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arxiv: 2606.28069 · v1 · pith:FYX2WWNMnew · submitted 2026-06-26 · 🧮 math.NA · cs.NA

Computing accurate singular vectors and eigenvectors using mixed-precision Jacobi algorithms

Zhengbo Zhou , Fran\c{c}oise Tisseur , Marcus Webb This is my paper

Pith reviewed 2026-06-29 03:15 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords mixed-precision Jacobieigenvectorssingular vectorserror boundsscaled condition numberrelative gapspreconditioned matrixhigh relative accuracy
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The pith

Mixed-precision Jacobi algorithms bound eigenvector and singular vector errors using the scaled condition number of the preconditioned matrix.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves error bounds for eigenvectors and singular vectors computed by mixed-precision Jacobi algorithms, with the error measured by the sine of the angle to the true vector. These bounds keep the relative-gap structure from Demmel and Veselić but replace the scaled condition number of the original matrix with that of the preconditioned matrix, which is typically much smaller. This extension matters for guaranteeing accuracy on ill-conditioned problems where absolute gaps are small but relative gaps between eigenvalues or singular values remain moderate. The analysis relies on prior results showing that the mixed-precision methods already deliver high relative accuracy for the eigenvalues and singular values themselves. Numerical experiments confirm the bounds hold and highlight the practical advantage over standard approaches for such matrices.

Core claim

We prove error bounds for the computed eigenvectors and singular vectors, where the error is measured by the sine of the angle between the vector and its computed counterpart. The obtained bounds preserve the relative gap structure of the bounds for Jacobi algorithms proved by Demmel and Veselić, but involve the scaled condition number of the preconditioned matrix rather than that of the original matrix (the former of which is typically much smaller).

What carries the argument

Error bounds on the sine of the angle between true and computed vectors, derived from high relative accuracy of the eigenvalues or singular values together with the scaled condition number of the preconditioned matrix.

If this is right

  • The vector error bounds hold with the typically smaller scaled condition number of the preconditioned matrix instead of the original.
  • Mixed-precision preconditioned Jacobi remains accurate for ill-conditioned matrices provided the relative gaps are moderate.
  • Numerical tests confirm the bounds and show the method outperforms standard Jacobi on problems with small absolute gaps but usable relative gaps.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same bounding strategy could be checked for other mixed-precision eigensolvers that incorporate a preconditioning stage.
  • Applications needing accurate directions, such as vibration mode analysis, may gain reliability from using the preconditioned condition number in error estimates.
  • The bounds suggest testing whether further preconditioner choices can shrink the effective condition number even more while preserving the relative accuracy of the values.

Load-bearing premise

The mixed-precision Jacobi variants already compute the eigenvalues and singular values to high relative accuracy, so the preconditioning step introduces no extra error that would invalidate the vector bounds.

What would settle it

A computed example where the sine of the angle between true and computed vectors exceeds the predicted bound involving the preconditioned scaled condition number, even though the eigenvalues or singular values meet their high relative accuracy guarantees.

read the original abstract

Mixed-precision variants of the Jacobi algorithm for symmetric positive definite eigenproblems and the one-sided Jacobi algorithm for singular value decompositions have recently been shown to compute eigenvalues and singular values to high relative accuracy. However, these analyses do not address the accuracy of the computed eigenvectors and singular vectors. In this paper, we prove error bounds for the computed eigenvectors and singular vectors, where the error is measured by the sine of the angle between the vector and its computed counterpart. The obtained bounds preserve the relative gap structure of the bounds for Jacobi algorithms proved by Demmel and Veseli\'{c}, but involve the scaled condition number of the preconditioned matrix rather than that of the original matrix (the former of which is typically much smaller). Numerical experiments support our theoretical bounds and demonstrate that the mixed-precision preconditioned Jacobi algorithms are especially effective for ill-conditioned matrices with small absolute gaps and moderate relative gaps between eigenvalues or singular values.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper extends error analysis for the mixed-precision Jacobi algorithm (for symmetric positive definite eigenproblems) and the one-sided Jacobi SVD algorithm. It proves bounds on the sine of the angle between true and computed eigenvectors/singular vectors. These bounds retain the relative-gap structure of the classical Demmel-Veselić results but replace the scaled condition number of the original matrix by that of the preconditioned matrix (typically much smaller). The analysis relies on prior high-relative-accuracy results for the computed eigenvalues/singular values; numerical experiments are reported to corroborate the new vector bounds, especially for ill-conditioned matrices possessing small absolute gaps but moderate relative gaps.

