spap2 (Spline Toolbox)

Least-squares spline approximation

Syntax

sp = spap2(knots,k,x,y)
sp = spap2(l,k,x,y)
sp = spap2(knots,k,x,y,w)

Description

spap2(knots,k,x,y) returns the B-form of the spline of order k with the given knot sequence knots for which

```
(*)   y(:,j) = f(x(j)), all j 
```

in the weighted mean-square sense, meaning that the sum

is minimized, with default weights equal to 1. If the sites x satisfy the (Schoenberg-Whitney) conditions

then there is a unique spline (of the given order and knot sequence) satisfying (*) exactly. No spline is returned unless (**) is satisfied for some subsequence of x.

sp = spap2(l,k,x,y) also returns the B-form of a least-squares spline approximant, but with the knot sequence chosen for you. The knot sequence is obtained by applying aptknt to an appropriate subsequence of x. The resulting piecewise-polynomial consists of l polynomial pieces and has k-2 continuous derivatives. If you feel that a different distribution of the interior knots might do a better job, follow this up with

```
sp1 = spap2(newknt(sp),k,x,y));
```

spap2(...,x,y,w) lets you specify the weights w in the error measure (given above). w must be a vector of the same size as x, with nonnegative entries.

spap2({korl1,...,korlm},k,{x1,...,xm},y) provides a least-squares spline approximation to gridded data. Here, each korli is either a knot sequence or a positive integer. Further, k must be an m-vector, and y must be an (m+1)-dimensional array, with y(:,i1,...,im) the datum to be fitted at the site [x{1}(i1),...,x{m}(im)], all i1, ..., im. However, if the spline is to be scalar-valued, then, in contrast to the univariate case, y is permitted to be an m-dimensional array, in which case y(i1,...,im) is the datum to be fitted at the site [x{1}(i1),...,x{m}(im)], all i1, ..., im.

spap2({korl1,...,korlm},k,{x1,...,xm},y,w) also lets you specify the weights. In this m-variate case, w must be a cell array with m entries, with w{i} a nonnegative vector of the same size as xi, or else w{i} must be empty, in which case the default weights are used in the ith variable.

Examples

```
sp = spap2(augknt([a,xi,b],4),4,x,y)
```

is the least-squares approximant to the data x, y, by cubic splines with two continuous derivatives, basic interval [a..b], and interior breaks xi, provided xi has all its entries in (a..b) and the conditions (**) are satisfied in some fashion. In that case, the approximant consists of length(xi)+1 polynomial pieces. If you do not want to worry about the conditions (**) but merely want to get a cubic spline approximant consisting of l polynomial pieces, use instead

```
sp = spap2(l,4,x,y); 
```

If the resulting approximation is not satisfactory, try using a larger l. Else use

```
sp = spap2(newknt(sp),4,x,y);
```

for a possibly better distribution of the knot sequence. In fact, if that helps, repeating it may help even more.

As another example, spap2(1, 2, x, y); provides the least-squares straight-line fit to data x,y, while

w = ones(size(x)); w([1 end]) = 100; spap2(1,2, x,y,w);

forces that fit to come very close to the first data point and to the last.

Algorithm

spcol is called on to provide the almost block-diagonal collocation matrix , and slvblk solves the linear system (*) in the (weighted) least-squares sense, using a block QR factorization.

Gridded data are fitted, in tensor-product fashion, one variable at a time, taking advantage of the fact that a univariate weighted least-squares fit depends linearly on the values being fitted.

See Also

slvblk, spapi, spcol

sorted spapi