Symmetric Tridiagonal Matrices Research Articles

The spectra of some trees and bounds for the largest eigenvalue of any tree

Let T be an unweighted tree of k levels such that in each level the vertices have equal degree. Let n k− j+1 and d k− j+1 be the number of vertices and the degree of them in the level j. We find the eigenvalues of the adjacency matrix and Laplacian matrix of T for the case of two vertices in level 1 ( n k = 2), including results concerning to their multiplicity. They are the eigenvalues of leading principal submatrices of nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for these matrices are d j - 1 , 2 ⩽ j ⩽ k, while the diagonal entries are 0, …, 0, ±1, in the case of the adjacency matrix, and d 1, d 2, …, d k−1 , d k ± 1, in the case of the Laplacian matrix. Finally, we use these results to find improved upper bounds for the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any given tree.

An Implicit Q Theorem for Hessenberg-like Matrices

The implicit Q theorem for Hessenberg matrices is a widespread and powerful theorem. It is used in the development of, for example, implicit QR algorithms to compute the eigendecomposition of Hessenberg matrices. Moreover it can also be used to prove the essential uniqueness of orthogonal similarity transformations of matrices to Hessenberg form. The theorem is also valid for symmetric tridiagonal matrices, proving thereby also in the symmetric case its power. Currently there is a growing interest to so-called semiseparable matrices. These matrices can be considered as the inverses of tridiagonal matrices. In a similar way, one can consider Hessenberg-like matrices as the inverses of Hessenberg matrices. In this paper, we formulate and prove an implicit Q theorem for the class of Hessenberg-like matrices. We introduce the notion of strongly unreduced Hessenberg-like matrices and also a method for transforming matrices via orthogonal transformations to this form is proposed. Moreover, as the theorem is valid for Hessenberg-like matrices it is also valid for symmetric semiseparable matrices.

On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of odd order—II

This paper is an extension of the work [J. Rimas, On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of odd order—I, Applied Mathematics and Computation, in press], in which the general expression of the lth power ( l ∈ N) for one type of symmetric tridiagonal matrices of order n = 2 p + 1 ( p ∈ N) is given. In this new paper, we present the complete derivation of this general expression. Expressions of eigenvectors and Jordan’s form of the matrix and of the transforming matrix and its inverse are given, too.

Skew-Symmetric Differential qd Algorithm

Differential qd (dqd) algorithm with shifts is probably the fastest known algorithm which computes eigenvalues of symmetric tridiagonal matrices with high relative accuracy. In this paper we will construct a similar algorithm for computing eigenvalues of skew-symmetric matrices, which is based on implicit usage of both the QR and the symplectic QR factorizations. If we apply this algorithm to tridiagonal skew-symmetric matrices, we obtain the skew-symmetric dqd algorithm. This algorithm also enjoys high relative stability. However, incorporation of shifts is much harder then in the symmetric case, and yet to be implemented. Finally, the standard algorithm for computing the eigenvalues of tridiagonal skew-symmetric matrices can also be interpreted in the context of the skew-symmetric dqd algorithm. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The spectra of the adjacency matrix and Laplacian matrix for some balanced trees

Let T be an unweighted rooted tree of k levels such that in each level the vertices have equal degree. Let d k− j+1 denotes the degree of the vertices in the level j. We find the eigenvalues of the adjacency matrix and of the Laplacian matrix of T . They are the eigenvalues of principal submatrices of two nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for both matrices are d j - 1 , 2 ⩽ j ⩽ k - 1 , and d k , while the diagonal entries are zeros, in the case of the adjacency matrix, and d j , 1 ⩽ j ⩽ k, in the case of the Laplacian matrix. Moreover, we give some results concerning to the multiplicity of the above mentioned eigenvalues.

