Computing accurate singular vectors and eigenvectors using mixed-precision Jacobi algorithms

Mixed-precision variants of the Jacobi algorithm for symmetric positive definite eigenproblems and the one-sided Jacobi algorithm for singular value decompositions have recently been shown to compute eigenvalues and singular values to high relative accuracy. However, these analyses do not address the accuracy of the computed eigenvectors and singular vectors. In this paper, we prove error bounds for the computed eigenvectors and singular vectors, where the error is measured by the sine of the angle between the vector and its computed counterpart. The obtained bounds preserve the relative gap structure of the bounds for Jacobi algorithms proved by Demmel and Veseli\'{c}, but involve the scaled condition number of the preconditioned matrix rather than that of the original matrix (the former of which is typically much smaller). Numerical experiments support our theoretical bounds and demonstrate that the mixed-precision preconditioned Jacobi algorithms are especially effective for ill-conditioned matrices with small absolute gaps and moderate relative gaps between eigenvalues or singular values.