A fast exponential decomposition algorithm and its applications to structured matrices

Abstract

The origins of these problems can be traced back to a paper published in 1795 by the French scientist Prony. A translation of this pioneering work is provided.

This thesis unifies all previously known linear algebraic exponential decomposition methods within a single matrix pencil based framework, a new result. The framework is further shown to suggest a novel, fast algorithm to compute a so-called Hankel-Vandermonde decomposition of a strongly nonsingular Hankel matrix.

We propose a new variant of the Hankel-Vandermonde decomposition and prove that it exists for any given Hankel matrix. Our proof confirms the strong ties between this factorization and the exponential decomposition problem of decomposing a signal into a sum of complex exponential sequences. By adapting an implicit Lanczos process, we demonstrate that for an important class of N x N Hankel matrices, these decomposition problems can be reduced to finding the eigenvalues of a tridiagonal matrix. The procedure thus solves the matrix pencil introduced by our unifying framework without explicitly inverting matrices. The total process requires only O(N2) operations and O(N) storage, compared to O(N3) operations and O(N2) storage requirements for previous methods.

We observe that this structured decomposition of a Hankel matrix is rank revealing. In particular, we derive from this factorization quantities which we call tau values and whose behavior resembles that of the singular values of the given Hankel matrix. The tau values thus provide us with a relatively accurate numerical rank estimator for Hankel matrices at a substantial savings in storage and operations. As with many other rank-revealing factorizations, the Hankel-Vandermonde decomposition can be used to find a low rank approximation to the given perturbed matrix. Besides the computational savings achieved by our algorithm, it has an additional advantage over many other popular methods in that it produces an approximation that exhibits both the exact desired structure and the exact desired rank.