Spline Toolbox

Use

Here are some operations you can perform on a piecewise-polynomial.

v=fnval(pp,x)	Evaluates
dpp=fnder(pp)	Differentiates
dirpp=fndir(pp,dir)	Differentiates in the direction dir
ipp=fnint(pp)	Integrates
pj=fnbrk(pp,j)	Pulls out the jth polynomial piece
pc=fnbrk(pp,[a b])	Restricts/extends to the interval [a..b]
fnplt(pp)	Plots piecewise-polynomial on its basic interval
sp = fn2fm(pp,'B-')	Converts to B-form
pr = fnrfn(pp,morebreaks)	Inserts additional breaks

Inserting additional breaks comes in handy when one wants to add two piecewise-polynomials with different breaks, as is done in the command fncmb.

To illustrate the use of some of these commands, here is a plot of the particular piecewise-polynomial we just made up. First, the basic plot:

• x = linspace(-5.5,-.5,101);
  plot(x, fnval(pp,x),'x')
  

Then add to the plot the breaklines:

• breaks=fnbrk(pp,'b'); yy=axis; hold on 
  for j=1:fnbrk(pp,'l')+1 
     plot(breaks([j j]),yy(3:4))
  end
  

Finally, superimpose on that plot the plot of the polynomial that supplies the third polynomial piece:

• plot(x,fnval(fnbrk(pp,3),x),'linew',1.3)
  set(gca,'ylim',[-60 -10]), hold off
  
  
  

Figure 2-9: A Piecewise-Polynomial Function, Its Breaks, and the Polynomial Giving Its Third Piece

Figure 2-9 is the final picture. It shows the piecewise-polynomial as a sequence of points and, solidly on top of it, the polynomial from which its third polynomial piece is taken. It is quite noticeable that the value of a piecewise-polynomial at a break is its limit from the right, and that the value of the piecewise-polynomial outside its basic interval is obtained by extending its leftmost, respectively its rightmost, polynomial piece.

While the ppform of a piecewise-polynomial is efficient for evaluation, the construction of a piecewise-polynomial from some data is usually more efficiently handled by determining first its B-form, i.e., its representation as a linear combination of B-splines.

Construction

The B-form