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## Multidimensional Arrays

### Overview

An array having more than two dimensions is called a multidimensional array in the MATLAB® application. Multidimensional arrays in MATLAB are an extension of the normal two-dimensional matrix. Matrices have two dimensions: the row dimension and the column dimension.

You can access a two-dimensional matrix element with two subscripts: the first representing the row index, and the second representing the column index.

Multidimensional arrays use additional subscripts for indexing. A three-dimensional array, for example, uses three subscripts:

• The first references array dimension 1, the row.

• The second references dimension 2, the column.

• The third references dimension 3. This illustration uses the concept of a page to represent dimensions 3 and higher.

To access the element in the second row, third column of page 2, for example, you use the subscripts (2,3,2).

As you add dimensions to an array, you also add subscripts. A four-dimensional array, for example, has four subscripts. The first two reference a row-column pair; the second two access the third and fourth dimensions of data.

Most of the operations that you can perform on matrices (i.e., two-dimensional arrays) can also be done on multidimensional arrays.

 Note   The general multidimensional array functions reside in the datatypes directory.

### Creating Multidimensional Arrays

You can use the same techniques to create multidimensional arrays that you use for two-dimensional matrices. In addition, MATLAB provides a special concatenation function that is useful for building multidimensional arrays.

This section discusses

#### Generating Arrays Using Indexing

One way to create a multidimensional array is to create a two-dimensional array and extend it. For example, begin with a simple two-dimensional array A.

```A = [5 7 8; 0 1 9; 4 3 6];
```

A is a 3-by-3 array, that is, its row dimension is 3 and its column dimension is 3. To add a third dimension to A,

```A(:,:,2) = [1 0 4; 3 5 6; 9 8 7]
```

MATLAB responds with

```A(:,:,1) =
5     7     8
0     1     9
4     3     6

A(:,:,2) =
1     0     4
3     5     6
9     8     7
```

You can continue to add rows, columns, or pages to the array using similar assignment statements.

#### Extending Multidimensional Arrays

To extend A in any dimension:

• Increment or add the appropriate subscript and assign the desired values.

• Assign the same number of elements to corresponding array dimensions. For numeric arrays, all rows must have the same number of elements, all pages must have the same number of rows and columns, and so on.

You can take advantage of the MATLAB scalar expansion capabilities, together with the colon operator, to fill an entire dimension with a single value:

```A(:,:,3) = 5;

A(:,:,3)
ans =
5     5     5
5     5     5
5     5     5
```

To turn A into a 3-by-3-by-3-by-2, four-dimensional array, enter

```A(:,:,1,2) = [1 2 3; 4 5 6; 7 8 9];
A(:,:,2,2) = [9 8 7; 6 5 4; 3 2 1];
A(:,:,3,2) = [1 0 1; 1 1 0; 0 1 1];
```

Note that after the first two assignments MATLAB pads A with zeros, as needed, to maintain the corresponding sizes of dimensions.

#### Generating Arrays Using MATLAB Functions

You can use MATLAB functions such as randn, ones, and zeros to generate multidimensional arrays in the same way you use them for two-dimensional arrays. Each argument you supply represents the size of the corresponding dimension in the resulting array. For example, to create a 4-by-3-by-2 array of normally distributed random numbers:

```B = randn(4,3,2)
```

To generate an array filled with a single constant value, use the repmat function. repmat replicates an array (in this case, a 1-by-1 array) through a vector of array dimensions.

```B = repmat(5, [3 4 2])

B(:,:,1) =
5     5     5     5
5     5     5     5
5     5     5     5

B(:,:,2) =
5     5     5     5
5     5     5     5
5     5     5     5
```
 Note   Any dimension of an array can have size zero, making it a form of empty array. For example, 10-by-0-by-20 is a valid size for a multidimensional array.

#### Building Multidimensional Arrays with the cat Function

The cat function is a simple way to build multidimensional arrays; it concatenates a list of arrays along a specified dimension:

```B = cat(dim, A1, A2...)
```

where A1, A2, and so on are the arrays to concatenate, and dim is the dimension along which to concatenate the arrays.

