QR Algorithm Research Articles

Structural properties of matrix unitary reduction to semiseparable form

This paper concerns the study of a unitary transformation of a generic real symmetric matrix A into a semiseparable matrix. The problem is studied both theoretically and from an algorithmic point of view. In particular, we first give a formal proof of the existence of such a transformation and then we discuss its uniqueness, proving an implicit-Q theorem for semiseparable matrices. Lastly, we study structural properties of the factors of the QR-decomposition of a semiseparable matrix. These properties allow us to design a method based on QR iterations applied to a semiseparable matrix for reducing a symmetric matrix to semiseparable form. This method has the same asymptotic cost of the reduction of a symmetric matrix to tridiagonal form. Once the transformation is accomplished, to compute the eigenvalues each further QR iteration can be done in linear time.

A Symmetric FEM-IE Formulation With a Single-Level IE-QR Algorithm for Solving Electromagnetic Radiation and Scattering Problems

This paper presents, for the first time in the engineering community, a symmetric coupling between the finite element and integral equation methods (FEM-IE) for solving three-dimensional unbounded radiation and scattering problems. The proposed FEM-IE is based on the E-field vector Helmholtz equation. Curl-conforming vector finite elements are used to discretize the interior region, whereas the divergence-conforming surface elements are utilized in the IE truncation surface. The symmetry in the IE part is restored through the application of the Calderon-projector. Moreover, the IE computations are accelerated using a single level QR algorithm. This reduces both memory and computational time. Furthermore it allows the use of different Green's functions for the exterior problem, with only minor modifications on the algorithm. The resulted system of equations is solved with a very efficient preconditioned conjugate gradient (PCCG) with a p-Multiplicative Schwarz preconditioner.

A QR?method for computing the singular values via semiseparable matrices

The standard procedure to compute the singular value decomposition of a dense matrix, first reduces it into a bidiagonal one by means of orthogonal transformations. Once the bidiagonal matrix has been computed, the QR–method is applied to reduce the latter matrix into a diagonal one. In this paper we propose a new method for computing the singular value decomposition of a real matrix. In a first phase, an algorithm for reducing the matrix A into an upper triangular semiseparable matrix by means of orthogonal transformations is described. A remarkable feature of this phase is that, depending on the distribution of the singular values, after few steps of the reduction, the largest singular values are already computed with a precision depending on the gaps between the singular values. An implicit QR–method for upper triangular semiseparable matrices is derived and applied to the latter matrix for computing its singular values. The numerical tests show that the proposed method can compete with the standard method (using an intermediate bidiagonal matrix) for computing the singular values of a matrix.

Numerical solution of Helmholtz equation of barotropic atmosphere using wavelets

The numerical solution of the Helmholtz equation for barotropic atmosphere is estimated by use of the wavelet-Galerkin method. The solution involves the decomposition of a circulant matrix consisting up of 2-term connection coefficients of wavelet scaling functions. Three matrix decompositions, i.e. fast Fourier transformation (FFT), Jacobian and QR decomposition methods, are tested numerically. The Jacobian method has the smallest matrix-reconstruction error with the best orthogonality while the FFT method causes the biggest errors. Numerical result reveals that the numerical solution of the equation is very sensitive to the decomposition methods and the QR and Jacobian decomposition methods, whose errors are of the order of 10−3, much smaller than that with the FFT method, are more suitable to the numerical solution of the equation. With the two methods the solutions are also proved to have much higher accuracy than the iteration solution with the finite difference approximation. In addition, the wavelet numerical method is very useful for the solution of a climate model in low resolution. The solution accuracy of the equation may significantly increase with the order of Daubechies wavelet.

On the computation of Lyapunov exponents for forced vibration of a Lennard–Jones oscillator

Abstract In this paper, we introduce a modified Lyapunov vector method for retrieving the leading Lyapunov exponent for a dynamical system with strong nonlinear forces. Performances of the proposed method and the classical QR method are compared. This method is applied to the computation of Lyapunov exponents of a mechanical oscillator under the influence of van der Waals forces. The advantage of this approach lies in the property of preserving the norm of the solution vector.

Finite element analysis of coupled electronic states in quantum dot nanostructures

Nanostructures, created by confinement of the motion of an electron from all three dimensions and known as quantum dots (QDs), provide materials scientists with a wide range of potential applications. These structures are produced today with advances of QD growth technology, and computational tools are fundamental in providing a better understanding of such structures. In this paper QD nanostructures are analysed with due account for coupling effects between electronic states in the dot and the wetting layer regions. The analysis, performed on the basis of the finite element methodology and Arnoldi iterations, demonstrates that the effect of coupling may be essential. The numerical procedure applied here is more efficient compared to the QR algorithm typically used in the context of modelling low-dimensional nanostructures. We report results of computational experiments for cylindrical and truncated conical QDs and compare them with the earlier results obtained for fully conical QD nanostructures.

