Mathematics 
Interpolation and Multidimensional Arrays
Several interpolation functions operate specifically on multidimensional data.
Function 
Description 

Threedimensional data interpolation. 

Multidimensional data interpolation. 

Multidimensional data gridding (ndfun directory). 
Interpolation of ThreeDimensional Data
The function interp3
performs threedimensional interpolation, finding interpolated values between points of a threedimensional set of samples V
. You must specify a set of known data points:
X
, Y
, and Z
matrices specify the points for which values of V
are given.
V
contains values corresponding to the points in X
, Y
, and Z
.
The most general form for interp3
is
XI
, YI
, and ZI
are the points at which interp3
interpolates values of V
. For outofrange values, interp3
returns NaN
.
There are three different interpolation methods for threedimensional data:
method = 'nearest'
). This method chooses the value of the nearest point.
method = 'linear'
). This method uses piecewise linear interpolation based on the values of the nearest eight points.
method = 'cubic'
). This method uses piecewise cubic interpolation based on the values of the nearest sixtyfour points.
All of these methods require that X
, Y
, and Z
be monotonic, that is, either always increasing or always decreasing in a particular direction. In addition, you should prepare these matrices using the meshgrid
function, or else be sure that the "pattern" of the points emulates the output of meshgrid
.
Each method automatically maps the input to an equally spaced domain before interpolating. If x
is already equally spaced, you can speed execution time by prepending an asterisk to the method
string, for example, '*cubic'
.
Interpolation of Higher Dimensional Data
The function interpn
performs multidimensional interpolation, finding interpolated values between points of a multidimensional set of samples V
. The most general form for interpn
is
1
, 2
, 3
, ...
are matrices that specify the points for which values of V
are given. V
is a matrix that contains the values corresponding to these points. 1
, 2
, 3
, ... are the points for which interpn
returns interpolated values of V
. For outofrange values, interpn
returns NaN
.
Y1
, Y2
, Y3
, ... must be either arrays of the same size, or vectors. If they are vectors of different sizes, interpn
passes them to ndgrid
and then uses the resulting arrays.
There are three different interpolation methods for multidimensional data:
method = 'nearest'
). This method chooses the value of the nearest point.
method = 'linear'
). This method uses piecewise linear interpolation based on the values of the nearest two points in each dimension.
method = 'cubic'
). This method uses piecewise cubic interpolation based on the values of the nearest four points in each dimension.
All of these methods require that X1
, X2,X3
be monotonic. In addition, you should prepare these matrices using the ndgrid
function, or else be sure that the "pattern" of the points emulates the output of ndgrid
.
Each method automatically maps the input to an equally spaced domain before interpolating. If X
is already equally spaced, you can speed execution time by prepending an asterisk to the method
string; for example, '*cubic'
.
Multidimensional Data Gridding
The ndgrid
function generates arrays of data for multidimensional function evaluation and interpolation. ndgrid
transforms the domain specified by a series of input vectors into a series of output arrays. The i
th dimension of these output arrays are copies of the elements of input vector x
_{i}.
For example, assume that you want to evaluate a function of three variables over a given range. Consider the function
for , , and . To evaluate and plot this function:
x1 =2:0.2:2;
x2 =2:0.25:2;
x3 = 2:0.16:2; [X1,X2,X3] = ndgrid(x1,x2,x3); z = X2.
*exp(X1
.^2 X2
.^2 X3.^2); slice(X2
,X1
,X3
,z,[1.2 0.8 2],2,[2 0.2])
Comparing Interpolation Methods  Triangulation and Interpolation of Scattered Data 