Abstract
A new scheme with a shift of origin for computing singular values σk is presented. A shift θ is introduced into the recurrence relation defined by the discrete integrable Lotka-Volterra system with variable step-size. A suitable shift strategy is given so that the singular value computation becomes numerically stable. It is proved that variables in the new scheme converge to σ 2 k - Σ θ2. A comparison of the zero-shift and the nonzero-shift routines is drawn. With respect to both the computational time and the numerical accuracy, it is shown that the nonzero-shift routine is more accurate and faster than a credible LAPACK routine for singular values at least in four different types of test matrices.
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Z. Bai, C. Bischof, L.S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Geenhaum, S. Hammarling, A. Mckenney, D. Sorensen and E. Anderson, Lapack Users’ Guide (third edition). SIAM, Philadelphia, 2000, http://www.netlib.org/lapack/.
J. Demmel, Accurate Singular Values of Bidiagonal Matrices. SIAM, Philadelphia, 1997.
J. Demmel and W. Kahan, Accurate singular values of bidiagonal matrices. SIAM Journal on Scientific Statistical Computing,11 (1990), 873–912.
K.V. Fernando and B.N. Parlett, Accurate singular values and differential qd algorithms. Numerische Mathematik,67 (1994), 191–229.
G.H. Golub and C. Reinsch, Singular value decomposition and least squares solutions. Numerische Mathematik,14 (1970), 403–420.
G.H. Golub and C.F. Van Loan, Matrix Computations (third edition). John Hopkins University Press, Baltimore, 1996.
A. Heck, Introduction to Maple (third edition). Springer-Verlag, New York, 2003.
R. Hirota, Conserved quantities of a “random-time Toda equation.” Journal of the Physical Society of Japan,66 (1997), 283–284.
M. Iwasaki and Y. Nakamura, On a convergence of solution of the discrete Lotka-Volterra system. Inverse Problems,18 (2002), 1569–1578.
M. Iwasaki and Y. Nakamura, An application of the discrete Lotka-Volterra system with variable step-size to singular value computation. Inverse Problems,20 (2004), 553–563.
C.R. Johnson, A Gersgorin-type lower bound for the smallest singular value. Linear Algebra and its Applications,112 (1989), 1–7.
H. Rutishauser, Lectures on Numerical Mathematics. Birkhäuser, Boston, 1990.
V. Spiridonov and A. Zhedanov, Discrete-time Volterra chain and classical orthogonal polynomial. Journal of Physics A: Mathematical and General,30 (1997), 8727–8737.
S. Tsujimoto, Y. Nakamura and M. Iwasaki, The discrete Lotka-Volterra system computes singular values. Inverse Problems,17 (2001), 53–58.
R.C. Whaley and J. J. Dongarra, Automatically tuned linear algebra software. Technical Report UT-CS-97-366, University of Tennessee, 1997, http://math-atlas.sourceforge.net/.
J.H. Wilkinson, The Algebraic Eigenvalue Problem. Clarendon Press, Oxford, 1965.
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Iwasaki, M., Nakamura, Y. Accurate computation of singular values in terms of shifted integrable schemes. Japan J. Indust. Appl. Math. 23, 239 (2006). https://doi.org/10.1007/BF03167593
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DOI: https://doi.org/10.1007/BF03167593