For example, to create a new array with cat:

```B = cat(3, [2 8; 0 5], [1 3; 7 9])

B(:,:,1) =
2     8
0     5

B(:,:,2) =
1     3
7     9
```

The cat function accepts any combination of existing and new data. In addition, you can nest calls to cat. The lines below, for example, create a four-dimensional array.

```A = cat(3, [9 2; 6 5], [7 1; 8 4])
B = cat(3, [3 5; 0 1], [5 6; 2 1])
D = cat(4, A, B, cat(3, [1 2; 3 4], [4 3;2 1]))
```

cat automatically adds subscripts of 1 between dimensions, if necessary. For example, to create a 2-by-2-by-1-by-2 array, enter

```C = cat(4, [1 2; 4 5], [7 8; 3 2])
```

In the previous case, cat inserts as many singleton dimensions as needed to create a four-dimensional array whose last dimension is not a singleton dimension. If the dim argument had been 5, the previous statement would have produced a 2-by-2-by-1-by-1-by-2 array. This adds additional 1s to indexing expressions for the array. To access the value 8 in the four-dimensional case, use

### Accessing Multidimensional Array Properties

You can use the following MATLAB functions to get information about multidimensional arrays you have created.

• size — Returns the size of each array dimension.

```size(C)
ans =
2      2      1      2
rows  columns  dim3   dim4
```
• ndims — Returns the number of dimensions in the array.

```ndims(C)
ans =
4
```
• whos — Provides information on the format and storage of the array.

```whos
Name     Size       Bytes   Class

A        2x2x2         64   double array
B        2x2x2         64   double array
C        4-D           64   double array
D        4-D          192   double array

Grand total is 48 elements using 384 bytes
```

### Indexing Multidimensional Arrays

Many of the concepts that apply to two-dimensional matrices extend to multidimensional arrays as well.

To access a single element of a multidimensional array, use integer subscripts. Each subscript indexes a dimension—the first indexes the row dimension, the second indexes the column dimension, the third indexes the first page dimension, and so on.

Consider a 10-by-5-by-3 array nddata of random integers:

```nddata = fix(8 * randn(10,5,3));
```

To access element (3,2) on page 2 of nddata, for example, use nddata(3,2,2).

You can use vectors as array subscripts. In this case, each vector element must be a valid subscript, that is, within the bounds defined by the dimensions of the array. To access elements (2,1), (2,3), and (2,4) on page 3 of nddata, use

```nddata(2,[1 3 4],3);
```

#### The Colon and Multidimensional Array Indexing

The MATLAB colon indexing extends to multidimensional arrays. For example, to access the entire third column on page 2 of nddata, use nddata(:,3,2).

The colon operator is also useful for accessing other subsets of data. For example, nddata(2:3,2:3,1) results in a 2-by-2 array, a subset of the data on page 1 of nddata. This matrix consists of the data in rows 2 and 3, columns 2 and 3, on the first page of the array.

The colon operator can appear as an array subscript on both sides of an assignment statement. For example, to create a 4-by-4 array of zeros:

```C = zeros(4, 4)
```

Now assign a 2-by-2 subset of array nddata to the four elements in the center of C.

```C(2:3,2:3) = nddata(2:3,1:2,2)
```

#### Linear Indexing with Multidimensional Arrays

MATLAB linear indexing also extends to multidimensional arrays. In this case, MATLAB operates on a page-by-page basis to create the storage column, again appending elements columnwise. See Linear Indexing for an introduction to this topic.

For example, consider a 5-by-4-by-3-by-2 array C.

Again, a single subscript indexes directly into this column. For example, C(4) produces the result

```ans =
0
```

If you specify two subscripts (i,j) indicating row-column indices, MATLAB calculates the offset as described above. Two subscripts always access the first page of a multidimensional array, provided they are within the range of the original array dimensions.

If more than one subscript is present, all subscripts must conform to the original array dimensions. For example, C(6,2) is invalid because all pages of C have only five rows.

If you specify more than two subscripts, MATLAB extends its indexing scheme accordingly. For example, consider four subscripts (i,j,k,l) into a four-dimensional array with size [d1 d2 d3 d4]. MATLAB calculates the offset into the storage column by

```(l-1)(d3)(d2)(d1)+(k-1)(d2)(d1)+(j-1)(d1)+i
```

For example, if you index the array C using subscripts (3, 4, 2, 1), MATLAB returns the value 5 (index 38 in the storage column).

