## Vector-Valued Functions

The toolbox supports *vector-valued* splines. For example, if you want a spline *curve* through given planar points , then the following
code defines some data and then creates and plots such a spline curve,
using chord-length parameterization and
cubic spline interpolation with the not-a-knot end condition.

x=[19 43 62 88 114 120 130 129 113 76 135 182 232 298 ... 348 386 420 456 471 485 463 444 414 348 275 192 106 ... 30 48 83 107 110 109 92 66 45 23 22 30 40 55 55 52 34 20 16]; y=[306 272 240 215 218 237 275 310 368 424 425 427 428 ... 397 353 302 259 200 148 105 77 47 28 17 10 12 23 41 43 ... 77 96 133 155 164 157 148 142 162 181 187 192 202 217 245 266 303]; xy = [x;y]; df = diff(xy,1,2); t = cumsum([0, sqrt([1 1]*(df.*df))]); cv = csapi(t,xy); fnplt(cv), hold on, plot(x,y,'o'), hold off

If you then wanted to know the area enclosed by this curve, you would want to evaluate
the integral , with the point on
the curve corresponding to the parameter value . For the spline
curve in `cv`

just constructed, this can be done
exactly in one (somewhat complicated) command:

area = diff(fnval(fnint( ... fncmb(fncmb(cv,[0 1]),'*',fnder(fncmb(cv,[1 0]))) ... ),fnbrk(cv,'interval')));

To explain, `y=fncmb(cv,[0 1])`

picks out the
second component of the curve in `cv`

, ```
Dx=fnder(fncmb(cv,[1
0]))
```

provides the derivative of the first component, and `yDx=fncmb(y,'*',Dx)`

constructs
their pointwise product. Then `IyDx=fnint(yDx)`

constructs
the indefinite integral of `yDx`

and, finally, `diff(fnval(IyDx,fnbrk(cv,'interval')))`

evaluates
that indefinite integral at the endpoints of the basic interval and
then takes the difference of the second from the first value, thus
getting the definite integral of `yDx`

over its basic
interval. Depending on whether the enclosed area is to the right or
to the left as the curve point travels with increasing parameter,
the resulting number is either positive or negative.

Further, all the values `Y`

(if any) for which
the point `(X,Y)`

lies on the spline curve in `cv`

just
constructed can be obtained by the following (somewhat complicated)
command:

X=250; %Define a value of X Y = fnval(fncmb(cv,[0 1]), ... mean(fnzeros(fncmb(fncmb(cv,[1 0]),'-',X))))

To explain: `x = fncmb(cv,[1 0])`

picks out
the first component of the curve in `cv`

; ```
xmX
= fncmb(x,'-',X)
```

translates that component by `X`

;```
t
= mean(fnzeros(xmX))
```

provides all the parameter values for
which `xmX is `

zero, i.e., for which the first component
of the curve equals `X`

; `y = fncmb(cv,[0,1])`

picks
out the second component of the curve in `cv`

; and,
finally, `Y = fnval(y,t)`

evaluates that second component
at those parameter sites at which the first component of the curve
in `cv`

equals `X`

.

As another example of the use of vector-valued functions, suppose
that you have solved the equations of motion of a particle in some
specified force field in the plane, obtaining, at discrete times , the position as well as the
velocity stored in the 4-vector , as you would
if, in the standard way, you had solved the equivalent first-order
system numerically. Then the following statement, which uses cubic Hermite interpolation, will produce a plot of the particle
path:`fnplt(spapi(augknt(t,4,2),t,reshape(z,2,2*n)))`

.