- Piecewise Polynomials
- Intro
- Basic idea, picture, and definition.
- Key questions:
- Where are the knots?
- What degree for each piece?
- How much continuity between pieces?

- Basic idea, picture, and definition.
- Splines
- Def: a spline is a degree k piecewise polynomial with k-1
continuous derivatives, i.e., it's as smooth as it can be.
- Piecewise linear spline interpolation: just
``connect-the-dots''.
- Quadratic splines are not used as often as cubic splines because
cubic splines look a little better and they can be constructed
for little more work.
- Cubic spline interpolation: the most common case.
- Some theory. There are theorems which show that the accuracy
of spline interpolation is better than polynomial interpolation
in the sense that the interpolant more closely matches the
(unknown) function from which the data is derived, across the
interval where the data lies. For example, see Equation (2) on
page 164 of our text (error estimate for polynomial
interpolation) and the theorem on linear splines on page 322 of
the text.
The bottom line is that, for sufficiently smooth functions f(x), the maximum difference between f(x) and the piecewise linear (spline) goes to zero like O(h

^{2}), where h is the interval size. We can make no similar statement about the error shrinking as n grows for polynomial interpolation (it may not even shrink at all, in some cases). For cubic splines, the corresponding theorem says the error shrinks like O(h^{4}).

- Def: a spline is a degree k piecewise polynomial with k-1
continuous derivatives, i.e., it's as smooth as it can be.

- Intro