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| 概要 |
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(((BTB)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))-1/(2M). In [4], simple recurrence relations for computing all the diagonals of ((BTB)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 over 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.続きを見る
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Mendeley出力
