Lifeng Wei, LI Xiao-mei · 2003
In this paper we present a new divide and conquer algorithm for solving a real symmetric band generalized eigenvalue problem under a distributed environment. This algorithm computes generalized eigenpairs of a real symmetric band matrix pair using the divide and conquer method combined with multisection and generalized Rayleigh quotient iteration. Theoretical analysis and numerical experiments prove that this algorithm is better than the classic method DSBGVX in LAPACK when the scale of matrices is large enough. During parallel computing, there is little communication between processors, so high speedup can be expected.
Cited by 0
Yusuke Hirota, Toshiyuki Imamura · Lecture notes in computer science · 2018
Cited by 2
Rutter, Jeffery D, Author · UC Berkeley
This report discusses a serial implementation of Cuppen's divide and conquer algorithm for computing all eigenvalues and eigenvectors of a real symmetric matrix <i>T</i> = <i>Q</i> Lambda <i>Q</i>^<i>T</i>. This method is compared with the LAPACK implementations of QR, bisection/inverse iteration, and root-free QR/inverse iteration to find all of the eigenvalues and eigenvectors. <p>On a DEC Alpha using optimized Basic Linear Algebra Subroutines (BLAS), divide and conquer was uniformly the fastest algorithm by a large margin for large tridiagonal eigenproblems. When Fortran BLAS were used, bisection/inverse iteration was somewhat faster (up to a factor of 2) for very large matrices (<i>n</i> >= 500) without clustered eigenvalues. When eigenvalues were clustered, divide and conquer was up to 80 times faster. The speedups over QR were so large in the tridiagonal case that the overall problem, including reduction to tridiagonal form, sped up by a factor of 2.5 over QR for <i>n</i> >= 500. <p>Nearly universally, the matrix of eigenvectors generated by divide and conquer suffered the least loss of orthogonality. The smallest eigensystem residual usually came from the eigensystem generated by bisection/inverse iteration, with divide and conquer coming a close second
Cited by 5
Françoise Tisseur, Jack Dongarra · OpenGrey (Institut de l'Information Scientifique et Technique) · 1998
We present a new parallel implementation of a divide and conquer algorithm for computing the spectral decomposition of a symmetric tridiagonal matrix on distributed memory architectures. The implementation we develop differs from other implementations in that we use a two dimensional block cyclic distribution of the data, we use the Lowner theorem approach to compute orthogonal eigenvectors, and we introduce permutations before the back transformation of each rank-one update in order to make good use of deflation. This algorithm yields the first scalable, portable and numerically stable parallel divide and conquer eigensolver. Numerical results confirm the effectiveness of our algorithm. We compare performance of the algorithm with that of the QR algorithm and of bisection followed by inverse iteration on an IBM SP2 and a cluster of Pentium PII&apos;s.
Cited by 16
Azzam Haidar, Hatem Ltaief, Jack Dongarra · SIAM Journal on Scientific Computing · 2012
Classical solvers for the dense symmetric eigenvalue problem suffer from the first step, which involves a reduction to tridiagonal form that is dominated by the cost of accessing memory during the panel factorization. The solution is to reduce the matrix to a banded form, which then requires the eigenvalues of the banded matrix to be computed. The standard divide and conquer algorithm can be modified for this purpose. The paper combines this insight with tile algorithms that can be scheduled via a dynamic runtime system to multicore architectures. A detailed analysis of performance and accuracy is included. Performance improvements of 14-fold and 4-fold speedups are reported relative to LAPACK and Intel's Math Kernel Library.
Cited by 21Open access
Francoise Tisseur Y · 1998
Jack Dongarra z We present a new parallel implementation of a divide and conquer algo-rithm for computing the spectral decomposition of a symmetric tridiagonal matrix on distributed memory architectures. The implementation we de-velop di ers from other implementations in that we use a two dimensional block cyclic distribution of the data, we use the Lowner theorem approach to compute orthogonal eigenvectors, and we introduce permutations before the back transformation of each rank-one update in order to make good
Cited by 0
Fran Coise, Jack Dongarra · 1999
We present a new parallel implementation of a divide and conquer algorithm for computing the spectral decomposition of a symmetric tridiagonal matrix on distributed memory architectures. The implementation we develop differs from other implementations in that we use a two dimensional block cyclic distribution of the data, we use the Lowner theorem approach to compute orthogonal eigenvectors, and we introduce permutations before the back transformation of each rank-one update in order to make good This work supported in part by Oak Ridge National Laboratory, managed by Lockheed Martin Energy Research Corp. for the U.S. Department of Energy under contract number DE-AC05-96OR2246 and by the Defense Advanced Research Projects Agency under contract DAAL03-91-C-0047, administered by the Army Research Office. y Department of Mathematics, University of Manchester, Manchester, M13 9PL, England (ftisseur@ma.man.ac.uk, http://www.ma.man.ac.uk/~ftisseur/). z Department of Computer Science, ...
