The Restrictively Preconditioned Conjugate Gradient Methods on Normal Residual for Block Two-by-Two Linear Systems

Abstract

The $restrictively$ $preconditioned$ $conjugate$ $gradient$ (RPCG) method is further developed to solve large sparse system of linear equations of a block two-by-two structure. The basic idea of this new approach is that we apply the RPCG method to the normal-residual equation of the block two-by-two linear system and construct each required approximate matrix by making use of the incomplete orthogonal factorization of the involved matrix blocks. Numerical experiments show that the new method, called the $restrictively$ $preconditioned$ $conjugate$ $gradient$ $on$ $normal$ $residual$ (RPCGNR), is more robust and effective than either the known RPCG method or the standard $conjugate$ $gradient$ $on$ $normal$ $residual$ (CGNR) method when being used for solving the large sparse saddle point problems.