A spectral trichotomy method for symplectic matrices

This paper presents an algorithm for the numerical approximation of spectral projectors onto the invariant subspaces corresponding to the eigenvalues inside, on, and outside the unit circle of a symplectic matrix. The algorithm constructs iteratively three matrix sequences from which the projectors are obtained. The convergence depends essentially on the gap between the unit circle and the eigenvalues inside it. A larger gap leads to faster convergence. Theoretical and algorithmic aspects of the algorithm are developed. Numerical results are reported.     

A criterion for the convergence of the Gauss-Seidel method

The solution of linear equations by iterative methods requires for convergence that the absolute magnitudes of all the eigenvalues of the iteration matrix should be less than unity. The test for convergence however is often difficult to apply because of the computation required. In this paper a method for determining the convergence of the Gauss-Seidel iteration is proposed. The method involves the numerical integration of initial value differential equations in the complex plane around the unit circle. The Gauss-Seidel method converges if the number of roots inside the unit circle is equal to the order of the iteration matrix.     

Fast solution of unsymmetric banded Toeplitz systems by means of spectral factorizations and Woodbury's formula

A fast algorithm for solving systems of linear equations with banded Toeplitz matrices is studied. An important step in the algorithm is a novel method for the spectral factorization of the generating function associated with the Toeplitz matrix. The spectral factorization is extracted from the right deflating subspaces corresponding to the eigenvalues inside and outside the open unit disk of a companion matrix pencil constructed from the coefficients of the generating function. The factorization is followed by the Woodbury inversion formula and solution of several banded triangular systems. Stability of the algorithm is discussed and its performance is demonstrated by numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

On equivalence of pencils from discrete-time and continuous-time control

We study two matrix pencils that arise, respectively, in discrete-time and continuous-time optimal and robust control. We introduce a one-to-one transformation between these two pencils. We show that for the pencils under the transformation, their regularity is preserved and their eigenvalues and deflating subspaces are equivalently related. The eigen-structures of the pencils under consideration have strong connections with the associated control problems. Our result may be applied to connect the discrete-time and continuous-time control problems and eventually lead to a unified treatment of these two types of control problems.     

Perturbation analysis of singular subspaces and deflating subspaces

Summary. Perturbation expansions for singular subspaces of a matrix and for deflating subspaces of a regular matrix pair are derived by using a technique previously described by the author. The perturbation expansions are then used to derive Fr\'echet derivatives, condition numbers, and th-order perturbation bounds for the subspaces. Vaccaro's result on second-order perturbation expansions for a special class of singular subspaces can be obtained from a general result of this paper. Besides, new perturbation bounds for singular subspaces and deflating subspaces are derived by applying a general theorem on solution of a system of nonlinear equations. The results of this paper reveal an important fact: Each singular subspace and each deflating subspace have individual perturbation bounds and individual condition numbers. Received July 26, 1994     

A subspace shift technique for nonsymmetric algebraic Riccati equations associated with an M‐matrix

The worst situation in computing the minimal nonnegative solution of a nonsymmetric algebraic Riccati equation associated with an M‐matrix occurs when the corresponding linearizing matrix has two very small eigenvalues, one with positive and one with negative real part. When both eigenvalues are exactly zero, the problem is called critical or null recurrent. Although in this case the problem is ill‐conditioned and the convergence of the algorithms based on matrix iterations is slow, there exist some techniques to remove the singularity and transform the problem to a well‐behaved one. Ill‐conditioning and slow convergence appear also in close‐to‐critical problems, but when none of the eigenvalues is exactly zero, the techniques used for the critical case cannot be applied. In this paper, we introduce a new method to accelerate the convergence properties of the iterations also in close‐to‐critical cases, by working on the invariant subspace associated with the problematic eigenvalues as a whole. We present numerical experiments that confirm the efficiency of the new method.Copyright © 2012 John Wiley & Sons, Ltd.     

An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems

Summary. We discuss an inverse-free, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix , or a pair of left and right deflating subspaces of a regular matrix pencil . This algorithm is based on earlier ones of Bulgakov, Godunov and Malyshev, but improves on them in several ways. This algorithm only uses easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition, but not matrix inversion. Similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which requires matrix inversion and is faster but can be less stable than the new algorithm. Received September 20, 1994 / Revised version received February 5, 1996     

