at-points: change at-points operators to use Chebyshev polynomials of the first kind (normalized)#1963
at-points: change at-points operators to use Chebyshev polynomials of the first kind (normalized)#1963zatkins-dev wants to merge 2 commits intomainfrom
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… the first kind (normalized)
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we could possibly squeeze an extra register out of the cuda and hip impls, since they use 3 doubles for the derivative calculation right now and we should only need two with this change. |
| static int CeedChebyshevPolynomialsAtPoint(CeedScalar x, CeedInt n, CeedScalar *chebyshev_x) { | ||
| chebyshev_x[0] = 1.0; | ||
| chebyshev_x[1] = 2 * x; | ||
| chebyshev_x[1] = x; |
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Can you toss references to Chebyshev of the first kind for the future grad student trying to grok what we're doing here? (Also for the derivative function)
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Yeah for sure, though tbh the options are wikipedia or a book (or both)
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Yea, Wikipedia is probably good for a quick reference to get people oriented, and I like the refs being here in this file. That makes the most sense I think
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| chebyshev_x[1] = 1.0; | ||
| chebyshev_x[2] = 2 * x; | ||
| chebyshev_x[0] = 1.0; |
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specifically we should note here that the second kind is used to make the derivatives of the first kind (probably why I stuck with the second kind throughout? I don't recall)
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(I know you have it there mathematically, but I'm trying to picture the theoretical new grad student)
| @@ -47,7 +47,7 @@ const CeedBasis CEED_BASIS_NONE = &ceed_basis_none; | |||
| **/ | |||
| static int CeedChebyshevPolynomialsAtPoint(CeedScalar x, CeedInt n, CeedScalar *chebyshev_x) { | |||
| chebyshev_x[0] = 1.0; | |||
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| chebyshev_x[0] = 1.0; | |
| // Chebyshev polynomials of the first kind, 3 term recurrence: T_0(x) = 1, T_1(x) = x, T_n = 2 x T_{n-1} - T_{n-2} | |
| // For more info, see e.g. https://en.wikipedia.org/wiki/Chebyshev_polynomials#Recurrence_definition | |
| chebyshev_x[0] = 1.0; |
| chebyshev_x[1] = 1.0; | ||
| chebyshev_x[2] = 2 * x; |
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| chebyshev_x[1] = 1.0; | |
| chebyshev_x[2] = 2 * x; | |
| // Note, these are Chebyshev polynomials of the 2nd kind, used for derivatives | |
| // See e.g. https://en.wikipedia.org/wiki/Chebyshev_polynomials#Differentiation_and_integration for the recurrence | |
| chebyshev_x[1] = 1.0; | |
| chebyshev_x[2] = 2 * x; |
2 participants
Purpose:
We're currently using Chebyshev polynomials of the second kind, which don't have their range bounded to the same [-1,1] as their domain. I'm really not sure why we would be using the second kind, since they're numerically less stable and have a more complicated derivative.
This doesn't cause any issues with tests here or in Ratel.
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