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# diff

Differences and Approximate Derivatives

## Description

example

Y = diff(X) calculates differences between adjacent elements of X along the first array dimension whose size does not equal 1:

• If X is a vector of length m, then Y = diff(X) returns a vector of length m-1. The elements of Y are the differences between adjacent elements of X.

`Y = [X(2)-X(1) X(3)-X(2) ... X(m)-X(m-1)]`
• If X is a nonempty, nonvector p-by-m matrix, then Y = diff(X) returns a matrix of size (p-1)-by-m, whose elements are the differences between the rows of X.

`Y = [X(2,:)-X(1,:); X(3,:)-X(2,:); ... X(p,:)-X(p-1,:)]`
• If X is a 0-by-0 empty matrix, then Y = diff(X) returns a 0-by-0 empty matrix.

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.

## Examples

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### Differences Between Vector Elements

Create a vector, then compute the differences between the elements.

```X = [1 1 2 3 5 8 13 21];
Y = diff(X)
```
```Y =

0     1     1     2     3     5     8```

Note 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 =

4     4     4
20    20    20```

The output Y is a 2-by-3 difference matrix.

### Multiple Differences

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 =

5     5     5     5     5```

### Differences Between Matrix Columns

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 =

2     2
4     2
2     4
```

The output Y is a 3-by-2 difference matrix.

### Approximate Derivatives with diff

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 size
X = -pi:h:pi;    %domain
f = sin(X);      %range
Y = diff(f)/h;   %first derivative
Z = diff(Y)/h;   %second derivative
plot(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.

## Input Arguments

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### X — Input arrayvector | matrix | multidimensional array

Input array, specified as a vector, matrix, or multidimensional array.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
Complex Number Support: Yes

### n — Difference orderpositive integer scalar | []

Difference order, specified as a positive integer scalar or []. The default value of n is 1.

It is possible to specify n sufficiently large so that dim reduces to a single (size(X,dim) = 1) dimension. When this happens, diff continues calculating along the next array dimension whose size does not equal 1. This process continues until a 0-by-0 empty matrix is returned.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

### dim — Dimension to operate alongpositive integer scalar

Dimension to operate along, specified as a positive integer scalar. If no value is specified, 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 of A and returns a (p-1)-by-m difference matrix.

• diff(A,1,2) works on successive elements in the rows of A and 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 arrayscalar | vector | matrix | multidimensional array

Difference array, returned as a scalar, vector, matrix, or multidimensional array. If X is a nonempty array, then the dimension of X acted on by diff is reduced in size by n in the output.