Low-rank approximation by QR decomposition with pivoting (pivoted QR) is known to be less accurate than singular value decomposition (SVD); however, the calculation amount is smaller than that of SVD. The least upper bound of the ratio of the truncation error, defined by \(\Vert A-BC\Vert _2\), using pivoted QR to that using SVD is proved to be \(\sqrt{\frac{4^k-1}{3}(n-k)+1}\) for \(A\in {\mathbb {R}}^{m\times n}\) \((m\ge n)\), approximated as a product of \(B\in {\mathbb {R}}^{m\times k}\) and \(C\in {\mathbb {R}}^{k\times n}\) in this study.

已知通过旋转旋转QR分解（旋转QR）进行的低秩逼近比奇异值分解（SVD）精度低；但是，计算量小于SVD。证明由\（\ Vert A-BC \ Vert _2 \）定义的使用旋转QR的截断误差与使用SVD的截断误差的比率的最小上限为\（\ sqrt {\ frac {4 ^ k- 1} {3}（NK）+1} \）为\（A \在{\ mathbb {R}} ^ {米\ n次} \） \（（M \ GE N）\），近似为产品本研究中\（B \ in {\ mathbb {R}} ^ {m \ times k} \）和\（C \ in {\ mathbb {R}} ^ {k \ times n} \）中的分布。