Abstract
Multipoint polynomial evaluation and interpolation are fundamental for modern numerical and symbolic computing. The known algorithms solve both problems over any field of constants in nearly linear arithmetic time, but the cost grows to quadratic for numerical solution. By combining a variant of the Multipole celebrated numerical techniques with transformations of matrix structures of [10] we achieve dramatic speedup and for a large class of inputs yield solution algorithms running in nearly linear time as well. The algorithms support similar speedup of approximation of the products of a Vandermonde matrix, its transpose, inverse, and the transpose of the inverse by a vector.
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