pp = mkpp(breaks,coefs) builds
a piecewise polynomial pp from its breaks and coefficients.
Use ppval to evaluate the piecewise
polynomial at specific points, or unmkpp to
extract details about the piecewise polynomial.

pp = mkpp(breaks,coefs,d) specifies
that the piecewise polynomial is vector-valued, such that the value
of each of its coefficients is a vector of length d.

Create a piecewise polynomial that has a cubic polynomial in the interval [0,4], a quadratic polynomial in the interval [4,10], and a quartic polynomial in the interval [10,15].

Evaluate the piecewise polynomial at many points in the interval [0,15] and plot the results. Plot vertical dashed lines at the break points where the polynomials meet.

Create and plot a piecewise polynomial with four intervals that alternate between two quadratic polynomials.

The first two subplots show a quadratic polynomial and its negation shifted to the intervals [-8,-4] and [-4,0]. The polynomial is

$$1-{(\frac{x}{2}-1)}^{2}=\frac{-{x}^{2}}{4}+x.$$

The third subplot shows a piecewise polynomial constructed by alternating these two quadratic pieces over four intervals. Vertical lines are added to show the points where the polynomials meet.

subplot(2,2,1)
cc = [-1/4 1 0];
pp1 = mkpp([-8 -4],cc);
xx1 = -8:0.1:-4;
plot(xx1,ppval(pp1,xx1),'k-')
subplot(2,2,2)
pp2 = mkpp([-4 0],-cc);
xx2 = -4:0.1:0;
plot(xx2,ppval(pp2,xx2),'k-')
subplot(2,1,2)
pp = mkpp([-8 -4 0 4 8],[cc;-cc;cc;-cc]);
xx = -8:0.1:8;
plot(xx,ppval(pp,xx),'k-')
hold on
line([-4 -4],ylim,'LineStyle','--')
line([0 0],ylim,'LineStyle','--')
line([4 4],ylim,'LineStyle','--')
hold off

Break points, specified as a vector of length L+1 with
strictly increasing elements that represent the start and end of each
of L intervals.

Data Types: single | double

coefs — Polynomial coefficients matrix

Polynomial coefficients, specified as an L-by-k matrix
with the ith row coefs(i,:) containing the local
coefficients of an order k polynomial on the ith
interval, [breaks(i), breaks(i+1)]. In other words,
the polynomial is coefs(i,1)*(X-breaks(i))^(k-1) + coefs(i,2)*(X-breaks(i))^(k-2)
+ ... + coefs(i,k-1)*(X-breaks(i)) + coefs(i,k).

Data Types: single | double

d — Dimension scalar | vector

Dimension, specified as a scalar or vector of integers. Specify d to
signify that the piecewise polynomial has coefficient values of size d.

Piecewise polynomial, returned as a structure. Use this structure
with the ppval function to
evaluate the piecewise polynomial at one or more query points. The
structure has these fields.

Field

Description

form

'pp' for piecewise polynomial

breaks

Vector of length L+1 with strictly
increasing elements that represent the start and end of each of L intervals

coefs

L-by-k matrix with
each row coefs(i,:) containing the local
coefficients of an order k polynomial on the ith
interval, [breaks(i),breaks(i+1)]

pieces

Number of pieces, L

order

Order of the polynomials

dim

Dimensionality of target

Since the polynomial coefficients in coefs are
local coefficients for each interval, you must subtract the lower
endpoint of the corresponding knot interval to use the coefficients
in a conventional polynomial equation. In other words, for the coefficients [a,b,c,d] on
the interval [x1,x2], the corresponding polynomial
is

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

The output structure pp differs
from the pp structure in MATLAB^{®}. In MATLAB, ppval cannot
use the pp structure from the code generator. For
code generation, ppval cannot use a pp structure
created by MATLAB. unmkpp can use a MATLAB pp structure
for code generation.

To create a MATLAB pp structure from
a pp structure created by the code generator:

In code generation, use unmkpp to
return the piecewise polynomial details to MATLAB.

In MATLAB, use mkpp to create
the pp structure.

If you do not provide d, then coefs must
be two-dimensional and have a fixed number of columns. In this case,
the number of columns is the order.

To define a piecewise constant polynomial, coefs must
be a column vector or d must have at least two
elements.

If you provide d and d is 1,
then d must be a constant. Otherwise, if the input
to ppval is nonscalar, then the shape of the
output of ppval can differ from ppval in MATLAB.

If you provide d, then it must
have a fixed length. One of the following sets of statements must
be true:

Suppose that m = length(d) and npieces
= length(breaks) - 1.

size(coefs,j) = d(j)
size(coefs,m+1) = npieces
size(coefs,m+2) = order

j = 1,2,...,m.
The dimension m+2 must be fixed length.

Suppose that m = length(d) and npieces
= length(breaks) - 1.

size(coefs,1) = prod(d)*npieces
size(coefs,2) = order

The second dimension
must be fixed length.

If you do not provide d, then the
following statements must be true:

Suppose that m = length(d) and npieces
= length(breaks) - 1.

size(coefs,1) = prod(d)*npieces
size(coefs,2) = order

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