Future Goals

[Levent-Batakci]

Here are some goals we set at the 4/15 meeting:

• "Nearby" function to get a close Toeplitz, Hankel, Cauchy, etc. approximation to a given matrix
• "Norms" fast norm computations for our structured matrices
• "Rank" fast eps-rank computations using theoretical results
• "IsLowRank" convenient check of whether a structured matrix is 'low' rank.
• "Toep2Cauchy" and "Hankel2Cauchy" to transform to cauchy-like matrices. The 'Toeps' code should have the Toeplitz -> Cauchylike transformation.
• Expand testing suite
• Gallery
• Rand functions for our structures
• Block compatibility

Big goal: write our own hierarchical code and build FMM

[Levent-Batakci]

PROGRESS:

Nearby I've added in the nearby function for Toeplitz, Hankel, and Circulant matrices. Currently, it finds the closest matrix of specified structure w.r.t the frobenius norm. It can be shown that the projection is obtained by simply averaging across the diagonals/antidiagonals/corresponding pairs of sub and super diagonals, respectively. I have yet to figure out how to find the best projections in the 2-norm.

Any kind of projection seems impossible for Vandermonde matrices, since they don't form a vector space. That being said, there are optimal Vandermonde approximants to a given matrix, at least in the Frobenius norm, but they are not unique in general, and computing them involves polynomial root-finding.

The progress on Nearby is currently on the experimental branch. (https://github.com/heatherw3521/structmats/tree/experimental)

NORMS I've added efficient 1, infinity, and frobenius norms for all of the diagonally structured matrices. The structure gives rise to very efficient computation, and interestingly, the 1 and infinity norm are always the same! The progress on these can be found in the TOEPLITZ branch, which really should be named DIAGONAL_STRUCTURE or something. I intend to merge the norms to main once I test them thoroughly. (https://github.com/heatherw3521/structmats/tree/TOEPLITZ)