A double optimal iterative algorithm in an affine Krylov subspace for solving nonlinear algebraic equations

For an nn-dimensional linear equations system, Liu (2014) has derived a double optimal solution in an affine mm-dimensional Krylov subspace with m≪nm≪n. An iterative algorithm, based on the double optimal solution of the Newton equation Bu=F in ẋ=λu, is proposed to solve a system of nonlinear algebraic equations F(x)=0 with dimension nn. By optimizing two merit functions, u can be explicitly solved in the affine Krylov subspace. The resulting double optimal iterative algorithm   (DOIA) is proven to be absolutely convergent with the square residual norm ‖F‖2 being reduced by ‖Bkuk‖2 at each iteration, and very time saving by merely inverting an m×mm×m positive definite matrix one time at each iterative step. We can prove that such an algorithm leads to the largest convergence rate without needing to invert the n×nn×n matrix B. Some numerical examples are used to evaluate the performance of the DOIA, where very fast convergence rates and saving the CPU time to find the solutions are observed.