Eigenvalues Of Symmetric Tridiagonal Matrices Research Articles

On the Roots of the Legendre Laguerre, and Hermite Polynomials

For several orthogonal polynomials, Cohen proved that their roots are the eigenvalues of symmetric tridiagonal matrices. In this paper, we give examples of this Cohen’s result for the Legendre, Laguerre, and Hermite polynomials, which are useful in applications to quantum mechanics and numerical analysis.

Mixed Precision Bisection

We discuss the implementation of the bisection algorithm for the computation of the eigenvalues of symmetric tridiagonal matrices in a context of mixed precision arithmetic. This approach is motivated by the emergence of processors which carry out floating-point operations much faster in single precision than they do in double precision. Perturbation theory results are used to decide when to switch from single to double precision. Numerical examples are presented.

Bethe graphs attached to the vertices of a connected graph - a spectral approach

A weighted Bethe graph B is obtained from a weighted generalized Bethe tree by identifying each set of children with the vertices of a graph belonging to a family F of graphs. The operation of identifying the root vertex of each of r weighted Bethe graphs to the vertices of a connected graph of order r is introduced as the -concatenation of a family of r weighted Bethe graphs. It is shown that the Laplacian eigenvalues (when F has arbitrary graphs) as well as the signless Laplacian and adjacency eigenvalues (when the graphs in F are all regular) of the -concatenation of a family of weighted Bethe graphs can be computed (in a unified way) using the stable and low computational cost methods available for the determination of the eigenvalues of symmetric tridiagonal matrices. Unlike the previous results already obtained on this topic, the more general context of families of distinct weighted Bethe graphs is herein considered.

Computing the Laplacian spectra of some graphs

In this paper we give a simple characterization of the Laplacian spectra of a family of graphs as the eigenvalues of symmetric tridiagonal matrices. In addition, we apply our result to obtain upper and lower bounds for the Laplacian-energy-like invariant of these graphs. The class of graphs considered are obtained from copies of modified generalized Bethe trees (obtained by joining the vertices at some level by paths), identifying their roots with the vertices of a regular graph or a path.

Spectra of weighted rooted graphs having prescribed subgraphs at some levels

Let B be a weighted generalized Bethe tree of k levels (k > 1) in which nj is the number of vertices at the level k j +1 (1 � jk). Let � � {1,2,...,k 1} and F= {Gj : j 2 �}, where Gj is a prescribed weighted graph on each set of children of B at the level k j+1. In this paper, the eigenvalues of a block symmetric tridiagonal matrix of order n1 +n2 +···+nk are characterized as the eigenvalues of symmetric tridiagonal matrices of order j, 1 � jk, easily constructed from the degrees of the vertices, the weights of the edges, and the eigenvalues of the matrices associated to the family of graphs F. These results are applied to characterize the eigenvalues of the Laplacian matrix, including their multiplicities, of the graph B (F) obtained from B and all the graphs in F = {Gj : j 2 �}; and also of the signless Laplacian and adjacency matrices whenever the graphs of the family F are regular.

Spectral characterization of some weighted rooted graphs with cliques

The level of a vertex in a rooted graph is one more than its distance from the root vertex. A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree. We characterize completely the eigenvalues of the Laplacian, signless Laplacian and adjacency matrices of a weighted rooted graph G obtained from a weighted generalized Bethe tree of k levels and weighted cliques in which (1) the edges connecting vertices at consecutive levels have the same weight, (2) each set of children, in one or more levels, defines a weighted clique, and (3) cliques at the same level are isomorphic. These eigenvalues are the eigenvalues of symmetric tridiagonal matrices of order j × j , 1 ⩽ j ⩽ k . Moreover, we give results on the multiplicity of the eigenvalues, on the spectral radii and on the algebraic conectivity. Finally, we apply the results to the unweighted case and some particular graphs are studied.

Benefits of IEEE‐754 Features in Modern Symmetric Tridiagonal Eigensolvers

Bisection is one of the most common methods used to compute the eigenvalues of symmetric tridiagonal matrices. Bisection relies on the Sturm count: For a given shift σ, the number of negative pivots in the factorization $T - \sigma I = LDL^T$ equals the number of eigenvalues of T that are smaller than σ. In IEEE‐754 arithmetic, the value ∞ permits the computation to continue past a zero pivot, producing a correct Sturm count when T is unreduced. Demmel and Li showed [IEEE Trans. Comput., 43 (1994), pp. 983–992] that using ∞ rather than testing for zero pivots within the loop could significantly improve performance on certain architectures. When eigenvalues are to be computed to high relative accuracy, it is often preferable to work with $LDL^T$ factorizations instead of the original tridiagonal T. One important example is the MRRR algorithm. When bisection is applied to the factored matrix, the Sturm count is computed from $LDL^T$ which makes differential stationary and progressive qds algorithms the methods of choice. While it seems trivial to replace T by $LDL^T$, in reality these algorithms are more complicated: In IEEE‐754 arithmetic, a zero pivot produces an overflow followed by an invalid exception (NaN, or “Not a Number”) that renders the Sturm count incorrect. We present alternative, safe formulations that are guaranteed to produce the correct result. Benchmarking these algorithms on a variety of platforms shows that the original formulation without tests is always faster provided that no exception occurs. The transforms see speed‐ups of up to $2.6\times$ over the careful formulations. Tests on industrial matrices show that encountering exceptions in practice is rare. This leads to the following design: First, compute the Sturm count by the fast but unsafe algorithm. Then, if an exception occurs, recompute the count by a safe, slower alternative. The new Sturm count algorithms improve the speed of bisection by up to $2\times$ on our test matrices. Furthermore, unlike the traditional tiny‐pivot substitution, proper use of IEEE‐754 features provides a careful formulation that imposes no input range restrictions.

