  It is shown in this paper that, almost all current prevalent iterative
\mbox{methods} for solving linear system of equations can be classified as what
we called extended Krylov subspace methods. In this paper a new type of
iterative methods are introduced which do not depend on any Krylov subspaces.
This type of methods are based on the so-called accumulated projection
technique proposed by authors. It overcomes some shortcomings of classical
Row-Projection technique and takes full advantages of the linear system.
Comparing with traditional Krylov subspace methods which always depend on the
matrix-vector multiplication with some fixed matrix, the newly introduced
method (SAP) uses different projection matrices which differ in each step in
the iteration process to form an approximate solution. More importantly some
particular accelerative schemes (named as MSAP1 and MSAP2) are introduced to
improve the convergence of the SAP method. Numerical experiments show some
surprisingly improved convergence behavior; some superior experimental behavior
of MSAP methods over GMRES and block-Jacobi are demonstrated in some
situations.
