Spline Toolbox

Approximation Space

We seek an approximate solution by collocation from piecewise cubics with a suitable break sequence; for instance,

• breaks = [0:4]/4;
  

Since cubics are of order 4, we have

• k = 4;
  

and since requires two smoothness conditions across each interior break, we want knot multiplicity = 4-2 = 2, hence use the knot sequence   

• knots = sort([0 0 breaks breaks 1 1]);
  

which we could also have obtained as knots = augknt(breaks,4,2). This gives a quadruple knot at both 0 and 1, which is consistent with the fact that we have cubics, i.e., have order 4.

This implies that we have

• n = length(knots)-k
  >> n = 10
  

i.e., 10 degrees of freedom.

Problem

Discretization