Fast approximate computations with Cauchy matrices and polynomials

Multipoint polynomial evaluation and interpolation are fundamental for modern algebraic and numerical computing. The known algorithms solve both problems over any field of constants in nearly linear arithmetic time, but the cost grew to quadratic for numerical solution. We fix this discrepancy: our new numerical algorithms run in nearly linear arithmetic time. At first we restate our goals as the multiplication of a Vandermonde matrix by a vector and the solution of a Vandermonde linear system of equations. Then we transform this matrix into a Cauchy matrix, which we approximate by a generalized hierarchically semiseparable matrix. Finally we achieve the desired algorithmic acceleration by applying the Fast Multipole Method to the latter matrix. Our resulting numerical algorithms run in nearly optimal arithmetic time for the above fundamental computations with polynomials and Vandermonde matrices as well as with transposed Vandermonde matrices and a large class of Cauchy and Cauchy-like matrices. Some of our techniques can be of independent interest.