MATLAB Function Reference 
Cubic spline data interpolation
Syntax
Description
pp = spline(x,Y)
returns the piecewise polynomial form of the cubic spline interpolant for later use with ppval
and the spline utility unmkpp
. x
must be a vector. Y
can be a scalar, a vector, or an array of any dimension, subject to the following conditions:
Y
is a scalar or vector, it must have the same length as x
. A scalar value for x
or Y
is expanded to have the same length as the other. See Exceptions (1) for an exception to this rule, in which the notaknot end conditions are used.
Y
is an array that is not a vector, the size of Y
must have the form [d1,d2,...,dk,n]
, where n
is the length of x
. The interpolation is performed for each d1
byd2
by...dk
value in Y
. See Exceptions (2) for an exception to this rule.
yy = spline(x,Y,xx)
is the same as yy = ppval(spline(x,Y),xx)
, thus providing, in yy
, the values of the interpolant at xx
. xx
can be a scalar, a vector, or a multidimensional array.The sizes of xx
and yy
are related as follows:
Exceptions
Y
is a vector that contains two more values than x
has entries, the first and last value in Y
are used as the endslopes for the cubic spline. If Y
is a vector, this means
Y
is a matrix or an N
dimensional array with size(Y,N)
equal to length(x)+2
, the following hold:
Note
You can also perform spline interpolation using the interp1 function with the command interp1(x,y,xx,'spline') . Note that while spline performs interpolation on rows of an input matrix, interp1 performs interpolation on columns of an input matrix.

Examples
Example 1. This generates a sine curve, then samples the spline over a finer mesh.
Example 2. This illustrates the use of clamped or complete spline interpolation where end slopes are prescribed. Zero slopes at the ends of an interpolant to the values of a certain distribution are enforced.
x = 4:4; y = [0 .15 1.12 2.36 2.36 1.46 .49 .06 0]; cs = spline(x,[0 y 0]); xx = linspace(4,4,101); plot(x,y,'o',xx,ppval(cs,xx),'');
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 ];
represent the census years from 1900 to 1990 and the corresponding United States population in millions of people. The expression
uses the cubic spline to extrapolate and predict the population in the year 2000. The result is
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
generate the plot of a circle, with the five data points y(:,2),...,y(:,6)
marked with o's. Note that this y
contains two more values (i.e., two more columns) than does x
, hence y(:,1)
and y(:,end)
are used as endslopes.
Example 5. The following code generates sine and cosine curves, then samples the splines 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;
Algorithm
A tridiagonal linear system (with, possibly, several right sides) is being solved for the information needed to describe the coefficients of the various cubic polynomials which 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 the Mfile help for these functions and the Spline Toolbox.
See Also
interp1
, ppval
, mkpp
, pchip
, unmkpp
References
[1] de Boor, C., A Practical Guide to Splines, SpringerVerlag, 1978.
spinmap  spones 
© 19942005 The MathWorks, Inc.