FUNCTIONS PRESERVING MATRIX GROUPS AND ITERATIONS FOR THE MATRIX SQUARE ROOT.
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| Title: | FUNCTIONS PRESERVING MATRIX GROUPS AND ITERATIONS FOR THE MATRIX SQUARE ROOT. |
|---|---|
| Authors: | HIGHAM, NICHOLAS J. (AUTHOR), MACKEY, D. STEVEN (AUTHOR), MACKEY, NILOUFER (AUTHOR), TISSEUR, FRANÇOISE (AUTHOR) |
| Source: | SIAM Journal on Matrix Analysis & Applications. 2004, Vol. 26 Issue 3, p849-877. 29p. 3 Charts. |
| Subjects: | Matrix groups, Iterative methods (Mathematics), Square root, Automorphism groups, Bilinear forms, Inner product spaces, Mathematical decomposition |
| Abstract: | For which functions f does A ∈ G ⇒ f(A) ∈ G when G is the matrix automorphism group associated with a bilinear or sesquilinear form? For example, if A is symplectic when is f(A) symplectic? We show that group structure is preserved precisely when f(A-1) = f(A)-1 for bilinear forms and when f(A-*) = f(A)-* for sesquilinear forms. Meromorphic functions that satisfy each of these conditions are characterized. Related to structure preservation is the condition f(Ā) = ..., and analytic functions and rational functions satisfying this condition are also characterized. These results enable us to characterize all meromorphic functions that map every G into itself as the ratio of a polynomial and its “reversal,” up to a monomial factor and conjugation. The principal square root is an important example of a function that preserves every automorphism group G. By exploiting the matrix sign function, a new family of coupled iterations for the matrix square root is derived. Some of these iterations preserve every G; all of them are shown, via a novel Fréchet derivative-based analysis, to be numerically stable. A rewritten form of Newton's method for the square root of A ∈ G is also derived. Unlike the original method, this new form has good numerical stability properties, and we argue that it is the iterative method of choice for computing A1/2 when A ∈ G. Our tools include a formula for the sign of a certain block 2 × 2 matrix, the generalized polar decomposition along with a wide class of iterations for computing it, and a connection between the generalized polar decomposition of I + A and the square root of A ∈ G. [ABSTRACT FROM AUTHOR] |
| Copyright of SIAM Journal on Matrix Analysis & Applications is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Items | – Name: Title Label: Title Group: Ti Data: FUNCTIONS PRESERVING MATRIX GROUPS AND ITERATIONS FOR THE MATRIX SQUARE ROOT. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22HIGHAM%2C+NICHOLAS+J%2E%22">HIGHAM, NICHOLAS J.</searchLink> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22MACKEY%2C+D%2E+STEVEN%22">MACKEY, D. STEVEN</searchLink> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22MACKEY%2C+NILOUFER%22">MACKEY, NILOUFER</searchLink> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22TISSEUR%2C+FRANÇOISE%22">TISSEUR, FRANÇOISE</searchLink> (AUTHOR) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22SIAM+Journal+on+Matrix+Analysis+%26+Applications%22">SIAM Journal on Matrix Analysis & Applications</searchLink>. 2004, Vol. 26 Issue 3, p849-877. 29p. 3 Charts. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Matrix+groups%22">Matrix groups</searchLink><br /><searchLink fieldCode="DE" term="%22Iterative+methods+%28Mathematics%29%22">Iterative methods (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Square+root%22">Square root</searchLink><br /><searchLink fieldCode="DE" term="%22Automorphism+groups%22">Automorphism groups</searchLink><br /><searchLink fieldCode="DE" term="%22Bilinear+forms%22">Bilinear forms</searchLink><br /><searchLink fieldCode="DE" term="%22Inner+product+spaces%22">Inner product spaces</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+decomposition%22">Mathematical decomposition</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: For which functions f does A ∈ G ⇒ f(A) ∈ G when G is the matrix automorphism group associated with a bilinear or sesquilinear form? For example, if A is symplectic when is f(A) symplectic? We show that group structure is preserved precisely when f(A-1) = f(A)-1 for bilinear forms and when f(A-*) = f(A)-* for sesquilinear forms. Meromorphic functions that satisfy each of these conditions are characterized. Related to structure preservation is the condition f(Ā) = ..., and analytic functions and rational functions satisfying this condition are also characterized. These results enable us to characterize all meromorphic functions that map every G into itself as the ratio of a polynomial and its “reversal,” up to a monomial factor and conjugation. The principal square root is an important example of a function that preserves every automorphism group G. By exploiting the matrix sign function, a new family of coupled iterations for the matrix square root is derived. Some of these iterations preserve every G; all of them are shown, via a novel Fréchet derivative-based analysis, to be numerically stable. A rewritten form of Newton's method for the square root of A ∈ G is also derived. Unlike the original method, this new form has good numerical stability properties, and we argue that it is the iterative method of choice for computing A1/2 when A ∈ G. Our tools include a formula for the sign of a certain block 2 × 2 matrix, the generalized polar decomposition along with a wide class of iterations for computing it, and a connection between the generalized polar decomposition of I + A and the square root of A ∈ G. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of SIAM Journal on Matrix Analysis & Applications is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1137/S0895479804442218 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 29 StartPage: 849 Subjects: – SubjectFull: Matrix groups Type: general – SubjectFull: Iterative methods (Mathematics) Type: general – SubjectFull: Square root Type: general – SubjectFull: Automorphism groups Type: general – SubjectFull: Bilinear forms Type: general – SubjectFull: Inner product spaces Type: general – SubjectFull: Mathematical decomposition Type: general Titles: – TitleFull: FUNCTIONS PRESERVING MATRIX GROUPS AND ITERATIONS FOR THE MATRIX SQUARE ROOT. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: HIGHAM, NICHOLAS J. – PersonEntity: Name: NameFull: MACKEY, D. STEVEN – PersonEntity: Name: NameFull: MACKEY, NILOUFER – PersonEntity: Name: NameFull: TISSEUR, FRANÇOISE IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 10 Text: 2004 Type: published Y: 2004 Identifiers: – Type: issn-print Value: 08954798 Numbering: – Type: volume Value: 26 – Type: issue Value: 3 Titles: – TitleFull: SIAM Journal on Matrix Analysis & Applications Type: main |
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