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A univariate *piecewise
polynomial* *f* is specified by its *break sequence* `breaks` and the *coefficient
array*** **`coefs` of
the local power form (see equation in Definition of ppform) of its polynomial pieces; see Multivariate Tensor Product Splines for a discussion of multivariate
piecewise-polynomials. The coefficients may be (column-)vectors, matrices,
even ND-arrays. For simplicity, the present discussion deals only
with the case when the coefficients are scalars.

The break sequence is assumed to be strictly increasing,

breaks(1) < breaks(2) < ... < breaks(l+1)

with `l` the number of polynomial pieces that
make up *f*.

While these polynomials may be of varying degrees, they are
all recorded as polynomials of the same *order* `k`,
i.e., the coefficient array `coefs` is of size `[l,k]`,
with `coefs(j,:)` containing the `k` coefficients
in the local power form for the `j`th
polynomial piece, from the highest to the lowest power; see equation
in Definition of ppform.

The items `breaks`, `coefs`, `l`,
and` k`, make up the *ppform* of *f*,
along with the dimension `d` of its coefficients;
usually `d` equals 1. The *basic interval* of this form is the interval
[`breaks(1) `.. `breaks(l`+`1)`].
It is the default interval over which a function in ppform is plotted
by the plot command `fnplt`.

In these terms, the precise description of the piecewise-polynomial *f* is

f(t) = polyval(coefs(j,:), t - breaks(j)) | (10-1) |

for breaks(*j*)≤*t*<breaks(*j*+1).

Here, `polyval`(`a`,`x`)
is the MATLAB^{®} function; it returns the number

This defines *f(t)* only for *t* in
the half-open interval `[breaks(1)`..`breaks(l+1))`.
For any other *t*, *f(t)* is defined
by

i.e., by extending the first, respectively
last, polynomial piece. In this way, a function in ppform has possible
jumps, in its value and/or its derivatives, only across the interior
breaks, `breaks(2:l)`. The end breaks, `breaks([1,l+1])`,
mainly serve to define the basic interval of the ppform.

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