Learn more about Spline Toolbox

Available Commands

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
fnmin(pp,[a,b])	Finds the minimum value in given interval
fnzeros(pp,[a,b])	Finds the zeros in the given interval
pj = fnbrk(pp,j)	Pulls out the jth polynomial piece
pc = fnbrk(pp,[a b])	Restricts/extends to the interval [a..b]
po = fnxtr(pp,order)	Extends outside its basic interval by polynomial of specified order
fnplt(pp,[a,b])	Plots on given 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 

A Piecewise-Polynomial Function, Its Breaks, and the Polynomial Giving Its Third Piece

The figure above 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

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