Conditioning Theory for <inline-formula><math display="inline"><semantics><mi mathvariant="script">ML</mi></semantics></math></inline-formula>-Weighted Pseudoinverse and <inline-formula><math display="inline"><semantics><mi mathvariant="script">ML</mi></semantics></math></inline-formula>-Weighted Least Squares Problem

Abstract

The conditioning theory of the ML-weighted least squares and ML-weighted pseudoinverse problems is explored in this article. We begin by introducing three types of condition numbers for the ML-weighted pseudoinverse problem: normwise, mixed, and componentwise, along with their explicit expressions. Utilizing the derivative of the ML-weighted pseudoinverse problem, we then provide explicit condition number expressions for the solution of the ML-weighted least squares problem. To ensure reliable estimation of these condition numbers, we employ the small-sample statistical condition estimation method for all three algorithms. The article concludes with numerical examples that highlight the results obtained.

Keywords

• ℳℒ-weighted pseudoinverse
• ℳℒ-weighted least squares problem
• normwise condition number
• mixed and componentwise condition numbers
• small-sample statistical condition estimation