# Documentation

### This is machine translation

Translated by
Mouseover text to see original. Click the button below to return to the English verison of the page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

# knt2brk, knt2mlt

Convert knots to breaks and their multiplicities

## Syntax

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

## Description

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

`knt2brk(knots) ` returns the distinct elements in `knots`, and in increasing order, hence is the same as `unique(knots)`.

`[breaks,mults] = knt2brk(knots) ` additionally provides, in `mults`, the multiplicity with which each distinct element occurs in `knots`. Explicitly, `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`.

`m = knt2mlt(t) ` returns a vector of the same length as `t`, with `m(i)` counting, in the vector `sort(t)`, the number of entries before its` `ith entry 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 sites 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 Dm(i)s(t(i)) equals z(i), all i.

`[m,sortedt] = knt2mlt(t) ` also returns the output from `sort(t)`.

Neither `knt2brk` nor `knt2mlt` is 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`.