A Challenge to Relative Errors by Numerical Algorithms with Positivity

本课题通过对浮点算法中的两种误差分析，研究了mdlv算法中出现的奇异值扰动和mdlv变量的前向误差。从而证明了mdlv算法的正向稳定性。考虑到离散化参数\delta (n)趋于无穷时，我们提出了一种新的相似变换，称为dLV＿ \infty相似变换。通过对dLV＿ \infty变换的反复应用，证明了双对角矩阵的奇异值是可计算的。因此，通过前后向混合误差分析，证明了奇异值的相对扰动足够小。最后证明了dLV＿ \infty变换是正向稳定的。

In this project, 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 is proved. Next we present a new similarity transformation named the dLV_\infty similarity transformation by considering the case where the discretization parameter \delta(n) goes to infinity in the integrable dLV system. It is proved that the singular values of bidiagonal matrix are computable by using the dLV_\infty transformation repeatedly. Therefore we clarify that the relative perturbations in the singular values are sufficiently small through the mixed forward-backward error analysis. We finally show that the dLV_\infty transformation is forward stable.