mkpp - Make piecewise polynomial (2025)

Table of Contents

Syntax Description Examples Create Piecewise Polynomial with Polynomials of Several Degrees Create Piecewise Polynomial with Repeated Pieces Input Arguments breaks — Break points vector coefs — Polynomial coefficients matrix d — Dimension scalar | vector Output Arguments pp — Piecewise polynomial structure Extended Capabilities C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™. Thread-Based Environment Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool. Version History See Also MATLAB Command Americas Europe Asia Pacific

Make piecewise polynomial

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Syntax

pp = mkpp(breaks,coefs)

pp = mkpp(breaks,coefs,d)

Description

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

example

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

Examples

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Create Piecewise Polynomial with Polynomials of Several Degrees

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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].

breaks = [0 4 10 15];coefs = [0 1 -1 1 1; 0 0 1 -2 53; -1 6 1 4 77];pp = mkpp(breaks,coefs)

pp = struct with fields: form: 'pp' breaks: [0 4 10 15] coefs: [3x5 double] pieces: 3 order: 5 dim: 1

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.

xq = 0:0.01:15;plot(xq,ppval(pp,xq))line([4 4],ylim,'LineStyle','--','Color','k')line([10 10],ylim,'LineStyle','--','Color','k')

Create Piecewise Polynomial with Repeated Pieces

Open Live Script

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 online([-4 -4],ylim,'LineStyle','--')line([0 0],ylim,'LineStyle','--')line([4 4],ylim,'LineStyle','--')hold off

Input Arguments

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`breaks` — Break points
vector

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

Data Types: single | double

`coefs` — Polynomial coefficients
matrix

Polynomial coefficients, specified as an L-by-k matrixwith the ith row coefs(i,:) containing the localcoefficients of an order k polynomial on the ithinterval, [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 tosignify that the piecewise polynomial has coefficient values of size d.

Data Types: single | double

Output Arguments

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`pp` — Piecewise polynomial
structure

Piecewise polynomial, returned as a structure. Use this structurewith the ppval function toevaluate the piecewise polynomial at one or more query points. Thestructure has these fields.

Field	Description
`form`	`'pp'` for piecewise polynomial
`breaks`	Vector of length `L+1` with strictlyincreasing elements that represent the start and end of each of `L`intervals
`coefs`	`L`-by-`k` matrix witheach row`coefs(i,:)`containing the localcoefficients of an order`k` polynomial on theithinterval,`[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 arelocal coefficients for each interval, you must subtract the lowerendpoint of the corresponding knot interval to use the coefficientsin a conventional polynomial equation. In other words, for the coefficients [a,b,c,d] onthe interval [x1,x2], the corresponding polynomialis

$f (x) = a {(x - x_{1})}^{3} + b {(x - x_{1})}^{2} + c (x - x_{1}) + d .$

Extended Capabilities

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

Usage notes and limitations:

The output structure pp differsfrom the pp structure in MATLAB^®. In MATLAB, ppval cannotuse the pp structure from the code generator. Forcode generation, ppval cannot use a pp structurecreated by MATLAB. unmkpp can use a MATLAB pp structurefor code generation.
To create a MATLAB pp structure froma pp structure created by the code generator:
- In code generation, use unmkpp toreturn the piecewise polynomial details to MATLAB.
- In MATLAB, use mkpp to createthe pp structure.
If you do not provide d, then coefs mustbe 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 mustbe a column vector or d must have at least twoelements.
If you provide d and d is 1,then d must be a constant. Otherwise, if the inputto ppval is nonscalar, then the shape of theoutput of ppval can differ from ppval in MATLAB.
If you provide d, then it musthave a fixed length. One of the following sets of statements mustbe true:
1. Suppose that m = length(d) and npieces= length(breaks) - 1.
```
size(coefs,j) = d(j) size(coefs,m+1) = npiecessize(coefs,m+2) = order 
```
  j = 1,2,...,m.The dimension m+2 must be fixed length.
2. Suppose that m = length(d) and npieces= length(breaks) - 1.
```
size(coefs,1) = prod(d)*npiecessize(coefs,2) = order 
```
  The second dimensionmust be fixed length.
If you do not provide d, then thefollowing statements must be true:
Suppose that m = length(d) and npieces= length(breaks) - 1.
```
size(coefs,1) = prod(d)*npiecessize(coefs,2) = order
```
The second dimension must befixed length.

Thread-Based Environment
Run code in the background using MATLAB® `backgroundPool` or accelerate code with Parallel Computing Toolbox™ `ThreadPool`.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

Version History

Introduced before R2006a

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