In general, the offset formula for an array with dimensions [d1 d2 d3 ... dn] using any subscripts (s1 s2 s3 ... sn) is

```(sn-1)(dn-1)(dn-2)...(d1)+(sn-1-1)(dn-2)...(d1)+...+(s2-1)(d1)+s1
```

Because of this scheme, you can index an array using any number of subscripts. You can append any number of 1s to the subscript list because these terms become zero. For example,

```C(3,2,1,1,1,1,1,1)
```

is equivalent to

```C(3,2)
```

#### Avoiding Ambiguity in Multidimensional Indexing

Some assignment statements, such as

```A(:,:,2) = 1:10
```

are ambiguous because they do not provide enough information about the shape of the dimension to receive the data. In the case above, the statement tries to assign a one-dimensional vector to a two-dimensional destination. MATLAB produces an error for such cases. To resolve the ambiguity, be sure you provide enough information about the destination for the assigned data, and that both data and destination have the same shape. For example:

```A(1,:,2) = 1:10;
```

### Reshaping Multidimensional Arrays

Unless you change its shape or size, a MATLAB array retains the dimensions specified at its creation. You change array size by adding or deleting elements. You change array shape by respecifying the array's row, column, or page dimensions while retaining the same elements. The reshape function performs the latter operation. For multidimensional arrays, its form is

```B = reshape(A,[s1 s2 s3 ...])
```

s1, s2, and so on represent the desired size for each dimension of the reshaped matrix. Note that a reshaped array must have the same number of elements as the original array (that is, the product of the dimension sizes is constant).

M

reshape(M, [6 5])

The reshape function operates in a columnwise manner. It creates the reshaped matrix by taking consecutive elements down each column of the original data construct.

C

reshape(C, [6 2])

Here are several new arrays from reshaping nddata:

```B = reshape(nddata, [6 25])
C = reshape(nddata, [5 3 10])
D = reshape(nddata, [5 3 2 5])
```

#### Removing Singleton Dimensions

MATLAB creates singleton dimensions if you explicitly specify them when you create or reshape an array, or if you perform a calculation that results in an array dimension of one:

```B = repmat(5, [2 3 1 4]);

size(B)
ans =
2     3     1     4
```

The squeeze function removes singleton dimensions from an array:

```C = squeeze(B);

size(C)
ans =
2     3     4
```

The squeeze function does not affect two-dimensional arrays; row vectors remain rows.

### Permuting Array Dimensions

The permute function reorders the dimensions of an array:

```B = permute(A, dims);
```

dims is a vector specifying the new order for the dimensions of A, where 1 corresponds to the first dimension (rows), 2 corresponds to the second dimension (columns), 3 corresponds to pages, and so on.

For a more detailed look at the permute function, consider a four-dimensional array A of size 5-by-4-by-3-by-2. Rearrange the dimensions, placing the column dimension first, followed by the second page dimension, the first page dimension, then the row dimension. The result is a 4-by-2-by-3-by-5 array.

You can think of permute's operation as an extension of the transpose function, which switches the row and column dimensions of a matrix. For permute, the order of the input dimension list determines the reordering of the subscripts. In the example above, element (4,2,1,2) of A becomes element (2,2,1,4) of B, element (5,4,3,2) of A becomes element (4,2,3,5) of B, and so on.

#### Inverse Permutation

The ipermute function is the inverse of permute. Given an input array A and a vector of dimensions v, ipermute produces an array B such that permute(B,v) returns A.

For example, these statements create an array E that is equal to the input array C:

```D = ipermute(C, [1 4 2 3]);
E = permute(D, [1 4 2 3])
```

You can obtain the original array after permuting it by calling ipermute with the same vector of dimensions.

### Computing with Multidimensional Arrays

Many of the MATLAB computational and mathematical functions accept multidimensional arrays as arguments. These functions operate on specific dimensions of multidimensional arrays; that is, they operate on individual elements, on vectors, or on matrices.

#### Operating on Vectors

Functions that operate on vectors, like sum, mean, and so on, by default typically work on the first nonsingleton dimension of a multidimensional array. Most of these functions optionally let you specify a particular dimension on which to operate. There are exceptions, however. For example, the cross function, which finds the cross product of two vectors, works on the first nonsingleton dimension having length 3.

 Note   In many cases, these functions have other restrictions on the input arguments — for example, some functions that accept multiple arrays require that the arrays be the same size. Refer to the online help for details on function arguments.

#### Operating Element-by-Element

MATLAB functions that operate element-by-element on two-dimensional arrays, like the trigonometric and exponential functions in the elfun directory, work in exactly the same way for multidimensional cases. For example, the sin function returns an array the same size as the function's input argument. Each element of the output array is the sine of the corresponding element of the input array.

