Accurate estimates for the speed of convergence of \(s\)-step iterative methods of variational type in a Hilbert space (Q1569392)

The authors consider an \(s\)-step steepest descent method for solving a linear operator equation \(Au=f\) in a Hilbert space. The successive approximations of this method are constructed as \[ u_{k+1}=u_{k}+\gamma_{1}w_{k}+\dots+\gamma_{s}A^{s-1}w_{k},\;k=0,1,\dots, \] where \(u_{0}\) is an arbitrary initial approximation and \(w_{k}=Au_{k}-f\) is the residual; the iteration parameters \(\gamma_{i}\), are chosen such that the quadratic functional \[ F(u_{k+1})=(Au_{k+1},u_{k+1})-2(u_{k+1},f) \] is minimized. It is obtained the accurate estimates for the convergence speed of \(s\)-step iterative methods of variational type for linear operator equations in a Hilbert space, are obtained.