The idea of the hybrid procedure is as follows. Let \(x'\), \(x''\) be two approximations of the solution of \(Ax = b\). Define \(r' = Ax' - b\), \(r'' = Ax'' - b\). For \(\alpha \in \mathbb{R}\) define \(y = \alpha x' + (1 - \alpha) x''\) and let \(r = Ay - b\). It is easy to see that \(r = \alpha r' + (1 - \alpha) r''\). Therefore \(\alpha\) may be chosen in a straightforward way so that \(r\) has minimal norm.
Typically \(x'\), \(x''\) are obtained from two iterative methods (sometimes they are two iterates of the same method). The computed \(y\) can be used to start the next iteration. The authors consider many applications and examples, in particular Lanczos, Gauss-Seidel, Jacobi, and Gastinel methods.