Algorithm 1019: A Task-based Multi-shift QR/QZ Algorithm with Aggressive Early Deflation

The QR algorithm is one of the three phases in the process of computing the eigenvalues and the eigenvectors of a dense nonsymmetric matrix. This paper describes a task-based QR algorithm for reducing an upper Hessenberg matrix to real Schur form. The task-based algorithm also supports generalized eigenvalue problems (QZ algorithm) but this paper concentrates on the standard case. The task-based algorithm adopts previous algorithmic improvements, such as tightly-coupled multi-shifts and Aggressive Early Deflation (AED) , and also incorporates several new ideas that significantly improve the performance. This includes, but is not limited to, the elimination of several synchronization points, the dynamic merging of previously separate computational steps, the shortening and the prioritization of the critical path, and experimental GPU support. The task-based implementation is demonstrated to be multiple times faster than multi-threaded LAPACK and ScaLAPACK in both single-node and multi-node configurations on two different machines based on Intel and AMD CPUs. The implementation is built on top of the StarPU runtime system and is part of the open-source StarNEig library.

А Gpu-based Orthogonal Matrix Factorization Algorithm that Produces a Two-Diagonal Shape

With the development of the Big Data sphere, as well as those fields of study that we can relate to artificial intelligence, the need for fast and efficient computing has become one of the most important tasks nowadays. That is why in the recent decade, graphics processing unit computations have been actively developing to provide an ability for scientists and developers to use thousands of cores GPUs have in order to perform intensive computations. The goal of this research is to implement orthogonal decomposition of a matrix by applying a series of Householder transformations in Java language using JCuda library to conduct a research on its benefits. Several related papers were examined. Malaschonok and Savchenko in their work have introduced an improved version of QR algorithm for this purpose [4] and achieved better results, however Householder algorithm is more promising for GPUs according to another team of researchers – Lahabar and Narayanan [6]. However, they were using Float numbers, while we are using Double, and apart from that we are working on a new BigDecimal type for CUDA. Apart from that, there is still no solution for handling huge matrices where errors in calculations might occur. The algorithm of orthogonal matrix decomposition, which is the first part of SVD algorithm, is researched and implemented in this work. The implementation of matrix bidiagonalization and calculation of orthogonal factors by the Hausholder method in the jCUDA environment on a graphics processor is presented, and the algorithm for the central processor for comparisons is also implemented. Research of the received results where we experimentally measured acceleration of calculations with the use of the graphic processor in comparison with the implementation on the central processor are carried out. We show a speedup up to 53 times compared to CPU implementation on a big matrix size, specifically 2048, and even better results when using more advanced GPUs. At the same time, we still experience bigger errors in calculations while using graphic processing units due to synchronization problems. We compared execution on different platforms (Windows 10 and Arch Linux) and discovered that they are almost the same, taking the computation speed into account. The results have shown that on GPU we can achieve better performance, however there are more implementation difficulties with this approach.

Node Generation for RBF-FD Methods by QR Factorization

Polyharmonic spline (PHS) radial basis functions (RBFs) have been used in conjunction with polynomials to create RBF finite-difference (RBF-FD) methods. In 2D, these methods are usually implemented with Cartesian nodes, hexagonal nodes, or most commonly, quasi-uniformly distributed nodes generated through fast algorithms. We explore novel strategies for computing the placement of sampling points for RBF-FD methods in both 1D and 2D while investigating the benefits of using these points. The optimality of sampling points is determined by a novel piecewise-defined Lebesgue constant. Points are then sampled by modifying a simple, robust, column-pivoting QR algorithm previously implemented to find sets of near-optimal sampling points for polynomial approximation. Using the newly computed sampling points for these methods preserves accuracy while reducing computational costs by mitigating stencil size restrictions for RBF-FD methods. The novel algorithm can also be used to select boundary points to be used in conjunction with fast algorithms that provide quasi-uniformly distributed nodes.

On the Kronecker Product of Matrices and Their Applications To Linear Systems Via Modified QR-Algorithm

This paper studies and supplements the proofs of the properties of the Kronecker Product of two matrices of different orders. We observe the relation between the singular value decomposition of the matrices and their Kronecker product and the relationship between the determinant, the trace, the rank and the polynomial matrix of the Kronecker products. We also establish the best least square solutions of the Kronecker product system of equations by using modified QR-algorithm.

Three Point Boundary Value Problems Associated with First Order Fuzzy Difference Systems-Existence and Uniqueness via the Best Least Square Solution

This paper presents a criteria for the existence and uniqueness of solutions to first order fuzzy difference system using QR-algorithm. Modified QR-algorithm is presented for fuzzy linear systems using singular value decomposition.

A Numerical Method to Solve Nonsymmetric Eigensystems Applied to Dynamics of Turbomachinery

In this paper, a canonical transformation is proposed to solve the eigenvalue problem related to the dynamics of rotor-bearing systems. In this problem, all matrices are real, but they may not be symmetric, which leads to the appearance of complex eigenvalues and eigenvectors. The bi-iteration method is selected to solve the original eigenproblem whereas the QR algorithm is adopted to solve the reduced or projected problem. A new canonical transformation of the global eigenproblem which reduces the quadratic eigenproblem to a linear eigenproblem, maintaining numerical stability since all that is required is that the stiffness matrix is well-conditioned, which is always true when it comes to applications in dynamic problems. The proposed technique is good for obtaining dominant eigenvalues and corresponding eigenvectors of real nonsymmetric matrices and it possesses the following properties: (i) the matrix is not transformed, therefore sparsity is maintained, (ii) partial eigensolutions can be obtained and (iii) use may be made of good eigenvectors predictions.

Features of practical application of electric power system’s stability estimation methods

Timeliness of the topic is conditioned by the need to keep up ample of static and transient stability margins in modern electrical power systems. The article object is to give a determination of present methods of evaluating damping of systems with synchronous machines, in the context of their effectiveness. And also to estimate the effectiveness of these methods in the performance of the task concerned with selecting best settings of automatic excitation regulators of synchronous generators. This analysis led us to the conclusions, in particular, about possibility to apply the root locus methods in estimating stability of electrical power systems, and also about genericity of the matrix method with the use of QR-algorithm, which is widely used in practice of calculating stability