Cited by 0
Jeffery D Rutter · 1994
This report discusses a serial implementation of Cuppen&apos;s divide and conquer algorithm for computing all eigenvalues and eigenvectors of a real symmetric matrix T = Q Q T. This method is compared with the LAPACK implementations of QR, bisection/inverse iteration, and root-free QR/inverse iteration to nd all of the eigenvalues and eigenvectors. On a DEC Alpha using optimized Basic Linear Algebra Subroutines (BLAS), divide and conquer was uniformly the fastest algorithm by a large margin for large tridiagonal eigenproblems. When Fortran BLAS were used, bisection/inverse iteration was somewhat faster (up to a factor of 2) for very large matrices (n 500) without clustered eigenvalues. When eigenvalues were clustered, divide and conquer was up to 80 times faster. The speedups over QR were so large in the tridiagonal case that the overall problem, including reduction to tridiagonal form, sped up by a factor of 2.5 over QR for n 500. Nearly universally, the matrix of eigenvectors generated by divide and con-
Cited by 0
Binh T. Nguyen, Anh-Duc Luong-Thanh, Nguyen Dinh Thuc et al. · 2012
We propose a stable and efficient divide-and-conquer algorithm for computing the eigendecomposition of a symmetric tri-block-diagonal matrix. The matrix can be derived from discretizing Laplace operator eigenvalue in some two-dimensional graphs. All numerical results show that our algorithm is competitive with other methods, such as e.g QR algorithm, Cuppen's divide-and-conquer algorithm. We also show how to improve our algorithm by Fast Multipole Method.
Cited by 1
K. E. Gates, Peter Arbenz · Repository for Publications and Research Data (ETH Zurich) · 1994
In this paper a new implementation of a divide and conquer algorithm will be considered.This algorithm, in contrast to the LAPACK algorithm, uses a di erent formulation of the update problem, and extended precision in order to maintain accuracy and orthogonality.O u r I n tel Paragon implementation shows, in contrast to the Hypercube implementation by Ipsen and Jessup 14], that good speedups can be obtained from a distributed memory parallel version of the divide and conquer eigenvalue algorithm .
Cited by 13Open access
Fraser K. Coutts, Jamie Corr, Keith Thompson et al. · 2017
A number of algorithms capable of iteratively calculating a polynomial matrix eigenvalue decomposition (PEVD) have been introduced. The PEVD is a generalisation of the ordinary EVD and will diagonalise a parahermitian matrix via paraunitary operations. Inspired by the existence of low complexity divide-and-conquer solutions to eigenproblems, this paper addresses a divide-and-conquer approach to the PEVD utilising the sequential matrix diagonalisation (SMD) algorithm. We demonstrate that with the proposed techniques, encapsulated in a novel algorithm titled divide-and-conquer sequential matrix diagonalisation (DC-SMD), algorithm complexity can be significantly reduced. This reduction impacts on a number of broadband multichannel problems, including those involving large arrays.
Cited by 11
Peter Arbenz, Walter Gander, Kevin Gates · 1994
. We investigate direct algorithms to solve linear banded systems of equations on MIMD multiprocessor computers with distributed memory. We show that it is hard to beat ordinary one-processor Gaussian elimination. Numerical computation results from the Intel Paragon are given. 1. Introduction In a project on divide and conquer algorithms in numerical linear algebra, the authors studied parallel algorithms to solve systems of linear equations and eigenvalue problems. The latter consisted in a study of the divide and conquer algorithm proposed by Cuppen [4] and stabilized by Sorensen and Tang [11]. This algorithm is evolving as the standard algorithm for solving the symmetric tridiagonal eigenvalue problem on sequential as on parallel computers. In [7], Gates and Arbenz report on the first successful parallel implementation of the algorithm. They observed almost optimal speedups on the Intel Paragon. The accuracy observed is as good as with any other known (fast) algorithm. The...
