Numerical Implementations of the Generalized Minimal Residual Method (GMRES)

Sammanfattning: The generalized minimal residual method (GMRES) is an iterative method used to find numerical solutions to non-symmetric linear systems of equations. The method relies on constructing an orthonormal basis of the Krylov space and is thus vulnerable to an imperfect basis caused by computational errors. There have been attempts to address this issue by devising variations of the method that are less sensitive to poorly conditioned problems. The GMRES algorithm is typically used when the dimensions of the problem are very large, thus it is of interest to investigate ways in which the computational and memory cost of running it can be reduced. One method for doing so involves replacing the matrix-vector multiplications with an approximating function. This work compares variations of the GMRES algorithm against each other by using it as a solver in simulations mirroring real-world applications.