Interpolation for 1-D, 2-D, 3-D, and N-D gridded data in ndgrid format
In a future release,
interpn will not accept
mixed combinations of row and column vectors for the sample and query
grids. For more information, and recommendations for updating your
code, see Functionality being removed or changed.
interpolated values of a function of n variables
at specific query points using linear interpolation. The results always
pass through the original sampling of the function.
Vq = interpn(
the coordinates of the sample points.
the corresponding function values at each sample point.
the coordinates of the query points.
a default grid of sample points. The default grid consists of the
points, 1,2,3,...ni in each dimension. The
value of ni is the length of the ith dimension
Vq = interpn(
V. Use this syntax to when you want to conserve
memory and are not concerned about the absolute distances between
an optional, trailing input argument that you can pass with any of
the previous syntaxes. The
Vq = interpn(___,
method argument can
be any of the following strings that specify alternative interpolation
'spline'. The default method is
If you omit the
extrapval argument for queries
outside the domain of the sample points, then based on the
one of the following:
The extrapolated values for the
NaN values for interpolation methods
Define the sample points and values.
x = [1 2 3 4 5]; v = [12 16 31 10 6];
Define the query points,
xq, and interpolate.
xq = (1:0.1:5); vq = interpn(x,v,xq,'cubic');
Plot the result.
figure plot(x,v,'o',xq,vq,'-'); legend('Samples','Cubic Interpolation');
Create a set of grid points and corresponding sample values.
[X1,X2] = ndgrid((-5:1:5)); R = sqrt(X1.^2 + X2.^2)+ eps; V = sin(R)./(R);
Interpolate over a finer grid using
Vq = interpn(V,'cubic'); mesh(Vq);
Create the grid vectors,
x3. These vectors define the points associated with the values in
x1 = 1:100; x2 = (1:50)'; x3 = 1:30;
Define the sample values to be a 100-by-50-by-30 random number array,
V. Use the
gallery function to create the array.
V = gallery('uniformdata',100,50,30,0);
V at three points outside the domain of
extrapval = -1.
xq1 = [0 0 0]; xq2 = [0 0 51]; xq3 = [0 101 102]; vq = interpn(x1,x2,x3,V,xq1,xq2,xq3,'linear',-1)
vq = -1 -1 -1
All three points evaluate to
-1 because they are outside the domain of
Define an anonymous function that represents .
f = @(x,y,z,t) t.*exp(-x.^2 - y.^2 - z.^2);
Create a grid of points in
. Then, pass the points through the function to create the sample values,
[x,y,z,t] = ndgrid(-1:0.2:1,-1:0.2:1,-1:0.2:1,0:2:10); V = f(x,y,z,t);
Now, create the query grid.
[xq,yq,zq,tq] = ... ndgrid(-1:0.05:1,-1:0.08:1,-1:0.05:1,0:0.5:10);
V at the query points.
Vq = interpn(x,y,z,t,V,xq,yq,zq,tq);
Create a movie to show the results.
figure('renderer','zbuffer'); nframes = size(tq, 4); for j = 1:nframes slice(yq(:,:,:,j),xq(:,:,:,j),zq(:,:,:,j),... Vq(:,:,:,j),0,0,0); caxis([0 10]); M(j) = getframe; end movie(M);
X1,X2,...,Xn— Sample grid pointsarrays | vectors
Sample grid points, specified as real arrays or vectors.
X1,X2,...,Xn are arrays, then
they contain the coordinates of a full grid (in ndgrid format).
ndgrid function to
X1,X2,...,Xn arrays together. These
arrays must be the same size.
[X1,X2,X3] = ndgrid(1:30,-10:10,1:5)
V— Sample valuesarray
Sample values, specified as a real or complex array. The size
V depend on the size of
X1,X2,...,Xn are arrays representing
a full grid (in
ndgrid format), then the size of
the size of any array,
X1,X2,...,Xn are grid vectors,
V is an array whose
dimension is the same length as grid vector
Complex Number Support: Yes
Xq1,Xq2,...,Xqn— Query pointsscalars | vectors | arrays
Query points, specified as a real scalars, vectors, or arrays.
