Vector-Valued Functions
The toolbox supports vector-valued splines. For example, if you want a spline curve through given planar points
, then the statements
xy = [x;y]; df = diff(xy,1,2);
t = cumsum([0, sqrt([1 1]*(df.*df))]);
cv = csapi(t,xy);
provide such a spline curve, using chord-length parametrization and cubic spline interpolation with the
not-a-knot end condition, as can be verified by the statements
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:
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)))
 | Using the Spline Fits | | Fitting Values at N-D Grid |  |
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