interpn

Interpolation for 1-D, 2-D, 3-D, and N-D gridded data in ndgrid format

Syntax

Vq = interpn(X1,X2,...,Xn,V,Xq1,Xq2,...,Xqn)

Vq = interpn(V,Xq1,Xq2,...,Xqn)

Vq = interpn(V)

Vq = interpn(V,k)

Vq = interpn(___,method)

Vq = interpn(___,method,extrapval)

Description

Vq = interpn(X1,X2,...,Xn,V,Xq1,Xq2,...,Xqn) returns 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. X1,X2,...,Xn contain the coordinates of the sample points. V contains the corresponding function values at each sample point. Xq1,Xq2,...,Xqn contain the coordinates of the query points.

Vq = interpn(V,Xq1,Xq2,...,Xqn) assumes a default grid of sample points. The default grid consists of the points, 1,2,3,...n_i in each dimension. The value of n_i is the length of the ith dimension in V. Use this syntax when you want to conserve memory and are not concerned about the absolute distances between points.

Vq = interpn(V) returns the interpolated values on a refined grid formed by dividing the interval between sample values once in each dimension.

Vq = interpn(V,k) returns the interpolated values on a refined grid formed by repeatedly halving the intervals k times in each dimension. This results in 2^k-1 interpolated points between sample values.

Vq = interpn(___,method) specifies an alternative interpolation method: 'linear', 'nearest', 'pchip','cubic', 'makima', or 'spline'. The default method is 'linear'.

Vq = interpn(___,method,extrapval) also specifies extrapval, a scalar value that is assigned to all queries that lie outside the domain of the sample points.

If you omit the extrapval argument for queries outside the domain of the sample points, then based on the method argument interpn returns one of the following:

• The extrapolated values for the 'spline' and 'makima' methods

• NaN values for other interpolation methods

Examples

1-D Interpolation

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');

2-D 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 ntimes=1.

Vq = interpn(V,'cubic');
mesh(Vq);

Evaluate Outside Domain of 3-D Function

Create the grid vectors, x1, x2, and x3. These vectors define the points associated with the values in V.

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.

rng('default')
V = rand(100,50,30);

Evaluate V at three points outside the domain of x1, x2, and x3. Specify 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 = 1×3

    -1    -1    -1

All three points evaluate to -1 because they are outside the domain of x1, x2, and x3.

4-D Interpolation

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, V.

[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);

Interpolate 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);
   clim([0 10]);
   M(j) = getframe;
end
movie(M);

Input Arguments

X1,X2,...,Xn — Sample grid points

arrays | vectors

Sample grid points, specified as real arrays or vectors. The sample grid points must be unique.

• If X1,X2,...,Xn are arrays, then they contain the coordinates of a full grid (in ndgrid format). Use the ndgrid function to create the X1,X2,...,Xn arrays together. These arrays must be the same size.

• If X1,X2,...,Xn are vectors, then they are treated as grid vectors. The values in these vectors must be strictly monotonic, either increasing or decreasing.

Example: [X1,X2,X3,X4] = ndgrid(1:30,-10:10,1:5,10:13)

Data Types: single | double

V — Sample values

array

Sample values, specified as a real or complex array. The size requirements for V depend on the size of X1,X2,...,Xn:

• If X1,X2,...,Xn are arrays representing a full grid (in ndgrid format), then the size of V matches the size of any array, X1,X2,...,Xn.

• If X1,X2,...,Xn are grid vectors, then V is an array whose ith dimension is the same length as grid vector Xi, where i= 1,2,...n.

If V contains complex numbers, then interpn interpolates the real and imaginary parts separately.

Example: rand(10,5,3,2)

Data Types: single | double
Complex Number Support: Yes

Xq1,Xq2,...,Xqn — Query points

scalars | vectors | arrays

Query points, specified as a real scalars, vectors, or arrays.

• If Xq1,Xq2,...,Xqn are scalars, then they are the coordinates of a single query point in Rⁿ.

• If Xq1,Xq2,...,Xqn are vectors of different orientations, then Xq1,Xq2,...,Xqn are treated as grid vectors in Rⁿ.

• If Xq1,Xq2,...,Xqn are vectors of the same size and orientation, then Xq1,Xq2,...,Xqn are treated as scattered points in Rⁿ.

• If Xq1,Xq2,...,Xqn are arrays of the same size, then they represent either a full grid of query points (in ndgrid format) or scattered points in Rⁿ.

Example: [X1,X2,X3,X4] = ndgrid(1:10,1:5,7:9,10:11)

Data Types: single | double

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 in 2^k-1 interpolated points between sample values.

If k is 0, then Vq is the same as V.

interpn(V,1) is the same as interpn(V).

The following illustration depicts k=2 in R². There are 72 interpolated values in red and 9 sample values in black.

Example: interpn(V,2)

Data Types: single | double

method — Interpolation method

'linear' (default) | 'nearest' | 'pchip' | 'cubic' | 'spline' | 'makima'

Interpolation method, specified as one of the options in this table.

