Two classes of methods for approximate matrix inversion with convergence orders p = 3 * 2 k +1 (Class 1) and p = 5 * 2 k −1 (Class 2), k ≥ 1 an integer, are given based on matrix multiplication and matrix addition. These methods perform less number of matrix multiplications compared to the known hyperpower method or pth-order method for the same orders and can be used to construct approximate inverse preconditioners for solving linear systems. Convergence, error, and stability analyses of the proposed classes of methods are provided. Theoretical results are justified with numerical results obtained by using the proposed methods of orders p = 7, 13 from Class 1 and the methods with orders p = 9, 19 from Class 2 to obtain polynomial preconditioners for preconditioning the biconjugate gradient (BICG) method for solving well-and ill-posed problems. From the literature, methods with orders p = 8, 16 belonging to a family developed by the effective representation of the pth-order method for orders p = 2 k , k is integer k ≥ 1, and other recently given high-order convergent methods of orders p = 6, 7, 8, 12 for approximate matrix inversion are also used to construct polynomial preconditioners for preconditioning the BICG method to solve the considered problems. Numerical comparisons are given to show the applicability, stability, and computational complexity of the proposed methods by paying attention to the asymptotic convergence rates. It is shown that the BICG method converges very quickly when applied to solve the preconditioned system. Therefore, the cost of constructing these preconditioners is amortized if the preconditioner is to be reused over several systems of same coefficient matrix with different right sides. KEYWORDS approximate inverse preconditioners, biconjugate gradient method, error bounds, Fredholm integral equation of the first kind, ill-posed problems, well-posed problems Numer Linear Algebra Appl. 2017;24:e2111.wileyonlinelibrary.com/journal/nla

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We consider the first type boundary value problem of the heat equation in two space dimensions on special polygons with interior angles α j π, j = 1, 2,. .. , M, where α j ∈ { 1 2 , 1 3 , 2 3 }. To approximate the solution we develop two difference problems on hexagonal grids using two layers with 14 points. It is proved that the given implicit schemes in both difference problems are unconditionally stable. It is also shown that the solutions of the constructed Difference Problem 1 and Difference Problem 2 converge to the exact solution on the grids of order O(h 2 + τ 2) and O(h 4 + τ) respectively, where h and √ 3 2 h are the step sizes in space variables x 1 and x 2 respectively and τ is the step size in time. Furthermore, theoretical results are justified by numerical examples on a rectangle, trapezoid and parallelogram.

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.