# interp1

1-D data interpolation (table lookup)

## Description

example

vq = interp1(x,v,xq) returns interpolated values of a 1-D function at specific query points using linear interpolation. Vector 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 1-D sample values.

example

vq = interp1(x,v,xq,method) specifies an alternative interpolation method: 'linear', 'nearest', 'next', 'previous', 'pchip', 'cubic', 'v5cubic', 'makima', or 'spline'. The default method is 'linear'.

example

vq = interp1(x,v,xq,method,extrapolation) specifies a strategy for evaluating points that lie outside the domain of 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.

example

vq = interp1(v,xq) returns interpolated values and assumes a default set of sample point coordinates. The default points are the sequence of numbers from 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.

vq = interp1(v,xq,method) specifies any of the alternative interpolation methods and uses the default sample points.

vq = interp1(v,xq,method,extrapolation) specifies an extrapolation strategy and uses the default sample points.

pp = interp1(x,v,method,'pp') returns the piecewise polynomial form of v(x) using the method algorithm.

Note

This syntax is not recommended. Use griddedInterpolant instead.

## Examples

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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');

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');

Define a set of sample points.

x = 1:10;

Define the values of the function, $v\left(x\right)=5x+{x}^{2}i$, 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');

Interpolate time-stamped 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')

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.

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

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

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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, then length(x) must equal length(v).

• If v is an array, then length(x) must equal size(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

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 1-D 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

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

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

Method

Description

Continuity

'linear'

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.

C0

• Requires at least 2 points

• Requires more memory and computation time than nearest neighbor

'nearest'

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

Discontinuous

• Requires at least 2 points

• Modest memory requirements

• Fastest computation time

'next'

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

Discontinuous

• Requires at least 2 points

• Same memory requirements and computation time as 'nearest'

'previous'

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

Discontinuous

• Requires at least 2 points

• Same memory requirements and computation time as 'nearest'

'pchip'

Shape-preserving piecewise cubic interpolation. 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 4 points

• Requires more memory and computation time than 'linear'

'cubic'

Cubic convolution used in MATLAB® 5.

C1

• Requires at least 3 points

• Points must be uniformly spaced

• This method falls back to 'spline' interpolation for irregularly-spaced data

• Similar memory requirements and computation time as 'pchip'

'v5cubic'

Same as 'cubic'.

C1

'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. The Akima formula is modified to avoid overshoots.

C1

• Requires at least 2 points

• Produces fewer undulations than 'spline', but does not flatten as aggressively as 'pchip'

• Computation is more expensive than 'pchip', but typically less than 'spline'

• Memory requirements are similar to those of 'spline'

'spline'

Spline interpolation using not-a-knot 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.

C2

• Requires at least 4 points

• Requires more memory and computation time than 'pchip'

Extrapolation strategy, specified as 'extrap' or a real scalar value.

• Specify 'extrap' when you want interp1 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

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Interpolated values, returned as a scalar, vector, matrix, or array. The size of vq depends on the shape of v and xq.

Shape of vShape of xqSize of VqExample
VectorVectorsize(xq)If size(v) = [1 100]
and size(xq) = [1 500],
then size(vq) = [1 500].
VectorMatrix
or N-D Array
size(xq)If size(v) = [1 100]
and size(xq) = [50 30],
then size(vq) = [50 30].
Matrix
or N-D 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 N-D Array
Matrix
or N-D 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].

Piecewise polynomial, returned as a structure that you can pass to the ppval function for evaluation.

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### Akima and Spline Interpolation

The Akima algorithm for one-dimensional interpolation, described in [1] and [2], performs cubic interpolation to produce piecewise polynomials with continuous first-order 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 second-order 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.

## Compatibility Considerations

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Behavior changed in R2020b

## 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. 589-602.

[2] Akima, Hiroshi. "A method of bivariate interpolation and smooth surface fitting based on local procedures." Communications of the ACM , 17.1, 1974, pp. 18-20.