  We propose an unconditionally robust and highly effective preconditioner for
general symmetric positive definite (SPD) matrices based on structured
incomplete factorization (SIF), called enhanced SIF (eSIF) preconditioner. The
original SIF strategy proposed recently derives a structured preconditioner by
applying block diagonal preprocessing to the matrix and then compressing
appropriate scaled off-diagonal blocks. Here, we use an enhanced
scaling-and-compression strategy to design the new eSIF preconditioner. Some
subtle modifications are made, such as the use of two-sided block triangular
preprocessing. A practical multilevel eSIF scheme is then designed. We give
rigorous analysis for both the enhanced scaling-and-compression strategy and
the multilevel eSIF preconditioner. The new eSIF framework has some significant
advantages and overcomes some major limitations of the SIF strategy. (i) With
the same tolerance for compressing the off-diagonal blocks, the eSIF
preconditioner can approximate the original matrix to a much higher accuracy.
(ii) The new preconditioner leads to much more significant reductions of
condition numbers due to an accelerated magnification effect for the decay in
the singular values of the scaled off-diagonal blocks. (iii) With the new
preconditioner, the eigenvalues of the preconditioned matrix are much better
clustered around $1$. (iv) The multilevel eSIF preconditioner is further
unconditionally robust or is guaranteed to be positive definite without the
need of extra stabilization, while the multilevel SIF preconditioner has a
strict requirement in order to preserve positive definiteness. Comprehensive
numerical tests are used to show the advantages of the eSIF preconditioner in
accelerating the convergence of iterative solutions.
