In this research, we solve polynomial, Sobolev polynomial, rational, and Sobolev rational least squares problems. Although the increase in the approximation degree allows us to fit the data better in attacking least squares problems, the ill-conditioning of the coefficient matrix fuels the dramatic decrease in the accuracy of the approximation at higher degrees. To overcome this drawback, we first show that the column space of the coefficient matrix is equivalent to a Krylov subspace. Then the connection between orthogonal polynomials or rational functions and orthogonal bases for Krylov subspaces in order to exploit Krylov subspace methods like Arnoldi orthogonalization is established. Furthermore, some examples are provided to illustrate the theory and the performance of the proposed approach.

翻译：本研究解决了多项式、Sobolev多项式、有理函数及Sobolev有理函数最小二乘问题。尽管提高逼近次数能在处理最小二乘问题时更好地拟合数据，但系数矩阵的病态性会导致高次逼近精度急剧下降。为克服此缺陷，我们首先证明系数矩阵的列空间等价于Krylov子空间，进而建立正交多项式/有理函数与Krylov子空间正交基之间的关联，以利用Arnoldi正交化等Krylov子空间方法。此外，本文通过若干算例验证了所提方法的理论基础与实际性能。