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Re: [Help-gsl] Question on Bsplines knot multiplicity

From: Rhys Ulerich
Subject: Re: [Help-gsl] Question on Bsplines knot multiplicity
Date: Wed, 9 Jul 2014 08:40:39 -0500

Hi Foivos,

> this is just to verify mainly, it is in my understanding that the sequence
> of breakpoints is non decreasing (but not strictly), i.e.
> \xi_1 <= \xi_2 <= ... <= \xi_n.
> thus intermediate breakpoints can have some multiplicity (say, if we allow
> some of them to be equal).

Yes, in for general B-spline theory the sequence of breakpoints only
need be increasing, not strictly increasing.  People tend to refer to
them as a "knot" sequence in that setting.  Repeated knots reduce
local continuity in the basis.  As far as a I know, when people use
"breakpoint" they tend to mean a strictly increasing sequence
producing a basis with the same continuity everywhere.

> Could please someone verify that GSL support
> this (I see it working - just for me ease of mind)?

I don't know about "support".  Certainly the PPPACK routines on which
the B-splines in the GSL are built should handle this correctly.  But
I'd suspect there may be edge cases in the code because, as far as I
know, they were intended for "breakpoints" and not more general

That said, it may all work fine.  Any contributions you wish to make
in this regard would be much appreciated.

- Rhys

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