Cited by 0
Vogel, James P. · Purdue · 2019
This PhD thesis is an important development in the theories, methods, and applications of eigenvalue algorithms for structured matrices. Though eigenvalue problems have been well-studied, the class of matrices that admit very fast (near-linear time) algorithms was quite small until very recently. We developed and implemented a generalization of the famous symmetric tridiagonal divide-and-conquer algorithm to a much larger class of rank structured matrices (symmetric hierarchically semisperable, or HSS) that appear frequently in applications. Altogether, this thesis makes valuable contributions to three different major areas of scientific computing: algorithmic development, numerical analysis, and applications. In addition to the previously stated divide-and-conquer algorithm, we generalize to larger classes of eigenvalue problems and provide several key new low-rank update algorithms. A major contribution the analysis of the structured eigenvalue problem. In addition to standard perturbation analysis, we elucidate some subtle and previously under-examined issues in structured matrix eigenvalue problems such as subspace contributions and secular equation conditioning. Finally, several applications are studied.
Cited by 0Open access
Victor Y. Pan, Elliot Linzer · 1999
We present new algorithms that accelerate the bisection method for the symmetric tridiagonal eigenvalue problem. The algorithms rely on some new techniques, including a new variant of Newton&apos;s iteration that reaches cubic convergence (right from the start) to the well separated eigenvalues and can be further applied to acceleration of The results of this paper ( apart from numerical experiments, designed and performed by the second author ) have been presented ( in the preliminary form of [P92] ) at the Copper Mountain Conference on Iterative Methods, Copper Mountain, Colorado, 1992. some other iterative processes, in particular, of the divide-and-conquer methods for the symmetric tridiagonal eigenvalue problem. Key words: symmetric eigenvalue problem, bisection algorithm, divide-and-conquer eigenvalue algorithms, Newton&apos;s iteration, convergence acceleration, approximating polynomial zeros. 1991 Mathematics subject classification: 65F15, 65Y20, 65B99. 1 Introduction The classica...
Cited by 0
Dario A. Bini, Luca Gemignani, Françoise Tisseur · 2003
Abstract. An algorithm based on the Ehrlich–Aberth iteration is presented for the computation of the zeros of p(λ) = det(T −λI), where T is a real irreducible nonsymmetric tridiagonal matrix. The algorithm requires the evaluation of p(λ)/p ′ (λ)=−1/trace(T − λI) −1, which is done by exploiting the QR factorization of T − λI and the semiseparable structure of (T − λI) −1. The choice of initial approximations relies on a divide-and-conquer strategy, and some results motivating this strategy are given. Guaranteed a posteriori error bounds based on a running error analysis are proved. A Fortran 95 module implementing the algorithm is provided and numerical experiments that confirm the effectiveness and the robustness of the approach are presented. In particular, comparisons with the LAPACK subroutine dhseqr show that our algorithm is faster for large dimensions. Key words. nonsymmetric eigenvalue problem, symmetric indefinite generalized eigenvalue problem, tridiagonal matrix, root finder, QR decomposition, divide and conquer AMS subject classification. 65F15 DOI. 10.1137/S0895479803429788
Cited by 0
James Demmel, Ioana Dumitriu, Ryan Schneider · ArXiv.org · 2025
This paper presents a fast, randomized divide-and-conquer algorithm for the definite generalized eigenvalue problem, which corresponds to pencils $(A,B)$ in which $A$ and $B$ are Hermitian and the Crawford number $γ(A,B) = \min_{||x||_2 = 1} |x^H(A+iB)x|$ is positive. Adapted from the fastest known method for diagonalizing arbitrary matrix pencils [Foundations of Computational Mathematics 2024], the algorithm is both inverse-free and highly parallel. As in the general case, randomization takes the form of perturbations applied to the input matrices, which regularize the problem for compatibility with fast, divide-and-conquer eigensolvers -- i.e., the now well-established phenomenon of pseudospectral shattering. We demonstrate that this high-level approach to diagonalization can be executed in a structure-aware fashion by (1) extending pseudospectral shattering to definite pencils under structured perturbations (either random diagonal or sampled from the Gaussian Unitary Ensemble) and (2) formulating the divide-and-conquer procedure in a way that maintains definiteness. The result is a specialized solver whose complexity, when applied to definite pencils, is provably lower than that of general divide-and-conquer.
Cited by 0Open access
Minki Ahn, Wookbong Lee, Eun-Sung Jeon et al. · 2019
In this paper, we present a low-complexity singular value decomposition (SVD) algorithm to generate a transmit beamforming matrix for multiple-input multiple-output (MIMO) systems. By introducing divide and conquer concept, we divide original channel matrix to sub-block channel matrices with a half of size. By using this form, we formulate eigenvalue problems according to the number of transmitted data streams and provide closed-form solutions. The proposed algorithm offers a good tradeoff between complexity and performance compared to the conventional iterative SVD algorithm. Simulation results demonstrate that the proposed SVD algorithm achieves almost the same performance of conventional iterative SVD algorithm with much reduced computational complexity.
