Computing

, Volume 92, Issue 3, pp 265–283 | Cite as

Numerical performance of hyperplane constrained method and its hybrid method for singular value decomposition

Article

Abstract

The hyperplane constrained method has been proposed in Yadani et al. (Appl Math Comp 216:779–790, 2010) computing singular value decomposition (SVD) of matrix. In the method, the SVD is replaced with solving nonlinear systems whose solutions are constrained on hyperplane, and then their solutions are computed with the help of Newton’s iterative method. In this paper, we present a new convergence theorem concerning the hyperplane constrained method in finite arithmetic. We also clarify the numerical performance of the hyperplane constrained method. In numerical experiments, we first show that the computed singular values and singular vectors are with high accuracy, even if the target matrix of SVD has small singular values, almost the same singular values, not small condition number. Though the hyperplane constrained method requires not small amount of computations, it fastens by combining other fast singular value decomposition method. We next propose a hybrid method which adopts the singular vectors computed by other fast method as the initial guess of the Newton type iteration in order to decrease the iteration number. By numerical experiments, we can see that the hybrid method runs faster than the original hyperplane constrained method with almost same accuracy.

Keywords

Singular value decomposition Newton’s iterative method Nonlinear system Inverse iteration 

Mathematics Subject Classification (2000)

65F15 65H10 

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References

  1. 1.
  2. 2.
    Chatelin F (1993) Eigenvalues of matrices. John Wiley, ChichesterMATHGoogle Scholar
  3. 3.
    Demmel J, McKenney A (1989) A test matrix generation suite. LAPACK working note no. 9, Courant Institute, New YorkGoogle Scholar
  4. 4.
    Demmel J (1997) Applied numerical linear algebra. SIAM, PhiladelphiaMATHGoogle Scholar
  5. 5.
    Elgindi MB, Kharab A (2000) The quadratic method for computing the eigenpairs of a matrix. Int J Comput Math 73: 517–530MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Fernando KV, Parlett BN (1994) Accurate singular values and differential qd algorithms. Numer Math 67: 191–229MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Golub GH, Van Loan CF (1996) Matrix Computations, 3rd edn. The Johns Hopkins University Press, BaltimoreMATHGoogle Scholar
  8. 8.
    Kondo K, Yasukouchi S, Iwasaki M (2009) Eigendecomposition algorithms solving sequentially quadratic systems by Newton method. JSIAM Lett 1: 40–43Google Scholar
  9. 9.
  10. 10.
    Matsumoto M, Nishimura T (1998) Mersenne twister: a 623-dimensionally equidistributed uniform pseudorandom number generator. ACM Trans Model Comp Simul 8(1): 3–30MATHCrossRefGoogle Scholar
  11. 11.
    Press WH et al (2007) Numerical recipes: the art of scientific computing, 3rd edn. Cambridge University PressGoogle Scholar
  12. 12.
    Wilkinson JH (1965) The algebraic eigenvalue problem. Oxford University PressGoogle Scholar
  13. 13.
    Yadani K, Kondo K, Iwasaki M (2010) A singular value decomposition algorithm based on solving hyperplane constrained nonlinear systems. Appl Math Comp 216: 779–790MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Graduate School of InformaticsKyoto UniversityKyotoJapan
  2. 2.Graduate School of EngineeringDoshisha UniversityKyotoJapan
  3. 3.Department of Informatics and Environmental ScienceKyoto Prefectural UniversityKyotoJapan

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