New fast divide‐and‐conquer algorithms for the symmetric tridiagonal eigenvalue problem

Abstract

SUMMARYIn this paper, two accelerated divide-and-conquer algorithms are proposed for the symmetric tridiagonal eigenvalue problem, which cost O(N 2 r) flops in the worst case, where N is the dimension of the matrix and r is a modest number depending on the distribution of eigenvalues. Both of these algorithms use hierarchically semiseparable (HSS) matrices to approximate some intermediate eigenvector matrices which are Cauchylike matrices and are off-diagonally low-rank. The difference of these two versions lies in using different HSS construction algorithms, one (denoted by ADC1) uses a structured low-rank approximation method and the other (ADC2) uses a randomized HSS construction algorithm. For the ADC2 algorithm, a method is proposed to estimate the off-diagonal rank. Numerous experiments have been done to show their stability and efficiency. These algorithms are implemented in parallel in a shared memory environment, and some parallel implementation details are included. Comparing the ADCs with highly optimized multithreaded libraries such as Intel MKL, we find that ADCs could be more than 6x times faster for some large matrices with few deflations.

A Parallel Structured Divide-and-Conquer Algorithm for Symmetric Tridiagonal Eigenvalue Problems

“…The main computational task of DC lies in computing the eigenvectors via matrix-matrix multiplications (MMM) (5), which costs O(N 3 ) flops. Since Q is a Cauchy-like matrix and off-diagonally low-rank, MMM (5) can be accelerated by using HSS matrix algorithms, and the computational complexity can be reduced significantly, see [15,12] for more details. The aim of this work is not only to reduce the computation cost of MMM (5) but also its communication cost in the distributed memory environment.…”

“…When there are few deflations, the size of matrix Q in (19) will be large, and most of the time spent by DC would correspond to the matrix-matrix multiplication in (19). Furthermore, it is well-known that matrix Q defined as in ( 4) is a Cauchy-like matrix with off-diagonally low rank property, see [2,12]. Therefore, we simply use the parallel structured matrix-matrix multiplication algorithm to compute the eigenvector matrix U in (19).…”

“…It is known that some Cauchy-like matrices with off-diagonally low-rank properties appear in the DC algorithm [1,2], which can be approximated by hierarchically semiseparable (HSS) matrices [10,11]. Then, the worst case complexity of DC can be reduced from O(N 3 ) to O(N 2 r), where r is a modest number and is usually much smaller than a large N , see [12]. This technique was extended for the bidiagonal and banded DC algorithms for the SVD problem on a shared memory multicore platform in [13,14].…”

An efficient hybrid tridiagonal divide-and-conquer algorithm on distributed memory architectures

“…Other rank-structured matrices include H-matrix [20,23], H 2 -matrix [22,21], quasiseparable matrices [14,40], and sequentially semiseparable (SSS) [5,6] matrices. We mostly follow the notation used in [35] and [41,27] to introduce HSS.…”

“…We expect HSS algorithms to have good performances for problems with large size. Therefore, we only use HSS algorithms when the problem size is large enough, just as in [26,27].…”