Rank-one modification of the symmetric eigenproblem

Summary An algorithm is presented for computing the eigensystem of the rank-one modification of a symmetric matrix with known eigensystem. The explicit computation of the updated eigenvectors and the treatment of multiple eigenvalues are discussed. The sensitivity of the computed eigenvectors to errors in the updated eigenvalues is shown by a perturbation analysis.Support for this research was provided by NSF grants MCS 75-06510 and MCS 76-03139Support for this research was provided by the Applied Mathematics Division, Argonne National Laboratory, Argonne, IL 60439, USA     

Using the Sherman-Morrison-Woodbury inversion formula for a fast solution of tridiagonal block Toeplitz systems

A fast numerical algorithm for solving systems of linear equations with tridiagonal block Toeplitz matrices is presented. The algorithm is based on a preliminary factorization of the generating quadratic matrix polynomial associated with the Toeplitz matrix, followed by the Sherman-Morrison-Woodbury inversion formula and solution of two bidiagonal and one diagonal block Toeplitz systems. Tight estimates of the condition numbers are provided for the matrix system and the main matrix systems generated during the preliminary factorization. The emphasis is put on rigorous stability analysis to rounding errors of the Sherman-Morrison-Woodbury inversion. Numerical experiments are provided to illustrate the theory.     

An eigenvalue localization theorem for pentadiagonal symmetric Toeplitz matrices

We express the eigenvalues of a pentadiagonal symmetric Toeplitz matrix as the zeros of explicitly given rational functions.     

A look-ahead algorithm for the solution of general Hankel systems

Summary The solution of systems of linear equations with Hankel coefficient matrices can be computed with onlyO(n ²) arithmetic operations, as compared toO(n ³) operations for the general cases. However, the classical Hankel solvers require the nonsingularity of all leading principal submatrices of the Hankel matrix. The known extensions of these algorithms to general Hankel systems can handle only exactly singular submatrices, but not ill-conditioned ones, and hence they are numerically unstable. In this paper, a stable procedure for solving general nonsingular Hankel systems is presented, using a look-ahead technique to skip over singular or ill-conditioned submatrices. The proposed approach is based on a look-ahead variant of the nonsymmetric Lanczos process that was recently developed by Freund, Gutknecht, and Nachtigal. We first derive a somewhat more general formulation of this look-ahead Lanczos algorithm in terms of formally orthogonal polynomials, which then yields the look-ahead Hankel solver as a special case. We prove some general properties of the resulting look-ahead algorithm for formally orthogonal polynomials. These results are then utilized in the implementation of the Hankel solver. We report some numerical experiments for Hankel systems with ill-conditioned submatrices.The research of the first author was supported by DARPA via Cooperative Agreement NCC 2-387 between NASA and the Universities Space Research Association (USRA).The research of the second author was supported in part by NSF grant DRC-8412314 and Cooperative Agreement NCC 2-387 between NASA and the Universities Space Research Association (USRA).     

Algebraic relations between the total least squares and least squares problems with more than one solution

Summary This paper completes our previous discussion on the total least squares (TLS) and the least squares (LS) problems for the linear systemAX=B which may contain more than one solution [12, 13], generalizes the work of Golub and Van Loan [1,2], Van Huffel [8], Van Huffel and Vandewalle [11]. The TLS problem is extended to the more general case. The sets of the solutions and the squared residuals for the TLS and LS problems are compared. The concept of the weighted squares residuals is extended and the difference between the TLS and the LS approaches is derived. The connection between the approximate subspaces and the perturbation theories are studied.It is proved that under moderate conditions, all the corresponding quantities for the solution sets of the TLS and the modified LS problems are close to each other, while the quantities for the solution set of the LS problem are close to the corresponding ones of a subset of that of the TLS problem.This work was financially supported by the Education Committee, People's Republic of China     

