Diagonalization in Parallel Space

Matrix diagonalization is an important component of many aspects of computational science. There are a variety of algorithms to accomplish this task. Jacobi’s algorithm is a good choice for parallel environments. Jacobi’s algorithm consists of a series of matrix plane rotations, the ordering of which can dramatically affect performance. We show a new ordering which cuts the number of necessary operations approximately in half. Additionally, Jacobi’s algorithm can be made parallel in space as well as time. This is an advantage when dealing with very large matrices, such as those found in quantum chemistry. 1 Matrix Diagonalization There are two major families of matrix diagonalization methods, those based on Jacobi rotations and those based on householder reflections. It has long been noted that while Householder methods are faster in a serial environment [1], Jacobi rotation-based methods have much more inherent parallelism, as well as better numerical properties and, therefore, greater accuracy [5]. Furthermore, there are refinements to the Jacobi algorithm available in a parallel environment that have been largely unused in serial approaches. The careful and selective use of these refinements has the potential to significantly improve the performance of the algorithm over the serial version, and makes the appearance of super-linear speedups for parallel diagonalization possible. The basic problem in matrix diagonalization is this: given a matrix A, to find matrices Q and S such that A = QSQ (1) where S has all zero off-diagonal elements. The diagonal elements of S are the eigenvalues of A, and the columns of Q are the corresponding eigenvectors. The eigenvalues and corresponding eigenvectors of a matrix are the key targets of many matrix operations. The efficient computation of these quantities is of great use in many applications, such as signal processing [4], computer graphics [8], and computational chemistry [3], among other fields. The motivation of this study is the use of this technique to calculate the quantum mechanical wave functions of noble gas complexes by solving the Schrödinger equation using basis set methods. The matrices for problems of this type can grow to be very large, on the order of 10 elements for non-linear molecules and 10 for simple linear molecules, so parallelization in space as well as in time is highly desirable.