Cubic spline data interpolation
Spline Interpolation of Sine Data
spline to interpolate a sine curve over unevenly-spaced sample points.
x = [0 1 2.5 3.6 5 7 8.1 10]; y = sin(x); xx = 0:.25:10; yy = spline(x,y,xx); plot(x,y,'o',xx,yy)
Spline Interpolation with Specified Endpoint Slopes
Use clamped or complete spline interpolation when endpoint slopes are known. To do this, you can specify the values vector with two extra elements, one at the beginning and one at the end, to define the endpoint slopes.
Create a vector of data and another vector with the -coordinates of the data.
x = -4:4; y = [0 .15 1.12 2.36 2.36 1.46 .49 .06 0];
Interpolate the data using
spline and plot the results. Specify the second input with two extra values
[0 y 0] to signify that the endpoint slopes are both zero. Use
ppval to evaluate the spline fit over 101 points in the interpolation interval.
cs = spline(x,[0 y 0]); xx = linspace(-4,4,101); plot(x,y,'o',xx,ppval(cs,xx),'-');
Extrapolation Using Cubic Spline
Extrapolate a data set to predict population growth.
Create two vectors to represent the census years from 1900 to 1990 (
t) and the corresponding United States population in millions of people (
t = 1900:10:1990; p = [ 75.995 91.972 105.711 123.203 131.669 ... 150.697 179.323 203.212 226.505 249.633 ];
Extrapolate and predict the population in the year 2000 using a cubic spline.
ans = 270.6060
Spline Interpolation of Angular Data
Generate the plot of a circle, with the five data points
y(:,2),...,y(:,6) marked with o's. The matrix
y contains two more columns than does
y(:,end) as the endslopes. The circle starts and ends at the point (1,0), so that point is plotted twice.
x = pi*[0:.5:2]; y = [0 1 0 -1 0 1 0; 1 0 1 0 -1 0 1]; pp = spline(x,y); yy = ppval(pp, linspace(0,2*pi,101)); plot(yy(1,:),yy(2,:),'-b',y(1,2:5),y(2,2:5),'or') axis equal
Spline Interpolation of Sine and Cosine Data
Use spline to sample a function over a finer mesh.
Generate sine and cosine curves for a few values between 0 and 1. Use spline interpolation to sample the functions over a finer mesh.
x = 0:.25:1; Y = [sin(x); cos(x)]; xx = 0:.1:1; YY = spline(x,Y,xx); plot(x,Y(1,:),'o',xx,YY(1,:),'-') hold on plot(x,Y(2,:),'o',xx,YY(2,:),':') hold off
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.
x — x-coordinates
x-coordinates, specified as a vector. The
x specifies the points at which the data
given. The elements of
x must be unique.
y — Function values at x-coordinates
vector | matrix | array
Function values at x-coordinates, specified as a numeric vector, matrix, or
y typically have the same
y also can have exactly two more elements
x to specify endslopes.
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
have exactly two more elements.
The endslopes of the cubic spline follow these rules:
yare vectors of the same size, then the not-a-knot end conditions are used.
yis a scalar, then it is expanded to have the same length as the other and the not-a-knot end conditions are used.
yis a vector that contains two more values than
xhas entries, then
splineuses the first and last values in
yas the endslopes for the cubic spline. For example, if
yis a vector, then:
y(2:end-1)gives the function values at each point in
y(1)gives the slope at the beginning of the interval located at
y(end)gives the slope at the end of the interval located at
yis a matrix or an
N-dimensional array with
y(:,...,:,j+1)gives the function values at each point in
j = 1:length(x)
y(:,:,...:,1)gives the slopes at the beginning of the intervals located at
y(:,:,...:,end)gives the slopes at the end of the intervals located at
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
s — 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
s is related to the sizes of
yis a vector, then
shas 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 piecewise polynomial 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
You also can perform spline interpolation using the
interp1function with the command
splineperforms interpolation on rows of an input matrix,
interp1performs interpolation on columns of an input matrix.
A tridiagonal linear system (possibly with several right-hand
sides) is solved for the information needed to describe the coefficients
of the various cubic polynomials that make up the interpolating spline.
unmkpp. These routines form a small suite
of functions for working with piecewise polynomials. For access to
more advanced features, see
the Curve Fitting Toolbox™ spline functions.
 de Boor, Carl. A Practical Guide to Splines. Springer-Verlag, New York: 1978.
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
Code generation does not report an error for infinite endslopes in
If you generate code for the
pp = spline(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
If you supply
xq, and if
yhas a variable-size and is not a variable-length vector, then the orientation of vector outputs in the generated code might not match the orientation in MATLAB.
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™.
Usage notes and limitations:
The input argument
ymust be non-sparse.
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).