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Accuracy and stability of quaternion Gaussian elimination

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Abstract

This paper is devoted to the accuracy and stability of quaternion Gaussian elimination (qGE). First, considering the noncommutativity of quaternion multiplications, we establish the rules of quaternion floating-point operations. After that, roundoff error analyses of quaternion LU and qGE with partial pivoting (qGEPP) are presented to show the backward stability of qGEPP is affected by the growth factor. Then, the upper bounds for growth factors of qGE with partial or complete pivoting, and the quaternion matrices with maximal growth factor are investigated. Finally, growth factors of some special matrices are also studied, and the theoretical results are confirmed through several numerical experiments. A conjecture about the growth factor of quaternion Hadamard matrix is given.

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References

  1. Chen, L.X.: Definition of determinant and Cramer solutions over the quatemion field. Acta Math. Sinica N. S. 7(2), 171–180 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cryer, C.W.: Pivot size in Gaussian elimination. Numer. Math. 12, 335–345 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  3. Day, J., Peterson, B.: Growth in Gaussian elimination. Amer. Math. Month. 95, 489–513 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. De Leo, S., Rotelli, P.: Quaternion scalar field. Phys. Rev. D 45, 575–579 (1992)

    Article  MathSciNet  Google Scholar 

  5. Dieudonné, J.: Les determinants sur un corp non-commutatif. Bull. Soc. Math. France 71, 27–45 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  6. Edelman, A.: The complete pivoting conjecture for Gaussian elimination is false. The Mathematical Journal 2(58), 58–61 (1992)

    Google Scholar 

  7. Edelman, A., Mascarenhas, W.: On the complete pivoting conjecture for a Hadamard matrix of order 12. Linear Multilinear Algebra 38, 181–187 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Finkelstein, D., Jauch, J.M., Schiminovich, S., Speiser, D.: Foundations of quaternion quantum mechanics. J. Math. Phys. 3, 207–220 (1962)

    Article  MathSciNet  Google Scholar 

  9. Golub, G.H., Van Loan, C.F.: Matrix Computation, 4th edn. Johns Hopkins University Press, Baltimore (2013)

    Book  MATH  Google Scholar 

  10. Gould, N.: On growth in Gaussian elimination with pivoting. SIAM J. Matrix Anal. Appl. 12, 354–361 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hamilton, W.R.: Elements of Quaternions. Chelsea, New York (1969)

    Google Scholar 

  12. Higham, N.J.: Accuracy and stability of numerical algorithms, 2nd edn. SIAM, Philadelphia, PA (2002)

  13. Higham, N.J., Higham, D.J.: Large growth factors in Gaussian elimination with pivoting. SIAM J. Matrix Anal. Appl. 10(2), 155–164 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Higham, D.J., Higham, N.J., Pranesh, S.: Random matrices generating large growth in LU factorization with pivoting. SIAM J. Matrix Anal. Appl. 42(1), 185–201 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  15. Horn, R.A., Johnson, C.R.: Matrix analysis. Cambridge University Press (1985)

  16. Jia, Z.G.: The eigenvalue problem of quaternion matrix: Structure-preserving algorithms and applications. Science Press, Beijing (2019)

    Google Scholar 

  17. Jia, Z.G., Ng, Michael K.: Structure preserving quaternion generalized minimal residual method. SIAM J. Matrix Anal. Appl. 42, 616–634 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jia, Z.G., Ng, Michael K., Song, G.J.: Lanczos method for large-scale quaternion singular value decomposition. Numer. Algorithms 82(2), 699–717 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jia, Z.G., Ng, Michael K., Song, G.J.: Robust quaternion matrix completion with applications to image inpainting. Numer. Linear Algebra Appl. 26(4), e2245 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jia, Z.G., Ng, Michael K., Wang, W.: Color image restoration by saturation-value total variation. SIAM J. Imaging Sci. 12(2), 972–1000 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jia, Z.G., Wei, M.S., Ling, S.T.: A new structure-preserving method for quaternion Hermitian eigenvalue problems. J. Comput. Appl. Math. 239, 12–24 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jia, Z.G., Wei, M.S., Zhao, M.X., Chen, Y.: A new real structure-preserving quaternion QR algorithm. J. Comput. Appl. Math. 343, 26–48 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jiang, T.S.: Algebraic methods for diagonalization of a quaternion matrix in quaternionic quantum theory. J. Math. Phys. 46, 052106–052108 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jiang, T.S., Zhao, J.L., Wei, M.S.: A new technique of quaternion equality constrained least squares problem. Comput. Appl. Math. 216(2), 509–513 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jiang, T.S., Jiang, Z.W., Zhang, Z.Z.: Two novel algebraic techniques for quaternion least squares problems in quaternionic quantum mechanics. Adv. Appl. Clifford Algebra. 26(1), 169–182 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kravvaritis, C. (2016) Hadamard matrices: insights into their growth factor and determinant computations. Math. Anal. Approx. Theo. Appl., pp. 383–415

