Piecewise Cubic Hermite Interpolating Polynomial (PCHIP)
Data Interpolation with
Compare the interpolation results produced by
makima for two different data sets. These functions all perform different forms of piecewise cubic Hermite interpolation. Each function differs in how it computes the slopes of the interpolant, leading to different behaviors when the underlying data has flat areas or undulations.
Compare the interpolation results on sample data that connects flat regions. Create vectors of
x values, function values at those points
y, and query points
xq. Compute interpolations at the query points using
makima. Plot the interpolated function values at the query points for comparison.
x = -3:3; y = [-1 -1 -1 0 1 1 1]; xq1 = -3:.01:3; p = pchip(x,y,xq1); s = spline(x,y,xq1); m = makima(x,y,xq1); plot(x,y,'o',xq1,p,'-',xq1,s,'-.',xq1,m,'--') legend('Sample Points','pchip','spline','makima','Location','SouthEast')
In this case,
makima have similar behavior in that they avoid overshoots and can accurately connect the flat regions.
Perform a second comparison using an oscillatory sample function.
x = 0:15; y = besselj(1,x); xq2 = 0:0.01:15; p = pchip(x,y,xq2); s = spline(x,y,xq2); m = makima(x,y,xq2); plot(x,y,'o',xq2,p,'-',xq2,s,'-.',xq2,m,'--') legend('Sample Points','pchip','spline','makima')
When the underlying function is oscillatory,
makima capture the movement between points better than
pchip, which is aggressively flattened near local extrema.
Interpolation with Piecewise Polynomial Structure
Create vectors for the
x values and function values
y, and then use
pchip to construct a piecewise polynomial structure.
x = -5:5; y = [1 1 1 1 0 0 1 2 2 2 2]; p = pchip(x,y);
Use the structure with
ppval to evaluate the interpolation at several query points. Plot the results.
xq = -5:0.2:5; pp = ppval(p,xq); plot(x,y,'o',xq,pp,'-.') ylim([-0.2 2.2])
x — Sample points
Sample points, specified as a vector. The vector
the points at which the data
y is given. The elements
x must be unique.
y — Function values at sample points
vector | matrix | array
Function values at sample points, specified as a numeric vector,
matrix, or array.
have the same length.
y is a matrix or array, then the values
in the last dimension,
y(:,...,:,j), are taken
as the values to match with
x. In that case, the
last dimension of
y must be the same length as
xq — Query points
scalar | vector | matrix | array
Query points, specified as a scalar, vector, matrix, or array. The points
xq are the x-coordinates
for the interpolated function values
yq computed by
p — Interpolated values at query points
scalar | vector | matrix | array
Interpolated values at query points, returned as a scalar, vector, matrix,
or array. The size of
p is related to the sizes of
yis a vector, then
phas the same size as
yis an array of size
Ny = size(y), then these conditions apply:
xqis a scalar or vector, then
xqis an array, then
pp — Piecewise polynomial
Piecewise polynomial, returned as a structure. Use this structure
ppval function to
evaluate the interpolating polynomials at one or more query points.
The structure has these fields.
Vector of length
Number of pieces,
Order of the polynomials
Dimensionality of target
Since the polynomial coefficients in
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
[x1,x2], the corresponding polynomial
Shape-Preserving Piecewise Cubic Interpolation
pchip interpolates using
a piecewise cubic polynomial with
On each subinterval , the polynomial is a cubic Hermite interpolating polynomial for the given data points with specified derivatives (slopes) at the interpolation points.
interpolates y, that is, , and the first derivative is continuous. The second derivative is probably not continuous so jumps at the are possible.
The cubic interpolant is shape preserving. The slopes at the are chosen in such a way that preserves the shape of the data and respects monotonicity. Therefore, on intervals where the data is monotonic, so is , and at points where the data has a local extremum, so does .
If y is a matrix, satisfies these properties for each row of y.
splineconstructs in almost the same way
pchipconstructs . However,
splinechooses the slopes at the differently, namely to make even continuous. This difference has several effects:
splineproduces a smoother result, such that is continuous.
splineproduces a more accurate result if the data consists of values of a smooth function.
pchiphas no overshoots and less oscillation if the data is not smooth.
pchipis less expensive to set up.
The two are equally expensive to evaluate.
 Fritsch, F. N. and R. E. Carlson. "Monotone Piecewise Cubic Interpolation." SIAM Journal on Numerical Analysis. Vol. 17, 1980, pp.238–246.
 Kahaner, David, Cleve Moler, Stephen Nash. Numerical Methods and Software. Upper Saddle River, NJ: Prentice Hall, 1988.
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
xmust be strictly increasing.
Code generation does not remove
If you generate code for the
pp = pchip(x,y)syntax, then you cannot input
ppvalfunction in MATLAB®. To create a MATLAB
ppstructure from a
ppstructure created by the code generator:
In code generation, use
unmkppto return the piecewise polynomial details to MATLAB.
In MATLAB, use
mkppto create the
Run code in the background using MATLAB®
backgroundPool or accelerate code with Parallel Computing Toolbox™
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.
Usage notes and limitations:
pchip(syntax is not supported for distributed arrays.
For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).