Generalized block Lanczos methods for large unsymmetric eigenproblems

Generalized block Lanczos methods for large unsymmetric eigenproblems are presented, which contain the block Arnoldi method, and the block Arnoldi algorithms are developed. The convergence of this class of methods is analyzed when the matrix A is diagonalizable. Upper bounds for the distances between normalized eigenvectors and a block Krylov subspace are derived, and a priori theoretical error bounds for Ritz elements are established. Compared with generalized Lanczos methods, which contain Arnoldi's method, the convergence analysis shows that the block versions have two advantages: First, they may be efficient for computing clustered eigenvalues; second, they are able to solve multiple eigenproblems. However, a deep analysis exposes that the approximate eigenvectors or Ritz vectors obtained by general orthogonal projection methods including generalized block methods may fail to converge theoretically for a general unsymmetric matrix A even if corresponding approximate eigenvalues or Ritz values do, since the convergence of Ritz vectors needs more sufficient conditions, which may be impossible to satisfy theoretically, than that of Ritz values does. The issues of how to restart and to solve multiple eigenproblems are addressed, and some numerical examples are reported to confirm the theoretical analysis. Received July 7, 1994 / Revised version received March 1, 1997     

Proper deflating subspaces: properties,algorithms and applications

An algorithm for computing proper deflating subspaces with specified spectrum for an arbitrary matrix pencil is presented. The method uses refined algorithms for computing the generalized Schur form of a matrix pencil and enlightens the connection that exists between reducing and proper deflating subspaces. The proposed algorithm can be applied for computing the stabilizing solution of the generalized algebraic Riccati equation, a recently introduced concept which extends the usual algebraic Riccati equation.     

A note on the eigenvalues of a special class of matrices

In the analysis of stability of a variant of the Crank-Nicolson (C-N) method for the heat equation on a staggered grid a class of non-symmetric matrices appear that have an interesting property: their eigenvalues are all real and lie within the unit circle. In this note we shall show how this class of matrices is derived from the C-N method and prove that their eigenvalues are inside [−1,1] for all values of m (the order of the matrix) and all values of a positive parameter σ, the stability parameter. As the order of the matrix is general, and the parameter σ lies on the positive real line this class of matrices turns out to be quite general and could be of interest as a test set for eigenvalue solvers, especially as examples of very large matrices.     

Computing eigenspaces with specified eigenvalues of a regular matrix pair (A,B) and condition estimation: theory,algorithms and software

Theory, algorithms and LAPACK-style software for computing a pair of deflating subspaces with specified eigenvalues of a regular matrix pair (A, B) and error bounds for computed quantities (eigenvalues and eigenspaces) are presented. Thereordering of specified eigenvalues is performed with a direct orthogonal transformation method with guaranteed numerical stability. Each swap of two adjacent diagonal blocks in the real generalized Schur form, where at least one of them corresponds to a complex conjugate pair of eigenvalues, involves solving a generalized Sylvester equation and the construction of two orthogonal transformation matrices from certain eigenspaces associated with the diagonal blocks. The swapping of two 1×1 blocks is performed using orthogonal (unitary) Givens rotations. Theerror bounds are based on estimates of condition numbers for eigenvalues and eigenspaces. The software computes reciprocal values of a condition number for an individual eigenvalue (or a cluster of eigenvalues), a condition number for an eigenvector (or eigenspace), and spectral projectors onto a selected cluster. By computing reciprocal values we avoid overflow. Changes in eigenvectors and eigenspaces are measured by their change in angle. The condition numbers yield bothasymptotic andglobal error bounds. The asymptotic bounds are only accurate for small perturbations (E, F) of (A, B), while the global bounds work for all (E, F.) up to a certain bound, whose size is determined by the conditioning of the problem. It is also shown how these upper bounds can be estimated. Fortran 77software that implements our algorithms for reordering eigenvalues, computing (left and right) deflating subspaces with specified eigenvalues and condition number estimation are presented. Computational experiments that illustrate the accuracy, efficiency and reliability of our software are also described.     

Deflated GMRES for systems with multiple shifts and multiple right-hand sides

We consider solution of multiply shifted systems of nonsymmetric linear equations, possibly also with multiple right-hand sides. First, for a single right-hand side, the matrix is shifted by several multiples of the identity. Such problems arise in a number of applications, including lattice quantum chromodynamics where the matrices are complex and non-Hermitian. Some Krylov iterative methods such as GMRES and BiCGStab have been used to solve multiply shifted systems for about the cost of solving just one system. Restarted GMRES can be improved by deflating eigenvalues for matrices that have a few small eigenvalues. We show that a particular deflated method, GMRES-DR, can be applied to multiply shifted systems.In quantum chromodynamics, it is common to have multiple right-hand sides with multiple shifts for each right-hand side. We develop a method that efficiently solves the multiple right-hand sides by using a deflated version of GMRES and yet keeps costs for all of the multiply shifted systems close to those for one shift. An example is given showing this can be extremely effective with a quantum chromodynamics matrix.     

