Ilan Bar-On

High performance solution of the complex symmetric eigenproblem

Complex symmetric matrices often appear in quantum physics in the solution methods of partial differential equations such as the Schrödinger equation. We have recently introduced a new fast and efficient direct eigensolver for this problem in [4], and reported its performance in the eigenvalue calculation in [3]. In this paper, we further report on some benchmark tests for computing the full and partial eigenspectrum on a variety of super computing machines, i.e., the Cray J-932, the DEC Alfa 8400, and the SGI Power Challenge 8000 and 10000. We observe that in all cases the new algorithm is much faster than codes available in standard state of the art eigensolver packages such as LAPACK.

Parallel wave-packet simulations of electron transmission through water

The dynamics of electron tunneling through water layers embedded between two metal plates is studied by electron wave-packet simulations. The tunneling flux is shown to increase by orders of magnitude due to resonances when the thermal motion of the water nuclei is “frozen” and transient molecular nanocavities dominate the tunneling mechanism. This enhancement is observed even when the energy width of the wave-packet is larger than the resonance width, and the transmission probability does not show resonance peaks as a function of the impact electron energy. The wave-packet simulations are based on a parallel solution of the multidimensional time-dependent Schrodinger equation, in which the N-dimensional Hilbert space is distributed into subspaces associated with an N-dimensional hypercube of processors. The propagated wave function is fully distributed at all times and the computation rate can increase linearly with the number of processors. The significant advantage of the present algorithm over serial ...

Checking robust nonsingularity of tridiagonal matrices in linear time

In this paper we present a linear time algorithm for checking whether a tridiagonal matrix will become singular under certain perturbations of its coefficients. The problem is known to be NP-hard for general matrices. Our algorithm can be used to get perturbation bounds on the solutions to tridiagonal systems.

A fast and stable parallel QR algorithm for symmetric tridiagonal matrices

Abstract We present a new, fast, and practical parallel algorithm for computing a few eigenvalues of a symmetric tridiagonal matrix by the explicit QR method. We present a new divide and conquer parallel algorithm which is fast and numerically stable. The algorithm is work efficient and of low communication overhead, and it can be used to solve very large problems infeasible by sequential methods.

Stable solution of tridiagonal systems

In this paper we present three different pivoting strategies for solving general tridiagonal systems of linear equations. The first strategy resembles the classical method of Gaussian elimination with no pivoting and is stable provided a simple and easily checkable condition is met. In the second strategy, the growth of the elements is monitored so as to ensure backward stability in most cases. Finally, the third strategy also uses the right‐hand side vector to make pivoting decisions and is proved to be unconditionally backward stable.

Interlacing Properties of Tridiagonal Symmetric Matrices with Applications to Parallel Computing

In this paper we present new interlacing properties for the eigenvalues of an unreduced tridiagonal symmetric matrix in terms of its leading and trailing submatrices. The results stated in Hill and Parlett [SIAM J. Matrix Anal. Appl., 13 (1992), pp. 239--247] are hereby improved. We further extend our results to reduced symmetric tridiagonal matrices and to specially structured full symmetric matrices. We then present new fast and efficient parallel algorithms for computing a few eigenvalues of symmetric tridiagonal matrices of very large order.

Reliable Solution of Tridiagonal Systems of Linear Equations

In this paper we present new formulas for characterizing the sensitivity of tridiagonal systems that are independent of the condition number of the underlying matrix. We also introduce efficient algorithms for solving tridiagonal systems of linear equations which are stable and reliable (namely, stable in the backward sense and little sensitive to perturbations in the coefficients).

Parallel solution of the multidimensional Helmhotz/Schroedinger equation using high order methods

Abstract We show that high order methods are useful in deriving fast and efficient parallel algorithms for solving multidimensional inhomogeneous Helmholtz/Schroedinger equations. Using high order methods we represent one-dimensional operators by small size matrices that serve together to construct an efficiently parallelizable preconditioner. The coupled multidimensional sparse system of equations is then solved iteratively on massively parallel systems with linear speedup. As an example, we demonstrate linear speed up in performance on the IBM SP2 massively parallel machine.

Paper: Efficient logarithmic time parallel algorithms for the cholesky decomposition and gram-schmidt process

We present a parallel algorithm for finding the Cholesky decomposition of a symmetric positive definite matrix of order n, in O(log^2n) time using p = n^3/log^2n processors. This algorithm can be used in order to find the QR factorization of a full rank matrix in M(m x n), m >= n. Alternatively, we give a more direct parallel algorithm of O(log n log m) time using p = m x n^2/(log n log m) processors.

Reliable parallel solution of bidiagonal systems

Summary. This paper presents a new efficient algorithm for solving bidiagonal systems of linear equations on massively parallel machines. We use a divide and conquer approach to compute a representative subset of the solution components after which we solve the complete system in parallel with no communication overhead. We address the numerical properties of the algorithm in two ways: we show how to verify the à posteriori backward stability at virtually no additional cost, and prove that the algorithm is à priori forward stable. We then show how we can use the algorithm in order to bound the possible perturbations in the solution components.