GMRES

In mathematics, the generalized minimal residual method (usually abbreviated GMRES) is an iterative method for the numerical solution of a nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual. The Arnoldi iteration is used to find this vector.

The GMRES method was developed by Yousef Saad and Martin H. Schultz in 1986. GMRES is a generalization of the MINRES method developed by Chris Paige and Michael Saunders in 1975. GMRES also is a special case of the DIIS method developed by Peter Pulay in 1980. DIIS is also applicable to non-linear systems.

Denote the Euclidean norm of any vector v by ${\displaystyle \|v\|}$. Denote the (square) system of linear equations to be solved by

The matrix A is assumed to be invertible of size m-by-m. Furthermore, it is assumed that b is normalized, i.e., that ${\displaystyle \|b\|=1}$.

...
Wikipedia