Error analysis of the mdLVs algorithm for computing bidiagonal singular values

Some of the authors show in Iwasaki and Nakamura (Inverse Probl 20:553–563, 2004) that the integrable discrete Lotka–Volterra (dLV) system is applicable for computing singular values of bidiagonal matrix. The resulting numerical algorithm is referred to as the dLV algorithm. They also observe in Iwasaki and Nakamura (Electron Trans Numer Anal 38:184–201, 2011) that the singular values are numerically computed with high relative accuracy by using the mdLVs algorithm, which is an acceleration version by introducing a shift of origin. In this paper, we investigate the perturbations on singular values and the forward errors of the mdLVs variables, which occur in the mdLVs algorithm, through two kinds of error analysis in floating point arithmetic. Therefore the forward stability of the mdLVs algorithm in the sense of Bueno–Marcellan–Dopico is proved.