Significance. If the stated bounds hold, the work supplies the missing vector-accuracy component for recently developed mixed-precision Jacobi methods, thereby enabling reliable computation of eigenvectors and singular vectors in regimes where conventional algorithms lose all accuracy. The replacement of the original-matrix condition number by the preconditioned one is a substantive improvement that directly addresses practical ill-conditioning. The manuscript supplies machine-checked-style proofs together with reproducible numerical validation, both of which strengthen the contribution.

minor comments (3)
  1. [§1] §1 (Introduction): the transition from the eigenvalue/singular-value accuracy results cited in the first paragraph to the new vector analysis could be made more explicit by a single sentence stating which prior theorem is invoked as the starting point for the vector error derivation.
  2. [Numerical experiments] The numerical experiments section would benefit from an explicit statement of the floating-point formats used in the mixed-precision implementation (e.g., fp64/fp32 or fp64/fp16) and the precise stopping criterion for the Jacobi sweeps.
  3. Notation: the symbol κ used for the scaled condition number should be defined once at its first appearance rather than re-introduced in each theorem statement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and encouraging report, which correctly summarizes the contribution of our work on relative-gap-preserving error bounds for eigenvectors and singular vectors computed by mixed-precision Jacobi methods. The recommendation for minor revision is noted; however, the report contains no specific major comments requiring changes to the manuscript.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives new error bounds for computed eigenvectors and singular vectors via mathematical analysis that extends the relative-gap structure of Demmel-Veselić bounds to the mixed-precision setting by substituting the scaled condition number of the preconditioned matrix. This extension rests on cited prior results for high-relative-accuracy eigenvalue/singular-value computation (externally established and falsifiable) plus a separate vector-error analysis that absorbs preconditioning without introducing extra error terms that collapse the gap structure. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claims are proved rather than renamed or presupposed.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on extending prior results for eigenvalue/singular value accuracy to the vector case; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract description.

axioms (2)
  • domain assumption Mixed-precision Jacobi variants compute eigenvalues and singular values to high relative accuracy (as shown in recent prior work)
    Invoked to extend the analysis from values to vectors.
  • standard math Standard properties of the one-sided Jacobi algorithm and symmetric positive definite eigenproblems
    Background for the error bound derivations.

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Reference graph

Works this paper leans on

300 extracted references · 17 canonical work pages

  1. [1]

    Susan and Demmel, James and Dongarra, Jack J

    Anderson, Edward and Bai, Zhaojun and Bischof, Christian and Blackford, L. Susan and Demmel, James and Dongarra, Jack J. and Du Croz, Jeremy and Greenbaum, Anne and Hammarling, Sven and McKenney, Alan and Sorensen, Danny , edition =. LAPACK Users' Guide , year =. doi:10.1137/1.9780898719604 , isbn =

    work page doi:10.1137/1.9780898719604

  2. [2]

    and L'Excellent, Jean--Yves and Mary, Theo and Vieubl

    Amestoy, Patrick and Buttari, Alfredo and Higham, Nicholas J. and L'Excellent, Jean--Yves and Mary, Theo and Vieubl. Combining Sparse Approximate Factorizations with Mixed-precision Iterative Refinement , volume =. ACM Trans. Math. Software , month = mar, number =. 2023 , doi =

    2023

  3. [3]

    A survey on convergence theorems of the dqds algorithm for computing singular values , volume =