Google PageRanking problem: The model and the analysis

The spectral and Jordan structures of the Web hyperlink matrix G(c)=cG+(1−c)ev^T have been analyzed when G is the basic (stochastic) Google matrix, c is a real parameter such that 0<c<1, v is a nonnegative probability vector, and e is the all-ones vector. Typical studies have relied heavily on special properties of nonnegative, positive, and stochastic matrices. There is a unique nonnegative vector y(c) such that y(c)^TG(c)=y(c)^T and y(c)^Te=1. This PageRank vector y(c) can be computed effectively by the power method.We consider a square complex matrix A and nonzero complex vectors x and v such that Ax=λx and v^∗x=1. We use standard matrix analytic tools to determine the eigenvalues, the Jordan blocks, and a distinguished left λ-eigenvector of A(c)=cA+(1−c)λxv^∗ as a function of a complex variable c. If λ is a semisimple eigenvalue of A, there is a uniquely determined projection N such that lim_c→1y(c)=Nv for all v; this limit may fail to exist for some v if λ is not semisimple. As a special case of our results, we obtain a complex analog of PageRank for the Web hyperlink matrix G(c) with a complex parameter c. We study regularity, limits, expansions, and conditioning of y(c) and we propose algorithms (e.g., complex extrapolation, power method on a modified matrix etc.) that may provide an efficient way to compute PageRank also with c close or equal to 1. An interpretation of the limit vector Nv and a related critical discussion on the model, on its adherence to reality, and possible ways for its improvement, represent the contribution of the paper on modeling issues.     

On nonsingularity of block two-by-two matrices

We derive necessary and sufficient conditions for guaranteeing the nonsingularity of a block two-by-two matrix by making use of the singular value decompositions and the Moore–Penrose pseudoinverses of the matrix blocks. These conditions are complete, and much weaker and simpler than those given by Decker and Keller [D.W. Decker, H.B. Keller, Multiple limit point bifurcation, J. Math. Anal. Appl. 75 (1980) 417–430], and may be more easily examined than those given by Bai [Z.-Z. Bai, Eigenvalue estimates for saddle point matrices of Hermitian and indefinite leading blocks, J. Comput. Appl. Math. 237 (2013) 295–306] from the computational viewpoint. We also derive general formulas for the rank of the block two-by-two matrix by utilizing either the unitarily compressed or the orthogonally projected sub-matrices.     

Some closer bounds of Perron root basing on generalized Perron complement

This paper is concerned with the bounds of the Perron root ρ(A) of a nonnegative irreducible matrix A. Two new methods utilizing the relationship between the Perron root of a nonnegative irreducible matrix and its generalized Perron complements are presented. The former method is efficient because it gives the bounds for ρ(A) only by calculating the row sums of the generalized Perron complement P_t(A/A[α]) or even the row sums of submatrices A[α],A[β],A[α,β] and A[β,α]. And the latter gives the closest bounds (just in this paper) of ρ(A). The results obtained by these methods largely improve the classical bounds. Numerical examples are given to illustrate the procedure and compare it with others, which shows that these methods are effective.     

Backward errors for eigenproblem of two kinds of structured matrices

In this paper, we consider backward errors in the eigenproblem of symmetric centrosymmetric and symmetric skew-centrosymmetric matrices. By making use of the properties of symmetric centrosymmetric and symmetric skew-centrosymmetric matrices, we derive explicit formulae for the backward errors of approximate eigenpairs.     

A fast algorithm for solving diagonally dominant symmetric pentadiagonal Toeplitz systems

Banded Toeplitz systems of linear equations arise in many application areas and have been well studied in the past. Recently, significant advancement has been made in algorithm development of fast parallel scalable methods to solve tridiagonal Toeplitz problems. In this paper we will derive a new algorithm for solving symmetric pentadiagonal Toeplitz systems of linear equations based upon a technique used in [J.M. McNally, L.E. Garey, R.E. Shaw, A split-correct parallel algorithm for solving tri-diagonal symmetric Toeplitz systems, Int. J. Comput. Math. 75 (2000) 303-313] for tridiagonal Toeplitz systems. A common example which arises in natural quintic spline problems will be used to demonstrate the algorithm’s effectiveness. Finally computational results and comparisons will be presented.     

Exponents of 2-coloring of symmetric digraphs

A 2-coloring (G₁,G₂) of a digraph is 2-primitive if there exist nonnegative integers h and k with h+k>0 such that for each ordered pair (u,v) of vertices there exists an (h,k)-walk in (G₁,G₂) from u to v. The exponent of (G₁,G₂) is the minimum value of h+k taken over all such h and k. In this paper, we consider 2-colorings of strongly connected symmetric digraphs with loops, establish necessary and sufficient conditions for these to be 2-primitive and determine an upper bound on their exponents. We also characterize the 2-colored digraphs that attain the upper bound and the exponent set for this family of digraphs on n vertices.     

Saving flops in LU based shift-and-invert strategy

The shift-and-invert method is very efficient in eigenvalue computations, in particular when interior eigenvalues are sought. This method involves solving linear systems of the form (A−σI)z=b. The shift σ is variable, hence when a direct method is used to solve the linear system, the LU factorization of (A−σI) needs to be computed for every shift change. We present two strategies that reduce the number of floating point operations performed in the LU factorization when the shift changes. Both methods perform first a preprocessing step that aims at eliminating parts of the matrix that are not affected by the diagonal change. This leads to about 43% and 50% flops savings respectively for the dense matrices.     

