newknt

New break distribution

Syntax

newknots = newknt(f,newl) newknt(f) [...,distfn] = newknt(...)

Description

newknots = newknt(f,newl) returns the knot sequence whose interior knots cut the basic interval of f into newl pieces, in such a way that a certain piecewise linear monotone function related to the high derivative of f is equidistributed.

The intent is to choose a knot sequence suitable to the fine approximation of a function g whose rough approximation in f is assumed to contain enough information about g to make this feasible.

newknt(f) uses for newl its default value, namely the number of polynomial pieces in f.

[...,distfn] = newknt(...) also returns, in distfn, the ppform of that piecewise linear monotone function being equidistributed.

Examples

If the error in the least-squares approximation sp to some data x,y by a spline of order k seems uneven, you might try for a more equitable distribution of knots by using

spap2(newknt(sp),k,x,y);

For another example, see Solving a Nonlinear ODE with a Boundary Layer by Collocation.

Algorithms

This is the Fortran routine NEWNOT in PGS. With k the order of the piecewise-polynomial function f in pp, the function |D^kf| is approximated by a piecewise constant function obtained by local, discrete, differentiation of the variation of D^k^–1f. The new break sequence is chosen to subdivide the basic interval of the piecewise-polynomial f in such a way that

$\int_{newknots (i)}^{newknots (i + 1)} | D^{k} f |^{1 / k} = const, i = k : k + newl - 1$