Shifting algorithms with Maple and implicit shift in the QR algorithm

We have written out all shifting algorithm in the QR corresponding to article [SIAM J. Matrix Anal. 12 (1991) 385], specially the block shift algorithm, and tested them with Maple8. A comparison table has been designed in each case to compare the implementation of the algorithms with each other. The differences of the shifts have been noted in the last section.

Inverse power and Durand-Kerner iterations for univariate polynomial root-finding

Univariate polynomial root-finding is the oldest classical problem of mathematics and computational mathematics, and is still an important research topic, due to its impact on computational algebra and geometry. The Weierstrass (Durand-Kerner) approach and its variations as well as matrix methods based on the QR algorithm are among the most popular practical choices for simultaneous approximation of all roots of a polynomial. We propose an alternative application of the inverse power iteration to generalized companion matrices for polynomial root-finding, demonstrate its effectiveness, and relate its study to unifying the derivation of the Weierstrass (Durand-Kerner) algorithm (having quadratic convergence) and its extensions having convergence rates 4, 6, 8, …. Our experiments show substantial improvement versus the latter algorithm, even though the inverse power iteration is most effective for the more limited tasks of approximating a single root or a few selected roots.

The periodic QR algorithm is a disguised QR algorithm

The periodic QR algorithm is a strongly backward stable method for computing the eigenvalues of products of matrices, or equivalently for computing the eigenvalues of block cyclic matrices. The main purpose of this paper is to show that this algorithm is numerically equivalent to the standard QR algorithm. It will be demonstrated how this connection may be used to develop a better understanding of the periodic QR algorithm.

Fast QR Algorithms Based on Backward Prediction Errors: A New Implementation and Its Finite Precision Performance

QR decomposition techniques are well known for their good numerical behavior and low complexity. Fast QRD recursive least squares adaptive algorithms benefit from these characteristics to offer robust and fast adaptive filters. This paper examines two different versions of the fast QR algorithm based on a priori backward prediction errors as well as two other corresponding versions of the fast QR algorithm based on a poste- riori backward prediction errors. The main matrix equations are presented with different versions derived from two distinct deployments of a particular matrix equation. From this study, a new algorithm is derived. The discussed algorithms are compared, and differences in computational complexity and in finite-precision behavior are shown. Ke yw ords: Adaptive system, fast RLS algorithm, QR decomposition.

Matrix Algorithms, Volume II: Eigensystems

1R3. Matrix Algorithms, Volume II: Eigensystems. - GW Stewart (Dept of Comput Sci, Univ of Maryland, College Park MD). SIAM, Philadelphia. 2001. 469 pp. (Softcover). ISBN 0-89871-503-2. $50.00. Reviewed by A Mahajan (Dept of Mech Eng and Energy Processes, S Illinois Univ, Carbondale IL 62901).This book contains a comprehensive presentation of computations involving the eigenvalues and eigenvectors of a matrix. It is the second volume in a projected five-volume series on matrix algorithms. The first volume was about basic decompositions. The second volume (this book) presents eigensystems. The next three books are projected to treat iterative methods for linear systems, sparse direct methods, and special topics, including fast algorithms for structured matrices. The author has managed to keep this volume fairly independent of the first volume, hence a basic knowledge of linear algebra is sufficient to understand most topics in the book is necessary. The book is really intended to be a reference book for scientists and engineers who need tools to solve problems in eigensystems. According to the author, the intended audience is the nonspecialist whose needs cannot be satisfied by black boxes. The book does live up to the author’s stated aim. The author gives detailed derivations that will help anyone who wants to adapt the methods to particular problems. The book really is not intended to be a textbook due to the lack of extensive solved and unsolved problems, but can be used as a self study by graduate students or even undergraduate students in honors programs. The book is divided into two parts: Dense Eigenproblems and Large Eigenproblems. The book has six chapters, a reference list of over 300 references, and an excellent subject index. The first chapter presents the underlying theory for eigensystems. The second chapter describes the widely used QR algorithm. Chapter 3 deals with symmetric matrices and the singular value decomposition. The fourth chapter deals with large matrices for which the computation of a complete eigensystem is not possible, hence an algebraic and analytic theory of eigensystems is presented. The fifth chapter deals with the Krylov sequence methods, in particular the Arnoldi and Lanczos methods. The sixth chapter presents alternative methods such as subspace iteration and the Jacobi-Davidson method. The book provides pseudo codes for many algorithms. The author does admit that the pseudo codes are for illustration only and should not be regarded as finished implementations, and it is difficult to verify their correctness. He also mentions that whereever possible he has checked the algorithms against MATLAB implementations. If there is a weakness in the book, it is right here, for the author could have provided these MATLAB programs on disk, or at the very least, made them available on a website. The reviewer did implement some of the algorithms in MATLAB, and it did take some time to develop the implementation. Of course, this in no way impacts the significance of the material to the serious reader who is looking for a detailed treatise on the subject, but does impact a student or practicing engineer who could be looking for a quick implementation of a particular algorithm. A recommendation to the author and publisher would be to make these MATLAB programs available on a website so that they can be easily downloaded. Matrix Algorithms, Volume II: Eigensystems is an excellent treatise on eigensystems. It should be purchased by all libraries and any individuals who make extensive use of computations in eigensystems.

