This example shows how to filter the elements of an array by applying conditions to the array. For instance, you can examine the even elements in a matrix, find the location of all 0s in a multidimensional array, or replace `NaN`

values in data. You can perform these tasks using a combination of the relational and logical operators. The relational operators (`>`

, `<`

, `>=`

, `<=`

, `==`

, `~=`

) impose conditions on the array, and you can apply multiple conditions by connecting them with the logical operators `and`

, `or`

, and `not`

, respectively denoted by the symbols `&`

, `|`

, and `~`

.

To apply a single condition, start by creating a 5-by-5 matrix that contains random integers between 1 and 15. Reset the random number generator to the default state for reproducibility.

```
rng default
A = randi(15,5)
```

`A = `*5×5*
13 2 3 3 10
14 5 15 7 1
2 9 15 14 13
14 15 8 12 15
10 15 13 15 11

Use the relational *less than* operator, `<`

, to determine which elements of `A`

are less than 9. Store the result in `B`

.

B = A < 9

`B = `*5x5 logical array*
0 1 1 1 0
0 1 0 1 1
1 0 0 0 0
0 0 1 0 0
0 0 0 0 0

The result is a logical matrix. Each value in `B`

represents a logical `1`

(`true`

) or logical `0`

(`false`

) state to indicate whether the corresponding element of `A`

fulfills the condition `A < 9`

. For example, `A(1,1)`

is `13`

, so `B(1,1)`

is logical `0`

(`false`

). However, `A(1,2)`

is `2`

, so `B(1,2)`

is logical `1`

(`true`

).

Although `B`

contains information about *which* elements in `A`

are less than 9, it doesn’t tell you what their *values* are. Rather than comparing the two matrices element by element, you can use `B`

to index into `A`

.

A(B)

`ans = `*8×1*
2
2
5
3
8
3
7
1

The result is a column vector of the elements in `A`

that are less than 9. Since `B`

is a logical matrix, this operation is called **logical indexing**. In this case, the logical array being used as an index is the same size as the other array, but this is not a requirement. For more information, see Array Indexing.

Some problems require information about the *locations* of the array elements that meet a condition rather than their actual values. In this example, you can use the `find`

function to locate all of the elements in `A`

less than 9.

I = find(A < 9)

`I = `*8×1*
3
6
7
11
14
16
17
22

The result is a column vector of linear indices. Each index describes the location of an element in `A`

that is less than 9, so in practice `A(I)`

returns the same result as `A(B)`

. The difference is that `A(B)`

uses logical indexing, whereas `A(I)`

uses linear indexing.

You can use the logical `and`

, `or`

, and `not`

operators to apply any number of conditions to an array; the number of conditions is not limited to one or two.

First, use the logical `and`

operator, denoted `&`

, to specify two conditions: the elements must be **less than 9 **and** greater than 2**. Specify the conditions as a logical index to view the elements that satisfy both conditions.

A(A<9 & A>2)

`ans = `*5×1*
5
3
8
3
7

The result is a list of the elements in `A`

that satisfy both conditions. Be sure to specify each condition with a separate statement connected by a logical operator. For example, you cannot specify the conditions above by `A(2<A<9)`

, since it evaluates to `A(2<A | A<9)`

.

Next, find the elements in `A`

that are **less than 9** and **even numbered**.

A(A<9 & ~mod(A,2))

`ans = `*3×1*
2
2
8

The result is a list of all even elements in `A`

that are less than 9. The use of the logical NOT operator, ~, converts the matrix `mod(A,2)`

into a logical matrix, with a value of logical `1`

(`true`

) located where an element is evenly divisible by 2.

Finally, find the elements in `A`

that are **less than 9** and **even numbered** and **not equal to 2**.

A(A<9 & ~mod(A,2) & A~=2)

ans = 8

The result, 8, is even, less than 9, and not equal to 2. It is the only element in `A`

that satisfies all three conditions.

Use the `find`

function to get the index of the element equal to 8 that satisfies the conditions.

find(A<9 & ~mod(A,2) & A~=2)

ans = 14

The result indicates that `A(14) = 8`

.

Sometimes it is useful to simultaneously change the values of several existing array elements. Use logical indexing with a simple assignment statement to replace the values in an array that meet a condition.

Replace all values in `A`

that are greater than 10 with the number 10.

A(A>10) = 10

`A = `*5×5*
10 2 3 3 10
10 5 10 7 1
2 9 10 10 10
10 10 8 10 10
10 10 10 10 10

Next, replace all values in `A`

that are not equal to 10 with a `NaN`

value.

A(A~=10) = NaN

`A = `*5×5*
10 NaN NaN NaN 10
10 NaN 10 NaN NaN
NaN NaN 10 10 10
10 10 NaN 10 10
10 10 10 10 10

Lastly, replace all of the `NaN`

values in `A`

with zeros and apply the logical NOT operator, `~A`

.

A(isnan(A)) = 0; C = ~A

`C = `*5x5 logical array*
0 1 1 1 0
0 1 0 1 1
1 1 0 0 0
0 0 1 0 0
0 0 0 0 0

The resulting matrix has values of logical `1`

(`true`

) in place of the `NaN`

values, and logical `0`

(`false`

) in place of the 10s. The logical NOT operation, `~A`

, converts the numeric array into a logical array such that `A&C`

returns a matrix of logical `0`

(`false`

) values and `A|C`

returns a matrix of logical `1`

(`true`

) values.

`Logical Operators: Short Circuit`

| `and`

| `find`

| `isnan`

| `nan`

| `not`

| `or`

| `xor`