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.

翻译：在最近几年中，涉及全正矩阵的病态线性代数问题的精确数值解的获取已引起研究人员的广泛关注。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基之间的转换矩阵的全正性，并给出了其双对角分解。此外，将对应于递增正节点的Vandermonde矩阵的著名高相对精度结果推广到了递减负节点的情形。这进一步允许以高相对精度求解关于$q$-Abel多项式的配置矩阵、Wronskian矩阵和Gramian矩阵的若干代数问题。最后，一系列数值测试支持了所提出的理论结果，并说明了该方法的优越性——在标准方法无法提供精确解的情况下，该方法依然有效。