id	summary	reporter	owner	description	type	status	priority	milestone	component	version	severity	resolution	keywords	cc
542	splines need work	Paul Price	gusciora@…	"psSpline1D has not yet crystallised.  There are many questions marked by 
""XXX"" in the code that should have been raised a long time ago.  This is not
really a bug against the psLib implementation (because the SDRS needs work as
well) but it serves as a central place where we can sort things out.

A large part of the problem came from my past misunderstanding of splines --- I
thought we could fit a spline with n knots to N points, where N > n.  This is
not so, but we need N == n.  I don't see much of a use for linear splines, so I
suggest we concentrate solely on cubic splines.  If we want linear splines in
the future, we can make a new type, or use an enum.  I think we also want to
remove the generality of the knot locations --- it's not done in general, and I
don't see at first glance how it might be done, since one needs the y values at
those points which is what we're trying to generate in the first place!  These
decisions will help simplify psSpline1D.

We want to fit a cubic spline to a set of points, (x,y).  Let's dispense with
the errors (yErr) because I haven't seen any implementation of splines that deal
with errors (because it's meant for interpolation, not fitting).  We shouldn't
be specifying the location of knots when we allocate the spline, because they're
set according to the particular data that's used to set the spline.

We need to decide on the type of spline (natural, clamped, not-a-knot).  (I am
disappointed that the current implementation does not mention what type of
spline it assumes.)  Unless there's a reasoned objection, let's go with natural
splines (I figure they look smoother than not-a-knot, but not overdamped like
clamped).

Now, the current definition of psSpline1D is sufficient for our needs.

We need to change the allocator.  All the allocator need do is allocate the
psSpline1D, because we dispensed with the order, and the number of spline pieces
is set when we do a psVectorFitSpline1D.
    psSpline1D *psSpline1DAlloc(void);

We get rid of psSpline1DAllocGeneric --- the knots are set by psVectorFitSpline1D.

psSpline1DEval and psSpline1DEvalVector are fine, but we need to change
psVectorFitSpline1D:

    psSpline1D psVectorFitSpline1D(const psVector *x, const psVector *y);

We have dispensed with yErr.  There's no need to feed in a spline, because
there's nothing to set; the function will call psSpline1DAlloc, set the knots
(from the x vector, or the indices if !x), and set the polynomials.

I think that's it.  Please let me know if you have concerns, questions or
recommendations.  I don't think the above changes need a great deal of more
work, but it's mainly a reorganisation and simplification of what's already there.

Some useful resources:
www.amath.washington.edu/~bloss/amath352_lectures/chapter4.pdf
www.amath.washington.edu/courses/352-winter-2002/spline_note.pdf"	defect	closed	high		math	0.7.0	normal	fixed		george.gusciora@…