Polynomial Evaluation and Interpolation and Transformations of Matrix Structures

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. We decrease this cost dramatically and for a large class of inputs yield nearly linear time as well. We first restate our tasks as multiplication of a Vandermonde matrix and its inverse by a vector, then transform this matrix into other structured matrices, and finally apply a variant of the Multipole celebrated techniques to achieve the desired speedup for the computations with polynomials, Vandermonde matrices and their transposes. An important impact of our work is a new demonstration of the power of the method of the transformation of matrix structures, which we proposed in [P90]. At the end we comment on further applications and extension of this method to computations with structured matrices, polynomials, and rational functions.