knt2brk, knt2mlt (Spline Toolbox)

Breaks and multiplicities from a knot sequence

Syntax

[breaks, mults] = knt2brk(knots)
[m, sortedt] = knt2mlt(t)

Description

The idea is to extract the distinct elements from a sequence, as well as their multiplicities in that sequence, with multiplicity taken in two slightly different senses.

The statement knt2brk(knots) returns the distinct elements in knots, and in increasing order. The optional second output argument provides the multiplicity with which each distinct element occurs in knots. In particular, the two outputs, breaks and mults, are of the same length, and knt2brk is complementary to brk2knt in that, for any knot sequence knots, the two commands [xi,mlts] = knt2brk(knots); knots1 = brk2knt(xi,mlts); give knots1 equal to knots.

The statement m = knt2mlt(t) returns a vector of the same length as t, with m(i) counting the number of entries to the left of the ith entry in sortedt = sort(t) that are equal to that entry. This kind of multiplicity vector is needed in spapi or spcol where such multiplicity is taken to specify which particular derivatives are to be matched at the points in t. Precisely, if t is nondecreasing and z is a vector of the same length, then sp = spapi(knots, t, z) attempts to construct a spline s (with knot sequence knots) for which D^m(i)s(t(i)) = z(i), all i. The optional second argument returns sort(t).

Neither knt2brk nor knt2mlt are likely to be used by the casual user of this toolbox.

Examples

[xi,mlts]=knt2brk([1 2 3 3 1 3]) returns [1 2 3] for xi and [2 1 3] for mlts.

[m,t]=knt2mlt([1 2 3 3 1 3]) returns [0 1 0 0 1 2] for m and
[1 1 2 3 3 3] for t.