Department of Applied Mathematics & Physics, Kyoto University

### Technical Report 2011-003 (January 21, 2011) Conserved quantities of the discrete finite Toda equation and lower bounds of the minimal singular value of upper bidiagonal matrices by Kinji Kimura, Takumi Yamashita and Yoshimasa Nakamura

Some numerical algorithms are known to be related to discrete-time integrable systems, where it is essential that quantities to be computed (for example, eigenvalues and singular values of a matrix, poles of a continued fraction) are conserved quantities. In this paper, a new application of conserved quantities of integrable systems to numerical algorithms is presented. For an $N \times N$ $(N \geq 2)$ real upper bidiagonal matrix $\bm{B}$ where all the diagonals and the upper subdiagonals are positive, conserved quantities $\textrm{Tr} ( ( ( \bm{B}^{T} \bm{B} )^{M} )^{-1} )$ $( M = 1, 2, \dotsb )$ of the discrete finite Toda equation give a sequence of lower bounds of the minimal singular value of $\bm{B}$. Recurrence relations for computing higher order conserved quantities $\textrm{Tr} ( ( ( \bm{B}^{T} \bm{B} )^{M} )^{-1} )$ are also derived.