Restrictively preconditioned conjugate gradient methods for systems of linear equations

A restrictively preconditioned conjugate gradient method is presented for solving a large sparse system of linear equations. This new method originates from the classical conjugate gradient method and its restrictively preconditioned variant, and covers many standard Krylov subspace iteration methods such as the conjugate gradient, conjugate residual, CGNR, CGNE and the corresponding preconditioned variants. In particular, the new method is applied to solve the linear systems of block two‐by‐two structures, and consequently, special but quite efficient preconditioned conjugate gradient‐like methods are obtained for these problems. Numerical computations show that the new methods are more effective than some standard preconditioned Krylov subspace methods like the preconditioned conjugate gradient and the preconditioned MINRES.