Mixed-Precision Algorithm for Finding Selected Eigenvalues and Eigenvectors of Symmetric and Hermitian Matrices1

The multi-precision methods commonly follow approximate-iterate scheme by first obtaining the approximate solution from a low-precision factorization and solve. Then, they iteratively refine the solution to the desired accuracy that is often as high as what is possible with traditional approaches. While targeting symmetric and Hermitian eigenvalue problems of the form Ax = λx, we revisit the SICE algorithm proposed by Dongarra et al. By applying the Sherman-Morrison formula on the diagonally-shifted tridiagonal systems, we propose an updated SICE-SM algorithm. By incorporating the latest two-stage algorithms from the PLASMA and MAGMA software libraries for numerical linear algebra, we achieved up to 3.6× speedup using the mixed-precision eigensolver with the blocked SICE-SM algorithm for iterative refinement when compared with full double complex precision solvers for the cases with a portion of eigenvalues and eigenvectors requested. 1