Put together spline in ppform
ppmak(breaks,coefs)
ppmak
ppmak(breaks,coefs,d)
ppmak(breaks,coefs,sizec)
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:
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)].
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:
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.
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.
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)]).
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
in the local polynomial representation of the function on the (hyper)rectangle with sides
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.
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.