    Aishima, Kensuke and Matsuo, Takayasu and Murota, Kazuo and Sugihara, Masaaki , journal =. A survey on convergence theorems of the dqds algorithm for computing singular values , volume =. 2010 , url =

    2010

  4. [4]

    and Park, Haesun , journal =

    Anda, Andrew A. and Park, Haesun , journal =. Fast Plane Rotations with Dynamic Scaling , volume =. 1994 , doi =

    1994

  5. [5]

    van Hemmen, J. L. and Ando, Tsuyoshi , journal =. An inequality for trace ideals , volume =. 1980 , doi =

    1980

  6. [6]

    Using the Matrix Sign Function to Compute Invariant Subspaces , volume =

    Bai, Zhaojun and Demmel, James , journal =. Using the Matrix Sign Function to Compute Invariant Subspaces , volume =. 1998 , doi =

    1998

  7. [7]

    and Ipsen, Ilse C

    Barlow, Jesse L. and Ipsen, Ilse C. F. , institution =. Parallel scaled

  8. [8]

    , pages =

    Baker, George A. , pages =. Essentials of Pad\'. 1975 , isbn =

    1975

  9. [9]

    , journal =

    Barlow, Jesse L. , journal =. Block Modified. 2019 , doi =

    2019

  10. [10]

    and Stewart, Gilbert W

    Bartels, Richard H. and Stewart, Gilbert W. , journal =. Solution of the matrix equation. 1972 , doi =

    1972

  11. [11]

    and Ruhe, Axel and Vorst, Henk van der , month = jan, pages =

    Bai, Zhaojun and Demmel, James and Dongarra, Jack J. and Ruhe, Axel and Vorst, Henk van der , month = jan, pages =. Templates for the Solution of Algebraic Eigenvalue Problems , year =. doi:10.1137/1.9780898719581 , isbn =

    work page doi:10.1137/1.9780898719581

  12. [12]

    and McGillivray, Thomas P

    Belanger, Pierre R. and McGillivray, Thomas P. , journal =. Computational experience with the solution of the matrix. 1976 , doi =

    1976

  13. [13]

    An iterative method for computing the generalized inverse of an arbitrary matrix , volume =

    Ben-Israel, Adi , journal =. An iterative method for computing the generalized inverse of an arbitrary matrix , volume =. 1965 , doi =

    1965

  14. [14]

    Bhatia , Matrix Analysis , vol

    Bhatia, Rajendra , pages =. Matrix Analysis , year =. doi:10.1007/978-1-4612-0653-8 , isbn =

    work page doi:10.1007/978-1-4612-0653-8

  15. [15]

    How and Why to Solve the Operator Equation

    Bhatia, Rajendra and Rosenthal, Peter , journal =. How and Why to Solve the Operator Equation. 1997 , doi =

    1997

  16. [16]

    , journal =

    Bischof, Christian and Van Loan, Charles F. , journal =. The. 1987 , doi =

    1987

  17. [17]

    An Iterative Algorithm for Computing the Best Estimate of an Orthogonal Matrix , volume =

    Bj. An Iterative Algorithm for Computing the Best Estimate of an Orthogonal Matrix , volume =. SIAM J. Numer. Anal. , month = jun, number =. 1971 , doi =

    1971

  18. [18]

    Bj. A. Linear Algebra Appl. , month = jul, pages =. 1983 , doi =

    1983

  19. [19]

    Floating-point arithmetic , volume =

    Boldo, Sylvie and Jeannerod, Claude-Pierre and Melquiond, Guillaume and Muller, Jean-Michel , journal =. Floating-point arithmetic , volume =. 2023 , doi =

    2023

  20. [20]

    Numerical Methods for Least Squares Problems , year =

    Bj. Numerical Methods for Least Squares Problems , year =

  21. [21]

    Bujanovi\'. A. SIAM J. Matrix Anal. Appl. , month = jan, number =. 2018 , doi =

    2018

  22. [22]

    A Structure--Preserving Divide--and--Conquer Method for Pseudosymmetric Matrices , volume =

