Conditioning Theory for Generalized Inverse CA‡ and Their Estimations

• Main Authors: Samar, M. (Author), Shakoor, A. (Author), Zhu, X. (Author)
• Format: Article
• Language: English
• Published: MDPI 2023
• Subjects: EILS problem; generalized inverse CA‡; mixed and componentwise condition numbers; normwise condition number; probabilistic spectral norm estimator; small-sample statistical condition estimation
• Online Access: View Fulltext in Publisher; View in Scopus
• ISBN: 22277390 (ISSN)
• DOI: 10.3390/math11092111

The conditioning theory of the generalized inverse (Formula presented.) is considered in this article. First, we introduce three kinds of condition numbers for the generalized inverse (Formula presented.), i.e., normwise, mixed and componentwise ones, and present their explicit expressions. Then, using the intermediate result, which is the derivative of (Formula presented.), we can recover the explicit condition number expressions for the solution of the equality constrained indefinite least squares problem. Furthermore, using the augment system, we investigate the componentwise perturbation analysis of the solution and residual of the equality constrained indefinite least squares problem. To estimate these condition numbers with high reliability, we choose the probabilistic spectral norm estimator to devise the first algorithm and the small-sample statistical condition estimation method for the other two algorithms. In the end, the numerical examples illuminate the obtained results. © 2023 by the authors.