TFQMR

TFQMR, which stands for Two-Fractional Quasi-Minimum Residual method, is a numerical algorithm used for solving large sparse systems of linear equations. It is an iterative method, meaning it generates a sequence of approximate solutions that ideally converge to the true solution. TFQMR is particularly well-suited for solving non-symmetric linear systems, which often arise in various scientific and engineering applications like computational fluid dynamics and structural analysis.

The algorithm is based on the minimum residual principle, aiming to minimize the norm of the residual vector, which is the difference between the right-hand side vector and the result of applying the matrix to the current approximate solution. TFQMR achieves this by constructing a sequence of Krylov subspaces, which are spanned by vectors derived from the matrix and the initial residual. A key feature of TFQMR is its use of two fractional steps to improve stability and convergence compared to some earlier methods.

While TFQMR can be effective, its performance can depend on the properties of the coefficient matrix. For certain matrices, other iterative solvers like GMRES (Generalized Minimal Residual method) or BiCGSTAB (Biconjugate Gradient Stabilized method) might offer better convergence rates or stability. Nevertheless, TFQMR remains a valuable tool in the arsenal of numerical analysts for tackling challenging linear systems.