发表机构
University of Valencia（瓦伦西亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种从标量列索引到全次数多元 Vandermonde 矩阵多指标的递归映射，支持随机访问任意列，无需构造完整矩阵，适用于流式构建和无矩阵求值。
AI 中文摘要
对于全次数至多为 n 的多元多项式基，Vandermonde 矩阵的一列由指数的多指标标识。本文给出一个从标量列索引到该多指标的直接、递归映射，其基础是一种规定的排序：在块级别上增加最后一个变量的指数，并对剩余变量递归应用相同规则。该映射首先在二元情形下显式导出，得到一元 Vandermonde 矩阵的闭式列索引，然后通过二项式计数函数递归推广到任意维数，并用归纳法证明其正确性。该过程无需构造完整的张量积矩阵即可随机访问任意全次数列，特别适用于流式构建和无矩阵求值。它不依赖于基，可直接应用于 Chebyshev–Vandermonde 矩阵。对于完整稠密矩阵，对指数多指标的嵌套循环达到相同的渐近代价；本方法的优势在于精确索引并避免多余的中间数组。
英文摘要
For multivariate polynomial bases of total degree at most n, a column of the Vandermonde matrix is identified by a multi-index of exponents. In this work, we give a direct, recursive map from the scalar column index to this multi-index, based on a prescribed ordering in which exponents of the last variable are increased at the block level and the same rule is applied recursively to the remaining variables. The map is first derived explicitly in the bivariate case, where it yields closed-form column indices for the univariate Vandermonde matrices, and is then extended recursively to arbitrary dimension by means of a binomial counting function, with a proof of correctness by induction. The procedure provides random access to any total-degree column without forming the full tensor-product matrix, and is particularly advantageous for streaming construction and matrix-free evaluation. It is basis independent and applies directly to Chebyshev--Vandermonde matrices. For the full dense matrix, nested loops over the exponent multi-indices achieve the same asymptotic cost; the gain of the present approach lies in exact indexing and the avoidance of superfluous intermediate arrays.
Comments16 pages
