  The eigenvalue problem is a fundamental problem in scientific computing. In
this paper, we propose a mixed precision Jacobi method for the symmetric
eigenvalue problem. We first compute the eigenvalue decomposition of a real
symmetric matrix by an eigensolver at low precision and we obtain a
low-precision matrix of eigenvectors. Then by using the modified Gram-Schmidt
orthogonalization process to the low-precision eigenvector matrix in high
precision, a high-precision orthogonal matrix is obtained, which is used as an
initial guess for the Jacobi method. We give the rounding error analysis for
the proposed method and the quadratic convergence of the proposed method is
established under some sufficient conditions. We also present a mixed precision
one-side Jacobi method for the singular value problem and the corresponding
rounding error analysis and quadratic convergence are discussed. Numerical
experiments on CPUs and GPUs are conducted to illustrate the efficiency of the
proposed mixed precision Jacobi method over the original Jacobi method.
