与$q$-Abel多项式相关的全正性及精确计算
Total positivity and accurate computations related to $q$-Abel polynomials
AI总结：
本文研究$q$-Abel多项式基的全正性，给出基变换矩阵的双对角分解，并将Vandermonde矩阵高相对精度结果推广到负节点情形，从而精确求解相关代数问题。
摘要：
近年来，求解涉及全正矩阵的病态线性代数问题以获得精确数值解已引起研究者的广泛关注。与此同时，$q$-微积分在文献中的关注度也在稳步增长。本文研究了Abel多项式基的$q$-模拟。刻画了单项式基与$q$-Abel基之间基变换矩阵的全正性，并给出了其双对角分解。此外，将关于递增正节点Vandermonde矩阵的已知高相对精度结果推广到递减负节点情形。这进一步使得能够以高相对精度求解与$q$-Abel多项式的配置矩阵、Wronskian矩阵和Gramian矩阵相关的若干代数问题。最后，一系列数值实验验证了所提出的理论结果，并说明了在标准方法无法给出精确解的情况下该方法的优越性。
英文摘要：
The attainment of accurate numerical solutions of ill-conditioned linear algebraic problems involving totally positive matrices has been gathering considerable attention among researchers over the last years. In parallel, the interest of $q$-calculus has been steadily growing in the literature. In this work the $q$-analogue of the Abel polynomial basis is studied. The total positivity of the matrix of change of basis between monomial and $q$-Abel bases is characterized, providing its bidiagonal factorization. Moreover, well-known high relative accuracy results of Vandermonde matrices corresponding to increasing positive nodes are extended to the decreasing negative case. This further allows to solve with high relative accuracy several algebraic problems concerning collocation, Wronskian and Gramian matrices of $q$-Abel polynomials. Finally, a series of numerical tests support the presented theoretical results and illustrate the goodness of the method where standard approaches fail to deliver accurate solutions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