Xq1,Xq2,...,Xqn are scalars,
then they are the coordinates of a single query point in Rn.
Xq1,Xq2,...,Xqn are vectors
of different orientations, then
treated as grid vectors in Rn.
Xq1,Xq2,...,Xqn are vectors
of the same size and orientation, then
treated as scattered points in Rn.
Xq1,Xq2,...,Xqn are arrays of
the same size, then they represent either a full grid of query points
ndgrid format) or scattered points in Rn.
[X1,X2,X3,X4] = ndgrid(1:10,1:5,7:9,10:11)
k— Refinement factor
1(default) | real, nonnegative, integer scalar
Refinement factor, specified as a real, nonnegative, integer
scalar. This value specifies the number of times to repeatedly divide
the intervals of the refined grid in each dimension. This results
2^k-1 interpolated points between sample values.
the same as
interpn(V,1) is the same as
The following illustration depicts
k=2 in R2.
There are 72 interpolated values in red and 9 sample values in black.
method— Interpolation method
Interpolation method, specified as a string from this table.
|The interpolated value at a query point is based on linear interpolation of the values at neighboring grid points in each respective dimension. This is the default interpolation method.||C0|
|The interpolated value at a query point is the value at the nearest sample grid point.||Discontinuous|
|Shape-preserving piecewise cubic interpolation (for 1-D only). The interpolated value at a query point is based on a shape-preserving piecewise cubic interpolation of the values at neighboring grid points.||C1|
|The interpolated value at a query point is based on a cubic interpolation of the values at neighboring grid points in each respective dimension. The interpolation is based on a cubic convolution.||C1|
|The interpolated value at a query point is based on a cubic interpolation of the values at neighboring grid points in each respective dimension. The interpolation is based on a cubic spline using not-a-knot end conditions.||C2|
Function value outside domain of
specified as a real or complex scalar.
this constant value for all points outside the domain of
Complex Number Support: Yes
Vq— Interpolated valuesscalar | vector | array
Interpolated values, returned as a real or complex scalar, vector,
or array. The size and shape of
Vq depends on the
syntax you use and, in some cases, the size and value of the input
|Syntaxes||Special Conditions||Size of Vq||Example|
and variations of these syntaxes that include
|Same as above||Vector of same size and orientation as ||In 3-D, if |
|Same as above||In 3-D, if |
|Same as above||Array of the same size as ||In 3-D, if |
and variations of this syntax that include
Array in which the length of the
|In 3-D, if |
A set of values that are always increasing
or decreasing, without reversals. For example, the sequence,
= [2 4 6 8] is strictly monotonic and increasing. The sequence,
= [2 4 4 6 8] is not strictly monotonic because there is
no change in value between
c = [2 4 6 8 6] contains a reversal
c(5), so it
is not monotonic at all.
interpn, the full
grid consists of n arrays,
whose elements represent a grid of points in Rn.
The ith array,
Xi, contains strictly monotonic,
increasing values that vary most rapidly along the ith dimension.
to create a full grid that you can pass to
For example, the following code creates a full grid in R2 for
the region, 1 ≤ X1 ≤ 3, 1≤ X2 ≤
[X1,X2] = ndgrid(-1:3,(1:4))
X1 = -1 -1 -1 -1 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 X2 = 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
interpn, grid vectors
consist of n vectors of mixed-orientation that
define the points of a grid in Rn.
For example, the following code creates the grid vectors in R3 for the region, 1 ≤ x1 ≤ 3, 4 ≤ x2 ≤ 5, and 6 ≤x3≤ 8:
x1 = 1:3; x2 = (4:5)'; x3 = 6:8;
points consist of n arrays or vectors,
that define a collection of points scattered in Rn.
the coordinates in the
For example, the following code specifies the points, (1, 19, 10), (6, 40, 1), (15, 33, 22), and (0, 61, 13) in R3.
Xq1 = [1 6; 15 0]; Xq2 = [19 40; 33 61]; Xq3 = [10 1; 22 13];