Differences and approximate derivatives
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Syntax
Y = diff(X)
Y = diff(X,n)
Y = diff(X,n,dim)
Description
Y = diff(X) calculates differences between adjacent elements of X. By default, diff operates along the first array dimension whose size does not equal 1.
If
Xis a vector of lengthm, thenY = diff(X)returns a vector of lengthm-1. The elements ofYare the differences between adjacent elements ofX.Y = [X(2)-X(1) X(3)-X(2) ... X(m)-X(m-1)]
If
Xis a nonempty, nonvector p-by-m matrix, thenY = diff(X)returns a matrix of size (p-1)-by-m, whose elements are the differences between the rows ofX.Y = [X(2,:)-X(1,:); X(3,:)-X(2,:); ... X(p,:)-X(p-1,:)]
If
Xis a 0-by-0 empty matrix, thenY = diff(X)returns a 0-by-0 empty matrix.If
Xis a p-by-m table or timetable, thenY = diff(X)returns a table or timetable of size (p-1)-by-m, whose elements are the differences between the rows ofX. IfXis a 1-by-m table or timetable, then the size ofYis 0-by-m. (since R2023a)
example
Y = diff(X,n) calculates the nth difference by applying the diff(X) operator recursively n times. In practice, this means diff(X,2) is the same as diff(diff(X)).
example
Y = diff(X,n,dim) is the nth difference calculated along the dimension specified by dim. The dim input is a positive integer scalar.
example
Examples
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Differences Between Vector Elements
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Create a vector, then compute the differences between the elements.
X = [1 1 2 3 5 8 13 21];Y = diff(X)
Y = 1×7 0 1 1 2 3 5 8Note that Y has one fewer element than X.
Differences Between Matrix Rows
Create a 3-by-3 matrix, then compute the first difference between the rows.
X = [1 1 1; 5 5 5; 25 25 25];Y = diff(X)
Y = 2×3 4 4 4 20 20 20Y is a 2-by-3 matrix.
Multiple Differences
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Create a vector and compute the second-order difference between the elements.
X = [0 5 15 30 50 75 105];Y = diff(X,2)
Y = 1×5 5 5 5 5 5Differences Between Matrix Columns
Open Live Script
Create a 3-by-3 matrix, then compute the first-order difference between the columns.
X = [1 3 5;7 11 13;17 19 23];Y = diff(X,1,2)
Y = 3×2 2 2 4 2 2 4Y is a 3-by-2 matrix.
Approximate Derivatives with diff
Open Live Script
Use the diff function to approximate partial derivatives with the syntax Y = diff(f)/h, where f is a vector of function values evaluated over some domain, X, and h is an appropriate step size.
For example, the first derivative of sin(x) with respect to x is cos(x), and the second derivative with respect to x is -sin(x). You can use diff to approximate these derivatives.
h = 0.001; % step sizeX = -pi:h:pi; % domainf = sin(X); % rangeY = diff(f)/h; % first derivativeZ = diff(Y)/h; % second derivativeplot(X(:,1:length(Y)),Y,'r',X,f,'b', X(:,1:length(Z)),Z,'k')
In this plot the blue line corresponds to the original function, sin. The red line corresponds to the calculated first derivative, cos, and the black line corresponds to the calculated second derivative, -sin.
Differences Between Datetime Values
Open Live Script
Create a sequence of equally-spaced datetime values, and find the time differences between them.
t1 = datetime('now');t2 = t1 + minutes(5);t = t1:minutes(1.5):t2t = 1x4 datetime 20-Jul-2024 15:09:36 20-Jul-2024 15:11:06 20-Jul-2024 15:12:36 20-Jul-2024 15:14:06dt = diff(t)
dt = 1x3 duration 00:01:30 00:01:30 00:01:30diff returns a duration array.
Input Arguments
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X — Input array
vector | matrix | multidimensional array | table | timetable
Input array, specified as a vector, matrix, multidimensional array, table, or timetable. X can be a numeric array, logical array, datetime array, or duration array, or a table or timetable whose variables have any of these data types.
Complex Number Support: Yes
n — Difference order
positive integer scalar | []
Difference order, specified as a positive integer scalar or []. The default value of n is 1.
The behavior of diff when n is larger than the dimension being operated on depends on the syntax being used:
With
diff(X,n), wherenis larger than the first non-singleton dimension,diffreduces the size of that dimension to 1 and then continues taking differences along the next non-singleton dimension.With
diff(X,n,dim), wheren >= size(X,dim),diffreturns an empty array.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
dim — Dimension to operate along
positive integer scalar
Dimension to operate along, specified as a positive integer scalar. If you do not specify the dimension, then the default is the first array dimension whose size does not equal 1.
Consider a two-dimensional p-by-m input array, A:
diff(A,1,1)works on successive elements in the columns ofAand returns a (p-1)-by-m difference matrix.diff(A,1,2)works on successive elements in the rows ofAand returns a p-by-(m-1) difference matrix.
Data Types: double | single | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
Output Arguments
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Y — Difference array
scalar | vector | matrix | multidimensional array | table | timetable
Difference array, returned as a scalar, vector, matrix, multidimensional array, table, or timetable. If X is a nonempty array, then the dimension of X acted on by diff is reduced in size by n in the output.
Extended Capabilities
Tall Arrays
Calculate with arrays that have more rows than fit in memory.
This function supports tall arrays with the limitations:
You must use the three-input syntax Y = diff(X,N,dim).
For more information, see Tall Arrays.
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
If supplied, the arguments representing the number of times to apply
diffand the dimension along which to calculate the difference must be constants.See Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).
Code generation does not support sparse matrix inputs for this function.
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™.
The diff function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). 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
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R2023a: Perform calculations directly on tables and timetables
The diff function can calculate on all variables within a table or timetable without indexing to access those variables. All variables must have data types that support the calculation. For more information, see Direct Calculations on Tables and Timetables.
R2022a: Improved performance with large number of elements
The diff function shows improved performance when operating on vectors with at least 105 elements or when operating along the first or second dimension of matrices and multidimensional arrays with at least 5 x 105 elements.
For example, this code constructs a double with 2.5 x 107 elements and calculates differences between adjacent elements. The code is approximately 2.4x faster than in the previous release:
function timingDiffrng defaultN = 5000;A = rand(N);ticfor k = 1:40 D = diff(A);endtocend
The approximate execution times are:
R2021b: 2.43 s
R2022a: 1.00 s
The code was timed on a Windows® 10, Intel® Xeon® CPU E5-1650 v4 @ 3.60 GHz test system by calling the timingDiff function.
See Also
gradient | prod | cumsum | sum
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