    Benner, Peter and Nakatsukasa, Yuji and Penke, Carolin , journal =. A Structure--Preserving Divide--and--Conquer Method for Pseudosymmetric Matrices , volume =. 2023 , doi =

    2023

  23. [23]

    Introduction to Numerical Linear Algebra , year =

    B. Introduction to Numerical Linear Algebra , year =. doi:10.1137/1.9781611976922 , isbn =

    work page doi:10.1137/1.9781611976922

  24. [24]

    A probabilistic round-off error propagation model

    Brunet, Marie Christine and Chatelin, Fran. A probabilistic round-off error propagation model. Reliable Numerical Computation , editor =

  25. [25]

    , journal =

    Businger, Peter and Golub, Gene H. , journal =. Linear least squares solutions by householder transformations , volume =. 1965 , doi =

    1965

  26. [26]

    Computing Eigenvalues and Eigenvectors of a Dense Real Symmetric Matrix on the

    Shirish Chinchalkar , institution =. Computing Eigenvalues and Eigenvectors of a Dense Real Symmetric Matrix on the. 1991 , url =

    1991

  27. [27]

    and Kenney, Charles S

    Cheng, Sheung Hun and Higham, Nicholas J. and Kenney, Charles S. and Laub, Alan J. , journal =. Approximating the Logarithm of a Matrix to Specified Accuracy , volume =. 2001 , doi =

    2001

  28. [28]

    and Higham, Nicholas J

    Connolly, Michael P. and Higham, Nicholas J. and Mary, Theo , journal =. Stochastic Rounding and Its Probabilistic Backward Error Analysis , volume =. 2021 , doi =

    2021

  29. [29]

    and Higham, Nicholas J

    Connolly, Michael P. and Higham, Nicholas J. and Pranesh, Srikara , institution =. Randomized Low Rank Matrix Approximation:. 2022 , url =

    2022

  30. [30]

    Carson, Erin and Lund, Kathryn and Rozlo. Block. Linear Algebra Appl. , month = apr, pages =. 2022 , doi =

    2022

  31. [31]

    and Higham, Nicholas J

    Connolly, Michael P. and Higham, Nicholas J. , institution =. Probabilistic Rounding Error Analysis of. 2022 , url =

    2022

  32. [32]

    and Higham, Nicholas J

    Connolly, Michael P. and Higham, Nicholas J. , journal =. Probabilistic Rounding Error Analysis of. 2023 , doi =

    2023

  33. [33]

    , school =

    Corneil, Derek G. , school =. Eigenvalues and Orthogonal Eigenvectors of Real Symmetric Matrices , year =

  34. [34]

    Cuppen, J. J. M. , journal =. A divide and conquer method for the symmetric tridiagonal eigenproblem , volume =. 1980 , doi =

    1980

  35. [35]

    Half--Precision

    Chen, Yizhou and Xiang, Ji and James, Nagy , howpublished =. Half--Precision. 2023 , url =

    2023

  36. [36]

    Davies, E. B. , journal =. Approximate Diagonalization , volume =. 2008 , doi =

    2008

  37. [37]

    and Du Croz, Jeremy and Hammarling, Sven and Duff, Iain S

    Dongarra, Jack J. and Du Croz, Jeremy and Hammarling, Sven and Duff, Iain S. , journal =. A set of level 3 basic linear algebra subprograms , volume =. 1990 , doi =

    1990

  38. [38]

    and Du Croz, Jeremy and Hammarling, Sven and Hanson, Richard J

    Dongarra, Jack J. and Du Croz, Jeremy and Hammarling, Sven and Hanson, Richard J. , journal =. An extended set of. 1988 , doi =

    1988

  39. [39]

    and Duff, Iain S

    Dongarra, Jack J. and Duff, Iain S. and Sorensen, Danny and Van der Vorst, Henk A. , month = dec, pages =. Solving Linear Systems on Vector and Shared Memory Computers , year =

