Iterative methods for the solution of large systems of linear equations (Q1114328)

The author presents projection-type iterative methods for solving large- scale systems of linear equations (1) \(Ax=b\) and \(A^ Ty=b\) with a diagonally dominant non-singular matrix A of order N. The key step of the algorithms considered in the paper consists in replacing (1) by the reduced systems \((2)\quad V^ T_ mAU_ m\tilde x_ m=V^ T_ mb\) and \(U^ T_ mA^ TV_ m\tilde y_ m=U^ T_ mb,\) where \(U_ m=(u_ 1,u_ 2,...,u_ m)\) and \(V_ m=(v_ 1,v_ 2,...,v_ m)\) are \(N\times m\) matrices satisfying the bi-orthogonality condition \(V^ T_ mU_ m=I_ m\). If \(m=N\) then (2) coincides with (1). However, even if m is much smaller than N, one can expect good approximate solutions \(x_ m=U_ m\tilde x_ m\) and \(y_ m=V_ m\tilde y_ m\) to (1) provided that the subspaces spanned by \((u_ i)_{i=1,...,m}\) and \((v_ i)_{i=1,...,m}\) are appropriately chosen. The author proposes a method to extend successively the subspaces onto which (1) are projected. If m becomes large then the process can be handled as an update procedure. Numerical experiments carried out for large systems arising in molecular calculations are reported.