On Symmetric Eigenproblems Induced by the Bidiagonal SVD

The relatively robust representations (RRR) algorithm is the method of choice to compute highly accurate eigenvector approximations for symmetric tridiagonal matrices. The task of computing singular vector pairs for a bidiagonal matrix $B = U \Sigma V ^T$ is closely connected to the RRR algorithm regarding BTB , BBT , or the Golub--Kahan matrix TGK . Nevertheless, separate application of the RRR algorithm to these matrices leads to poor results regarding either numerical orthogonality or the residual $\| {B}{V} - {U}{\Sigma} \|$. It turns out that the coupling strategy proposed in [B. Grosser and B. Lang, Linear Algebra Appl., 358 (2003), pp. 45--70] resolves this problem. This article provides the corresponding perturbation theory: We compare the eigenvalues of the separate and coupled decompositions and explain why singular vector pairs approximated via couplings are of superior quality.

QL method for symmetric tridiagonal matrices

QL(QR) method is an efficient method to find eigenvalues of a matrix. Especially we use QL(QR) method to find eigenvalues of a symmetric tridiagonal matrix. In this case it only costs O(n 2) flops, to find all eigenvalues. So it is one of the most efficient method for symmetric tridiagonal matrices. Many experts have researched it. Even the method is mature, it still has many problems need to be researched. We put forward five problems here. They are: (1) Convergence and convergence rate; (2) The convergence of diagonal elements; (3) Shift designed to produce the eigenvalues in monotone order; (4) QL algorithm with multi-shift; (5) Error bound. We intoduce our works on these problems, some of them were published and some are new.

On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of even order

In this paper we derive the general expression of the lth power ( l ∈ N) for one type of symmetric tridiagonal matrices of order n = 2 p( p ∈ N).

The Godunov-inverse iteration: A fast and accurate solution to the symmetric tridiagonal eigenvalue problem

We present a new algorithm for computing eigenvectors of real symmetric tridiagonal matrices based on Godunov's two-sided Sturm sequence method and inverse iteration, which we call the Godunov-inverse iteration. We use eigenvector approximations computed recursively from two-sided Sturm sequences as starting vectors in inverse iteration, replacing any nonnumeric elements of these approximate eigenvectors with uniform random numbers. We use the left-hand bounds of the smallest machine presentable eigenvalue intervals found by the bisection method as inverse iteration shifts, while staying within guaranteed error bounds. In most test cases convergence is reached after only one or two iterations, producing accurate residuals.

Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices

In this paper we present an O( nk) procedure, Algorithm MR 3 , for computing k eigenvectors of an n× n symmetric tridiagonal matrix T. A salient feature of the algorithm is that a number of different LDL t products ( L unit lower triangular, D diagonal) are computed. In exact arithmetic each LDL t is a factorization of a translate of T. We call the various LDL t products representations (of T) and, roughly speaking, there is a representation for each cluster of close eigenvalues. The unfolding of the algorithm, for each matrix, is well described by a representation tree. We present the tree and use it to show that if each representation satisfies three prescribed conditions then the computed eigenvectors are orthogonal to working accuracy and have small residual norms with respect to the original matrix T.

Stability of the Matrix Factorization for Solving Block Tridiagonal Symmetric Indefinite Linear Systems

In this paper, we propose a new factorization method for block tridiagonal symmetric indefinite matrices. We also discuss the stability of the factorization method. As a measurement of stability, an effective condition number is derived by using backward error analysis and perturbation analysis. It shows that under some suitable assumptions, the solution obtained by this factorization method is acceptable. Numerical results demonstrate that the factorization is stable if its condition number is not too large.

Graphs whose minimal rank is two

Let F be a field, F= (V,E) be an undirected graph on n vertices, and let S(F,G) be the set of all symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the eduges of G. For example, if G is a path, S(F,G) consists of the symmetric irreducible tridiagonal matrices. Let mr(F, G) be the minimum rank over all matrices in S(F,G). Then mr(F,G)=1 if and only if G is the union of a clique with at least 2 vertices and an independent set. If F is an infinite field such that char F≠2, then mr(F,G)≤ 2 if and only if the complement of G is the join of a clique and a graph that is the union of at most two cliques and any number of complete bipartite graphs. A similar result is obtained in the case that F is an infinite field with charF=2. Furthermore, in each case, such graphs are characterized as those for which 6 specific graphs do not occur as induced subgraphs. The number of forbiddern subgraphs is reduced to 4 if the graph is connected. Finally, similar criteria is obtained for the minimum rank of a Hermitian matrix to be less than or equal to two. The complement is the join of a clique and a graph that is the union of any number of cliques and any number of complete bipartite graphs. The number of forbidder subgraphs is now 5, or in the connected case, 3. 

