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Relationship Between the Stiffly Weighted Pseudoinverse and Multi-Level Constrained Rseudoinverse

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@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.

Mu-sheng Wei. (1970). Relationship Between the Stiffly Weighted Pseudoinverse and Multi-Level Constrained Rseudoinverse.

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*Journal of Computational Mathematics*.*22*(3). 427-436. doi: