spmak

Put together spline in B-form

Syntax

spmak(knots,coefs)
spmak(knots,coefs,sizec)
spmak
sp = spmak(knots,coeffs)


Description

The command spmak(...) puts together a spline function in B-form, from minimal information, with the rest inferred from the input. fnbrk returns all the parts of the completed description. In this way, the actual data structure used for the storage of this form is easily modified without any effect on the various fn... commands that use this construct.

spmak(knots,coefs) returns the B-form of the spline specified by the knot information in knots and the coefficient information in coefs.

The action taken by spmak depends on whether the function is univariate or multivariate, as indicated by knots being a sequence or a cell array. For the description, let sizec be size(coefs).

If knots is a sequence (required to be non-decreasing), then the spline is taken to be univariate, and its order k is taken to be length(knots)-sizec(end). This means that each `column' coefs(:,j) of coefs is taken to be a B-spline coefficient of the spline, hence the spline is taken to be sizec(1:end-1)-valued. The basic interval of the B-form is [knots(1) .. knots(end)].

Knot multiplicity is held to be ≤ k. This means that the coefficient coefs(:,j) is simply ignored in case the corresponding B-spline has only one distinct knot, i.e., in case knots(j) equals knots(j+k).

If knots is a cell array, of length m, then the spline is taken to be m-variate, and coefs must be an (r+m)-dimensional array, – except when the spline is to be scalar-valued, in which case, in contrast to the univariate case, coefs is permitted to be an m-dimensional array, but sizec is reset by

sizec = [1, sizec]; r = 1;

The spline is sizec(1:r)-valued. This means the output of the spline is an array with r dimensions, e.g., if sizec(1:2) = [2, 3] then the output of the spline is a 2-by-3 matrix.

The spline is sizec(1:r)-valued, the ith entry of the m-vector k is computed as length(knots{i}) - sizec(r+i), i=1:m, and the ith entry of the cell array of basic intervals is set to [knots{i}(1), knots{i}(end)].

spmak(knots,coefs,sizec) lets you supply the intended size of the array coefs. Assuming that coefs is correctly sized, this is of concern only in the rare case that coefs has one or more trailing singleton dimensions. For, MATLAB^® suppresses trailing singleton dimensions, hence, without this explicit specification of the intended size of coefs, spmak would interpret coefs incorrectly.

spmak prompts you for knots and coefs.

sp = spmak(knots,coeffs) returns the spline sp.

Examples

spmak(1:6,0:2) constructs a spline function with basic interval [1. .6], with 6 knots and 3 coefficients, hence of order 6 - 3 = 3.

spmak(t,1) provides the B-spline B(·|t) in B-form.

The coefficients may be d-vectors (e.g., 2-vectors or 3-vectors), in which case the resulting spline is a curve or surface (in R² or R³).

If the intent is to construct a 2-vector-valued bivariate polynomial on the rectangle [–1..1] × [0..1], linear in the first variable and constant in the second, say

 coefs = zeros([2 2 1]); coefs(:,:,1) = [1 0;0 1];

then the straightforward

sp = spmak({[-1 -1 1 1],[0 1]},coefs);

will result in the error message 'There should be no more knots than coefficients', because the trailing singleton dimension of coefs will not be perceived by spmak, while proper use of that third argument, as in

sp = spmak({[-1 -1 1 1],[0 1]},coefs,[2 2 1]);

will succeed. Replacing here [2 2 1] by size(coefs) would not work.

See the example “Intro to B-form” for other examples.

Diagnostics

There will be an error return if the proposed knot sequence fails to be nondecreasing, or if the coefficient array is empty, or if there are not more knots than there are coefficients. If the spline is to be multivariate, then this last diagnostic may be due to trailing singleton dimensions in coefs.