Method	Description	Continuity	Comments
'linear'	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	Requires at least two grid points in each dimension Requires more memory than 'nearest'
'nearest'	The interpolated value at a query point is the value at the nearest sample grid point.	Discontinuous	Requires two grid points in each dimension. Fastest computation with modest memory requirements
'pchip'	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	Requires at least four points Requires more memory and computation time than 'linear'
'cubic'	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	Grid must have uniform spacing in each dimension, but the spacing does not have to be the same for all dimensions Requires at least four points in each dimension Requires more memory and computation time than 'linear'
'makima'	Modified Akima cubic Hermite interpolation. The interpolated value at a query point is based on a piecewise function of polynomials with degree at most three evaluated using the values of neighboring grid points in each respective dimension. The Akima formula is modified to avoid overshoots.	C1	Requires at least 2 points in each dimension Produces fewer undulations than 'spline' Computation time is typically less than 'spline', but the memory requirements are similar
'spline'	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	Requires four points in each dimension Requires more memory and computation time than 'cubic'

extrapval — Function value outside domain of X1,X2,...,Xn

scalar

Function value outside domain of X1,X2,...,Xn, specified as a real or complex scalar. interpn returns this constant value for all points outside the domain of X1,X2,...,Xn.

Example: 5

Example: 5+1i

Data Types: single | double
Complex Number Support: Yes

Output Arguments

Vq — Interpolated values

scalar | 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 arguments.

Syntaxes	Special Conditions	Size of Vq	Example
interpn(X1,...,Xn,V,Xq1,...,Xqn) interpn(V,Xq1,...,Xqn) and variations of these syntaxes that include method or extrapval	Xq1,...,Xqn are scalars	Scalar	size(Vq) = [1 1] when you pass Xq1,...,Xqn as scalars.
Same as above	Xq1,...,Xqn are vectors of the same size and orientation	Vector of same size and orientation as Xq1,...,Xqn	In 3-D, if size(Xq1) = [100 1], and size(Xq2) = [100 1], and size(Xq3) = [100 1], then size(Vq) = [100 1].
Same as above	Xq1,...,Xqn are vectors of mixed orientation	size(Vq) = [length(Xq1),...,length(Xqn)]	In 3-D, if size(Xq1) = [1 100], and size(Xq2) = [50 1], and size(Xq3) = [1 5], then size(Vq) = [100 50 5].
Same as above	Xq1,...,Xqn are arrays of the same size	Array of the same size as Xq1,...,Xqn	In 3-D, if size(Xq1) = [50 25], and size(Xq2) = [50 25], and size(Xq3) = [50 25], then size(Vq) = [50 25].
interpn(V,k) and variations of this syntax that include method or extrapval	None	Array in which the length of the ith dimension is 2^k * (size(V,i)-1)+1,	In 3-D, if size(V) = [10 12 5], and k = 3, then size(Vq) = [73 89 33].

More About

Strictly Monotonic

A set of values that are always increasing or decreasing, without reversals. For example, the sequence, a = [2 4 6 8] is strictly monotonic and increasing. The sequence, b = [2 4 4 6 8] is not strictly monotonic because there is no change in value between b(2) and b(3). The sequence, c = [2 4 6 8 6] contains a reversal between c(4) and c(5), so it is not monotonic at all.

Full Grid (in ndgrid Format)

For interpn, the full grid consists of n arrays, X1,X2,...,Xn, 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.

Use the ndgrid function to create a full grid that you can pass to interpn. For example, the following code creates a full grid in R² for the region, 1 ≤ X1 ≤ 3, 1≤ X2 ≤ 4.

[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

Grid Vectors

For 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;

Scattered Points

For interpn, scattered points consist of n arrays or vectors, Xq1,Xq2,...,Xqn, that define a collection of points scattered in Rⁿ. The ith array, Xi, contains the coordinates in the ith dimension.

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];

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• For best results, provide X1,X2,...,Xn as vectors. The values in these vectors must be strictly monotonic and increasing.

• Code generation does not support the 'makima' interpolation method.

• The interpolation method must be a constant character vector.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

• A maximum of five dimensions is supported.

• V must be a double or single N-D array. V can be real or complex.

• X1,X2,...,Xn, Y must:

  • Have the same type (double or single).

  • Be finite vectors or N-D arrays with increasing and nonrepeating elements in corresponding dimensions.

  • Align with Cartesian axes when X1,X2,...,Xn are N-D arrays (as if they were produced by ndgrid).

  • Have dimensions consistent with V.

• X1,X2,...,Xn must be vectors or arrays of the same type (double or single). If X1,X2,...,Xn are arrays, then they must have the same size. If they are vectors with different lengths, then one of them must have a different orientation.

• method must be 'linear' or'nearest'.

• The extrapolation for the out-of-boundary input is not supported.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a