interp1
1D data interpolation (table lookup)
Syntax
Description
returns
interpolated values of a 1D function at specific query points using
linear interpolation. Vector vq
= interp1(x
,v
,xq
)x
contains the sample
points, and v
contains the corresponding values, v(x).
Vector xq
contains the coordinates of the query
points.
If you have multiple sets of data that are sampled at the same
point coordinates, then you can pass v
as an array.
Each column of array v
contains a different set
of 1D sample values.
specifies
a strategy for evaluating points that lie outside the domain of vq
= interp1(x
,v
,xq
,method
,extrapolation
)x
.
Set extrapolation
to 'extrap'
when
you want to use the method
algorithm for extrapolation.
Alternatively, you can specify a scalar value, in which case, interp1
returns
that value for all points outside the domain of x
.
returns
interpolated values and assumes a default set of sample point coordinates.
The default points are the sequence of numbers from vq
= interp1(v
,xq
)1
to n
,
where n
depends on the shape of v
:
When v is a vector, the default points are
1:length(v)
.When v is an array, the default points are
1:size(v,1)
.
Use this syntax when you are not concerned about the absolute distances between points.
specifies
an extrapolation strategy and uses the default sample points.vq
= interp1(v
,xq
,method
,extrapolation
)
returns the piecewise polynomial form of
v(x) using the
pp
= interp1(x
,v
,method
,'pp')method
algorithm.
Note
This syntax is not recommended. Use griddedInterpolant
instead.
Examples
Interpolation of Coarsely Sampled Sine Function
Define the sample points, x
, and corresponding sample values, v
.
x = 0:pi/4:2*pi; v = sin(x);
Define the query points to be a finer sampling over the range of x
.
xq = 0:pi/16:2*pi;
Interpolate the function at the query points and plot the result.
figure vq1 = interp1(x,v,xq); plot(x,v,'o',xq,vq1,':.'); xlim([0 2*pi]); title('(Default) Linear Interpolation');
Now evaluate v
at the same points using the 'spline'
method.
figure vq2 = interp1(x,v,xq,'spline'); plot(x,v,'o',xq,vq2,':.'); xlim([0 2*pi]); title('Spline Interpolation');
Interpolation Without Specifying Points
Define a set of function values.
v = [0 1.41 2 1.41 0 1.41 2 1.41 0];
Define a set of query points that fall between the default points, 1:9
. In this case, the default points are 1:9
because v
contains 9
values.
xq = 1.5:8.5;
Evaluate v
at xq
.
vq = interp1(v,xq);
Plot the result.
figure plot((1:9),v,'o',xq,vq,'*'); legend('v','vq');
Interpolation of Complex Values
Define a set of sample points.
x = 1:10;
Define the values of the function, $$v(x)=5x+{x}^{2}i$$, at the sample points.
v = (5*x)+(x.^2*1i);
Define the query points to be a finer sampling over the range of x
.
xq = 1:0.25:10;
Interpolate v
at the query points.
vq = interp1(x,v,xq);
Plot the real part of the result in red and the imaginary part in blue.
figure plot(x,real(v),'*r',xq,real(vq),'r'); hold on plot(x,imag(v),'*b',xq,imag(vq),'b');
Interpolation of Dates and Times
Interpolate timestamped data points.
Consider a data set containing temperature readings that are measured every four hours. Create a table with one day's worth of data and plot the data.
x = (datetime(2016,1,1):hours(4):datetime(2016,1,2))'; x.Format = 'MMM dd, HH:mm'; T = [31 25 24 41 43 33 31]'; WeatherData = table(x,T,'VariableNames',{'Time','Temperature'})
WeatherData=7×2 table
Time Temperature
_____________ ___________
Jan 01, 00:00 31
Jan 01, 04:00 25
Jan 01, 08:00 24
Jan 01, 12:00 41
Jan 01, 16:00 43
Jan 01, 20:00 33
Jan 02, 00:00 31
plot(WeatherData.Time, WeatherData.Temperature, 'o')
Interpolate the data set to predict the temperature reading during each minute of the day. Since the data is periodic, use the 'spline'
interpolation method.
xq = (datetime(2016,1,1):minutes(1):datetime(2016,1,2))';
V = interp1(WeatherData.Time, WeatherData.Temperature, xq, 'spline');
Plot the interpolated points.
hold on plot(xq,V,'r')
Extrapolation Using Two Different Methods
Define the sample points, x
, and corresponding sample values, v
.
x = [1 2 3 4 5]; v = [12 16 31 10 6];
Specify the query points, xq
, that extend beyond the domain of x
.
xq = [0 0.5 1.5 5.5 6];
Evaluate v
at xq
using the 'pchip'
method.
vq1 = interp1(x,v,xq,'pchip')
vq1 = 1×5
19.3684 13.6316 13.2105 7.4800 12.5600
Next, evaluate v
at xq
using the 'linear'
method.
vq2 = interp1(x,v,xq,'linear')
vq2 = 1×5
NaN NaN 14 NaN NaN
Now, use the 'linear'
method with the 'extrap'
option.
vq3 = interp1(x,v,xq,'linear','extrap')
vq3 = 1×5
8 10 14 4 2
'pchip'
extrapolates by default, but 'linear'
does not.
Designate Constant Value for All Queries Outside the Domain of x
Define the sample points, x
, and corresponding sample values, v
.
x = [3 2 1 0 1 2 3]; v = 3*x.^2;
Specify the query points, xq
, that extend beyond the domain of x
.
xq = [4 2.5 0.5 0.5 2.5 4];
Now evaluate v
at xq
using the 'pchip'
method and assign any values outside the domain of x
to the value, 27
.
vq = interp1(x,v,xq,'pchip',27)
vq = 1×6
27.0000 18.6562 0.9375 0.9375 18.6562 27.0000
Interpolate Multiple Sets of Data in One Pass
Define the sample points.
x = (5:5)';
Sample three different parabolic functions at the points defined in x
.
v1 = x.^2; v2 = 2*x.^2 + 2; v3 = 3*x.^2 + 4;
Create matrix v
, whose columns are the vectors, v1
, v2
, and v3
.
v = [v1 v2 v3];
Define a set of query points, xq
, to be a finer sampling over the range of x
.
xq = 5:0.1:5;
Evaluate all three functions at xq
and plot the results.
vq = interp1(x,v,xq,'pchip'); figure plot(x,v,'o',xq,vq); h = gca; h.XTick = 5:5;
The circles in the plot represent v
, and the solid lines represent vq
.
Input Arguments
x
— Sample points
vector
Sample points, specified as a row or column vector of real numbers.
The values in x
must be distinct. The length of x
must
conform to one of the following requirements:
If
v
is a vector, thenlength(x)
must equallength(v)
.If
v
is an array, thenlength(x)
must equalsize(v,1)
.
Example: [1 2 3 4 5 6 7 8 9 10]
Example: 1:10
Example: [3
7 11 15 19 23 27 31]'
Data Types: single
 double
 duration
 datetime
v
— Sample values
vector  matrix  array
Sample values, specified as a vector, matrix, or array of real
or complex numbers. If v
is a matrix or an array,
then each column contains a separate set of 1D values.
If v
contains complex numbers, then interp1
interpolates
the real and imaginary parts separately.
Example: rand(1,10)
Example: rand(10,1)
Example: rand(10,3)
Data Types: single
 double
 duration
 datetime
Complex Number Support: Yes
xq
— Query points
scalar  vector  matrix  array
Query points, specified as a scalar, vector, matrix, or array of real numbers.
Example: 5
Example: 1:0.05:10
Example: (1:0.05:10)'
Example: [0 1 2 7.5 10]
Data Types: single
 double
 duration
 datetime
method
— Interpolation method
'linear'
(default)  'nearest'
 'next'
 'previous'
 'pchip'
 'cubic'
 'v5cubic'
 'makima'
 'spline'
Interpolation method, specified as one of the options in this table.
Method 
Description 
Continuity 
Comments 


