>>>>> Abby Spurdle
>>>>> on Sun, 3 May 2020 16:15:17 +1200 writes:
>> and just today a colleague asked me about spline interpolation
>> with general 2nd derivative boundary conditions
>> s''(x_1) = s2_1, s''(x_n) = s2_2
actually I was wrong... I *read* it as the above, but what he
really wanted was what the wikipedia page class "clamped"
boundary conditions, i.e., for the *first* derivative
s'(x_1) = s1_1, s'(x_n) = s1_2
> It should possible via cubic Hermite splines.
and indeed that is available via cubic Hermite splines available
in R via stats package's splinefunH()
{which I wrote when implementing "monoH.FC"}
> A nontrivial design decision in my package was the computation of
> slopes at the endpoints.
> (Something which I got wrong, twice...)
The "wikiversity" has a very nice small math lecture on this
https://en.wikiversity.org/wiki/Cubic_Spline_Interpolation
which derives *both* cases (first and 2nd derivative boundary conditions),
the only draw back to quickly do it with ('base R') is that I
need to "translate" that (2nd derivative values $M_i$) parametrization
into either the one used into the (a,b,c,y)-parametrizat of the
default spline() / splinefun() methods or the Hilbert spline
form for splinefunH(x[],y[],m[]).
Martin
> My guess is that I could write a function in about 60 to 90 minutes,
> including all the testing and calculus.
> However, I need to note two things:
> (1) Cubic Hermite splines do not have a continuous second derivative.
well, many don't if you allow any slopes at node. However, if
you only set 2 boundary conditions *instead* of the natural
spline ones f''(x_1) = f''(x_1) = 0,
you can still remain in C_2 (i.e. continuous 2nd derivative).
(And that's also what the above wikiversity lecture provides).
> (2) Specifying the second derivatives (at the endpoints) would prevent
> the user from specifying the first derivatives (aka the slopes).
> Could you please confirm if the function would still be of interest...?
Well, as I know spent enough time reading and thinking, I'd
really like to add method = "clamped" to splinefun() and also
the other one where fix the 2nd derivatives (to arbitrary values
instead of zero).
So if you already have the R code leading up to one of the 2
"parametrizations" we use in spline() / splinefun(), I'd be
grateful, if you have the time.
Martin
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