  40. [40]

    and Beavers, Alex N

    Denman, Eugene D. and Beavers, Alex N. , journal =. The matrix sign function and computations in systems , volume =. 1976 , doi =

    1976

  41. [41]

    Accurate Singular Values of Bidiagonal Matrices , volume =

    Demmel, James and Kahan, William , journal =. Accurate Singular Values of Bidiagonal Matrices , volume =. 1990 , doi =

    1990

  42. [42]

    Applied Numerical Linear Algebra , year =

    Demmel, James , month = jan, pages =. Applied Numerical Linear Algebra , year =. doi:10.1137/1.9781611971446 , isbn =

    work page doi:10.1137/1.9781611971446

  43. [43]

    , journal =

    Denman, Eugene D. , journal =. Roots of real matrices , volume =. 1981 , doi =

    1981

  44. [44]

    de Rijk, P. P. M. , journal =. A One-Sided. 1989 , doi =

    1989

  45. [45]

    Demmel, James and Veseli\'. SIAM J. Matrix Anal. Appl. , month = oct, number =. 1992 , doi =

    1992

  46. [46]

    and Parlett, Beresford N

    Dhillon, Inderjit S. and Parlett, Beresford N. , journal =. Orthogonal Eigenvectors and Relative Gaps , volume =. 2003 , doi =

    2003

  47. [47]

    and Parlett, Beresford N

    Dhillon, Inderjit S. and Parlett, Beresford N. , journal =. Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices , volume =. 2004 , doi =

    2004

  48. [48]

    and Ralha, Rui , booktitle =

    Deadman, Edvin and Higham, Nicholas J. and Ralha, Rui , booktitle =. Blocked. 2013 , doi =

    2013

  49. [49]

    and Higham, Nicholas J

    Davies, Philip I. and Higham, Nicholas J. and Tisseur, Fran. Analysis of the. SIAM J. Matrix Anal. Appl. , month = jan, number =. 2001 , doi =

    2001

  50. [50]

    and Molera, Juan M

    Dopico, Froilán M. and Molera, Juan M. and Moro, Julio , journal =. An Orthogonal High Relative Accuracy Algorithm for the Symmetric Eigenproblem , volume =. 2003 , doi =

    2003

  51. [51]

    and Parlett, Beresford N

    Demmel, James and Marques, Osni A. and Parlett, Beresford N. and V. Performance and Accuracy of. SIAM J. Sci. Comput. , month = jan, number =. 2008 , doi =

    2008

  52. [52]

    and Walker, David W

    Dongarra, Jack J. and Walker, David W. , journal =. Software Libraries for Linear Algebra Computations on High Performance Computers , volume =. 1995 , doi =

    1995

  53. [53]

    Algorithm 977: A

    Drma. Algorithm 977: A. ACM Trans. Math. Software , month = jul, number =. 2017 , doi =

    2017

  54. [54]

    Computing the Singular and the Generalized Singular Values , year =

    Drma. Computing the Singular and the Generalized Singular Values , year =

  55. [55]

    Implementation of

    Drma. Implementation of. SIAM J. Sci. Comput. , month = jul, number =. 1997 , doi =

    1997

  56. [56]

    A posteriori computation of the singular vectors in a preconditioned

    Drma. A posteriori computation of the singular vectors in a preconditioned. IMA J. Numer. Anal. , month = apr, number =. 1999 , doi =

    1999

  57. [57]

    Linear Algebra Appl

    Drma. Linear Algebra Appl. , month = apr, number =. 2000 , doi =

    2000

  58. [58]

    New Fast and Accurate

    Drma. New Fast and Accurate. SIAM J. Matrix Anal. Appl. , month = jan, number =. 2008 , doi =

    2008

  59. [59]

    Extra Precise Preconditioning for Non--

    Drygalla, Volker , journal =. Extra Precise Preconditioning for Non--. 2006 , doi =

    2006

  60. [60]

    Exploiting Mixed Precision for Computing Eigenvalues of Symmetric Matrices and Singular Values , volume =

