  In a Jacobi--Davidson (JD) type method for singular value decomposition (SVD)
problems, called JDSVD, a large symmetric and generally indefinite correction
equation is approximately solved iteratively at each outer iteration, which
constitutes the inner iterations and dominates the overall efficiency of JDSVD.
In this paper, a convergence analysis is made on the minimal residual (MINRES)
method for the correction equation. Motivated by the results obtained, a
preconditioned correction equation is derived that extracts useful information
from current searching subspaces to construct effective preconditioners for the
correction equation and is proved to retain the same convergence of outer
iterations of JDSVD. The resulting method is called inner preconditioned JDSVD
(IPJDSVD) method. Convergence results show that MINRES for the preconditioned
correction equation can converge much faster when there is a cluster of
singular values closest to a given target, so that IPJDSVD is more efficient
than JDSVD. A new thick-restart IPJDSVD algorithm with deflation and purgation
is proposed that simultaneously accelerates the outer and inner convergence of
the standard thick-restart JDSVD and computes several singular triplets of a
large matrix. Numerical experiments justify the theory and illustrate the
considerable superiority of IPJDSVD to JDSVD.
