Abstract not found

Bisection is a parallelizable method for finding the eigenvalues of real symmetric tridiagonal matrices, or more generally symmetric acyclic matrices.

Abstract not found

Bisection is an easily parallelizable method for finding the eigenvalues of real symmetric tridiagonal matrices, or more generally symmetric acyclic matrices. It requires a function Count(x) which counts the number of eigenvalues less than x. In exact arithmetic Count(x) is an increasing function of x, but this is not necessarily the case with roundoff. Our first result is that as long as the floating point arithmetic is monotonic, the computed function Count(x) implemented appropriately will also be monotonic; this extends an unpublished 1966 result of Kahan to the larger class of symmetric acyclic matrices. Second, we analyze the impact of nonmonotonicity of Count(x) on the serial and parallel implementations of bisection. We present simple and natural implementations which can fail because of nonmonotonicity; this includes the routine bisect in EISPACK. We also show how to implement bisection correctly despite nonmonotonicity; this is important because the fastest known ...

In this report we list diagonalisation routines available for parallel computers. The methodology of each routine is outlined together with benchmark results on a typical matrix where available. Storage requirements and advantages and disadvantages of the method are also compared. The vast majority of these routines are available for real dense symmetric matrices only, although there is a known requirement for other data types -- such as Hermitian or structured sparse matrices. We will report on new codes as they become available. This report is available from http://www.dl.ac.uk/TCSC/HPCI/ c fl1996, Daresbury Laboratory. We do not accept any responsibility for loss or damage arising from the use of information contained in any of our reports or in any communication about our tests or investigations. ii CONTENTS iii Contents 1 Summary 1 1.1 Test Results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 Recommendations : : : : : : : : : : :...

We present a new parallel algorithm for the dense symmetric eigenvalue/eigenvector problem that is based upon the tridiagonal eigensolver, Algorithm MR 3, recently developed by Dhillon and Parlett. Algorithm MR 3 has a complexity of O(n 2) operations for computing all eigenvalues and eigenvectors of a symmetric tridiagonal problem. Moreover the algorithm requires only O(n) extra workspace and can be adapted to compute any subset of k eigenpairs in O(nk) time. In contrast, all earlier stable parallel algorithms for the tridiagonal eigenproblem require O(n 3) operations in the worst case, while some implementations, such as divide and conquer, have an extra O(n 2) memory requirement. The proposed parallel algorithm balances the workload equally among the processors by traversing a matrix-dependent representation tree which captures the sequence of computations performed by Algorithm MR 3. The resulting implementation allows problems of very large size to be solved efficiently—the largest dense eigenproblem solved in-core on a 256 processor machine with 2 GBytes of memory per processor is for a matrix of size 128,000 × 128,000, which required about 8 hours of CPU time. We present comparisons with other eigensolvers and results on matrices that arise in the applications of computational quantum chemistry and finite element modeling of automobile bodies.

Abstract not found