roots

[MikaelSlevinsky]

Calculating roots leads to the ability to find max, min, and inf- and 1-norms. This can be done in O(n^2) complexity via the algorithm in http://epubs.siam.org/doi/abs/10.1137/130904508 since the roots of a sincfun in barycentric form are the same as the barycentric polynomial. (The other part is just a double exponential envelope.) Additionally, something similar could be used to calculate the barycentric form's poles.

[MikaelSlevinsky]

Also with roots, it would be possible to override division (/).

[MikaelSlevinsky]

A basic routine is now implemented via @1fa0a20eb3a4dde5bdb1f814. This now allows for:

using SincFun
f(x) = exp(x)*cospi(5x)/(1+x^2); sf = sincfun(f)
roots(sf)
norm(sf[roots(sf)])

[MikaelSlevinsky]

From what I understand, I will divide the Lawrence algorithm into two steps:

• tranform the matrix A to upper Hessenberg-plus-rank-one matrix
• deflate two spurious infinite eigenvalues Then, the result is a regular eigenvalue problem and should be quicker.

[MikaelSlevinsky]

It turns out that, so far as I understand it, Lawrence's Algorithm 1. leads to a significant loss in the orthogonality of the vectors q, affecting the roots. Will investigate the second approach based on Givens rotations.