High performance parallel approximate eigensolver for real symmetric matrices

In the first-principles calculation of electronic structures, one of the most timeconsuming tasks is that of computing the eigensystem of a large symmetric nonlinear eigenvalue problem. The standard approach is to use an iterative scheme involving the solution to a large symmetric linear eigenvalue problem in each iteration. In the early and intermediate iterations, significant gains in efficiency may result from solving the eigensystem to reduced accuracy. As the iteration nears convergence, the eigensystem can be computed to the required accuracy. The main contribution of this dissertation is an efficient parallel approximate eigensolver that computes eigenpairs of a real symmetric matrix to reduced accuracy. This eigensolver consists of three major parts: (1) a parallel block divide-and-conquer algorithm that computes the approximate eigenpairs of a block tridiagonal matrix to prescribed accuracy; (2) a parallel block tridiagonalization algorithm that constructs a block tridiagonal matrix from a sparse matrix or "effectively" sparse matrix---matrix with many small elements that can be regarded as zeros without affecting the prescribed accuracy of the eigenvalues; (3) a parallel orthogonal block tridiagonal reduction algorithm that reduces a dense real symmetric matrix to block tridiagonal form using similarity transformations with a high ratio of level 3 BLAS operations. The parallel approximate eigensolver chooses a proper combination of the three algorithms depending on the structure of the input matrix and computes all the eigenpairs of the input matrix to prescribed accuracy. Numerical results show that the parallel approximate eigensolver is efficient and accurate to the prescribed tolerance. The time required for computing the approximate eigenpairs decreases significantly as the accuracy tolerance becomes larger.

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