scholarly journals A Seventh-Order Scheme for Computing the Generalized Drazin Inverse

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 622 ◽  
Author(s):  
Dilan Ahmed ◽  
Mudhafar Hama ◽  
Karwan Hama Faraj Jwamer ◽  
Stanford Shateyi
Keyword(s):  
Iterative Method ◽  
Rate Of Convergence ◽  
Generalized Inverses ◽  
Drazin Inverse ◽  
Computational Tool ◽  
Initial Matrix ◽  
Matrix Products ◽  
Seventh Order ◽  
Order Scheme ◽  
Generalized Drazin Inverse

One of the most important generalized inverses is the Drazin inverse, which is defined for square matrices having an index. The objective of this work is to investigate and present a computational tool in the form of an iterative method for computing this task. This scheme reaches the seventh rate of convergence as long as a suitable initial matrix is chosen and by employing only five matrix products per cycle. After some analytical discussions, several tests are provided to show the efficiency of the presented formulation.

scholarly journals New extensions of Cline’s formula for generalized inverses

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 1973-1980 ◽  
Author(s):  
Qingping Zeng ◽  
Zhenying Wu ◽  
Yongxian Wen
Keyword(s):  
Banach Space ◽  
Generalized Inverses ◽  
Drazin Inverse ◽  
Space Operator ◽  
Cline’S Formula ◽  
Generalized Drazin Inverse

In this paper, Cline?s formula for the well-known generalized inverses such as Drazin inverse, pseudo Drazin inverse and generalized Drazin inverse is extended to the case when ( acd = dbd dba = aca. Also, applications are given to some interesting Banach space operator properties like algebraic, meromorphic, polaroidness and B-Fredholmness.

Jacobson’s Lemma via Gröbner-Shirshov Bases

2017 ◽  
Vol 24 (02) ◽  
pp. 309-314
Author(s):  
Xiangui Zhao
Keyword(s):  
Generalized Inverses ◽  
Drazin Inverse ◽  
Generalized Drazin Inverse ◽  
Jacobson’S Lemma

Let R be a ring with identity 1. Jacobson’s lemma states that for any [Formula: see text], if 1− ab is invertible then so is 1 − ba. Jacobson’s lemma has suitable analogues for several types of generalized inverses, e.g., Drazin inverse, generalized Drazin inverse, and inner inverse. In this note we give a constructive way via Gröbner-Shirshov basis theory to obtain the inverse of 1 − ab in terms of (1 − ba)−1, assuming the latter exists.

scholarly journals A hyperpower iterative method for computing the generalized Drazin inverse of Banach algebra element

Sadhana ◽  
2017 ◽  
Vol 42 (5) ◽  
pp. 625-630 ◽  
Author(s):  
Shwetabh Srivastava ◽  
Dharmendra K Gupta ◽  
Predrag Stanimirović ◽  
Sukhjit Singh ◽  
Falguni Roy
Keyword(s):  
Iterative Method ◽  
Banach Algebra ◽  
Drazin Inverse ◽  
Generalized Drazin Inverse

scholarly journals On the Rate of Convergence of P-Iteration, SP-Iteration, and D-Iteration Methods for Continuous Nondecreasing Functions on Closed Intervals

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Jukkrit Daengsaen ◽  
Anchalee Khemphet
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Rate Of Convergence ◽  
Convergence Theorem ◽  
Computational Cost ◽  
Closed Intervals ◽  
Iteration Methods ◽  
Nondecreasing Functions ◽  
New Iterative Method

We introduce a new iterative method called D-iteration to approximate a fixed point of continuous nondecreasing functions on arbitrary closed intervals. The purpose is to improve the rate of convergence compared to previous work. Specifically, our main result shows that D-iteration converges faster than P-iteration and SP-iteration to the fixed point. Consequently, we have that D-iteration converges faster than the others under the same computational cost. Moreover, the analogue of their convergence theorem holds for D-iteration.

scholarly journals Generalized Inverses Estimations by Means of Iterative Methods with Memory

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 2
Author(s):  
Santiago Artidiello ◽  
Alicia Cordero ◽  
Juan R. Torregrosa ◽  
María P. Vassileva
Keyword(s):  
Iterative Methods ◽  
Generalized Inverses ◽  
Drazin Inverse ◽  
Nonsingular Matrix ◽  
Type Method ◽  
The Matrix ◽  
Nonlinear Matrix ◽  
First Time ◽  
With Memory ◽  
Theoretical Results

A secant-type method is designed for approximating the inverse and some generalized inverses of a complex matrix A. For a nonsingular matrix, the proposed method gives us an approximation of the inverse and, when the matrix is singular, an approximation of the Moore–Penrose inverse and Drazin inverse are obtained. The convergence and the order of convergence is presented in each case. Some numerical tests allowed us to confirm the theoretical results and to compare the performance of our method with other known ones. With these results, the iterative methods with memory appear for the first time for estimating the solution of a nonlinear matrix equations.

scholarly journals Jacobson’s lemma for the generalized Drazin inverse

2012 ◽  
Vol 436 (3) ◽  
pp. 742-746 ◽  
Author(s):  
Guifen Zhuang ◽  
Jianlong Chen ◽  
Jian Cui
Keyword(s):  
Drazin Inverse ◽  
Generalized Drazin Inverse ◽  
Jacobson’S Lemma

scholarly journals Some Results on the Symmetric Representation of the Generalized Drazin Inverse in a Banach Algebra

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 105 ◽  
Author(s):  
Yonghui Qin ◽  
Xiaoji Liu ◽  
Julio Benítez
Keyword(s):  
Banach Algebra ◽  
Drazin Inverse ◽  
Symmetric Representation ◽  
Generalized Drazin Inverse

Based on the conditions a b 2 = 0 and b π ( a b ) ∈ A d , we derive that ( a b ) n , ( b a ) n , and a b + b a are all generalized Drazin invertible in a Banach algebra A , where n ∈ N and a and b are elements of A . By using these results, some results on the symmetry representations for the generalized Drazin inverse of a b + b a are given. We also consider that additive properties for the generalized Drazin inverse of the sum a + b .

Sign in / Sign up

Export Citation Format

Share Document