DQGMRES: a Direct Quasi‐minimal Residual Algorithm Based on Incomplete Orthogonalization

Hybrid methods based on LCG and GMRES

“…Regardless of finite precision, the mathematical equivalence of LCG and FOM can be easily established as follows. Actually, the similar review has been discussed by Saad et al (1996). He presented the direct implementation of truncated GMRES by combining quasiminimization and incomplete orthogonalization.…”

The effect of non-optimal bases on the convergence of Krylov subspace methods

“…• The nonsymmetric Lanczos method [19] and its variations, e.g., QMR [10]; • Incomplete (truncated) methods [26], [27, §6.4.2 and §6.5.6], [29], and restarted methods [27], [31];…”

“…We refer to the number k as the truncation parameter. In this case, the upper Hessenberg matrix T m+1,m in (2.1) is banded with k nonzero superdiagonals; see, e.g., [26], [27, §6.4.2 and §6.5.6], [29].…”

High-order ILU preconditioners for CFD problems

“…It was argued in Reference [9] that if scaling results in a deterioration of the degree of normality of the matrix, measured by the number AA T −A T A/AA T , this will result in worse performances. Diagonal scaling consists of scaling all the rows of the matrix and then scaling all the columns (or the same operations done in reverse order).…”