In recent years many efforts have been devoted to finding bidiagonal factorizations of nonsingular totally positive matrices, since their accurate computation allows to numerically solve several important algebraic problems with great precision, even for large ill-conditioned matrices. In this framework, the present work provides the factorization of the collocation matrices of Newton bases -- of relevance when considering the Lagrange interpolation problem -- together with an algorithm that allows to numerically compute it to high relative accuracy. This further allows to determine the coefficients of the interpolating polynomial and to compute the singular values and the inverse of the collocation matrix. Conditions that guarantee high relative accuracy for these methods and, in the former case, for the classical recursion formula of divided differences, are determined. Numerical errors due to imprecise computer arithmetic or perturbed input data in the computation of the factorization are analyzed. Finally, numerical experiments illustrate the accuracy and effectiveness of the proposed methods with several algebraic problems, in stark contrast with traditional approaches.

翻译：近年来，大量研究致力于寻找非奇异全正矩阵的双对角分解，因为其精确计算能够高精度地数值求解若干重要代数问题，即使对于大型病态矩阵也是如此。在此框架下，本文提供了牛顿基的配置矩阵的分解——该矩阵在拉格朗日插值问题中具有重要相关性——并给出了一种能够以高相对精度数值计算该分解的算法。这进一步使得能够确定插值多项式的系数，并计算配置矩阵的奇异值及其逆矩阵。本文确定了保证这些方法（以及在前一种情形中用于经典差商递推公式）达到高相对精度的条件，并分析了由于不精确的计算机算术或扰动输入数据在分解计算中导致的数值误差。最后，数值实验通过若干代数问题展示了所提方法的精度与有效性，这与传统方法形成了鲜明对比。