Fast algorithms for multivariate interpolation and evaluation at special points

Abstract

In this paper we present explicit formulae for the multivariate Lagrange–Newton transformation T : K n 1 × n 2 × ⋯ × n d → K n 1 × n 2 × ⋯ × n d and its inverse T − 1 with respect to points x i , j = λ i x i , j − 1 + δ i ( i = 1 , 2 , … , d , j = 1 , 2 , … , n i − 1 ) , where λ i ≠ 0 , δ i and x i , 0 = ϰ i belong to the field K . Moreover, we derive fast algorithms for computing these transformations. The running time of them is O ( ∏ j = 1 d n j ⋅ log ∏ j = 1 d n j ) + O ( d ∏ j = 1 d n j ) base operations from K .