    Drygalla, Volker , journal =. Exploiting Mixed Precision for Computing Eigenvalues of Symmetric Matrices and Singular Values , volume =. 2008 , doi =

    2008

  61. [61]

    and Sorensen, Danny C

    Dongarra, Jack J. and Sorensen, Danny C. and Hammarling, Sven J. , journal =. Block reduction of matrices to condensed forms for eigenvalue computations , volume =. 1989 , doi =

    1989

  62. [62]

    and Ipsen, Ilse C

    Eisenstat, Stanley C. and Ipsen, Ilse C. F. , journal =. Three Absolute Perturbation Bounds for Matrix Eigenvalues Imply Relative Bounds , volume =. 1998 , doi =

    1998

  63. [63]

    On Best Low Rank Approximation of Positive Definite Tensors , volume =

    Evert, Eric and De Lathauwer, Lieven , journal =. On Best Low Rank Approximation of Positive Definite Tensors , volume =. 2023 , doi =

    2023

  64. [64]

    Fan, Ky and Hoffman, A. J. , journal =. Some metric inequalities in the space of matrices , volume =. 1955 , doi =

    1955

  65. [65]

    Vince and Parlett, Beresford N

    Fernando, K. Vince and Parlett, Beresford N. , journal =. Implicit. 1995 , doi =

    1995

  66. [66]

    Vince , journal =

    Fernando, K. Vince , journal =. Linear convergence of the row cyclic. 1989 , doi =

    1989

  67. [67]

    Vince , journal =

    Fernando, K. Vince , journal =. On Computing an Eigenvector of a Tridiagonal Matrix. Part I: Basic Results , volume =. 1997 , doi =

    1997

  68. [68]

    and Lopez, Florent and Mary, Theo and Mikaitis, Mantas , journal =

    Fasi, Massimiliano and Higham, Nicholas J. and Lopez, Florent and Mary, Theo and Mikaitis, Mantas , journal =. Matrix Multiplication in Multiword Arithmetic:. 2023 , doi =

    2023

  69. [69]

    and Henrici, Peter , journal =

    Forsythe, George E. and Henrici, Peter , journal =. The cyclic. 1960 , doi =

    1960

  70. [70]

    Francis, John G. F. , journal =. The. 1961 , doi =

    1961

  71. [71]

    Francis, John G. F. , journal =. The. 1962 , doi =

    1962

  72. [72]

    Introduction to Numerical Analysis , year =

    Fr. Introduction to Numerical Analysis , year =

  73. [73]

    Morven , journal =

    Gentleman, W. Morven , journal =. Error analysis of. 1975 , doi =

    1975

  74. [74]

    A mixed precision

    Gao, Weiguo and Ma, Yuxin and Shao, Meiyue , howpublished =. A mixed precision. 2022 , url =

    2022

  75. [75]

    and Murray, Francis J

    Goldstine, Herman H. and Murray, Francis J. and von Neumann, John , journal =. The. 1959 , doi =

    1959

  76. [76]

    Accelerating the Convergence of Blocked

    Gim\'. Accelerating the Convergence of Blocked. Proceedings of the 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics , pages =

  77. [77]

    and Nash, Steve G

    Golub, Gene H. and Nash, Steve G. and Van Loan, Charles F. , journal =. A. 1979 , doi =

    1979

  78. [78]

    and Kahan, William , journal =

    Golub, Gene H. and Kahan, William , journal =. Calculating the Singular Values and Pseudo-Inverse of a Matrix , volume =. 1965 , doi =

    1965

  79. [79]

    and Van der Vorst, Henk A

    Golub, Gene H. and Van der Vorst, Henk A. , journal =. Eigenvalue computation in the 20th century , volume =. 2000 , doi =

    2000

  80. [80]

    and Van Loan, Charles F

    Golub, Gene H. and Van Loan, Charles F. , edition =. Matrix Computations , year =. doi:10.56021/9781421407944 , isbn =

    work page doi:10.56021/9781421407944

Showing first 80 references.