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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`j`

th coefficient of the`(i+1)`

st polynomial piece (with the first coefficient the highest and the`k`

th 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`i`

th entry and, correspondingly, the`m`

-cell array of its basic intervals has the interval`[breaks{i}(1) .. breaks{i}(end)]`

as its`i`

th 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

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

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

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

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 `j`

th 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*|→*x*^{2} +
2*x* + 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)*x*^{2} +
(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*|→*x*^{3} +
2*x*^{2} + 3*x* +
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.