aveknt

Provide knot averages

Syntax

tstar = aveknt(t,k)


Description

tstar = aveknt(t,k) returns the averages of successive k-1 knots, i.e., the sites

$$t_i*:=(t_{i+1}+\cdot \cdot \cdot +t_{i+k-1})/(k-1),i=1:n$$

which are recommended as good interpolation site choices when interpolating from splines of order k with knot sequence $$t={\left(t_i\right)}_{i=1}^{n+k}$$.

Examples

aveknt([1 2 3 3 3],3) returns the vector [2.5000 3.0000], while aveknt([1 2 3],3) returns the empty vector.

With k and the strictly increasing sequence breaks given, the statements

t = augknt(breaks,k); x = aveknt(t);
sp = spapi(t,x,sin(x));

provide a spline interpolant to the sine function on the interval [breaks(1)..breaks(end)].

For sp the B-form of a scalar-valued univariate spline function, and with tstar and a computed as

tstar = aveknt(fnbrk(sp,'knots'),fnbrk(sp,'order'));
a = fnbrk(sp,'coefs');

the points (tstar(i), a(i)) constitute the control points of the spline, i.e., the vertices of the spline's control polygon.