(interp1 __ Functions _(MATLAB®_))

interp1 :: Functions (MATLAB®) text://0

1 of 5 21/8/2012 17:01

interp11-D data interpolation (table lookup)

Syntaxyi = 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')

Descriptionyi = interp1(x,Y,xi) interpolates to find yi , the values of the underlying function Y at the points in the vector orarray xi . x must be a vector. Y can be a scalar, a vector, or an array of any dimension, subject to the following conditions:

If Y is a vector, it must have the same length as x . A scalar value for Y is expanded to have the same length as x . xi can be a scalar, a vector, or a multidimensional array, and yi has the same size as xi .If 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 asfollows:

If xi is a scalar or vector, size(yi) equals [length(xi), d1, d2, ..., dk] .If xi is an array of size [m1,m2,...,mj] , yi has size [m1,m2,...,mj,d1,d2,...,dk] .

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.

yi = interp1(x,Y,xi,method) interpolates using alternative methods:

'nearest ' Nearest neighbor interpolation

'linear ' Linear interpolation (default)

'spline ' Cubic spline interpolation

'pchip ' Piecewise cubic Hermite interpolation

'cubic ' (Same as 'pchip' )

'v5cubic ' Cubic interpolation used in MATLAB 5. This method does not extrapolate. Also, if x is not equally spaced, 'spline' is used/

For the 'nearest' , 'linear' , and 'v5cubic' methods, interp1(x,Y,xi,method) returns NaN for any element ofxi that is outside the interval spanned by x . For all other methods, interp1 performs extrapolation for out of rangevalues.

yi = interp1(x,Y,xi,method,'extrap') uses the specified method to perform extrapolation for out of rangevalues.

yi = interp1(x,Y,xi,method,extrapval) returns the scalar extrapval for out of range values. NaN and 0 areoften used for extrapval .

pp = interp1(x,Y,method,'pp') uses the specified method to generate the piecewise polynomial form (ppform) ofY. You can use any of the methods in the preceding table, except for 'v5cubic' . pp can then be evaluated via ppval . ppval(pp,xi) is the same as interp1(x,Y,xi,method,'extrap') .

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 .


2 of 5 21/8/2012 17:01

Interpolation is the same operation as table lookup. Described in table lookup terms, the table is [x,Y] and interp1looks up the elements of xi in x , and, based upon their locations, returns values yi interpolated within the elements of Y.

Note interp1q is quicker than interp1 on non-uniformly spaced data because it does no input checking. Forinterp1q to work properly, x must be a monotonically increasing column vector and Y must be a column vector or matrix with length(X) rows. Type help interp1q at the command line for more information.

ExamplesExample 1

Generate a coarse sine curve and interpolate over a finer abscissa.

x = 0:10; y = sin(x); xi = 0:.25:10; yi = interp1(x,y,xi); plot(x,y,'o',xi,yi)

Example 2

The following multidimensional example creates 2-by-2 matrices of interpolated function values, one matrix for each of thethree functions x , x , and x .

x = [1:10]'; y = [ x.^2, x.^3, x.^4 ]; xi = [1.5, 1.75; 7.5, 7.75]; yi = interp1(x,y,xi);

2 3 4


3 of 5 21/8/2012 17:01

The result yi has size 2-by-2-by-3.

size(yi)

ans =

2 2 3

Example 3

Here are two vectors representing the census years from 1900 to 1990 and the corresponding United States population inmillions 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 resultis

ans = 214.8585

Now interpolate within the data at every year from 1900 to 2000, and plot the result.

x = 1900:1:2000; y = interp1(t,p,x,'spline'); plot(t,p,'o',x,y)

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,

tab = 1950 150.697 1960 179.323 1970 203.212 1980 226.505 1990 249.633

then the population in 1975, obtained by table lookup within the matrix tab , is

p = interp1(tab(:,1),tab(:,2),1975)p = 214.8585

Example 4


4 of 5 21/8/2012 17:01

The following example uses the 'cubic' method to generate the piecewise polynomial form (ppform) of Y, and thenevaluates 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

AlgorithmThe interp1 command is a MATLAB M-file. The 'nearest' and 'linear' methods have straightforwardimplementations.

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 cubicspline interpolation. For access to more advanced features, see the spline reference page, the M-file help for thesefunctions, and the Spline Toolbox™ .

For the 'pchip' and 'cubic' methods, interp1 calls a function pchip that performs piecewise cubic interpolationwithin the vectors x and y . This method preserves monotonicity and the shape of the data. See the pchip reference pagefor more information.

Interpolating Complex Data

For Real x and Complex Y. For interp1(x,Y,...) where x is real and Y is complex, you can use any interp1method except for 'pchip' . The shape-preserving aspect of the 'pchip' algorithm involves the signs of the slopesbetween the data points. Because there is no notion of sign with complex data, it is impossible to talk about whether afunction is increasing or decreasing. Consequently, the 'pchip' algorithm does not generalize to complex data.

The 'spline' method is often a good choice because piecewise cubic splines are derived purely from smoothnessconditions. The second derivative of the interpolant must be continuous across the interpolating points. This does notinvolve any notion of sign or shape and so generalizes to complex data.

For Complex x. For interp1(x,Y,...) where x is complex and Y is either real or complex, use the two-dimensionalinterpolation routine interp2(REAL(x), IMAG(x),Y,...) instead.

See Alsointerp1q , interpft , interp2 , interp3 , interpn , pchip , spline


5 of 5 21/8/2012 17:01

References[1] de Boor, C., A Practical Guide to Splines, Springer-Verlag, 1978.

Was this topic helpful? Yes No

© 1984-2010 The MathWorks, Inc. • Terms of Use • Patents • Trademarks • Acknowledgments

Documents

(interp1 __ Functions _(MATLAB®_))