A DIRECT METHOD FOR THE UNIT CIRCLE PROBLEM

The unit circle problem is the problem of finding the number of eigenvalues of a non-Hermitian matrix inside and outside the unit circle . To reduce the cost of computing eigenvalues for the problem, a direct method, which is analogous to that given in [5], is proposed in this paper.     

On Some Spectral Characteristics of Multiparameter Polynomial Matrices

Definitions of certain spectral characteristics of polynomial matrices (such as the analytical (algebraic) and geometric multiplicities of a point of the spectrum, deflating subspaces, matrix solvents, and block eigenvalues and eigenvectors) are generalized to the multiparameter case, and properties of these characteristics are analyzed. Bibliogrhaphy: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 166–173.     

Recursive estimation for ordered eigenvectors of symmetric matrix with observation noise

The principal component analysis is to recursively estimate the eigenvectors and the corresponding eigenvalues of a symmetric matrix A based on its noisy observations Ak=A+Nk, where A is allowed to have arbitrary eigenvalues with multiplicity possibly bigger than one. In the paper the recursive algorithms are proposed and their ordered convergence is established: It is shown that the first algorithm a.s. converges to a unit eigenvector corresponding to the largest eigenvalue, the second algorithm a.s. converges to a unit eigenvector corresponding to either the second largest eigenvalue in the case the largest eigenvalue is of single multiplicity or the largest eigenvalue if the multiplicity of the largest eigenvalue is bigger than one, and so on. The convergence rate is also derived.     

Matrices with prescribed Ritz values

On the way to establishing a commutative analog to the Gelfand-Kirillov theorem in Lie theory, Kostant and Wallach produced a decomposition of M(n) which we will describe in the language of linear algebra. The “Ritz values” of a matrix are the eigenvalues of its leading principal submatrices of order m=1,2,…,n. There is a unique unit upper Hessenberg matrix H with those eigenvalues. For real symmetric matrices with interlacing Ritz values, we extend their analysis to allow eigenvalues at successive levels to be equal. We also decide whether given Ritz values can come from a tridiagonal matrix.     

On proximity of Rayleigh quotients for different vectors and Ritz values generated by different trial subspaces

The Rayleigh quotient is unarguably the most important function used in the analysis and computation of eigenvalues of symmetric matrices. The Rayleigh-Ritz method finds the stationary values of the Rayleigh quotient, called Ritz values, on a given trial subspace as optimal, in some sense, approximations to eigenvalues.In the present paper, we derive upper bounds for proximity of the Ritz values in terms of the proximity of the trial subspaces without making an assumption that the trial subspace is close to an invariant subspace. The main result is that the absolute value of the perturbations in the Ritz values is bounded by a constant times the gap between the original trial subspace and its perturbation. The constant is the spread in the matrix spectrum, i.e. the difference between the largest and the smallest eigenvalues of the matrix. It’s shown that the constant cannot be improved. We then generalize this result to arbitrary unitarily invariant norms, but we have to increase the constant by a factor of .Our results demonstrate, in particular, the stability of the Ritz values with respect to a perturbation in the trial subspace.     

A Cache-Aware Implementation of the Spectral Divide-and-Conquer Approach for the Non-Symmetric Generalized Eigenvalue Problem

We propose a new implementation of the sign function based spectral divide-and-conquer method for the generalized non-symmetric eigenvalue problem. The basic idea is to use the generalized matrix sign function to split the spectrum and the corresponding deflating subspaces and to build a recursive scheme on top of this. The method focuses on an extensive usage of scalable level 3 BLAS operations in order to achieve a good performance on modern computer architectures which makes it computationally superior to current implementations of the classical QZ algorithm. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lur’e equations and even matrix pencils

In this work, we consider the so-called Lur’e matrix equations that arise e.g. in model reduction and linear-quadratic infinite time horizon optimal control. We characterize the set of solutions in terms of deflating subspaces of even matrix pencils. In particular, it is shown that there exist solutions which are extremal in terms of definiteness. It is shown how these special solutions can be constructed via deflating subspaces of even matrix pencils.     

Numerical Determination of a Canonical Form of a Symplectic Matrix

We propose an algorithm that transforms a real symplectic matrix with a stable structure to a block diagonal form composed of three main blocks. The two extreme blocks of the same size are associated respectively with the eigenvalues outside and inside the unit circle. Moreover, these eigenvalues are symmetric with respect to the unit circle. The central block is in turn composed of several diagonal blocks whose eigenvalues are on the unit circle and satisfy a modification of the Krein-Gelfand-Lidskii criterion. The proposed algorithm also gives a qualitative criterion for structural stability.