Conserved quantities of the discrete finite Toda equation and lower bounds of the minimal singular value of upper bidiagonal matrices
Kento Kimura, Takahiro Yamashita, Yusuke Nakamura · Journal of Physics A Mathematical and Theoretical · 2011
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 × N ( N ⩾ 2) real upper bidiagonal matrix B where all the diagonals and the upper subdiagonals are positive, conserved quantities Tr((( B T B ) M ) −1 ) ( M = 1, 2, …) of the discrete finite Toda equation give a sequence of lower bounds of the minimal singular value of B . Recurrence relations for computing higher order conserved quantities Tr((( B T B ) M ) −1 ) are also derived.