Where is the bug in the Toeplitz-dot-Hankel approach? about fasttransforms.jl HOT 10 CLOSED

I just checked with @aaustin141 in the chebfun implementation and the CXN versions appear to have 1e-12 error at length 100_000.

Will investigate.

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Yes, the MATLAB implementation in Chebfun of CXN seems to far more accurate than presented here.

We were seeing a error growth like this in MATLAB when the ratio of gamma functions was not computed by the recurrence...

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if somebody could find out why, that would be amazing

from fasttransforms.jl.

I have a test for the use of Stirling's series for a ratio of gamma functions compared with BigFloat. It appears to compute with high relative accuracy.

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It appears that the chebfun code for cheb2leg is more involved than the other cases probably for reasons of numerical stability. From what I understand,

• the right diagonal scaling is built into the columns of the Hankel matrix
• the diagonal of the Toeplitz matrix is zeroed and the diagonal of the connection matrix, with its opposite sign, is computed and applied separately

@ajt60gaibb, it would be straightforward to translate the code, but is there a way to use the existing modular, functional approach in this repository?

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I don't remember writing cheb2leg, did someone else do this?

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On 21 May 2016, at 7:54 AM, Richard Mikael Slevinsky [email protected] wrote:

It appears that the chebfun code for cheb2leg is more involved than the other cases probably for reasons of numerical stability. From what I understand,

the right diagonal scaling is built into the columns of the Hankel matrix
the diagonal of the Toeplitz matrix is zeroed and the diagonal of the connection matrix, with its opposite sign, is computed and applied separately
@ajt60gaibb, it would be straightforward to translate the code, but is there a way to use the existing modular, functional approach in this repository?

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I did... Kuan wanted a new tag for the package, and your paper says v0.0.4 will have the leg2cheb, cheb2leg, jac2jac methods via the T.*H approach. So i tried adding them as described in the paper (S 5.1 in particular for cheb2leg), and ended up with this issue.

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I've introduced diagonal scaling into the partial Cholesky decomposition of the Hankel part in cf090e3 (thanks to @marcusdavidwebb for pointing this out!). Running the same gist now results in much better accuracy. To me, the Chebfun code has obfuscated this detail, but I think the methods are now equivalent. Are we satisfied?

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OK. Great! Is there something we should change to make the diagonal scaling more clear in the Chebfun code?

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Maybe a short description of the incorporation of two-sided diagonal scaling (by num) in the comments in lines 78-82 of https://github.com/chebfun/chebfun/blob/development/cheb2leg.m would suffice.

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