Highly accurate symmetric eigenvalue decomposition and hyperbolic SVD

Let G be a m×n real matrix with full column rank and let J be a n×n diagonal matrix of signs, J ii∈{−1,1} . The hyperbolic singular value decomposition (HSVD) of the pair (G,J) is defined as G=UΣV −1 , where U is orthogonal, Σ is positive definite diagonal, and V is J-orthogonal matrix, V T JV=J . We analyze when it is possible to compute the HSVD with high relative accuracy. This essentially means that each computed hyperbolic singular value is guaranteed to have some correct digits, even if they have widely varying magnitudes. We show that one-sided J-orthogonal Jacobi method method computes the HSVD with high relative accuracy. More precisely, let B=GD −1 , where D is diagonal such that the columns of B have unit norms. Essentially, we show that the computed hyperbolic singular values of the pair (G,J) will have log 10(ε/σ min (B)) correct decimal digits, where ε is machine precision. We give the necessary relative perturbation bounds and error analysis of the algorithm. Our numerical tests confirmed all theoretical results. For the symmetric non-singular eigenvalue problem Hx=λx, we analyze the two-step algorithm which consists of factorization H=GJG T followed by the computation of the HSVD of the pair (G,J). Here G is square and non-singular. Let B ̂ = D ̂ G , where D ̂ is diagonal such that the rows of B ̂ have unit norms, and let B be defined as above. Essentially, we show that the computed eigenvalues of H will have log 10(ε/σ 2 min ( B ̂ )+ε/σ min (B)) correct decimal digits. This accuracy can be much higher then the one obtained by the classical QR and Jacobi methods applied to H, where the accuracy depends on the spectral condition number of H, particularly if the matrices B and B ̂ are well conditioned, and we are interested in the accurate computation of tiny eigenvalues. Again, we give the perturbation and error bounds, and our theoretical predictions are confirmed by a series of numerical experiments.We also give the corresponding results for eigenvectors and hyperbolic singular vectors.

Reduced-Order Realization of a Nonlinear Power Network Using Companion-Form State Equations with Periodic Coefficients

This paper presents a methodology for the identification of a reduced-order dynamic equivalent of a nonlinear power network for the simulation of electromagnetic transients. The equivalent is deduced from the observer companion form of the state equations with periodic coefficients and includes the effects of the nonlinearity of the power network at its operating point. A previously proposed concept, the harmonic domain dynamic transfer function (HDDTF), is used to characterize the network's transient behavior, superimposed on the steady state. The HDDTF is obtained by linearization of the nonlinear state equations of the network corresponding to harmonic perturbations applied to the steady-state operating point. Then reduced-order companion-form state equations with periodic coefficients are fitted to the HDDTF in the frequency domain using a least-squares procedure based on the SVD and QR algorithms. The fitting procedure includes sequential weighting, column scaling, and vertical partitioning to improve computational accuracy and efficiency. The SVD algorithm serves to determine an appropriate model order. A test network with nonlinear inductances is used to demonstrate the performance of the identification method as well as the time-domain simulation results.

Convergence of the unitary $QR$ algorithm with a unimodular Wilkinson shift

In applying the QR algorithm to compute the eigenvalues of a unitary Hessenberg matrix, a projected Wilkinson shift of unit modulus is proposed and proved to give global convergence with (at least) a quadratic asymptotic rate for the QR iteration. Experimental testing demonstrates that the unimodular shift produces more efficient numerical convergence.

