Conditioning Theory for Generalized Inverse <inline-formula><math display="inline"><semantics><msubsup><mi>C</mi><mi>A</mi><mo>‡</mo></msubsup></semantics></math></inline-formula> and Their Estimations

Abstract

The conditioning theory of the generalized inverse CA‡ is considered in this article. First, we introduce three kinds of condition numbers for the generalized inverse CA‡, i.e., normwise, mixed and componentwise ones, and present their explicit expressions. Then, using the intermediate result, which is the derivative of CA‡, 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.

Keywords

• C A ‡ %22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> generalized inverse <named-content content-type="equation"><inline-formula> <mml:math id="mm631"> <mml:semantics> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>A</mml:mi> <mml:mo>‡</mml:mo> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>
• normwise condition number
• mixed and componentwise condition numbers
• EILS problem
• probabilistic spectral norm estimator
• small-sample statistical condition estimation