This example shows how to construct and work with the ppform of a spline in Curve Fitting Toolbox™.

A (univariate) piecewise polynomial, or *pp* for short, is characterized by its *break sequence*, `breaks`

say, and its *coefficient array*, `coefs`

say, of the local power form of its polynomial pieces. The break sequence is assumed to be strictly increasing,

breaks(1) < breaks(2) < ... < breaks(l+1),

with `l`

the number of polynomial pieces that make up the pp. In the figure below, `breaks`

is [0,1,4,6], hence `l`

is 3.

While these polynomials may be of varying degrees, they are all recorded as polynomials of the same *order* `k`

, i.e., the coefficient array `coefs`

is of size `[l,k]`

, with `coefs(j,:)`

containing the `k`

coefficients in the local power form for the `j`

-th polynomial piece.

Here is an example of a pp made up of three quadratic polynomials, i.e., `l`

= `k`

= 3. The breaks are marked in red.

sp = spmak([0 1 4 4 6],[2 -1]); pp = fn2fm(sp,'pp') ; breaks = fnbrk(pp,'b'); coefs = fnbrk(pp,'c'); coefs(3,[1 2]) = [0 1]; pp = ppmak(breaks,coefs,1); fnplt(pp,[breaks(1)-1 breaks(2)],'g',1.8) hold on fnplt(pp, breaks([2 3]),'b',1.8) fnplt(pp,[breaks(3),breaks(4)+1],'m',1.8) lp1 = length(breaks); xb = repmat(breaks,3,1); yb = repmat([2;-2.2;NaN],1,lp1); plot(xb(:),yb(:),'r') axis([-1 7 -2.5 2.3]) hold off

The precise description of the pp in terms of the break sequence `breaks`

and the coefficient array `coefs`

is

pp(t) = polyval(coefs(j,:), t-breaks(j))

for breaks(j) <= t < breaks(j+1)

where, to recall,

polyval(a,x) = a(1)*x^(k-1) + a(2)*x^(k-2) + ... + a(k)*x^0.

For the pp in the figure above, `breaks(1)`

is 0, and `coefs(1,:)`

is [-1/2 0 0], while `breaks(3)`

is 4, and `coefs(3,:)`

is [0 1 -1]. For `t`

not in `[breaks(1) .. breaks(l+1))`

, `pp(t)`

is defined by extending the first or last polynomial piece.

A pp is usually constructed through a process of interpolation or approximation. But it is also possible to make one up in ppform from scratch, using the command `ppmak`

. For example, the pp above can be obtained as

breaks = [0 1 4 6]; coefs = [1/2 0 0 -1/2 1 1/2 0 1 -1]; fn = ppmak(breaks,coefs)

fn = struct with fields: form: 'pp' breaks: [0 1 4 6] coefs: [3×3 double] pieces: 3 order: 3 dim: 1

This returns, in `fn`

, a complete description of this pp function in the so-called ppform.

Various commands in Curve Fitting Toolbox can operate on this form. The remaining sections show some examples.

To evaluate a pp, use the `fnval`

command.

fnval(fn, -1:7)

ans = Columns 1 through 7 0.5000 0 0.5000 1.0000 0.5000 -1.0000 0 Columns 8 through 9 1.0000 2.0000

To differentiate a pp, use the `fnder`

command.

dfn = fnder ( fn ); hold on fnplt(dfn, 'jumps','y', 3) hold off h1 = findobj(gca,'Color','y'); legend(h1,{'First Derivative'},'location','SW')

Note that the derivative of the example pp is continuous at 1 but discontinuous at 4. Also note that, by default, `fnplt`

plots a ppform on its *basic interval*, i.e., on the interval `[breaks(1) .. breaks(end)]`

.

You can also use `fnder`

to take the second derivative of a pp.

ddfn = fnder(fn, 2); hold on fnplt( ddfn ,'j', 'k', 1.6) hold off h2 = findobj(gca,'Color','k'); legend([h1 h2],{'First Derivative' 'Second Derivative'},'location','SW')

The second derivative is piecewise constant.

Note that differentiation via `fnder`

is done separately for each polynomial piece. For example, although the first derivative has a jump discontinuity across 4, the second derivative is not infinite there. This has consequences when we integrate the second derivative.

To integrate a pp, use the `fnint`

command.

iddfn = fnint(ddfn); hold on fnplt(iddfn, 'b', .5) hold off h3 = findobj(gca,'Color','b', 'LineWidth',.5); legend([h1 h2 h3],{'First Derivative' 'Second Derivative' ... 'Integral of Second Derivative'},'location','SW')

Note that integration of the second derivative does recover the first derivative, except for the jump across 4, which cannot be recovered, since the integral of any pp function is continuous.

You can obtain parts with the aid of the command `fnbrk`

. For example

```
breaks = fnbrk(fn, 'breaks')
```

breaks = 0 1 4 6

recovers the break sequence of the pp in `fn`

, while

piece2 = fnbrk(fn, 2);

recovers the second polynomial piece, as this plot confirms.

fnplt(pp,[breaks(1)-1 breaks(2)],'g',1.8) hold on fnplt(piece2, 'b', 2.5, breaks([2 3])+[-1 .5]) fnplt(pp,[breaks(3),breaks(4)+1],'m',1.8) plot(xb(:),yb(:),'r') title('The Polynomial that Supplies the Second Polynomial Piece') hold off axis([-1 7 -2.5 2.3])

A pp can also be vector-valued, to describe a curve, in 2-space or 3-space. In that case, each local polynomial coefficient is a vector rather than a number, but nothing else about the ppform changes. There is one additional part of the ppform to record this, the *dimension* of its target.

For example, here is a 2-vector-valued pp describing the unit square, as its plot shows. It is a 2D-curve, hence its dimension is 2.

square = ppmak(0:4, [1 0 0 1 -1 1 0 0 ; 0 0 1 0 0 1 -1 1]); fnplt(square,'r',2) axis([-.5 1.5 -.5 1.5]) axis equal title('A Vector-Valued PP that Describes a Square')

A pp in Curve Fitting Toolbox can also be multivariate, namely, a tensor product of univariate pp functions. The ppform of such a multivariate pp is only slightly more complicated, with `breaks`

now a cell array containing the break sequence for each variable, and `coefs`

now a multidimensional array. It is much harder to make up a non-random such function from scratch, so we won't try that here, particularly since the toolbox is meant to help with the construction of pp functions from some conditions about them. For example, the sphere in this figure is constructed with the aid of `csape`

.

x = 0:4; y = -2:2; s2 = 1/sqrt(2); v = zeros(3,7,5); v(3,:,:) = [0 1 s2 0 -s2 -1 0].'*[1 1 1 1 1]; v(2,:,:) = [1 0 s2 1 s2 0 -1].'*[0 1 0 -1 0]; v(1,:,:) = [1 0 s2 1 s2 0 -1].'*[1 0 -1 0 1]; sph = csape({x,y},v,{'clamped','periodic'}); fnplt(sph) axis equal axis off title('A Sphere Described by a Bicubic 3-Vector-Valued Spline')

While the ppform of a pp is efficient for *evaluation*, the *construction* of a pp from some data is usually handled more efficiently by first determining its *B-form*, i.e., its representation as a linear combination of B-splines.

For this, look at the example Introduction to the B-form.

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