Construction of B-form

Usually, a spline is constructed from some information, like function values and/or derivative values, or as the approximate solution of some ordinary differential equation. But it is also possible to make up a spline from scratch, by providing its knot sequence and its coefficient sequence to the commandspmak.

For example, if you enter

sp = spmak(1:10,3:8);

you supply the uniform knot sequence1:10and the coefficient sequence3:8. Because there are 10 knots and 6 coefficients, the order must be 4(= 10 – 6), i.e., you get a cubic spline. The command

fnbrk(sp)

prints out the constituent parts of the B-form of this cubic spline, as follows:

knots(1:n+k) 1 2 3 4 5 6 7 8 9 10 coefficients(d,n) 3 4 5 6 7 8 number n of coefficients 6 order k 4 dimension d of target 1

Further,fnbrkcan be used to supply each of these parts separately.

But the point of the Curve Fitting Toolbox™ spline functionality is that there shouldn't be any need for you to look up these details. You simply usespas an argument to commands that evaluate, differentiate, integrate, convert, or plot the spline whose description is contained insp.

Working With B-form Splines

The following commands are available for spline work. There isspmakandfnbrkto make up a spline and take it apart again. Usefn2fmto convert from B-form to ppform. You can evaluate, differentiate, integrate, minimize, find zeros of, plot, refine, or selectively extrapolate a spline with the aid offnval,fnder,fndir,fnint,fnmin,fnzeros,fnplt,fnrfn, andfnxtr.

There are five commands for generating knot sequences:

In addition, there is:

To display a splinecurvewith given two-dimensional coefficient sequence and a uniform knot sequence, usespcrv.

You can also write your own spline construction commands, in which case you will need to know the following. The construction of a spline satisfying some interpolation or approximation conditions usually requires acollocation matrix, i.e., the matrix that, in each row, contains the sequence of numbersD^rB_j,k(τ), i.e., therth derivative at τ of thejth B-spline, for allj, for somerand some site τ. Such a matrix is provided byspcol. An optional argument allows for this matrix to be supplied byspcolin a space-saving spline-almost-block-diagonal-form or as a MATLAB^®sparse matrix. It can be fed toslvblk, a command for solving linear systems with an almost-block-diagonal coefficient matrix. If you are interested in seeing howspcolandslvblkare used in this toolbox, have a look at the commandsspapi,spap2, andspaps.

此外,还有为构建例程cubicsplines.csapiandcsapeprovide the cubic spline interpolant at knots to given data, using the not-a-knot and various other end conditions, respectively. A parametric cubic spline curve through given points is provided bycscvn. The cubicsmoothingspline is constructed incsaps.

Example: B-form Spline Approximation to a Circle

As another simple example,

points = .95*[0 -1 0 1;1 0 -1 0]; sp = spmak(-4:8,[points points]);

provides a planar,quartic, spline curve whose middle part is a pretty good approximation to a circle, as the plot on the next page shows. It is generated by a subsequent

plot(points(1,:),points(2,:),'x'), hold on fnplt(sp,[0,4]), axis equal square, hold off

Insertion of additionalcontrol points $$\left(\pm 0.95,\pm 0.95\right)/\sqrt{1.9}$$would make a visually perfect circle.

Here are more details. The splinecurve generated has the form Σ⁸_j=1B_j,5a(:,j), with -4:8the uniform knot sequence, and with its control pointsa(:,j) the sequence(0,α),(–α,0),(0,–α),(α,0),(0,α),(–α,0),(0,–α),(α,0)withα=0.95. Only the curve part between the parameter values 0 and 4 is actually plotted.

To get a feeling for how close to circular this part of the curve actually is, compute its unsignedcurvature. The curvature κ(t) at the curve point γ(t) = (x(t), y(t)) of a space curve γ can be computed from the formula

in which x', x″, y', and y” are the first and second derivatives of the curve with respect to the parameter used (t). Treat the planar curve as a space curve in the (x,y)-plane, hence obtain the maximum and minimum of its curvature at 21 points as follows:

t = linspace(0、4、21);zt型= 0(大小(t));dsp = fnder(sp); dspt = fnval(dsp,t); ddspt = fnval(fnder(dsp),t); kappa = abs(dspt(1,:).*ddspt(2,:)-dspt(2,:).*ddspt(1,:))./... (sum(dspt.^2)).^(3/2); [min(kappa),max(kappa)] ans = 1.6747 1.8611

So, while the curvature is not quite constant, it is close to 1/radius of the circle, as you see from the next calculation:

1/norm(fnval(sp,0)) ans = 1.7864