Cubic spline data interpolation - MATLAB spline - MathWorks Deutschland (2024)

Cubic spline data interpolation

collapse all in page


s = spline(x,y,xq)

pp = spline(x,y)



s = spline(x,y,xq) returnsa vector of interpolated values s correspondingto the query points in xq. The values of s aredetermined by cubic spline interpolation of x and y.


pp = spline(x,y) returnsa piecewise polynomial structure for use by ppval andthe spline utility unmkpp.


collapse all

Spline Interpolation of Sine Data

Open Live Script

Use 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)

Cubic spline data interpolation - MATLAB spline- MathWorks Deutschland (1)

Spline Interpolation with Specified Endpoint Slopes

Open Live Script

Use clamped or complete spline interpolation when endpoint slopes are known. To do this, you can specify the values vector y with two extra elements, one at the beginning and one at the end, to define the endpoint slopes.

Create a vector of data y and another vector with the 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),'-');

Cubic spline data interpolation - MATLAB spline- MathWorks Deutschland (2)

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.

ans = 270.6060

Spline Interpolation of Angular Data

Open Live Script

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

Cubic spline data interpolation - MATLAB spline- MathWorks Deutschland (3)

Spline Interpolation of Sine and Cosine Data

Open Live Script

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 onplot(x,Y(2,:),'o',xx,YY(2,:),':')hold off

Cubic spline data interpolation - MATLAB spline- MathWorks Deutschland (4)

Data Interpolation with spline, pchip, and makima

Open Live Script

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')

Cubic spline data interpolation - MATLAB spline- MathWorks Deutschland (5)

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')

Cubic spline data interpolation - MATLAB spline- MathWorks Deutschland (6)

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

collapse all


x-coordinates, specified as a vector. Thevector x specifies the points at which the data y isgiven. The elements of x must be unique.

Data Types: single | double

yFunction values at x-coordinates
vector | matrix | array

Function values at x-coordinates, 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 and y arevectors of the same size, then the not-a-knot end conditions are used.

  • If x or y isa scalar, then it is expanded to have the same length as the otherand the not-a-knot end conditions are used.

  • If y is a vector that contains two more values than x has entries, then spline uses the first and last values in y as the endslopes for the cubic spline. For example, if y is a vector, then:

    • y(2:end-1) gives the function values at each point in x

    • y(1) gives the slope at the beginning of the interval located at min(x)

    • y(end) gives the slope at the end of the interval located at max(x)

  • Similarly, if y is a matrix or an N-dimensional array with size(y,N) equal to length(x)+2, then:

    • y(:,...,:,j+1) gives the function values at each point in x for j = 1:length(x)

    • y(:,:,...:,1) gives the slopes at the beginning of the intervals located at min(x)

    • y(:,:,...:,end) gives the slopes at the end of the intervals located at max(x)

Data Types: single | double

xqQuery points
scalar | vector | matrix | array

Query points, specified as a scalar, vector, matrix, or array. The points specified in xq are the x-coordinates for the interpolated function values yq computed by spline.

Data Types: single | double

Output Arguments

collapse all

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, then s has the same size as xq.

  • If y is an array of size Ny = size(y), then these conditions apply:

    • If xq is a scalar or vector, then size(s) returns [Ny(1:end-1) length(xq)].

    • If xq is an array, then size(s) returns [Ny(1:end-1) size(xq)].

pp — Piecewise polynomial

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.


'pp' for piecewise polynomial


Vector of lengthL+1 withstrictly increasing elements that represent the start and end of eachofLintervals


L-by-k matrixwith each rowcoefs(i,:)containing thelocal coefficients of an orderkpolynomialon theith interval,[breaks(i),breaks(i+1)]


Number of pieces, L


Order of the polynomials


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



  • You also can perform spline interpolation using the interp1 function with the command interp1(x,y,xq,'spline').While spline performs interpolation on rows ofan input matrix, interp1 performs interpolationon columns of an input matrix.


A tridiagonal linear system (possibly with several right-handsides) is solved for the information needed to describe the coefficientsof the various cubic polynomials that make up the interpolating spline. spline usesthe functions ppval, mkpp,and unmkpp. These routines form a small suiteof functions for working with piecewise polynomials. For access tomore advanced features, see interp1 orthe Curve Fitting Toolbox™ spline functions.


[1] de Boor, Carl. A PracticalGuide to Splines. Springer-Verlag, New York: 1978.

Extended Capabilities

Version History

Introduced before R2006a

See Also

interp1 | pchip | makima | ppval


Sie haben auf einen Link geklickt, der diesem MATLAB-Befehl entspricht:


Führen Sie den Befehl durch Eingabe in das MATLAB-Befehlsfenster aus. Webbrowser unterstützen keine MATLAB-Befehle.

Cubic spline data interpolation - MATLAB spline- MathWorks Deutschland (7)

Select a Web Site

Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select: .

You can also select a web site from the following list:


  • América Latina (Español)
  • Canada (English)
  • United States (English)


  • Belgium (English)
  • Denmark (English)
  • Deutschland (Deutsch)
  • España (Español)
  • Finland (English)
  • France (Français)
  • Ireland (English)
  • Italia (Italiano)
  • Luxembourg (English)
  • Netherlands (English)
  • Norway (English)
  • Österreich (Deutsch)
  • Portugal (English)
  • Sweden (English)
  • Switzerland
    • Deutsch
    • English
    • Français
  • United Kingdom (English)

Asia Pacific

  • Australia (English)
  • India (English)
  • New Zealand (English)
  • 中国
  • 日本 (日本語)
  • 한국 (한국어)

Contact your local office

Cubic spline data interpolation - MATLAB spline
- MathWorks Deutschland (2024)
Top Articles
Latest Posts
Article information

Author: Sen. Emmett Berge

Last Updated:

Views: 5545

Rating: 5 / 5 (80 voted)

Reviews: 95% of readers found this page helpful

Author information

Name: Sen. Emmett Berge

Birthday: 1993-06-17

Address: 787 Elvis Divide, Port Brice, OH 24507-6802

Phone: +9779049645255

Job: Senior Healthcare Specialist

Hobby: Cycling, Model building, Kitesurfing, Origami, Lapidary, Dance, Basketball

Introduction: My name is Sen. Emmett Berge, I am a funny, vast, charming, courageous, enthusiastic, jolly, famous person who loves writing and wants to share my knowledge and understanding with you.