A note on condition numbers for generalized inverse A T , S ( 2 ) and constrained linear systems
In this paper, we use the Schur decomposition to measure the sensitivity of the general inverse A T , S ( 2 ) and the constrained singular linear system Ax = b with regard to 2-norm and F-norm, rather than PQ-norm in [12], where P and Q are nonsingular matrices. The explicit forms of the condition n...
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| Published in | Applied mathematics and computation Vol. 217; no. 7; pp. 3199 - 3206 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Amsterdam
Elsevier Inc
2010
Elsevier |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0096-3003 1873-5649 |
| DOI | 10.1016/j.amc.2010.08.052 |
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| Abstract | In this paper, we use the Schur decomposition to measure the sensitivity of the general inverse
A
T
,
S
(
2
)
and the constrained singular linear system
Ax
=
b with regard to 2-norm and
F-norm, rather than
PQ-norm in
[12], where
P and
Q are nonsingular matrices. The explicit forms of the condition numbers approximate the sensitivity of
A
T
,
S
(
2
)
and the constrained singular linear system pretty well. Furthermore, since Schur decomposition is well-posed, the evaluation of the sensitivity can be numerically easy and stable. |
|---|---|
| AbstractList | In this paper, we use the Schur decomposition to measure the sensitivity of the general inverse
A
T
,
S
(
2
)
and the constrained singular linear system
Ax
=
b with regard to 2-norm and
F-norm, rather than
PQ-norm in
[12], where
P and
Q are nonsingular matrices. The explicit forms of the condition numbers approximate the sensitivity of
A
T
,
S
(
2
)
and the constrained singular linear system pretty well. Furthermore, since Schur decomposition is well-posed, the evaluation of the sensitivity can be numerically easy and stable. |
| Author | Liu, Gang Chen, Juping Lu, Shengqi Xu, Wei |
| Author_xml | – sequence: 1 givenname: Gang surname: Liu fullname: Liu, Gang email: liu_gang@fudan.edu.cn organization: Software School, Fudan University, Shanghai 200433, China – sequence: 2 givenname: Shengqi surname: Lu fullname: Lu, Shengqi email: shengqilu@fudan.edu.cn organization: Software School, Fudan University, Shanghai 200433, China – sequence: 3 givenname: Juping surname: Chen fullname: Chen, Juping organization: School of Mathematics Science, Fudan University, Shanghai 200433, China – sequence: 4 givenname: Wei surname: Xu fullname: Xu, Wei email: wdxu@tongji.edu.cn, xu_wei8@hotmail.com organization: Department of Mathematics, Tongji University, Shanghai, China |
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| Keywords | Condition number Schur decomposition Index Perturbation Generalized inverse A T , S Numerical linear algebra Generalized inverse A Decomposition method Iterative method Direct method Numerical analysis Linear system Matrix inversion Applied mathematics |
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| References | Campbell, Meyer (bib2) 1991 Wei, Zhang (bib13) 2003; 142 Golub, Van Loan (bib4) 1996 Djordjević, Stanimirovic, Wei (bib3) 2004; 104 Rice (bib7) 1966; 3 Wei, Diao, Qin (bib14) 2006; 174 Wei, Djordjević (bib11) 2003; 142 Wang, Wei, Qiao (bib9) 2004 Xu, Wei, Qiao (bib16) 2006; 46 Ben-Israel, Greville (bib1) 2003 Higham (bib5) 1995; 214 Wei, Zhang (bib12) 2003; 157 Wei, Xu (bib15) 2003; 142 Wei (bib10) 1998; 280 Sheng, Chen, Gong (bib8) 2008; 3 Mosic, Djordjevic (bib6) 2009; 215 |
| References_xml | – volume: 157 start-page: 57 year: 2003 end-page: 72 ident: bib12 article-title: Condition number related with generalized inverse publication-title: J. Comput. Math. – volume: 142 start-page: 79 year: 2003 end-page: 97 ident: bib15 article-title: Condition number of Bott–Duffin inverse and their condition numbers publication-title: Appl. Math. Comput. – volume: 104 start-page: 1 year: 2004 end-page: 26 ident: bib3 article-title: The representation and approximation of outer generalized inverses publication-title: Acta. Math. Hungar. – year: 2003 ident: bib1 article-title: Generalized Inverses: Theory and Applications – volume: 46 start-page: 203 year: 2006 end-page: 225 ident: bib16 article-title: Condition Numbers for Structured Least Squares publication-title: BIT – volume: 215 start-page: 2826 year: 2009 end-page: 2834 ident: bib6 article-title: Condition number related to the outer inverse of a complex matrix publication-title: Appl. Math. Comput. – volume: 142 start-page: 189 year: 2003 end-page: 194 ident: bib11 article-title: On integral representation of the generalized inverse publication-title: Appl. Math. Comput. – volume: 142 start-page: 63 year: 2003 end-page: 78 ident: bib13 article-title: Perturbation bounds for the generalized inverse publication-title: Appl. Math. Comput. – year: 1991 ident: bib2 article-title: Generalized Inverses of Linear transformations – volume: 3 start-page: 287 year: 1966 end-page: 310 ident: bib7 article-title: A theory of condition publication-title: SIAM J. Numer. Anal. – year: 1996 ident: bib4 article-title: Matrix Computations – volume: 174 start-page: 588 year: 2006 end-page: 612 ident: bib14 article-title: Condition numbers for the outer inverse and constrained singular linear system publication-title: Appl. Math. Comput. – year: 2004 ident: bib9 article-title: Generalized Inverses: Theory and Computations – volume: 280 start-page: 87 year: 1998 end-page: 96 ident: bib10 article-title: A characterization and representation of the generalized inverse publication-title: Linear Algebr. Appl. – volume: 214 start-page: 193 year: 1995 end-page: 213 ident: bib5 article-title: Condition numbers and their condition numbers publication-title: Linear Algebr. Appl. – volume: 3 start-page: 248 year: 2008 end-page: 257 ident: bib8 article-title: The representation and computation of generalized inverse publication-title: J. Comput. Appl. Math. |
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| Snippet | In this paper, we use the Schur decomposition to measure the sensitivity of the general inverse
A
T
,
S
(
2
)
and the constrained singular linear system
Ax
=
b... |
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| StartPage | 3199 |
| SubjectTerms | Algebra Algebraic geometry Calculus of variations and optimal control Condition number Exact sciences and technology Generalized inverse [formula omitted] Index Mathematical analysis Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Perturbation Schur decomposition Sciences and techniques of general use |
| Title | A note on condition numbers for generalized inverse A T , S ( 2 ) and constrained linear systems |
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