A doubly optimized solution of linear equations system expressed in an affine Krylov subspace

A mathematical procedure for finding a closed-form double optimal solution   (DOS) of an nn-dimensional linear equations system Bx=b is developed, which expresses the solution in an mm-dimensional affine Krylov subspace with undetermined coefficients, and two optimization techniques are used to determine these coefficients in closed-form. To find the DOS, it is very time saving without the need of any iteration; in practice, we only need to invert an m×mm×m matrix one time, where m≪nm≪n. The DOS is not exactly equal to the exact solution, but it can provide an acceptable approximate solution of linear equations system, whose applicable range is identified. Some properties are analyzed that the DOS is an exact solution of a projected linear system of Bx=b onto the affine Krylov subspace. The tests for large scale problems demonstrate the efficiency of DOS on non-sparse linear systems.