Multi-Index Recovery for Total-Degree Multivariate Vandermonde Matrices

Abstract

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.