1 edition of **A parallel QR-algorithm for tridiagonal symmetric matrices** found in the catalog.

A parallel QR-algorithm for tridiagonal symmetric matrices

Ahmed Sameh

- 103 Want to read
- 33 Currently reading

Published
**1975**
by Dept. of Computer Science, University of Illinois at Urbana-Champaign in Urbana, Illinois
.

Written in English

- Eigenvalues,
- Symmetric matrices,
- Data processing,
- Parallel processing (Electronic computers)

**Edition Notes**

Statement | by Ahmed H. Sameh and David J. Kuck |

Series | Report (University of Illinois at Urbana-Champaign. Dept. of Computer Science) -- no. 700, Report (University of Illinois at Urbana-Champaign. Dept. of Computer Science) -- no. 700. |

Contributions | Kuck, David J. author |

Classifications | |
---|---|

LC Classifications | QA76 .I4 no.700, QA193 .I4 no.700 |

The Physical Object | |

Pagination | 9 p. ; |

ID Numbers | |

Open Library | OL25454958M |

OCLC/WorldCa | 1307290 |

A new algorithm for the orthogonal reduction of a symmetric matrix to tridiagonal form is developed and analysed. It uses a Cholesky factorization of the original matrix and the rotations are. In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations.A tridiagonal system for n unknowns may be written as − + + + =, where = and. [⋱ ⋱ ⋱ −] [⋮] = [⋮].For such systems, the solution can be obtained.

In this paper we discuss the parallel implementation of the Cholesky factorization of a positive definite symmetric matrix when that matrix is block tridiagonal. While parallel implementations for. () A New Parallel Symmetric Tridiagonal Eigensolver Based on Bisection and Inverse Iteration Algorithms for Shared-Memory Multi-core Processors. 10th International Conference on P2P, Parallel, Grid, Cloud and Internet Computing (3PGCIC),

suitable Givens matrix G. Note that the Givens operation preserves the symmetry and the eigenvalues of A. To compute the eigenvalues of a general symmetric matrix economically, one first transforms it to tridiagonal form. Then it is easier to calculate the eigenvalue of the tridiagonal matrix by using the QR algorithm, for example. $\begingroup$ @vanvu: A common strategy (that gives local quadratic convergence (cubic for symmetric matrices)) is to use the last entry in the matrix (negated) as the shift. A better strategy is to use one of the eigenvalues of the last 2x2 block of the matrix. (IIRC, this leads to global convergence on symmetric matrices.) It makes sense if you think of the matrix as being already close to.

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[Published as Parallel Computing 21 ()] Abstract. A new algorithm is presented, designed to solve tridiagonal matrix problems efficiently with parallel computers (multiple instruction stream, multiple data stream (MIMD) machines with distributed memory). The algorithm is designed to be extendable to higher order banded diagonal systems.

We show that if the size of the tridiagonal matrix in any given iteration is n, then the parallel QR algorithm requires 0(log2n) steps with 0(n) processors per iteration and no square roots. This results in a speedup of 0(n/log2n) over the sequential.

-HOLLAND A Fast and Stable Parallel QR Algorithm for Symmetric Tridiagonal Matrices Ilan Bar-On Department of Computer Science Technion HaifaIsrael and Bruno Codenotti* LEI-CNR Via S. Maria 46 Pisa, Italy Submitted by Daniel Hershkowitz ABSTRACT We present a new, fast, and practical parallel algorithm for computing a few eigenvalues of a symmetric tridiagonal matrix Cited by: 7.

An efficient parallel algorithm, which we dubbed farm- zeroinNR, for the eigenvalue problem of a symmetric tridiagonal matrix has been implemented in a distributed memory multiprocessor with The QR algorithm has been the method of choice for computing the eigen-values of a matrix on a single processor.

However, the choice is less clear in the context of parallel computing. Nevertheless, there is a noteworthy paper by Sameh and Kuch [13] which describes a parallel implementation of the QR algorithm for symmetric tridiagonal matrices.

For solving systems of linear algebraic equations with block-tridiagonal matrices arising in geoelectrics problems, the parallel matrix sweep algorithm, conjugate gradient method with preconditioner, and square root method are proposed and implemented numerically on multi-core CPU Intel with graphics processors NVIDIA.