On convergence of the inexact Rayleigh quotient iteration with MINRES

For the Hermitian inexact Rayleigh quotient iteration (RQI), we present a new general theory, independent of iterative solvers for shifted inner linear systems. The theory shows that the method converges at least quadratically under a new condition, called the uniform positiveness condition, that may allow the residual norm _ξk≥1${\xi }_k\ge 1$ of the inner linear system at outer iteration k+1$k+1$ and can be considerably weaker than the condition _ξk≤ξ<1${\xi }_k\le \xi <1$ with ξ$\xi$ a constant not near one commonly used in the literature. We consider the convergence of the inexact RQI with the unpreconditioned and tuned preconditioned MINRES methods for the linear systems. Some attractive properties are derived for the residuals obtained by MINRES. Based on them and the new general theory, we make a refined analysis and establish a number of new convergence results. Let ‖_rk‖$\Vert r_k\Vert$ be the residual norm of approximating eigenpair at outer iteration k$k$. Then all the available cubic and quadratic convergence results require _ξk=O(‖_rk‖)${\xi }_k=O\left(\Vert r_k\Vert \right)$ and _ξk≤ξ${\xi }_k\le \xi$ with a fixed ξ$\xi$ not near one, respectively. Fundamentally different from these, we prove that the inexact RQI with MINRES generally converges cubically, quadratically and linearly provided that _ξk≤ξ${\xi }_k\le \xi$ with a constant ξ<1$\xi <1$ not near one, _ξk=1−O(‖_rk‖)${\xi }_k=1-O\left(\Vert r_k\Vert \right)$ and _ξk=1−O(‖_rk‖²)${\xi }_k=1-O\left({\Vert r_k\Vert }^2\right)$, respectively. The new convergence conditions are much more relaxed than ever before. The theory can be used to design practical stopping criteria to implement the method more effectively. Numerical experiments confirm our results.     

A Hamiltonian Krylov-Schur-type method based on the symplectic Lanczos process

We discuss a Krylov-Schur like restarting technique applied within the symplectic Lanczos algorithm for the Hamiltonian eigenvalue problem. This allows us to easily implement a purging and locking strategy in order to improve the convergence properties of the symplectic Lanczos algorithm. The Krylov-Schur-like restarting is based on the SR algorithm. Some ingredients of the latter need to be adapted to the structure of the symplectic Lanczos recursion. We demonstrate the efficiency of the new method for several Hamiltonian eigenproblems.     

The simultaneous computation of a few of the algebraically largest and smallest eigenvalues of a large,sparse, symmetric matrix

A heuristic argument and supporting numerical results are given to demonstrate that a block Lanczos procedure can be used to compute simultaneously a few of the algebraically largest and smallest eigenvalues and a corresponding eigenspace of a large, sparse, symmetric matrixA. This block procedure can be used, for example, to compute appropriate parameters for iterative schemes used in solving the equationAx=b. Moreover, if there exists an efficient method for repeatedly solving the equation (A–I)X=B, this procedure can be used to determine the interior eigenvalues (and corresponding eigenvectors) ofA closest to .     

A backward error for the inverse singular value problem

A backward error for inverse singular value problems with respect to an approximate solution is defined, and an explicit expression for the backward error is derived by extending the approach described in [J.G. Sun, Backward errors for the inverse eigenvalue problem, Numer. Math. 82 (1999) 339-349]. The expression may be useful for testing the stability of practical algorithms.     

On the Z-eigenvalues of the adjacency tensors for uniform hypergraphs

The adjacency matrices for graphs are generalized to the adjacency tensors for uniform hypergraphs, and some fundamental properties for the adjacency tensor and its Z-eigenvalues of a uniform hypergraph are obtained. In particular, some bounds on the smallest and the largest Z-eigenvalues of the adjacency tensors for uniform hypergraphs are presented.     

Efficient numerical solution of the generalized Dirichlet-Neumann map for linear elliptic PDEs in regular polygon domains

A new and novel approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was recently introduced. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet-Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to this direction is computed without solving on the interior of the domain. For this computation, a collocation-type numerical method has been recently developed. Here, we study the collocation’s coefficient matrix properties. We prove that, for the Laplace’s equation on regular polygon domains with the same type of boundary conditions on each side, the collocation matrix is block circulant, independently of the choice of basis functions. This leads to the deployment of the FFT for the solution of the associated collocation linear system, yielding significant computational savings. Numerical experiments are included to demonstrate the efficiency of the whole computation.