Similarly, the arithmetic, logical, and relational operators all work with corresponding elements of multidimensional arrays that are the same size in every dimension. If one operand is a scalar and one an array, the operator applies the scalar to each element of the array.

#### Operating on Planes and Matrices

Functions that operate on planes or matrices, such as the linear algebra and matrix functions in the matfun directory, do not accept multidimensional arrays as arguments. That is, you cannot use the functions in the matfun directory, or the array operators *, ^, \, or /, with multidimensional arguments. Supplying multidimensional arguments or operands in these cases results in an error.

You can use indexing to apply a matrix function or operator to matrices within a multidimensional array. For example, create a three-dimensional array A:

```A = cat(3, [1 2 3; 9 8 7; 4 6 5], [0 3 2; 8 8 4; 5 3 5], ...
[6 4 7; 6 8 5; 5 4 3]);
```

Applying the eig function to the entire multidimensional array results in an error:

```eig(A)
??? Undefined function or method 'eig' for input
arguments of type 'double' and attributes 'full 3d real'.```

You can, however, apply eig to planes within the array. For example, use colon notation to index just one page (in this case, the second) of the array:

```eig(A(:,:,2))
ans =
12.9129
-2.6260
2.7131
```
 Note   In the first case, subscripts are not colons; you must use squeeze to avoid an error. For example, eig(A(2,:,:)) results in an error because the size of the input is [1 3 3]. The expression eig(squeeze(A(2,:,:))), however, passes a valid two-dimensional matrix to eig.

### Organizing Data in Multidimensional Arrays

You can use multidimensional arrays to represent data in two ways:

• As planes or pages of two-dimensional data. You can then treat these pages as matrices.

• As multivariate or multidimensional data. For example, you might have a four-dimensional array where each element corresponds to either a temperature or air pressure measurement taken at one of a set of equally spaced points in a room.

For example, consider an RGB image. For a single image, a multidimensional array is probably the easiest way to store and access data.

To access an entire plane of the image, use

```redPlane = RGB(:,:,1);
```

To access a subimage, use

```subimage = RGB(20:40,50:85,:);
```

The RGB image is a good example of data that needs to be accessed in planes for operations like display or filtering. In other instances, however, the data itself might be multidimensional. For example, consider a set of temperature measurements taken at equally spaced points in a room. Here the location of each value is an integral part of the data set—the physical placement in three-space of each element is an aspect of the information. Such data also lends itself to representation as a multidimensional array.

Now to find the average of all the measurements, use

```mean(mean(mean(TEMP)));
```

To obtain a vector of the "middle" values (element (2,2)) in the room on each page, use

```B = TEMP(2,2,:);
```

### Multidimensional Cell Arrays

Like numeric arrays, the framework for multidimensional cell arrays in MATLAB is an extension of the two-dimensional cell array model. You can use the cat function to build multidimensional cell arrays, just as you use it for numeric arrays.

For example, create a simple three-dimensional cell array C:

```A{1,1} = [1 2;4 5];
A{1,2} = 'Name';
A{2,1} = 2-4i;
A{2,2} = 7;
B{1,1} = 'Name2';
B{1,2} = 3;
B{2,1} = 0:1:3;
B{2,2} = [4 5]';
C = cat(3, A, B);
```

The subscripts for the cells of C look like

### Multidimensional Structure Arrays

Multidimensional structure arrays are extensions of rectangular structure arrays. Like other types of multidimensional arrays, you can build them using direct assignment or the cat function:

```patient(1,1,1).name = 'John Doe';
patient(1,1,1).billing = 127.00;
patient(1,1,1).test = [79 75 73; 180 178 177.5; 220 210 205];
patient(1,2,1).name = 'Ann Lane';
patient(1,2,1).billing = 28.50;
patient(1,2,1).test = [68 70 68; 118 118 119; 172 170 169];
patient(1,1,2).name = 'Al Smith';
patient(1,1,2).billing = 504.70;
patient(1,1,2).test = [80 80 80; 153 153 154; 181 190 182];
patient(1,2,2).name = 'Dora Jones';
patient(1,2,2).billing = 1173.90;
patient(1,2,2).test = [73 73 75; 103 103 102; 201 198 200];
```

#### Applying Functions to Multidimensional Structure Arrays

To apply functions to multidimensional structure arrays, operate on fields and field elements using indexing. For example, find the sum of the columns of the test array in patient(1,1,2):

```sum((patient(1,1,2).test));
```

Similarly, add all the billing fields in the patient array:

```total = sum([patient.billing]);
```