Given a system of functions, we introduce the concept of weighted φ‐transformed system, which will include a very large class of useful representations in Statistics and Computer Aided Geometric Design. An accurate bidiagonal decomposition of the collocation matrices of these systems is obtained. This decomposition is used to present computational methods with high relative accuracy for solving algebraic problems with collocation matrices of weighted φ‐transformed systems such as the computation of eigenvalues, singular values, and the solution of some linear systems. Numerical examples illustrate the accuracy of the performed computations.

中文翻译：

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加权ϕ变换系统的配置矩阵的精确对角分解

给定一个功能系统，我们介绍加权φ变换系统的概念，其中将包括统计和计算机辅助几何设计中的一类非常有用的表示形式。获得了这些系统的搭配矩阵的精确对角分解。该分解用于提供具有较高相对精度的计算方法，以解决带有加权φ变换系统的并置矩阵的代数问题，例如特征值，奇异值的计算以及某些线性系统的求解。数值示例说明了所执行计算的准确性。