Rescaling the GSVD with application to ill-posed problems
Abstract The generalized singular value decomposition (GSVD) of a pair of matrices expresses each matrix as a product of an orthogonal, a diagonal, and a nonsingular matrix. The nonsingular matrix, which we denote by XT, is the same in both products. Available software for computing the GSVD scales...
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Gespeichert in:
| Autor*in: |
Dykes, L. [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2014 |
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| Schlagwörter: |
Generalized singular value decomposition |
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| Anmerkung: |
© Springer Science+Business Media New York 2014 |
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| Übergeordnetes Werk: |
Enthalten in: Numerical algorithms - Springer US, 1991, 68(2014), 3 vom: 08. Mai, Seite 531-545 |
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| Übergeordnetes Werk: |
volume:68 ; year:2014 ; number:3 ; day:08 ; month:05 ; pages:531-545 |
| Links: |
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| DOI / URN: |
10.1007/s11075-014-9859-3 |
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OLC2051164150 |
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| 520 | |a Abstract The generalized singular value decomposition (GSVD) of a pair of matrices expresses each matrix as a product of an orthogonal, a diagonal, and a nonsingular matrix. The nonsingular matrix, which we denote by XT, is the same in both products. Available software for computing the GSVD scales the diagonal matrices and XT so that the squares of corresponding diagonal entries sum to one. This paper proposes a scaling that seeks to minimize the condition number of XT. The rescaled GSVD gives rise to new truncated GSVD methods, one of which is well suited for the solution of linear discrete ill-posed problems. Numerical examples show this new truncated GSVD method to be competitive with the standard truncated GSVD method as well as with Tikhonov regularization with regard to the quality of the computed approximate solution. | ||
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| 700 | 1 | |a Reichel, L. |4 aut | |
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10.1007/s11075-014-9859-3 doi (DE-627)OLC2051164150 (DE-He213)s11075-014-9859-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Dykes, L. verfasserin aut Rescaling the GSVD with application to ill-posed problems 2014 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2014 Abstract The generalized singular value decomposition (GSVD) of a pair of matrices expresses each matrix as a product of an orthogonal, a diagonal, and a nonsingular matrix. The nonsingular matrix, which we denote by XT, is the same in both products. Available software for computing the GSVD scales the diagonal matrices and XT so that the squares of corresponding diagonal entries sum to one. This paper proposes a scaling that seeks to minimize the condition number of XT. The rescaled GSVD gives rise to new truncated GSVD methods, one of which is well suited for the solution of linear discrete ill-posed problems. Numerical examples show this new truncated GSVD method to be competitive with the standard truncated GSVD method as well as with Tikhonov regularization with regard to the quality of the computed approximate solution. Generalized singular value decomposition Truncated generalized singular value decomposition Ill-posed problem Noschese, S. aut Reichel, L. aut Enthalten in Numerical algorithms Springer US, 1991 68(2014), 3 vom: 08. Mai, Seite 531-545 (DE-627)130981753 (DE-600)1075844-6 (DE-576)029154111 1017-1398 nnns volume:68 year:2014 number:3 day:08 month:05 pages:531-545 https://doi.org/10.1007/s11075-014-9859-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2088 AR 68 2014 3 08 05 531-545 |
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Abstract The generalized singular value decomposition (GSVD) of a pair of matrices expresses each matrix as a product of an orthogonal, a diagonal, and a nonsingular matrix. The nonsingular matrix, which we denote by XT, is the same in both products. Available software for computing the GSVD scales the diagonal matrices and XT so that the squares of corresponding diagonal entries sum to one. This paper proposes a scaling that seeks to minimize the condition number of XT. The rescaled GSVD gives rise to new truncated GSVD methods, one of which is well suited for the solution of linear discrete ill-posed problems. Numerical examples show this new truncated GSVD method to be competitive with the standard truncated GSVD method as well as with Tikhonov regularization with regard to the quality of the computed approximate solution. © Springer Science+Business Media New York 2014 |
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Abstract The generalized singular value decomposition (GSVD) of a pair of matrices expresses each matrix as a product of an orthogonal, a diagonal, and a nonsingular matrix. The nonsingular matrix, which we denote by XT, is the same in both products. Available software for computing the GSVD scales the diagonal matrices and XT so that the squares of corresponding diagonal entries sum to one. This paper proposes a scaling that seeks to minimize the condition number of XT. The rescaled GSVD gives rise to new truncated GSVD methods, one of which is well suited for the solution of linear discrete ill-posed problems. Numerical examples show this new truncated GSVD method to be competitive with the standard truncated GSVD method as well as with Tikhonov regularization with regard to the quality of the computed approximate solution. © Springer Science+Business Media New York 2014 |
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Abstract The generalized singular value decomposition (GSVD) of a pair of matrices expresses each matrix as a product of an orthogonal, a diagonal, and a nonsingular matrix. The nonsingular matrix, which we denote by XT, is the same in both products. Available software for computing the GSVD scales the diagonal matrices and XT so that the squares of corresponding diagonal entries sum to one. This paper proposes a scaling that seeks to minimize the condition number of XT. The rescaled GSVD gives rise to new truncated GSVD methods, one of which is well suited for the solution of linear discrete ill-posed problems. Numerical examples show this new truncated GSVD method to be competitive with the standard truncated GSVD method as well as with Tikhonov regularization with regard to the quality of the computed approximate solution. © Springer Science+Business Media New York 2014 |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">OLC2051164150</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230503232956.0</controlfield><controlfield tag="007">tu</controlfield><controlfield tag="008">200820s2014 xx ||||| 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s11075-014-9859-3</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)OLC2051164150</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-He213)s11075-014-9859-3-p</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">510</subfield><subfield code="q">VZ</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">17,1</subfield><subfield code="2">ssgn</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Dykes, L.</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Rescaling the GSVD with application to ill-posed problems</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2014</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">ohne Hilfsmittel zu benutzen</subfield><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Band</subfield><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© Springer Science+Business Media New York 2014</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract The generalized singular value decomposition (GSVD) of a pair of matrices expresses each matrix as a product of an orthogonal, a diagonal, and a nonsingular matrix. The nonsingular matrix, which we denote by XT, is the same in both products. Available software for computing the GSVD scales the diagonal matrices and XT so that the squares of corresponding diagonal entries sum to one. This paper proposes a scaling that seeks to minimize the condition number of XT. The rescaled GSVD gives rise to new truncated GSVD methods, one of which is well suited for the solution of linear discrete ill-posed problems. Numerical examples show this new truncated GSVD method to be competitive with the standard truncated GSVD method as well as with Tikhonov regularization with regard to the quality of the computed approximate solution.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Generalized singular value decomposition</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Truncated generalized singular value decomposition</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Ill-posed problem</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Noschese, S.</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Reichel, L.</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Numerical algorithms</subfield><subfield code="d">Springer US, 1991</subfield><subfield code="g">68(2014), 3 vom: 08. 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