On the two classes of high‐order convergent methods of approximate inverse preconditioners for solving linear systems

Abstract

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

High order iterative methods for matrix inversion and regularized solution of fredholm integral equation of first kind with noisy data

“…When the two factorizations of p-th order hyperpower method have same asymptotic convergence factor ACF [10] values then it is necessary to use an other quantity which measures the efficiency of the hyperpower method both with respect to matrix by matrix multiplications (MMs) and matrix by matrix additions (Mas). Let κ be the number of MMs and p be the order of the hyperpower method.…”

“…We denote the methods (10) Table 1. ACV for the proposed family of methods (10) and for the methods from the literature [10], [11] and [12] of the orders p=7, 11,15,19.…”

“…The p-th order hyperpower methods 2 p  are a family of well known methods used to approximate the inverse of a nonsingular matrix [9]. Recent studies on iterative methods for computing approximate inverse are but not limited to [10]- [12]. The factorization of hyperpower method of order 7 for computing generalized inverse (2) ,…”

“… and a predescribed accuracy 0   , the approximate inverse m V of A in (12) is found by using the methods (10) with m iterations to reach the accuracy m  is the regularized solutions. This algorithm is given analogous to Algorithm 1 in[10].…”

Approximate Schur‐Block ILU Preconditioners for Regularized Solution of Discrete Ill‐Posed Problems

“…"For matrices with irregular structure, it is unclear how to choose the nonzero pattern of the FSAI preconditioner at least in a nearly optimal way and the serial costs of constructing this preconditioner can be very high if its nonzero pattern is sufficiently dense" [9] and the authors proposed two new approaches to rise the efficiency. Most recently, in [10] two classes of iterative methods for matrix inversion are proposed and these methods are used as approximate inverse preconditioners to precondition BICG method for solving linear systems (1) with the same coefficient matrix and multiple right sides.…”