2 research outputs found
Structured condition numbers and small sample condition estimation of symmetric algebraic Riccati equations
This paper is devoted to a structured perturbation analysis of the symmetric
algebraic Riccati equations by exploiting the symmetry structure. Based on the
analysis, the upper bounds for the structured normwise, mixed and componentwise
condition numbers are derived. Due to the exploitation of the symmetry
structure, our results are improvements of the previous work on the
perturbation analysis and condition numbers of the symmetric algebraic Riccati
equations. Our preliminary numerical experiments demonstrate that our condition
numbers provide accurate estimates for the change in the solution caused by the
perturbations on the data. Moreover, by applying the small sample condition
estimation method, we propose a statistical algorithm for practically
estimating the condition numbers of the symmetric algebraic Riccati equations
Structured Condition Numbers of Structured Tikhonov Regularization Problem and their Estimations
Both structured componentwise and structured normwise perturbation analysis
of the Tikhonov regularization are presented. The structured matrices under
consideration include: Toeplitz, Hankel, Vandermonde, and Cauchy matrices.
Structured normwise, mixed and componentwise condition numbers for the Tikhonov
regularization are introduced and their explicit expressions are derived. For
the general linear structure, we prove the structured condition numbers are
smaller than their corresponding unstructured counterparts based on the derived
expressions. By means of the power method and small sample condition
estimation, the fast condition estimation algorithms are proposed. Our
estimation methods can be integrated into Tikhonov regularization algorithms
that use the generalized singular value decomposition (GSVD). The structured
condition numbers and perturbation bounds are tested on some numerical examples
and compared with their unstructured counterparts. Our numerical examples
demonstrate that the structured mixed condition numbers give sharper
perturbation bounds than existing ones, and the proposed condition estimation
algorithms are reliable