| Abstract: | Most current prevalent iterative methods can be classified into the socalled
extended Krylov subspace methods, a class of iterative methods which do not
fall into this category are also proposed in this paper. Comparing with traditional
Krylov subspace methods which always depend on the matrix-vector multiplication
with a fixed matrix, the newly introduced methods (the so-called (progressively) accumulated
projection methods, or AP (PAP) for short) use a projection matrix which
varies in every iteration to form a subspace from which an approximate solution is
sought. More importantly, an accelerative approach (called APAP) is introduced to
improve the convergence of PAP method. Numerical experiments demonstrate some
surprisingly improved convergence behaviors. Comparisons between benchmark extended
Krylov subspace methods (Block Jacobi and GMRES) are made and one can
also see remarkable advantage of APAP in some examples. APAP is also used to
solve systems with extremely ill-conditioned coefficient matrix (the Hilbert matrix)
and numerical experiments shows that it can bring very satisfactory results even
when the size of system is up to a few thousands. |