Linear interpolation. 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. 
C^{0} 


Nearest neighbor interpolation. The interpolated value at a query point is the value at the nearest sample grid point. 
Discontinuous 


Next neighbor interpolation. The interpolated value at a query point is the value at the next sample grid point. 
Discontinuous 


Previous neighbor interpolation. The interpolated value at a query point is the value at the previous sample grid point. 
Discontinuous 


Shapepreserving piecewise cubic interpolation. The interpolated value at a query point is based on a shapepreserving piecewise cubic interpolation of the values at neighboring grid points. 
C^{1} 

 Cubic convolution used in MATLAB^{®} 5. 
C^{1} 

 Same as

C^{1}  

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. The Akima formula is modified to avoid overshoots. 
C^{1} 


Spline interpolation using notaknot end conditions. The interpolated value at a query point is based on a cubic interpolation of the values at neighboring grid points in each respective dimension. 
C^{2} 

extrapolation
— Extrapolation strategy
'extrap'
 scalar value
Extrapolation strategy, specified as 'extrap'
or
a real scalar value.
Specify
'extrap'
when you wantinterp1
to evaluate points outside the domain using the same method it uses for interpolation.Specify a scalar value when you want
interp1
to return a specific constant value for points outside the domain.
The default behavior depends on the input arguments:
If you specify the
'pchip'
,'spline'
, or'makima'
interpolation methods, then the default behavior is'extrap'
.All other interpolation methods return
NaN
by default for query points outside the domain.
Example: 'extrap'
Example: 5
Data Types: char
 string
 single
 double
Output Arguments
vq
— Interpolated values
scalar  vector  matrix  array
Interpolated values, returned as a scalar, vector, matrix, or
array. The size of vq
depends on the shape of v
and xq
.
Shape of v  Shape of xq  Size of Vq  Example 

