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Mixed-Precision Orthogonalization-Free Projection Methods for Eigenvalue and Singular Value Problems.

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Title: Mixed-Precision Orthogonalization-Free Projection Methods for Eigenvalue and Singular Value Problems.
Authors: Xu, Tianshi1 (AUTHOR) tianshi.xu@emory.edu, Zhang, Zechen2 (AUTHOR) zhan5260@umn.edu, Chen, Jie3 (AUTHOR) jiechenj@amazon.com, Saad, Yousef2 (AUTHOR) saad@umn.ed, Xi, Yuanzhe1 (AUTHOR) yuanzhe.xi@emory.edu
Source: SIAM Journal on Scientific Computing. 2026, Vol. 48 Issue 3, pB544-B568. 25p.
Subjects: Eigenvalues, Singular value decomposition, Arithmetic, Rayleigh-Ritz method, Linear dependence (Mathematics), Matrices (Mathematics), Orthogonalization
Abstract: Mixed-precision arithmetic offers significant computational advantages for large-scale matrix computation tasks, yet preserving accuracy and stability in eigenvalue problems and the singular value decomposition remains challenging. This paper introduces an approach that eliminates orthogonalization requirements in traditional Rayleigh–Ritz projection methods. The proposed method employs nonorthogonal bases computed at reduced precision, resulting in bases computed without inner products. A primary focus is on maintaining the linear independence of the basis vectors. Through extensive evaluation with both synthetic test cases and real-world applications, we demonstrate that the proposed approach achieves the desired accuracy while fully taking advantage of mixed-precision arithmetic. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Scientific Computing 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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Header DbId: egs
DbLabel: Engineering Source
An: 195222014
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
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  Label: Title
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  Data: Mixed-Precision Orthogonalization-Free Projection Methods for Eigenvalue and Singular Value Problems.
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  Data: <searchLink fieldCode="AR" term="%22Xu%2C+Tianshi%22">Xu, Tianshi</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> tianshi.xu@emory.edu</i><br /><searchLink fieldCode="AR" term="%22Zhang%2C+Zechen%22">Zhang, Zechen</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> zhan5260@umn.edu</i><br /><searchLink fieldCode="AR" term="%22Chen%2C+Jie%22">Chen, Jie</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> jiechenj@amazon.com</i><br /><searchLink fieldCode="AR" term="%22Saad%2C+Yousef%22">Saad, Yousef</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> saad@umn.ed</i><br /><searchLink fieldCode="AR" term="%22Xi%2C+Yuanzhe%22">Xi, Yuanzhe</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> yuanzhe.xi@emory.edu</i>
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  Data: <searchLink fieldCode="JN" term="%22SIAM+Journal+on+Scientific+Computing%22">SIAM Journal on Scientific Computing</searchLink>. 2026, Vol. 48 Issue 3, pB544-B568. 25p.
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  Data: <searchLink fieldCode="DE" term="%22Eigenvalues%22">Eigenvalues</searchLink><br /><searchLink fieldCode="DE" term="%22Singular+value+decomposition%22">Singular value decomposition</searchLink><br /><searchLink fieldCode="DE" term="%22Arithmetic%22">Arithmetic</searchLink><br /><searchLink fieldCode="DE" term="%22Rayleigh-Ritz+method%22">Rayleigh-Ritz method</searchLink><br /><searchLink fieldCode="DE" term="%22Linear+dependence+%28Mathematics%29%22">Linear dependence (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Matrices+%28Mathematics%29%22">Matrices (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Orthogonalization%22">Orthogonalization</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Mixed-precision arithmetic offers significant computational advantages for large-scale matrix computation tasks, yet preserving accuracy and stability in eigenvalue problems and the singular value decomposition remains challenging. This paper introduces an approach that eliminates orthogonalization requirements in traditional Rayleigh–Ritz projection methods. The proposed method employs nonorthogonal bases computed at reduced precision, resulting in bases computed without inner products. A primary focus is on maintaining the linear independence of the basis vectors. Through extensive evaluation with both synthetic test cases and real-world applications, we demonstrate that the proposed approach achieves the desired accuracy while fully taking advantage of mixed-precision arithmetic. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of SIAM Journal on Scientific Computing 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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        Value: 10.1137/25M1756922
    Languages:
      – Code: eng
        Text: English
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      Pagination:
        PageCount: 25
        StartPage: B544
    Subjects:
      – SubjectFull: Eigenvalues
        Type: general
      – SubjectFull: Singular value decomposition
        Type: general
      – SubjectFull: Arithmetic
        Type: general
      – SubjectFull: Rayleigh-Ritz method
        Type: general
      – SubjectFull: Linear dependence (Mathematics)
        Type: general
      – SubjectFull: Matrices (Mathematics)
        Type: general
      – SubjectFull: Orthogonalization
        Type: general
    Titles:
      – TitleFull: Mixed-Precision Orthogonalization-Free Projection Methods for Eigenvalue and Singular Value Problems.
        Type: main
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            NameFull: Xu, Tianshi
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            NameFull: Zhang, Zechen
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            NameFull: Chen, Jie
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            NameFull: Saad, Yousef
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            NameFull: Xi, Yuanzhe
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            – D: 01
              M: 05
              Text: 2026
              Type: published
              Y: 2026
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