diferentes clases de matrices complejas conocidas en la literatura, a saber: matrices {K}-hermíticas, proyectores {s+1}-generalizados, matrices unitarias, matrices normales, centrosimétricas {K}-generalizadas, etc. Con la intención de construir de manera efectiva matrices de esta clase, en el capítulo 4 se han diseñado algoritmos tanto en el caso s mayor o igual a 1 y el caso s=0. Primero se construyen matrices en esta clase a partir de información espectral de la matriz involutiva K. Utilizando este algoritmo se pueden construir más ejemplos. Concretamente, se hallan matrices {K,s+1}-potentes que conmutan con las encontradas anteriormente, y mediante estos dos algoritmos, se puede realizar el análisis de combinaciones lineales de matrices de este tipo. Por otra parte, para los casos s mayor o igual a 1 y s=0 se ha resuelto el problema inverso de calcular las matrices involutivas K que satisfacen la ecuación matricial que se está tratando. También en este caso se han presentado métodos numéricos que lo resuelven. Por último, en este capítulo se incluyen ejemplos numéricos para mostrar las prestaciones de los métodos desarrollados. En el capítulo 5, se extiende el estudio anterior al caso de matrices {K,-(s+1)}-potentes, completando así todos los valores de s enteros posibles. Especial énfasis se ha puesto en el análisis espectral de estas clases de matrices. La tesis finaliza con un anexo en el que se indican las conclusiones finales y las líneas futuras.Romero Martínez, JO. (2012). Estudio de la clase de matrices {K,s+1}-potentes [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/15974Palanci