The QR Algorithm

In this section, we will consider two methods for computing an eigenvector and in addition the associated eigenvalue of a matrix A

How long does it take to compute the eigenvalues of a random symmetric matrix?

We present the results of an empirical study of the performance of the QR algorithm (with and without shifts) and the Toda algorithm on random symmetric matrices. The random matrices are chosen from six ensembles, four of which lie in the Wigner class. For all three algorithms, we observe a form of universality for the deflation time statistics for random matrices within the Wigner class. For these ensembles, the empirical distribution of a normalized deflation time is found to collapse onto a curve that depends only on the algorithm, but not on the matrix size or deflation tolerance provided the matrix size is large enough (see Figure 4, Figure 7 and Figure 10). For the QR algorithm with the Wilkinson shift, the observed universality is even stronger and includes certain non-Wigner ensembles. Our experiments also provide a quantitative statistical picture of the accelerated convergence with shifts.Comment: 20 Figures; Revision includes a treatment of the QR algorithm with shift

Parallel-vector unsymmetric Eigen-Solver on high performance computers

The popular QR algorithm for solving all eigenvalues of an unsymmetric matrix is reviewed. Among the basic components in the QR algorithm, it was concluded from this study, that the reduction of an unsymmetric matrix to a Hessenberg form (before applying the QR algorithm itself) can be done effectively by exploiting the vector speed and multiple processors offered by modern high-performance computers. Numerical examples of several test cases have indicated that the proposed parallel-vector algorithm for converting a given unsymmetric matrix to a Hessenberg form offers computational advantages over the existing algorithm. The time saving obtained by the proposed methods is increased as the problem size increased

The QR algorithm for unitary Hessenberg matrices

AbstractLet H be an n × n unitary right Hessenberg matrix with positive subdiagonal elements. Using what we call the Schur parameterization of H, we show how one step of the shifted QR algorithm for H can be carried out in O(n) arithmetic operations. Coupled with the shift strategy of Eberlein and Huang [3], this will permit computation of the spectrum of H, to machine precision, in O(n2) operations. One potential application is the computation of Gauss-Szegö quadrature formulas [12], given the associated Schur parameters [7]. The weights can also be computed, by direct analogy with [6]

Efficient method for calculating the eigenvalue of the Zakharov-Shabat system

In this paper, a direct method is proposed to calculate the eigenvalue of the Zakharov-Shabat system. The main tools of our method are Chebyshev polynomials and the QR algorithm. After introducing the hyperbolic tangent mapping, the eigenfunctions and potential function defined in the real field can be represented by Chebyshev polynomials. Using Chebyshev nodes, the Zakharov-Shabat eigenvalue problem is transformed into a matrix eigenvalue problem. The matrix eigenvalue problem is solved by the QR algorithm. Our method is used to calculate eigenvalues of the Zakharov-Shabat equation with three potentials, the rationality of our method is verified by comparison with analytical results

STAR adaptation of QR algorithm

The QR algorithm used on a serial computer and executed on the Control Data Corporation 6000 Computer was adapted to execute efficiently on the Control Data STAR-100 computer. How the scalar program was adapted for the STAR-100 and why these adaptations yielded an efficient STAR program is described. Program listings of the old scalar version and the vectorized SL/1 version are presented in the appendices. Execution times for the two versions applied to the same system of linear equations, are compared

Improved Accuracy and Parallelism for MRRR-based Eigensolvers -- A Mixed Precision Approach

The real symmetric tridiagonal eigenproblem is of outstanding importance in numerical computations; it arises frequently as part of eigensolvers for standard and generalized dense Hermitian eigenproblems that are based on a reduction to tridiagonal form. For its solution, the algorithm of Multiple Relatively Robust Representations (MRRR) is among the fastest methods. Although fast, the solvers based on MRRR do not deliver the same accuracy as competing methods like Divide & Conquer or the QR algorithm. In this paper, we demonstrate that the use of mixed precisions leads to improved accuracy of MRRR-based eigensolvers with limited or no performance penalty. As a result, we obtain eigensolvers that are not only equally or more accurate than the best available methods, but also -in most circumstances- faster and more scalable than the competition

An FPGA implementation of givens rotation based digital architecture for computing eigenvalues of asymmetric matrix

This paper proposes the digital circuit design that performs the eigenvalue calculation of asymmetric matrices with realvalued elements. Eigenvalues are computed iteratively through the QR algorithm. In the QR algorithm, the input matrix is factorized into orthogonal Q and upper triangular R matrix, then the RQ product is calculated to obtain an iterated matrix. For a time-efficient QR decomposition process, the Givens Rotation (GR) Principle is utilized to benefit from the parallelization feature. Parallelization is managed by the Systolic Array (SA) architecture that is created by placing Givens Generation (GG) and Row Updates (RU) blocks in a triangle array. In this paper, 4×4 input matrix is used to create a TSA architecture including n-1 diagonal (GG), and (n ∗ (n−1))/2 off-diagonal (RU) modules. In the results section, Givens Rotation is compared with the Gram Schmidt algorithm used in our previous study [1] in terms of error, and area usage.Scopus - Affiliation ID: 60105072Oca

Matrix Bruhat decompositions with a remark on the QR (GR) algorithm

AbstractIn a simple and systematic way we present matrix Bruhat decompositions of two kinds: basic and modified. We show that it is the modified Bruhat decomposition that governs the eigenvalue disorder in the QR (GR) algorithm. This paper can be considered as a commentary on a previous observation about the QR algorithm made by Wilkinson