Efforts on developing mixed precision algorithms in the numerical linear algebra and high performance computing communities have mainly focussed on linear systems and least squares problems. Eigenvalue problems are considerably more challenging to solve and have a larger solution space that cannot be computed in a finite number of steps.

In this talk we present a mixed-precision preconditioned Jacobi algorithm, which uses low precision to compute the preconditioner, applies it in high precision (amounting to two matrix-matrix multiplications) and solves the eigenproblem using the Jacobi algorithm in working precision. Our analysis yields meaningfully smaller relative forward error bounds for the computed eigenvalues compared with those of the standard Jacobi algorithm. We further prove that, after preconditioning, if the off-diagonal entries of the preconditioned matrix are sufficiently small relative to its smallest diagonal entry, the relative forward error bound is independent of the condition number of the original matrix. We present two constructions for the preconditioner that exploit low precision, along with their error analyses. Our numerical experiments confirm our theoretical results.