spline
Cubic spline data interpolation
Description
Examples
Spline Interpolation of Sine Data
Use spline
to interpolate a sine curve over unevenlyspaced 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 $\mathit{y}$ with two extra elements, one at the beginning and one at the end, to define the endpoint slopes.
Create a vector of data $\mathit{y}$ and another vector with the $\mathit{x}$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 (p
).
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.
spline(t,p,2000)
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 x
. Therefore, spline
uses y(:,1)
and 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 spline
, pchip
, and makima
Compare the interpolation results produced by spline
, pchip
, and 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 spline
, pchip
, and 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, pchip
and 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, spline
and makima
capture the movement between points better than pchip
, which is aggressively flattened near local extrema.
Input Arguments
x
— xcoordinates
vector
xcoordinates, specified as a vector. The
vector x
specifies the points at which the data y
is
given. The elements of x
must be unique.
Data Types: single
 double
y
— Function values at xcoordinates
vector  matrix  array
Function values at xcoordinates, specified as a numeric vector, matrix, or
array. x
and y
typically have the same
length, but y
also can have exactly two more elements
than x
to specify endslopes.
If 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 x
or
have exactly two more elements.
The endslopes of the cubic spline follow these rules:
If
x
andy
are vectors of the same size, then the notaknot end conditions are used.If
x
ory
is a scalar, then it is expanded to have the same length as the other and the notaknot end conditions are used.If
y
is a vector that contains two more values thanx
has entries, thenspline
uses the first and last values iny
as the endslopes for the cubic spline. For example, ify
is a vector, then:y(2:end1)
gives the function values at each point inx
y(1)
gives the slope at the beginning of the interval located atmin(x)
y(end)
gives the slope at the end of the interval located atmax(x)
Similarly, if
y
is a matrix or anN
dimensional array withsize(y,N)
equal tolength(x)+2
, then:y(:,...,:,j+1)
gives the function values at each point inx
forj = 1:length(x)
y(:,:,...:,1)
gives the slopes at the beginning of the intervals located atmin(x)
y(:,:,...:,end)
gives the slopes at the end of the intervals located atmax(x)
Data Types: single
 double
xq
— Query points
scalar  vector  matrix  array
Query points, specified as a scalar, vector, matrix, or array. The points
specified in xq
are the xcoordinates
for the interpolated function values yq
computed by
spline
.
Data Types: single
 double
Output Arguments
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 y
and
xq
:
If
y
is a vector, thens
has the same size asxq
.If
y
is an array of sizeNy = size(y)
, then these conditions apply:If
xq
is a scalar or vector, thensize(s)
returns[Ny(1:end1) length(xq)]
.If
xq
is an array, thensize(s)
returns[Ny(1:end1) size(xq)]
.
pp
— Piecewise polynomial
structure
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 

breaks  Vector of length 
coefs 

pieces  Number of pieces, 
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
$$f\left(x\right)=a{\left(x{x}_{1}\right)}^{3}+b{\left(x{x}_{1}\right)}^{2}+c\left(x{x}_{1}\right)+d\text{\hspace{0.17em}}.$$
Tips
You also can perform spline interpolation using the
interp1
function with the commandinterp1(x,y,xq,'spline')
. Whilespline
performs interpolation on rows of an input matrix,interp1
performs interpolation on columns of an input matrix.
Algorithms
A tridiagonal linear system (possibly with several righthand
sides) is solved for the information needed to describe the coefficients
of the various cubic polynomials that make up the interpolating spline. spline
uses
the functions ppval
, mkpp
,
and unmkpp
. These routines form a small suite
of functions for working with piecewise polynomials. For access to
more advanced features, see interp1
or
the Curve Fitting Toolbox™ spline functions.
References
[1] de Boor, Carl. A Practical Guide to Splines. SpringerVerlag, New York: 1978.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
Input
x
must be strictly increasing.Code generation does not remove
y
entries withNaN
values.Code generation does not report an error for infinite endslopes in
y
.If you generate code for the
pp = spline(x,y)
syntax, then you cannot inputpp
to theppval
function in MATLAB^{®}. To create a MATLABpp
structure from app
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 thepp
structure.
If you supply
xq
, and ify
has a variablesize and is not a variablelength vector, then the orientation of vector outputs in the generated code might not match the orientation in MATLAB.
ThreadBased Environment
Run code in the background using MATLAB® backgroundPool
or accelerate code with Parallel Computing Toolbox™ ThreadPool
.
This function fully supports threadbased environments. For more information, see Run MATLAB Functions in ThreadBased Environment.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
Usage notes and limitations:
The input argument
y
must be nonsparse.
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced before R2006a
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