On-line key, Exam 2, Fall 2005

Some of you never considered symmetry: the whole thrust of this problem had to do with symmetry -- if you just write down the "old cubic spline rules" (e.g. matching first and second derivatives, and interpolation), you've missed the whole point! The point is that we're fitting an even function, so we'd better make sure that our spline is even!
This student deduced everything possible: a values zero, b values zero; c₀=c₁; d₀=-d₁. The derivative must be zero at zero, to preserve symmetry (hence the condition on the b's).
Furthermore, whatever is deduced here must be in agreement with what you find for natural and clamped splines: these constraints are more general than any specific spline.
The following elegant solution was from the in-class exam: aside from a subtraction error (which might have been caught with more time), it's as succinct as possible!
These papers from the redo show two different forward difference schemes: you could go with a three-point, but your life is easier with two-point! Don't make your life harder than it needs to be....
The tough part was the Taylor polynomial: you need to show more than the leading order error, since we want to use this formula in the Richardson's extrapolation to follow. And your analysis should make some reference to the forward difference scheme you use. If it's independent of scheme, then you're undoubtably in trouble....

This student makes reference to the orders in the error expansion when choosing coefficients for the extrapolation (the fours and the eights in the extrapolation terms). Without the form, these values are only right if you get lucky!
Using two-point forward difference, it's possible to get to O(h⁵).
Some used the Lagrange form, which was really ugly: Newton's form was a lot more elegant. The plot is really only of interest on the interval [1,1.05], where the Hermite is approximating the function. Outside that we already can guess what's going to happen!
The error term is a variant of the Taylor formula, as I mentioned: it should have a fourth derivative, if it's multiplying a quartic (and it does!).

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