  27. Kravvaritis, C., Mitrouli, M.: The growth factor of a Hadamard matrix of order 16 is 16. Numer. Linear Algebra Appl. 16, 715–743 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, Y., Wei, M.S., Zhang, F.X., Zhao, J.L.: A fast structure-preserving method for computing the singular value decomposition of quaternion matrix. Appl. Math. Comput. 235, 157–167 (2014)

    MathSciNet  MATH  Google Scholar 

  29. Li, Y., Wei, M.S., Zhang, F.X., Zhao, J.L.: Real structure-preserving algorithms of Householder based transformations for quaternion matrices. J. Comput. Appl. Math. 305, 82–91 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Li, Y., Wei, M.S., Zhang, F.X., Zhao, J.L.: A real structure-preserving method for the quaternion LU decomposition, revisited. Calcolo 54(4), 1553–1563 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ling, S.T., Cheng, X.H., Jiang, T.S.: Consimilarity of quaternions and coneigenvalues of quaternion matrices. Appl. Math. Comput. 270(1), 984–992 (2015)

    MathSciNet  MATH  Google Scholar 

  32. Pratt, W.K.: Digital signal processing. John Wiley and Sons (1978)

    Google Scholar 

  33. Rodman, L.: Topics in quaternion linear algebra. Princeton University Press (2014)

  34. Rutledge, W.A.: Quaternions and Hardmard matrices. Proc. Amer. Math. Soc. 3, 625–630 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ryser, H.J.: Combinatorial mathematics. Wiley (1963)

  36. Sangwine, S., Le, Bihan N. (2005) Quaternion toolbox for Matlab. http://qtfm.sourceforge.net/

  37. Scolarici, G., Solombrino, L.: Notes on quaternionic groups representation. Int. J. Theor. Phys. 34, 2491–2500 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  38. Scolarici, G., Solombrino, L.: Quaternionic representation of magnetic groups. J. Math. Phys. 38, 1147–1160 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  39. Study, E.: Zur Theorie der Linearen Gleichungen. Acta Math. 42, 1–61 (1920)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang, M.H., Ma, W.H.: A structure-preserving method for the quaternion LU decomposition in quaternionic quantum theory. Comput. Phys. Commun. 184, 2182–2186 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wei, M.S., Li, Y., Zhang, F.X., Zhao, J.L. (2018) Quaternion matrix computations. Nova Science Publishers

  42. Wilkinson, J.H. (1961) Error analysis of direct methods of matrix inversion. J. ACM, pp. 281-330

  43. Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Oxford University Press, Oxford (1965)

    MATH  Google Scholar 

  44. Wilkinson, J.H. (1965) Rounding errors in algebraic processes. Notes Appl. Sci., vol. 32. Her Majesty’s Stationary Office, London

  45. Zhang, F.Z.: Quaternions and matrices of quaternion. Linear Algebra Appl. 251, 21–57 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhang, Y.Z., Li, Y., Wei, M.S., Zhao, H.: An algorithm based on QSVD for the quaternion equality constrained least squares problem. Numer. Algorithms 87, 1563–1576 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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Funding

This research was partially supported by the National Natural Science Foundation of China (11971294), Natural Science Foundation of Shanghai, China (23ZR1422400), Natural Science Foundation of Shandong Province, China (ZR2022MA030), and Doctoral Research Fund of Nantong University, China (135420602001).

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Qiaohua Liu & Qian Zhang

  2. School of Science, Nantong University, Nantong, Jiangsu, 226019, People’s Republic of China

    Xiangjian Xu

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  1. Qiaohua Liu
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  2. Qian Zhang
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Correspondence to Qiaohua Liu.

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Liu, Q., Zhang, Q. & Xu, X. Accuracy and stability of quaternion Gaussian elimination. Numer Algor 94, 1159–1183 (2023). https://doi.org/10.1007/s11075-023-01531-y

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