Nearly optimal computations with structured matrices

Nearly Optimal Bit Complexity Bounds for Computations with Structured Matrices

We present optimal, up to poly-logarithmic factors, bit complexity results for basic operations, matrix-vector multiplication and solving non-singular linear systems, with structured matrices.

Unified Nearly Optimal Algorithms for Structured Integer Matrices and Polynomials

We seek the solution of banded, Toeplitz, Hankel, Vandermonde, Cauchy and other structured linear systems of equations with integer coefficients. By combining Hensel’s symbolic lifting with either divide-and-conquer algorithms or numerical iterative refinement, we unify the solution for all these structures. We yield the solution in nearly optimal randomized Boolean time, which covers both solu...

On Computations with Dense Structured Matrices

We reduce several computations with Hilbert and Vandermonde type matrices to matrix computations of the Hankel-Toeplitz type (and vice versa). This unifies various known algorithms for computations with dense structured matrices and enables us to extend any progress in computations with matrices of one class to the computations with other classes of matrices. In particular, this enables us to c...

TR-2002017: Nearly Optimal Toeplitz/Hankel Computations

I Improving computations with structured matrices 7 1

Designing e cient code in practice for a given computation is a hard task. In this thesis, we tacklethis issue in two di erent situations.The rst part of the thesis introduces some algorithmic improvements in structured linear algebra.We rst show how to extend an algorithm by Cardinal for inverting Cauchy-like matrices to the othercommon structures. This approach, which mainly r...