A double optimal iterative algorithm in an affine Krylov subspace for solving nonlinear algebraic equations. (November 2015)

• Record Type: Journal Article
• Authors: Liu, Chein-Shan
• Is Part Of: Computers & mathematics with applications. Volume 70:issue 10(2015)
• Page Start: 2376
• Page End: 2400
• Publication Date: 2015-11
• Subjects: Nonlinear algebraic equations -- Differentiable invariant manifold -- Descent vector -- Newton equation -- Double optimal iterative algorithm (DOIA) -- Affine Krylov subspace; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2015.09.002
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: For an n -dimensional linear equations system, Liu (2014) has derived a double optimal solution in an affine m -dimensional Krylov subspace with m ≪ n . An iterative algorithm, based on the double optimal solution of the Newton equation B u = F in x ̇ = λ u, is proposed to solve a system of nonlinear algebraic equations F ( x ) = 0 with dimension n . 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 ‖ B k u k ‖ 2 at each iteration, and very time saving by merely inverting an m × 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 × 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.