In this study, we develop a new parallel algorithm for solving systems of linear algebraic equations with the same block-tridiagonal matrix but with different right-hand sides. The method is a generalization of the parallel dichotomy algorithm for solving systems of linear equations with tridiagonal matrices.

parallel versions. The QR algorithm, independently invented by Francis [26, 27] and Kublanov-skaya [36], is an iteration that produces a sequence of similar matrices that converges to diagonal form. When the starting matrix is symmetric and tridiagonal, each iter-ate produced by the QR algorithm is also symmetric and tridiagonal.

Convergence. Bini and Pan () give a method for computing the eigenvalues of a real symmetric tridiagonal (rst) matrix. This problem is related to solving a polynomial with only real roots, for given the coefficients of an n'th degree polynomial p(z) having only real zeros ζ 1, ζ 2,ζ n, we may compute an n × n rst matrix.

1 Properties and structure of the algorithm General description. The Cholesky decomposition algorithm was first proposed by Andre-Louis Cholesky (Octo - Aug ) at the end of the First World War shortly before he was killed in battle.

He was a French military officer and mathematician. The idea of this algorithm was published in by his fellow officer and, later.

This feature makes the method suited for implementation on a multivector computer with two vector processors. When A is a block tridiagonal matrix, it is possible to derive another form of the arithmetic mean method, having a higher degree of parallelism within its structure.

Let A be a q q block tridiagonal matrix A = B1 C1 D2 B2 /)3 B3 C3 ". The algorithm Input a Hermitian matrix of size ×, and optionally a number of iterations (as default, let =). Strictly speaking, the algorithm does not need access to the explicit matrix, but only a function ↦ that computes the product of the matrix by an arbitrary vector.

This function is called at most times.; Output an × matrix with orthonormal columns and a tridiagonal real symmetric. from book Parallel Processing of the symmetric tridiagonal QR algorithm.

QR algorithm for computing eigenvalues and eigenvectors of real symmetric tridiagonal matrices. This approach is. Abstract. Multishift QR algorithms are efficient for solving the symmetric tridiagonal eigenvalue problem on a parallel computer. In this paper, we focus on three variants of the multishift QR algorithm, namely, the conventional multishift QR algorithm, the deferred shift QR algorithm and the fully pipelined multishift QR algorithm, and construct performance models for them.

THE SEQUENTIAL QR ALGORITHM FOR SYMMETRIC TRIDIAGONAL MATRICES In this section we review the basic features of the sequential QR algorithm, and the method by which it can be adapted to parallel machines. Let A be an n X n real symmetric tridiagonal matrix, and let us seek some of.

A parallel implementation of the symmetric tridiagonal QR algorithm. [Proceedings ] The Fourth Symposium on the Frontiers of Massively Parallel Computation, Design and performance evaluation of a distributed eigenvalue solver on a workstation cluster.

A parallel QR algorithm for the symmetrical tridiagonal eigenvalue problem Abstract: A parallel/pipelined algorithm and its architecture are proposed to solve the symmetric eigenvalue problem.

This algorithm is based on Given's rotations, and it is associated with the initial reduction of the dense matrix to a tridiagonal one using Householder.

Divide-and-conquer eigenvalue algorithms are a class of eigenvalue algorithms for Hermitian or real symmetric matrices that have recently (circa s) become competitive in terms of stability and efficiency with more traditional algorithms such as the QR basic concept behind these algorithms is the divide-and-conquer approach from computer science.

We present a hybrid MPI/OpenMP parallel implementation for the eigenvalues of symmetric tridiagonal matrices on cluster of SMP’s environments. The algorithm is based on a divide-and-conquer method which uses the split-merge technique and Laguerre’s iteration. Buy A parallel QR-algorithm for tridiagonal symmetric matrices (Report - UIUCDCS-R ; no.

) on FREE SHIPPING on qualified orders. For linear systems with coefficient matrices of classical structure, WZ factorizations for matrices are basic mathematical theories to design a class of parallel algorithms.

Based on WZ factorization, a parallel algorithm is provided for symmetric tridiagonal linear systems. The method estimates the computation task carefully so that it assigns the system skillfully to get even load balance.We compare four algorithms from the latest LAPACK release for computing eigenpairs of a symmetric tridiagonal matrix.

These include QR iteration, bisection and inverse iteration (BI), the divide-and-conquer method (DC), and the method of multiple relatively robust representations (MR). Our evaluation considers speed and accuracy when computing all eigenpairs and additionally subset.A novel variant of the parallel QR algorithm for solving dense nonsymmetric eigenvalue problems on hybrid distributed high performance computing (HPC) systems is presented.