On the spectra of certain rooted trees

Let T be an unweighted tree with vertex root v which is the union of two trees T 1 = ( V 1 , E 1 ) , T 2 = ( V 2 , E 2 ) such that V 1 ∩ V 2 = { v} and T 1 and T 2 have the property that the vertices in each of their levels have equal degree. We characterize completely the eigenvalues of the adjacency matrix and of the Laplacian matrix of T . They are the eigenvalues of symmetric tridiagonal matrices whose entries are given in terms of the vertex degrees. Moreover, we give some results about the multiplicity of the eigenvalues. Applications to some particular trees are developed.

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 quasi-Laguerre iteration

The quasi-Laguerre iteration has been successfully established, by the same authors, in the spirit of Laguerre's iteration for solving the eigenvalues of symmetric tridiagonal matrices. The improvement in efficiency over Laguerre's iteration is drastic. This paper supplements the theoretical background of this new iteration, including the proofs of the convergence properties.

Interlacing Properties of Tridiagonal Symmetric Matrices with Applications to Parallel Computing

In this paper we present new interlacing properties for the eigenvalues of an unreduced tridiagonal symmetric matrix in terms of its leading and trailing submatrices. The results stated in Hill and Parlett [SIAM J. Matrix Anal. Appl., 13 (1992), pp. 239–247] are hereby improved. We further extend our results to reduced symmetric tridiagonal matrices and to specially structured full symmetric matrices. We then present new fast and efficient parallel algorithms for computing a few eigenvalues of symmetric tridiagonal matrices of very large order.

A parallel algorithm for evaluating general linear recurrence equations

A block parallel partitioning method for evaluating general linearmth order recurrence equations is presented and applied to solve the eigenvalues of symmetric tridiagonal matrices. The algorithm is based on partitioning, in a way that ensures load balance during computation. This method is applicable to both shared memory and distributed memory MIMD systems. The algorithm achieves a speedup ofO(p) on a parallel computer withp-fold parallelism, which is linear and is greater than the existing results, and the data communication between processors is less than that required for other methods. For solving symmetric tridiagonal eigenvalue problems, our method is faster than the best previous results. The results were tested and evaluated on an MIMD machine, and were within 79% to 96% of the predicted performance for the second order linear recurrence problem.

Bisection is not Optimal on Vector Processors

Recently there has been revived interest in the bisection method for computing eigenvalues of symmetric tridiagonal matrices, since this method lends itself easily to a parallel implementation. A natural extension of the bisection method is the multisection method. The relative advantages of these two methods have been discussed in several publications ([H. Bernstein and M. Goldstein, SIAM J. Sci. Statist. Comput., 9(1988), pp. 601–602], [I. Ipsen and E. Jessup, Solving the symmetric tridiagonal eigenvalue problem on the hypercube, Tech. Report YALEU/DCS/RR-548, Yale Univ., Dept. of Computer Science, July 1987], [S. Lo, B. Philippe, and A. Sameh, SIAM J. Sci. Statist. Comput., 8(1987), pp. s155–s165]). The purpose of this note is to contribute another argument in favor of using the multisection method, which did not arise explicitly in the past discussion. A simple analysis and some numerical examples show that the bisection method is in general not optimal in the class of multisection methods for the extraction of one eigenvalue on vector processors. Numerical results on a CRAY-2, a Convex C1-XP, and an Alliant FX/8 show that the optimal multisection section method can be several times faster than the bisection method.

Calculation of Gauss quadratures with multiple free and fixed knots

Algorithms are derived for the evaluation of Gauss knots in the presence of fixed knots by modification of the Jacobi matrix for the weight function of the integral. Simple Gauss knots are obtained as eigenvalues of symmetric tridiagonal matrices and a rapidly converging simple iterative process, based on the merging of free and fixed knots, of quadratic convergence is presented for multiple Gauss knots. The procedures also allow for the evaluation of the weights of the quadrature corresponding to the simple Gauss knots. A new characterization of simple Gauss knots as a solution of a partial inverse eigenvalue problem is derived.

Eigenvalues of Symmetric Tridiagonal Matrices: A Fast, Accurate and Reliable Algorithm

An algorithm is developed for obtaining eigenvalues of real, symmetric, tridiagonal matrices. It combines dynamically Given's method of bisection and the use of Sturm sequences with various acceleration devices. A FORTRAN IV computer implementation of the algorithm was used on ten test matrices found in the literature. The new method is as precise and reliable as the best published program (Kahan & Varah, 1966), it is never slower, and in at least one case is two and half times faster than the Kahan and Varah program.