Cited by 2
Schneider, Ryan · eScholarship (California Digital Library) · 2024
\\indent Over the last two decades, randomization has emerged as a leading tool for pursuing efficiency in numerical linear algebra. Its benefits can be explained in part by smoothed analysis, where an algorithm that fails spectacularly in certain cases may be unlikely to do so on random -- or randomly perturbed -- inputs. This observation implies a simple framework for developing accurate and efficient randomized algorithms:\\ apply a random perturbation and run an existing method, whose worst-case error (or run-time) can be avoided with high probability. \\\\\\indent Recent work of Banks, Garza-Vargas, Kulkarni, and Srivastava applied this framework to the standard eigenvalue problem, developing a randomized algorithm that (with high probability) diagonalizes a matrix in nearly matrix multiplication time [Foundations of Computational Math 2022]. Central to their work is the phenomenon of \\textit{pseudospectral shattering}, in which a small Gaussian perturbation regularizes the pseudospectrum of a matrix, with high probability breaking it into disjoint components and allowing classical, divide-and-conquer eigensolvers to run successfully. Prior to their work, no way of accessing the benefits of divide-and-conquer’s natural parallelization was known in general. \\\\\n\\indent In this thesis, we extend the work of Banks et al.\\ to the generalized eigenvalue problem -- e.g., $Av = \\lambda Bv$ for matrices $A,B \\in {\\mathbb C}^{n \\times n}$. Our main contributions can be summarized as follows.\n\\begin{enumerate}[itemsep=0em]\n \\item First, we show that pseudospectral shattering generalizes directly:\\ randomly perturbing $A$ and $B$ has a similar regularizing effect on the pseudospectra of the corresponding matrix pencil $(A,B)$ with high probability.\n \\item Building on pseudospectral shattering, we construct and analyze a fast, randomized, divide-and-conquer algorithm for diagonalizing $(A,B)$, which begins by randomly perturbing the inputs.\n \\item Finally, we demonstrate that both pseudospectral shattering and the corresponding diagonalization algorithm can be adapted to definite pencils, further pursuing efficiency by preserving and exploiting symmetry. \n\\end{enumerate}\\indent The resulting algorithm, which we call pseudospectral divide-and-conquer, is the first general, sub-$O(n^3)$ solver for the generalized eigenvalue problem. It is not only highly parallel and capable of accommodating structure, but also promotes stability by avoiding matrix inversion. In essence, this thesis is a handbook for understanding, adapting, and implementing the method.
Cited by 0
Calvetti Golub, D. Calvetti, G. H. Golub et al. · 2007
. Recently Laurie presented a new algorithm for the computation of (2n + 1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order 2n + 1 from certain mixed moments, and then computes a partial spectral factorization. We describe a new algorithm that does not require the entries of the tridiagonal matrix to be determined, and thereby avoids computations that can be sensitive to perturbations. Our algorithm uses the consolidation phase of a divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. We also discuss how the algorithm can be applied to compute Kronrod extensions of Gauss-Radau and Gauss-Lobatto quadrature rules. Throughout the paper we emphasize how the structure of the algorithm makes efficient implementation on parallel computers possible. Numerical examples illustrate the performance of the algorithm. Key words. Jacobi matrix, inverse eigenvalue problem, divide-and...
Cited by 0
T. Y. Li, Zhonggang Zeng · 1992
. In this paper we present an algorithm for the eigenvalue problem of symmetric tridiagonal matrices. Our algorithm employs the determinant evaluation, split-and-merge strategy and Laguerre&apos;s iteration. The method directly evaluates eigenvalues and uses inverse iteration as an option when eigenvectors are needed. This algorithm combines the advantages of existing algorithms such as QR, bisection/multisection and Cuppen&apos;s divide-and-conquer method. It is fully parallel, and competitive in speed with the most efficient QR algorithm in serial mode. On the other hand, our algorithm is as accurate as any standard algorithm for the symmetric tridiagonal eigenproblem and enjoys the flexibility in evaluating partial spectrum. Key words. eigenvalue, Laguerre&apos;s iteration, symmetric tridiagonal matrix 1. Introduction. For a symmetric tridiagonal matrix T with nonzero subdiagonal entries, the eigenvalues of T or the zeros of its characteristic polynomial f() = det[T \\Gamma I ] (1.1) are all rea...
Cited by 0