Calculation of induced electron states in three-dimensional semiconductor artificial molecules

The energy levels calculation of electrons confined in small three-dimensional (3D) coupled quantum In x Ga 1-x As dots embedded in GaAs semiconductor matrix is presented. The quantum dots have disk shapes and are separated (in the disk symmetry axis direction) by a certain distance. Based on the effective one electronic band Hamiltonian, the energy and position dependent electron effective mass approximation, a finite height hard-wall 3D confinement potential, and the Ben Daniel-Duke boundary conditions, the problem is formulated and solved for the disk-shaped coupled quantum dots. To calculate the ground and induced state energy levels, the nonlinear 3D Schrodinger equation (SE) is solved with a developed nonlinear iterative method to obtain the final self-consistent solutions. In the iteration loops, the Schrodinger equation is discretized with a nonuniform mesh finite difference method, and the matrix eigenvalue problem is solved with the balanced and shifted QR method. Our complete 3D approach demonstrates a principal possibility that the number of bound electronic states in the system can be changed when the interdot (vertical) distance is modified. However, it is impossible to produce an additional possibility to manipulate the system electronic properties within only a two-dimensional (2D) simulation.

Efficient time-independent wave packet scattering calculations within a Lanczos subspace: H+O2 (J=0) state-to-state reaction probabilities

An efficient Lanczos subspace method has been devised for calculating state-to-state reaction probabilities. The method recasts the time-independent wave packet Lippmann–Schwinger equation [Kouri et al., Chem. Phys. Lett. 203, 166 (1993)] inside a tridiagonal (Lanczos) representation in which action of the causal Green’s operator is affected easily with a QR algorithm. The method is designed to yield all state-to-state reaction probabilities from a given reactant-channel wave packet using a single Lanczos subspace; the spectral properties of the tridiagonal Hamiltonian allow calculations to be undertaken at arbitrary energies within the spectral range of the initial wave packet. The method is applied to a H+O2 system (J=0), and the results indicate the approach is accurate and stable.

Rounding error and perturbation bounds for the symplectic QR factorization

To compute the eigenvalues of a skew-symmetric matrix A, we can use a one-sided Jacobi-like algorithm to enhance accuracy. This algorithm begins by a suitable Cholesky-like factorization of A, A=G T JG . In some applications, A is given implicitly in that form and its natural Cholesky-like factor G is immediately available, but “tall”, i.e., not of full row rank. This factor G is unsuitable for the Jacobi-like process. To avoid explicit computation of A, and possible loss of accuracy, the factor has to be preprocessed by a QR-like factorization. In this paper we present the symplectic QR algorithm to achieve such a factorization, together with the corresponding rounding error and perturbation bounds. These bounds fit well into the relative perturbation theory for skew-symmetric matrices given in factorized form.

A Parallel Implementation of the Nonsymmetric QR Algorithm for Distributed Memory Architectures

One approach to solving the nonsymmetric eigenvalue problem in parallel is to parallelize the QR algorithm. Not long ago, this was widely considered to be a hopeless task. Recent efforts have led to significant advances, although the methods proposed up to now have suffered from scalability problems. This paper discusses an approach to parallelizing the QR algorithm that greatly improves scalability. A theoretical analysis indicates that the algorithm is ultimately not scalable, but the nonscalability does not become evident until the matrix dimension is enormous. Experiments on the Intel Paragon system, the IBM SP2 supercomputer, the SGI Origin 2000, and the Intel ASCI Option Red supercomputer are reported.

The Multishift QR Algorithm. Part II: Aggressive Early Deflation

Aggressive early deflation is a QR algorithm deflation strategy that takes advantage of matrix perturbations outside of the subdiagonal entries of the Hessenberg QR iterate. It identifies and deflates converged eigenvalues long before the classic small-subdiagonal strategy would. The new deflation strategy enhances the performance of conventional large-bulge multishift QR algorithms, but it is particularly effective in combination with the small-bulge multishift QR algorithm. The small-bulge multishift QR sweep with aggressive early deflation maintains a high rate of execution of floating point operations while significantly reducing the number of operations required.

The Multishift QR Algorithm. Part I: Maintaining Well-Focused Shifts and Level 3 Performance

This paper presents a small-bulge multishift variation of the multishift QR algorithm that avoids the phenomenon of shift blurring, which retards convergence and limits the number of simultaneous shifts. It replaces the large diagonal bulge in the multishift QR sweep with a chain of many small bulges. The small-bulge multishift QR sweep admits nearly any number of simultaneous shifts---even hundreds---without adverse effects on the convergence rate. With enough simultaneous shifts, the small-bulge multishift QR algorithm takes advantage of the level 3 BLAS, which is a special advantage for computers with advanced architectures.