fnrfn

Refine partition of form

Syntax

```g = fnrfn(f,addpts) ```

Description

`g = fnrfn(f,addpts) ` describes the same function as does `f`, but uses more terms to do it. This is of use when the sum of two or more functions of different forms is wanted or when the number of degrees of freedom in the form is to be increased to make fine local changes possible. The precise action depends on the form in `f`.

If the form in `f` is a B-form or BBform, then the entries of `addpts` are inserted into the existing knot sequence, subject to the following restriction: The multiplicity of no knot exceed the order of the spline. The equivalent B-form with this refined knot sequence for the function given by `f` is returned.

If the form in `f` is a ppform, then the entries of `addpts` are inserted into the existing break sequence, subject to the following restriction: The break sequence be strictly increasing. The equivalent ppform with this refined break sequence for the function in `f` is returned.

`fnrfn` does not work for functions in stform.

If the function in `f` is m-variate, then `addpts` must be a cell array, ```{addpts1,..., addptsm}```, and the refinement is carried out in each of the variables. If the ith entry in this cell array is empty, then the knot or break sequence in the ith variable is unchanged.

Examples

Construct a spline in B-form, plot it, then apply two midpoint refinements, and also plot the control polygon of the resulting refined spline, expecting it to be quite close to the spline itself:

```k = 4; sp = spapi( k, [1,1:10,10], [cos(1),sin(1:10),cos(10)] ); fnplt(sp), hold on sp3 = fnrfn(fnrfn(sp)); plot( aveknt( fnbrk(sp3,'knots'),k), fnbrk(sp3,'coefs'), 'r') hold off ```
A third refinement would have made the two curves indistinguishable.

Use `fnrfn` to add two B-splines of the same order:

```B1 = spmak([0:4],1); B2 = spmak([2:6],1); B1r = fnrfn(B1,fnbrk(B2,'knots')); B2r = fnrfn(B2,fnbrk(B1,'knots')); B1pB2 = spmak(fnbrk(B1r,'knots'),fnbrk(B1r,'c')+fnbrk(B2r,'c')); fnplt(B1,'r'),hold on, fnplt(B2,'b'), fnplt(B1pB2,'y',2) hold off ```

Algorithms

The standard knot insertion algorithm is used for the calculation of the B-form coefficients for the refined knot sequence, while Horner's method is used for the calculation of the local polynomial coefficients at the additional breaks in the refined break sequence.