Volume 22, Issue 3
Relationship Between the Stiffly Weighted Pseudoinverse and Multi-Level Constrained Rseudoinverse
DOI:

J. Comp. Math., 22 (2004), pp. 427-436

Published online: 2004-06

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• Abstract

It is known that for a given matrix A of rank r, and a set D of positive diagonal matrices, $\sup_{W\in D}||(W^{\frac{1}{2}})^{\dag}W^{\frac{1}{2}}||_2=(\min_i \sigma_{\dag}(A^{(i)})^{-1}$,, in which $(A^{(i)})^{-1}$is a submatrix of A formed with r (rank(A)) rows of A, such that $(A^{(i)})^{-1}$ has full row rank r. In many practical applications this value is too large to be used. In this paper we consider the case that both A and W(G$\in D$) are fixed with W severely stiff. We show that in this case the weighted pseudoinverse $W^{\frac{1}{2}})^{\dag}W^{\frac{1}{2}}$is close to a multi- level constrained weighted pseudoinverse therefore $||(W^{\frac{1}{2}})^{\dag}W^{\frac{1}{2}}||_2$ is uniformly bounded. We also prove that in this case the solution set the stiffly weighted least squares problem is close to that of corresponding multi-level constrained least squares problem.

• Keywords

Weighted least squares Stiff Multi-Level constrained pseudoinverse

@Article{JCM-22-427, author = {}, title = {Relationship Between the Stiffly Weighted Pseudoinverse and Multi-Level Constrained Rseudoinverse}, journal = {Journal of Computational Mathematics}, year = {2004}, volume = {22}, number = {3}, pages = {427--436}, abstract = { It is known that for a given matrix A of rank r, and a set D of positive diagonal matrices, $\sup_{W\in D}||(W^{\frac{1}{2}})^{\dag}W^{\frac{1}{2}}||_2=(\min_i \sigma_{\dag}(A^{(i)})^{-1}$,, in which $(A^{(i)})^{-1}$is a submatrix of A formed with r (rank(A)) rows of A, such that $(A^{(i)})^{-1}$ has full row rank r. In many practical applications this value is too large to be used. In this paper we consider the case that both A and W(G$\in D$) are fixed with W severely stiff. We show that in this case the weighted pseudoinverse $W^{\frac{1}{2}})^{\dag}W^{\frac{1}{2}}$is close to a multi- level constrained weighted pseudoinverse therefore $||(W^{\frac{1}{2}})^{\dag}W^{\frac{1}{2}}||_2$ is uniformly bounded. We also prove that in this case the solution set the stiffly weighted least squares problem is close to that of corresponding multi-level constrained least squares problem. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/10316.html} }
TY - JOUR T1 - Relationship Between the Stiffly Weighted Pseudoinverse and Multi-Level Constrained Rseudoinverse JO - Journal of Computational Mathematics VL - 3 SP - 427 EP - 436 PY - 2004 DA - 2004/06 SN - 22 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/10316.html KW - Weighted least squares KW - Stiff KW - Multi-Level constrained pseudoinverse AB - It is known that for a given matrix A of rank r, and a set D of positive diagonal matrices, $\sup_{W\in D}||(W^{\frac{1}{2}})^{\dag}W^{\frac{1}{2}}||_2=(\min_i \sigma_{\dag}(A^{(i)})^{-1}$,, in which $(A^{(i)})^{-1}$is a submatrix of A formed with r (rank(A)) rows of A, such that $(A^{(i)})^{-1}$ has full row rank r. In many practical applications this value is too large to be used. In this paper we consider the case that both A and W(G$\in D$) are fixed with W severely stiff. We show that in this case the weighted pseudoinverse $W^{\frac{1}{2}})^{\dag}W^{\frac{1}{2}}$is close to a multi- level constrained weighted pseudoinverse therefore $||(W^{\frac{1}{2}})^{\dag}W^{\frac{1}{2}}||_2$ is uniformly bounded. We also prove that in this case the solution set the stiffly weighted least squares problem is close to that of corresponding multi-level constrained least squares problem.