Orthogonal Eigenvectors and Relative Gaps

This paper presents and analyzes a new algorithm for computing eigenvectors of symmetric tridiagonal matrices factored as LDLt, with D diagonal and L unit bidiagonal. If an eigenpair is well behaved in a certain sense with respect to the factorization, the algorithm is shown to compute an approximate eigenvector which is accurate to working precision. As a consequence, all the eigenvectors computed by the algorithm come out numerically orthogonal to each other without making use of any reorthogonalization process. The key is first running a qd-type algorithm on the factored matrix LDLt and then applying a fine-tuned version of inverse iteration especially adapted to this situation. Since the computational cost is O(n) per eigenvector for an n × n matrix, the proposed algorithm is the central step of a more ambitious algorithm which, at best (i.e., when all eigenvectors are well-conditioned), would compute all eigenvectors of an n × n symmetric tridiagonal at O(n2 ) cost, a great improvement over existing algorithms.

Inner deflation for symmetric tridiagonal matrices

Suppose that one knows an accurate approximation to an eigenvalue of a real symmetric tridiagonal matrix. A variant of deflation by the Givens rotations is proposed in order to split off the approximated eigenvalue. Such a deflation can be used instead of inverse iteration to compute the corresponding eigenvector.

On Green's Matrices of Trees

The inverse C = [ci,j] of an irreducible nonsingular symmetric tridiagonal matrix is a so-called Green's matrix. A Green's matrix is a symmetric matrix which is given by two sequences of real numbers {ui} and {vi} such that ci,j = uivj for $i \leq j$. A similar result holds for nonsymmetric matrices. An open problem on nonsingular sparse matrices is whether there exists a similar structure for their inverses as in the tridiagonal case. Here we positively answer this question for irreducible acyclic matrices, i.e., matrices whose undirected graphs are trees. We prove that the inverses of irreducible acyclic symmetric matrices are given as the Hadamard product of three matrices, a type D matrix, a flipped type D matrix, and a matrix of tree structure which is closely related to the graph of the original matrix itself. For nonsymmetric matrices we obtain a similar structure. Moreover, our results include the result for symmetric and nonsymmetric tridiagonal matrices.

On an inverse problem in Cauer networks

In this paper a new procedure is established to obtain the element values of capacitors and resistances of a first Cauer network directly from the poles and zeros of the impedance function using the technique for constructing tridiagonal symmetric matrices from spectral data. This method is illustrated numerically.

Stability of block LDLT factorization of a symmetric tridiagonal matrix

For symmetric indefinite tridiagonal matrices, block LDLT factorization without interchanges is shown to have excellent numerical stability when a pivoting strategy of Bunch is used to choose the dimension (1 or 2) of the pivots.

The generalized eigenvalue problem for tridiagonal symmetric interval matrices

An algorithm to compute the eigenintervals of the generalized eigenvalue problem for tridiagonal symmetric matrices (A,B) with interval entries is developed. The algorithm is based on the Sturm Sequences and requires twice as much computational effort as for point matrices. The algorithm is applied to the problem of computing the natural frequencies of a vibrating beam where the manufacturing tolerance is taken into consideration.

Analytical inversion of general tridiagonal matrices

In this paper we give a complete analysis for general tridiagonal matrix inversion for both non-block and block cases, and provide some very simple analytical formulae which immediately lead to closed forms for some special cases such as symmetric or Toeplitz tridiagonal matrices.

A divide-and-conquer method for the tridiagonal generalized eigenvalue problem

We introduce a divide-and-conquer method for the generalized eigenvalue problem Ax = λBx, where A and B are real symmetric tridiagonal matrices and B is positive-definite. It is a generalization of Cuppen's method for the standard eigenvalue problem, B = I, which is based on rank-one modifications. Our method is an alternative to a method developed by Borges and Gragg using restrictions and extensions.