Vector  Vector  size(xq)  If size(v) = [1 100] and size(xq)
= [1 500] , then size(vq) = [1 500] . 
Vector  Matrix or ND Array  size(xq)  If size(v) = [1 100] and size(xq)
= [50 30] , then size(vq) = [50
30] . 
Matrix or ND Array  Vector  [length(xq) size(v,2),...,size(v,n)]  If size(v) = [100 3] and size(xq)
= [1 500] , then size(vq) = [500
3] . 
Matrix or ND Array  Matrix or ND Array  [size(xq,1),...,size(xq,n),... size(v,2),...,size(v,m)]  If size(v) = [4 5 6] and size(xq)
= [2 3 7] , then size(vq) = [2 3
7 5 6] . 
pp
— Piecewise polynomial
structure
Piecewise polynomial, returned as a structure that you can pass
to the ppval
function for evaluation.
More About
Akima and Spline Interpolation
The Akima algorithm for onedimensional interpolation, described in [1] and [2], performs cubic interpolation to produce piecewise polynomials with continuous firstorder derivatives (C1). The algorithm preserves the slope and avoids undulations in flat regions. A flat region occurs whenever there are three or more consecutive collinear points, which the algorithm connects with a straight line. To ensure that the region between two data points is flat, insert an additional data point between those two points.
When two flat regions with different slopes meet, the modification made to the original Akima algorithm gives more weight to the side where the slope is closer to zero. This modification gives priority to the side that is closer to horizontal, which is more intuitive and avoids the overshoot. (The original Akima algorithm gives equal weights to the points on both sides, thus evenly dividing the undulation.)
The spline algorithm, on the other hand, performs cubic interpolation to produce piecewise polynomials with continuous secondorder derivatives (C2). The result is comparable to a regular polynomial interpolation, but is less susceptible to heavy oscillation between data points for high degrees. Still, this method can be susceptible to overshoots and oscillations between data points.
Compared to the spline algorithm, the Akima algorithm produces fewer undulations and is better suited to deal with quick changes between flat regions. This difference is illustrated below using test data that connects multiple flat regions.
References
[1] Akima, Hiroshi. "A new method of interpolation and smooth curve fitting based on local procedures." Journal of the ACM (JACM) , 17.4, 1970, pp. 589602.
[2] Akima, Hiroshi. "A method of bivariate interpolation and smooth surface fitting based on local procedures." Communications of the ACM , 17.1, 1974, pp. 1820.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
The input argument
x
(sample points) must be strictly increasing or strictly decreasing. Indices are not reordered.If the input argument
v
(sample values) is a variablelength vector (1by:
or:
by1), then the shape of the outputvq
matches the shape in MATLAB.If the input argument
v
is variablesize, is not a variablelength vector, and becomes a row vector at run time, then an error occurs.If the sample values or query points contain
Inf
orInf
, the output of the generated code might not match the output in MATLAB.If the input argument
xq
(query points) is variablesize, is not a variablelength vector, and becomes a row or column vector at run time, then an error occurs.See VariableSizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).
GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.
Usage notes and limitations:
The input argument
x
(sample points) must be strictly increasing or strictly decreasing. Indices are not reordered.If the input argument
v
(sample values) is a variablelength vector (1by:
or:
by1), then the shape of the outputvq
matches the shape in MATLAB.If the input argument
v
is variablesize, is not a variablelength vector, and becomes a row vector at run time, then an error occurs.If the input argument
xq
(query points) is variablesize, is not a variablelength vector, and becomes a row or column vector at run time, then an error occurs.See VariableSizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).
ThreadBased Environment
Run code in the background using MATLAB® backgroundPool
or accelerate code with Parallel Computing Toolbox™ ThreadPool
.
This function fully supports threadbased environments. For more information, see Run MATLAB Functions in ThreadBased Environment.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
Usage notes and limitations:
The interpolation methods
'pchip'
and'makima'
are not supported.The
'spline'
interpolation method is not supported for sample valuesv
containingNaN
.
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 R2006aR2020b: 'cubic'
method of interp1
performs cubic convolution
In R2020b, the 'cubic'
interpolation method of
interp1
performs cubic convolution. The
'v5cubic'
and 'cubic'
interpolation
methods now perform the same type of interpolation, which is consistent with the
behavior of interp2
, interp3
, and
interpn
. The cubic convolution interpolation method is
intended for uniformlyspaced data, and it falls back to 'spline'
interpolation for irregularlyspaced data.
In previous releases, 'cubic'
was the same as
'pchip'
, and only 'v5cubic'
performed
cubic convolution.
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