ppmak

Put together spline in ppform

Syntax

ppmak(breaks,coefs)
ppmak
ppmak(breaks,coefs,d)
ppmak(breaks,coefs,sizec)


Description

The command ppmak(...) puts together a spline in ppform from minimal information, with the rest inferred from that information. fnbrk provides any or all of the parts of the completed description. In this way, the actual data structure used for the storage of the ppform is easily modified without any effect on the various fn... commands that use this construct. However, the casual user is not likely to use ppmak explicitly, relying instead on the various spline construction commands in the toolbox to construct particular splines.

ppmak(breaks,coefs) returns the ppform of the spline specified by the break information in breaks and the coefficient information in coefs. How that information is interpreted depends on whether the function is univariate or multivariate, as indicated by breaks being a sequence or a cell array.

If breaks is a sequence, it must be nondecreasing, with its first entry different from its last. Then the function is assumed to be univariate, and the various parts of its ppform are determined as follows:

1. The number l of polynomial pieces is computed as length(breaks)-1, and the basic interval is, correspondingly, the interval [breaks(1) .. breaks(l+1)].

2. The dimension d of the function's target is taken to be the number of rows in coefs. In other words, each column of coefs is taken to be one coefficient. More explicitly, coefs(:,i*k+j) is assumed to contain the jth coefficient of the (i+1)st polynomial piece (with the first coefficient the highest and the kth coefficient the lowest, or constant, coefficient). Thus, with kl the number of columns of coefs, the order k of the piecewise-polynomial is computed as fix(kl/l).

After that, the entries of coefs are reordered, by the command

 coefs = reshape(permute(reshape(coefs,[d,k,l]),[1 3 2]),[d*l,k])

in order to conform with the internal interpretation of the coefficient array in the ppform for a univariate spline. This only applies when you use the syntax ppmak(breaks,coefs) where breaks is a sequence (row vector), not when it is a cell-array. The permutation is not made when you use the three-argument forms of ppmak. For the three-argument forms only a reshape is done, not a permute.

If breaks is a cell array, of length m, then the function is assumed to be m-variate (tensor product), and the various parts of its ppform are determined from the input as follows:

1. The m-vector l has length(breaks{i})-1 as its ith entry and, correspondingly, the m-cell array of its basic intervals has the interval [breaks{i}(1) .. breaks{i}(end)] as its ith entry.

2. The dimension d of the function's target and the m-vector k of (coordinate-wise polynomial) orders of its pieces are obtained directly from the size of coefs, as follows.

   1. If coefs is an m-dimensional array, then the function is taken to be scalar-valued, i.e., d is 1, and the m-vector k is computed as size(coefs)./l. After that, coefs is reshaped by the command coefs = reshape(coefs,[1,size(coefs)]).

   2. If coefs is an (r+m)-dimensional array, with sizec = size(c) say, then d is set to sizec(1:r), and the vector k is computed as sizec(r+(1:m))./l. After that, coefs is reshaped by the command coefs = reshape(coefs,[prod(d),sizec(r+(1:m))]).

Then, coefs is interpreted as an equivalent array of size [d,l(1),k(1),l(2),k(2),...,l(m),k(m)], with its (:,i(1),r(1),i(2),r(2),...,i(m),r(m))th entry the coefficient of

$${\displaystyle \prod _{\mu =1}^m{\left(x(\mu )-\text{breaks|}\mu ](i(\mu ))\right)}^{\left(k(\mu )-r(\mu )\right)}}$$

in the local polynomial representation of the function on the (hyper)rectangle with sides

$$[\text{breaks}|\mu ]\left(i(\mu )\right)\mathrm{..}\text{ breaks}|\mu ]\left(i(\mu )+1\right)],\mu =1:m$$

This is, in fact, the internal interpretation of the coefficient array in the ppform of a multivariate spline.

ppmak prompts you for breaks and coefs.

ppmak(breaks,coefs,d) with d a positive integer, also puts together the ppform of a spline from the information supplied, but expects the function to be univariate. In that case, coefs is taken to be of size [d*l,k], with l obtained as length(breaks)-1, and this determines the order, k, of the spline. With this, coefs(i*d+j,:) is taken to be the jth components of the coefficient vector for the (i+1)st polynomial piece.

ppmak(breaks,coefs,sizec) with sizec a row vector of positive integers, also puts together the ppform of a spline from the information supplied, but interprets coefs to be of size sizec (and returns an error when prod(size(coefs)) differs from prod(sizec)). This option is important only in the rare case that the input argument coefs is an array with one or more trailing singleton dimensions. For, MATLAB^® suppresses trailing singleton dimensions, hence, without this explicit specification of the intended size of coefs, ppmak would interpret coefs incorrectly.

Examples

The two splines

p1 = ppmak([1 3 4],[1 2 5 6;3 4 7 8]);
p2 = ppmak([1 3 4],[1 2;3 4;5 6;7 8],2);

have exactly the same ppform (2-vector-valued, of order 2). But the second command provides the coefficients in the arrangement used internally.

ppmak([0:2],[1:6]) constructs a piecewise-polynomial function with basic interval [0..2] and consisting of two pieces of order 3, with the sole interior break 1. The resulting function is scalar, i.e., the dimension d of its target is 1. The function happens to be continuous at that break since the first piece is x|→x² + 2x + 3, while the second piece is x|→4(x – 1)² + 5(x–1) + 6.

When the function is univariate and the dimension d is not explicitly specified, then it is taken to be the row number of coefs. The column number should be an integer multiple of the number l of pieces specified by breaks. For example, the statement ppmak([0:2],[1:3;4:6]) leads to an error, since the break sequence [0:2] indicates two polynomial pieces, hence an even number of columns are expected in the coefficient matrix. The modified statement ppmak([0:1],[1:3;4:6]) specifies the parabolic curve x|→(1,4)x² + (2,5)x + (3,6). In particular, the dimension d of its target is 2. The differently modified statement ppmak([0:2],[1:4;5:8]) also specifies a planar curve (i.e., d is 2), but this one is piecewise linear; its first polynomial piece is x|→(1,5)x + (2,6).

Explicit specification of the dimension d leads, in the univariate case, to a different interpretation of the entries of coefs. Now the column number indicates the polynomial order of the pieces, and the row number should equal d times the number of pieces. Thus, the statement ppmak([0:2],[1:4;5:8],2) is in error, while the statement ppmak([0:2],[1:4;5:8],1) specifies a scalar piecewise cubic whose first piece is x|→x³ + 2x² + 3x + 4.

If you wanted to make up the constant polynomial, with basic interval [0..1] say, whose value is the matrix eye(2), then you would have to use the full optional third argument, i.e., use the command

pp = ppmak(0:1,eye(2),[2,2,1,1]);

Finally, if you want to construct a 2-vector-valued bivariate polynomial on the rectangle [–1 .. 1] x [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

   pp = ppmak({[-1 1],[0 1]},coefs);

will fail, producing a scalar-valued function of order 2 in each variable, as will

   pp = ppmak({[-1 1],[0 1]},coefs,size(coefs));

while the following command will succeed:

   pp = ppmak({[-1 1],[0 1]},coefs,[2 2 1]);

See the example “Intro to ppform” for other examples.