Opened 5 years ago
Last modified 5 years ago
#21639 new enhancement
Implement derivative of gegenbauer(n,a,x) wrt to a
Reported by: | mafra | Owned by: | |
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Priority: | minor | Milestone: | sage-7.4 |
Component: | calculus | Keywords: | gegenbauer, ultraspherical, derivative |
Cc: | rws | Merged in: | |
Authors: | Carlos R. Mafra | Reviewers: | |
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | #21645 | Stopgaps: |
Description (last modified by )
Implement
In [3]: gegenbauer(n,m,x) Out[3]: gegenbauer(n, m, x) In [4]: _.diff(m) Out[4]: n - 1 ____ ╲ ╲ ⎛⎛ -k + n ⎞ ╲ ⎜⎝2⋅(-1) + 2⎠⋅(k + m)⋅gegenbauer(k, m, x) ⎛ 2⋅k + 2 ╱ ⎜────────────────────────────────────────────── + ⎜────────────────────── ╱ ⎝ (-k + n)⋅(k + 2⋅m + n) ⎝(k + 2⋅m)⋅(2⋅k + 2⋅m + ╱ ‾‾‾‾ k = 0 ⎞ 2 ⎞ ⎟ ─── + ───────────⎟⋅gegenbauer(n, m, x) ⎟ 1) k + 2⋅m + n ⎠ ⎠
Previous description was (now implemented):
I noticed that the derivative of the gegenbauer polynomial wrt x was not implemented, so I wrote a patch for it.
I used the formula C'(n,a,x) = 2*a*C(n-1,a+1,x)
With the patch applied I get, for example:
sage: var('a'); sage: derivative(gegenbauer(2,a,x),x) 4*(a + 1)*a*x
Attachments (1)
Change History (13)
Changed 5 years ago by
comment:1 Changed 5 years ago by
- Description modified (diff)
comment:2 Changed 5 years ago by
- Cc rws added
comment:3 Changed 5 years ago by
comment:4 Changed 5 years ago by
Thanks. I would like to extend this ticket with derivatives on the second index:
In [3]: gegenbauer(n,m,x) Out[3]: gegenbauer(n, m, x) In [4]: _.diff(m) Out[4]: n - 1 ____ ╲ ╲ ⎛⎛ -k + n ⎞ ╲ ⎜⎝2⋅(-1) + 2⎠⋅(k + m)⋅gegenbauer(k, m, x) ⎛ 2⋅k + 2 ╱ ⎜────────────────────────────────────────────── + ⎜────────────────────── ╱ ⎝ (-k + n)⋅(k + 2⋅m + n) ⎝(k + 2⋅m)⋅(2⋅k + 2⋅m + ╱ ‾‾‾‾ k = 0 ⎞ 2 ⎞ ⎟ ─── + ───────────⎟⋅gegenbauer(n, m, x) ⎟ 1) k + 2⋅m + n ⎠ ⎠
I'll both implement in Pynac directly. See also #21644.
comment:5 Changed 5 years ago by
The second case would depend on #21645.
comment:6 Changed 5 years ago by
The first case is https://github.com/pynac/pynac/commit/6587fdc83ef1b002b583339bca70bcf94e50535c
Doctests will be in the Pynac upgrade ticket, can be already seen at https://github.com/pynac/sage/commit/85726b14a96eb91c8fd70c1e76e729230e606a15
comment:7 follow-up: ↓ 8 Changed 5 years ago by
Thanks for adding the derivative wrt m, I wasn't aware of this identity (where did you get it?).
Btw, what is the guiding principle to decide if things should go into pynac or pure sage?
I was under the impression that 'performance' was the primary factor to move things to pynac. Is this the case here?
comment:8 in reply to: ↑ 7 ; follow-up: ↓ 9 Changed 5 years ago by
Replying to mafra:
Thanks for adding the derivative wrt m, I wasn't aware of this identity (where did you get it?).
This is from SymPy.
Btw, what is the guiding principle to decide if things should go into pynac or pure sage?
I was under the impression that 'performance' was the primary factor to move things to pynac. Is this the case here?
You are right, in many cases this is the reason. One other reason can be that when all the function code is already in Pynac then for clarity add the new functionality there.
comment:9 in reply to: ↑ 8 Changed 5 years ago by
Replying to rws:
Replying to mafra:
Btw, what is the guiding principle to decide if things should go into pynac or pure sage?
I was under the impression that 'performance' was the primary factor to move things to pynac. Is this the case here?
You are right, in many cases this is the reason. One other reason can be that when all the function code is already in Pynac then for clarity add the new functionality there.
IMHO, Pynac should be reserved for performance reasons only.
Adding the derivative functions to Pynac only complicates the situation here as we lose a unified handling of the derivatives within the Python file, like in this case with 'hermite' needing a fix in Pynac and gen_laguerre in Sage. I don't think this adds to 'clarity'.
So I disagree with moving the derivative of gegenbauer to Pynac, as it is not performance motivated and could be easily done in Sage.
comment:10 follow-up: ↓ 11 Changed 5 years ago by
You would not believe how much slower any Python (and Cython) code is versus C/C++. That's for example why SymPy has extraordinary performance problems for even simple computations, and why they urgently push the SymEngine project. I therefore feel justified to replace *any Python/Cython code with C++, and contrarily to SymPy we are in the fortunate position that Pynac is already integrated in Sage. While the time when to do this transcription may certainly be a matter of debate, the necessity itself is not.
comment:11 in reply to: ↑ 10 Changed 5 years ago by
Replying to rws:
You would not believe how much slower any Python (and Cython) code is versus C/C++. That's for example why SymPy has extraordinary performance problems for even simple computations, and why they urgently push the SymEngine project. I therefore feel justified to replace *any Python/Cython code with C++, and contrarily to SymPy we are in the fortunate position that Pynac is already integrated in Sage. While the time when to do this transcription may certainly be a matter of debate, the necessity itself is not.
Fair enough, thanks!
comment:12 Changed 5 years ago by
- Dependencies set to #21645
- Description modified (diff)
- Summary changed from [PATCH] Implement derivative of gegenbauer(n,a,x) wrt to x to Implement derivative of gegenbauer(n,a,x) wrt to a
Patch implementing the derivative of gegenbauer(n,a,x) wrt x