On a Class of High Order Methods for Inverting Matrices

Stickel, Eberhard

doi:10.1002/zamm.19870670712

Cited by 23 publications

(41 citation statements)

References 8 publications

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“…But this evaluation can substantially influence the efficiency index of the method. For some examples see Herzberger [4] or Stickel [12]. We should mention here that the expression for ~p (A, X) in (5) is still defined for p = 1 as the identity mapping.…”

Section: Error-bounds For Hyperpower Methodsmentioning

confidence: 98%

Efficient algorithms for the inclusion of the inverse matrix using error-bounds for hyperpower methods

Herzberger

1991

Computing

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--ZusammenfassungEfficient Algorithms for the Inclusion of the Inverse Matrix Using Error-Bounds for Hyperpower Methods.By exploiting generalized error-bounds for the well-known hyperpower methods for approximating the inverse of a matrix we derive inclusion methods for the inverse matrix. These methods make use of interval operations in order to give guaranteed inclusions whenever the convergence of the applied hyperpower method can be shown. The efficiency index of some of the new methods is greater than that of the optimal methods in [2] or [5]. A numerical example is given.

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Section: Error-bounds For Hyperpower Methodsmentioning

confidence: 98%

Efficient algorithms for the inclusion of the inverse matrix using error-bounds for hyperpower methods

Herzberger

1991

Computing

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“…is proposed by Amat et al, 19 which has at least third order of convergence. 21 Based on Neumann's series, methods (7)-(9) can be generalized to a larger class of high-order methods called the hyperpower method, or the pth-order method [15][16][17][18] for p ≥ 2, performing p matrix by matrix multiplications for each iteration as follows:…”

Section: Survey Of Iterative Methods For Matrix Inversionmentioning

confidence: 99%

“…is valid, then for the sequence {V m } m=∞ m=1 obtained by using Class 1 methods (17) or Class 2 methods (18),…”

Section: Two Classes Of High-order Iterative Methods For Matrix Invermentioning

confidence: 99%

“…Proof. We give the proof for the methods belonging to Class 1 (17). The order of convergence for these methods is q k = 3 * 2 k + 1, for an integer k ≥ 1.…”

Section: Two Classes Of High-order Iterative Methods For Matrix Invermentioning

confidence: 99%

“…11 In recent years, there has been great interest in algebraic methods of rapid convergence rate to compute matrix inversion, where algorithms are defined by matrix multiplications and additions. Some of the approaches that have been published to propose and investigate iterative methods for computing the generalized inverses and approximate inverse of a square matrix are by Schulz, 12 Ben-Israel et al, 13 Soderstorm et al, 14 Householder, 15 Isaacson et al, 16 Stickel, 17 Herzberger, 18 Amat et al, 19 Li et al, 20,21 Krishnamurthy et al, 22 Soleymani,23,24 and Haghani et al 25 (see also references therein).…”

Section: Introductionmentioning

confidence: 99%

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

Buranay

Subasi

Iyikal

2017

Numerical Linear Algebra App

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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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On finding strong approximate inverses for tensors

Dehdezi

Karimi

2022

Numerical Linear Algebra App

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This article investigates a fast and highly efficient algorithm to find the strong approximation inverse of an invertible tensor. The convergence analysis shows that the proposed method is of ten order of convergence using only six tensor-tensor multiplications per iteration. Also, we obtain a bound for the perturbation error in each iteration. We show that the proposed algorithm can be used for finding the Moore-Penrose and outer inverses of tensors. We obtain the relationship between the singular values of an arbitrary tensor  and eigenvalues of the  * ⋆ N . We give the computational complexity of our algorithm and prove the theoretical aspects of the article. The generalized Moore-Penrose inverse of tensors is defined. As an application, we use the iteration obtained by the algorithm as preconditioning of the Krylov subspace methods to solve the multilinear system  ⋆ N  = . Several numerical experiments are proposed to show the effectiveness and accuracy of the method. Finally, we give some concluding remarks.

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On a Class of High Order Methods for Inverting Matrices

Cited by 23 publications

References 8 publications

Efficient algorithms for the inclusion of the inverse matrix using error-bounds for hyperpower methods

Efficient algorithms for the inclusion of the inverse matrix using error-bounds for hyperpower methods

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

On finding strong approximate inverses for tensors

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