Typically, parametrization captures the essence of a class of matrices, and its potential advantage is to make accurate computations possible. But, in general, parametrization suitable for accurate computations is not always easy to find. In this paper, we introduce a parametrization of a class of negative matrices to accurately solve the singular value problem. It is observed that, given a set of parameters, the associated nonsingular negative matrix can be orthogonally transformed into a totally nonnegative matrix in an implicit and subtraction-free way, which implies that such a set of parameters determines singular values of the associated negative matrix accurately. Based on this observation, a new O(n³) algorithm is designed to compute all the singular values, large and small, to high relative accuracy.

通常，参数化捕获了一类矩阵的本质，其潜在优势是使精确计算成为可能。但是，一般来说，适合精确计算的参数化并不总是很容易找到。在本文中，我们引入了一类负矩阵的参数化来精确解决奇异值问题。观察到，给定一组参数，相关联的非奇异负矩阵可以以隐式和免减法的方式正交变换为完全非负矩阵，这意味着这组参数决定了相关联的负矩阵的奇异值准确。基于这一观察，一个新的O ( n ³) 算法旨在以较高的相对精度计算所有奇异值，无论大小。