MATLAB Function Reference |
One-dimensional data interpolation (table lookup)
Syntax
yi = interp1(x,Y,xi) yi = interp1(Y,xi) yi = interp1(x,Y,xi,method) yi = interp1(x,Y,xi,method,'extrap') yi = interp1(x,Y,xi,method,extrapval) pp = interp1(x,Y,method,'pp')
Description
yi = interp1(x,Y,xi)
interpolates to find yi
, the values of the underlying function Y
at the points in the vector or array xi
. 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. xi
can be a scalar, a vector, or a multidimensional array, and yi
has the same size as xi
.
Y
is an array that is not a vector, the size of Y
must have the form [n,d1,d2,...,dk]
, where n
is the length of x
. The interpolation is performed for each d1
-by-d2
-by-...-dk
value in Y
. The sizes of xi
and yi
are related as follows:
yi = interp1(Y,xi)
assumes that x = 1:N
, where N
is the length of Y
for vector Y
, or size(Y,1)
for matrix Y
.
interpolates using alternative methods:yi = interp1(x,Y,xi,
method
)
For the 'nearest'
, 'linear'
, and 'v5cubic'
methods, interp1(x,Y,xi,method)
returns NaN
for any element of xi
that is outside the interval spanned by x
. For all other methods, interp1
performs extrapolation for out of range values.
yi = interp1(x,Y,xi,method,'extrap')
uses the specified method to perform extrapolation for out of range values.
yi = interp1(x,Y,xi,method,extrapval)
returns the scalar extrapval
for out of range values. NaN
and 0
are often used for extrapval
.
pp = interp1(x,Y,method,'pp')
uses the specified method to generate the piecewise polynomial form (ppform) of Y
. You can use any of the methods in the preceding table, except for 'v5cubic'
.
The interp1
command interpolates between data points. It finds values at intermediate points, of a one-dimensional function that underlies the data. This function is shown below, along with the relationship between vectors x
, Y
, xi
, and yi
.
Interpolation is the same operation as table lookup. Described in table lookup terms, the table is [x,Y]
and interp1
looks up the elements of xi
in x
, and, based upon their locations, returns values yi
interpolated within the elements of Y
.
Examples
Example 1. Generate a coarse sine curve and interpolate over a finer abscissa.
Example 2. The following multidimensional example creates 2-by-2 matrices of interpolated function values, one matrix for each of the three functions x2, x3, and x4.
The result yi
has size 2-by-2-by-3.
Example 3. Here are two vectors representing the census years from 1900 to 1990 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];
The expression interp1(t,p,1975)
interpolates within the census data to estimate the population in 1975. The result is
Now interpolate within the data at every year from 1900 to 2000, and plot the result.
Sometimes it is more convenient to think of interpolation in table lookup terms, where the data are stored in a single table. If a portion of the census data is stored in a single 5-by-2 table,
then the population in 1975, obtained by table lookup within the matrix tab
, is
Example 4. The following example uses the 'cubic'
method to generate the piecewise polynomial form (ppform) of Y
, and then evaluates the result using ppval
.
x = 0:.2:pi; y = sin(x); pp = interp1(x,y,'cubic','pp'); xi = 0:.1:pi; yi = ppval(pp,xi); plot(x,y,'ko'), hold on, plot(xi,yi,'r:'), hold off
Algorithm
The interp1
command is a MATLAB M-file. The 'nearest'
and 'linear'
methods have straightforward implementations.
For the 'spline'
method, interp1
calls a function spline
that uses the functions ppval
, mkpp
, and unmkpp
. These routines form a small suite of functions for working with piecewise polynomials. spline
uses them to perform the cubic spline interpolation. For access to more advanced features, see the spline
reference page, the M-file help for these functions, and the Spline Toolbox.
For the 'pchip'
and 'cubic'
methods, interp1
calls a function pchip
that performs piecewise cubic interpolation within the vectors x
and y
. This method preserves monotonicity and the shape of the data. See the pchip
reference page for more information.
See Also
interpft
, interp2
, interp3
, interpn
, pchip
, spline
References
[1] de Boor, C., A Practical Guide to Splines, Springer-Verlag, 1978.
int8, int16, int32, int64 | interp2 |
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