Abstract
High order iterative methods with a recurrence formula for approximate matrix inversion are proposed such that the matrix multiplications and additions in the calculation of matrix polynomials for the hyperpower methods of orders of convergence p=4k+3, where k≥1 is integer, are reduced through factorizations and nested loops in which the iterations are defined using a recurrence formula. Therefore, the computational cost is lowered from κ=4k+3 to κ=k+4 matrix multiplications per step. An algorithm is proposed to obtain regularized solution of ill-posed discrete problems with noisy data by constructing approximate Schur-Block Incomplete LU (Schur-BILU) preconditioner and by preconditioning the one step stationary iterative method. From the proposed methods of approximate matrix inversion, the methods of orders p=7,11,15,19 are applied for approximating the Schur complement matrices. This algorithm is applied to solve two problems of Fredholm integral equation of first kind. The first example is the harmonic continuation problem and the second example is Phillip’s problem. Furthermore, experimental study on some nonsymmetric linear systems of coefficient matrices with strong indefinite symmetric components from Harwell-Boeing collection is also given. Numerical analysis for the regularized solutions of the considered problems is given and numerical comparisons with methods from the literature are provided through tables and figures.
Highlights
The numerical solution of many scientific and engineering problems requires the solution of large linear systems of equations in the form Ax = b, (1)where x, b ∈ Rn and A ∈ Rn×n is nonsingular and is usually unsymmetric and unstructured matrix
The motivation of this study is firstly to propose high order iterative methods for approximate matrix inversion of a real nonsingular matrix A such that the matrix multiplications and additions in the calculation of matrix polynomials for the hyperpower methods of orders of convergence p = 4k + 3, where k ≥ 1 is integer, are reduced through factorizations and nested loops of which the iterations are defined using a recurrence formula, and secondly to give an algorithm that constructs 2 × 2 block incomplete LU decomposition based on approximate Schur complement for the nonsingular coefficient matrix à ∈ Rn×n of the algebraic linear system of equations arising from the ill-posed discrete problems with noisy data
It is proven that these methods require κ = k + 4, matrix by matrix multiplications per iteration, which are fewer than κ = p = 4k + 3, for the standard hyperpower method of same order
Summary
Where x, b ∈ Rn and A ∈ Rn×n is nonsingular and is usually unsymmetric and unstructured matrix. The motivation of this study is firstly to propose high order iterative methods for approximate matrix inversion of a real nonsingular matrix A such that the matrix multiplications and additions in the calculation of matrix polynomials for the hyperpower methods of orders of convergence p = 4k + 3, where k ≥ 1 is integer, are reduced through factorizations and nested loops of which the iterations are defined using a recurrence formula, and secondly to give an algorithm that constructs 2 × 2 block incomplete LU decomposition based on approximate Schur complement for the nonsingular coefficient matrix à ∈ Rn×n (when n is even) of the algebraic linear system of equations arising from the ill-posed discrete problems with noisy data In this algorithm the methods of orders p = 7, 11, 15, 19 are applied for approximating the Schur complement matrices and the obtained preconditioners are used to precondition the one step stationary iterative method. In last section concluding remarks are given based on theoretical and numerical analysis
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
