# fn2fm

Convert to specified form

## Syntax

`g = fn2fm(f,form) `

sp = fn2fm(f,'B-',sconds)

fn2fm(f)

## Description

`g = fn2fm(f,form) `

describes the same
function as is described by `f`

, but in the form specified by the
character vector or string scalar `form`

. Choices for
`form`

are `'B-'`

, `'pp'`

,
`'BB'`

, `'rB'`

, `'rp'`

, for
the B-form, the ppform, the BBform, and the two rational spline forms,
respectively.

The B-form describes a function as a weighted sum of the B-splines of a given order k for a given knot sequence, and the BBform (or, Bernstein-Bézier form) is the special case when each knot in that sequence appears with maximal multiplicity, k. The ppform describes a function in terms of its local polynomial coefficients. The B-form is good for constructing and/or shaping a function, while the ppform is cheaper to evaluate.

Conversion from a polynomial form to the corresponding rational form is possible only if the function in the polynomial form is vector-valued, in which case its last component is designated as the denominator. Converting from a rational form to the corresponding polynomial form simply reverses this process by reinterpreting the denominator of the function in the rational form as an additional component of the piecewise polynomial function.

Conversion to or from the stform is not possible at present.

If `form`

is `'B-'`

(and `f`

is
in ppform), then the actual smoothness of the function in ```
f
```

across each of its interior breaks has to be guessed. This is done by
looking, for each interior break, for the first derivative whose jump across that
break is not *small* compared to the size of that derivative
nearby. The default tolerance used in this is `1.e-12`

.

`sp = fn2fm(f,'B-',sconds) `

permits you to
supply, as the input argument `sconds`

, a tolerance (strictly
between 0 and 1) to be used in the conversion from ppform to B-form.

Alternatively, you can input `sconds`

as a vector with integer
entries, with at least as many entries as the ppform in `f`

has
*interior* breaks. In that case, `sconds(i)`

specifies the number of smoothness conditions to be used across the
`i`

th *interior* break. If the function in
`f`

is a tensor product, then `sconds`

, if
given, must be a cell array.

`fn2fm(f) `

converts a possibly old version
of a form into its present version.

## Examples

`sp = fn2fm(spline(x,y),'B-')`

gives the interpolating cubic
spline provided by the MATLAB^{®} command `spline`

, but in B-form rather than in
ppform.

p0 = ppmak([0 1],[3 0 0]); p1 = fn2fm(fn2fm(fnrfn(p0,[.4 .6]),'B-'),'pp');

gives `p1`

identical to `p0`

(up to round-off in
the coefficients) since the spline has no discontinuity in any derivative across the
additional breaks introduced by `fnrfn`

, hence conversion to B-form
ignores these additional breaks, and conversion to ppform does not retain any knot
multiplicities (like the knot multiplicities introduced, by conversion to B-form, at
the endpoints of the spline's basic interval).

## Cautionary Note

When going from B-form to ppform, any jump discontinuity at the first and last
knot, `t(1)`

or `t(end)`

, will be lost since the
ppform considers *f* to be defined outside its basic interval by extension of the first, respectively, the last polynomial piece. For
example, while `sp=spmak([0 1],1)`

gives the characteristic
function of the interval [`0`

..`1`

],
`pp=fn2fm(spmak([0 1],1),'pp')`

is the constant polynomial,
*x*|→1.

## Algorithms

For a multivariate (tensor-product) function, univariate algorithms are applied in each variable.

For the conversion from B-form (or BBform) to ppform, the utility command
`sprpp`

is used to convert the B-form of all polynomial pieces
to their local power form, using repeated knot insertion at the left endpoint.

The conversion from B-form to BBform is accomplished by inserting each knot enough times to increase its multiplicity to the order of the spline.

The conversion from ppform to B-form makes use of the dual functionals discussed in . Without further information, such a conversion has to ascertain the
actual smoothness across each interior break of the function in
`f`

.