Subtraction-free recurrence relations for lower bounds of the minimal singular value of an upper bidiagonal matrix

On an N × N upper bidiagonal matrix B, where all the diagonals and the upper subdiagonals are positive, and its transpose BT , it is shown in the recent paper [4] that quantities JM(B) ≡ Tr(((BT B)M)−1) (M = 1, 2, . . . ) gives a sequence of lower bounds θM(B) of the minimal singular value of B through θM(B) ≡ (JM(B)). In [4], simple recurrence relations for computing all the diagonals of ((BT B)M)−1 and ((BBT )M)−1 are also presented. The square of θM(B) can be used as a shift of origin in numerical algorithms for computing all the singular values of B. In this paper, new recurrence relations which have advantages to the old ones in [4] are presented. The new recurrence relations consist of only addition, multiplication and division among positive quantities. Namely, they are subtraction-free. This property excludes any possibility of cancellation error in numerical computation of the traces JM(B). Computational cost for the trace JM(B) (M = 1, 2, . . . ) and efficient implementations for J2(B) and J3(B) are also discussed.