When it is necessary to extract subsets of entries out of an array whose indices cannot be written as a Cartesian product of components, linear indexing together with the function `sub2ind` can be used. For example:

```A = reshape (1:8, 2, 2, 2)  # Create 3-D array
A =

ans(:,:,1) =

1   3
2   4

ans(:,:,2) =

5   7
6   8

A(sub2ind (size (A), [1, 2, 1], [1, 1, 2], [1, 2, 1]))
⇒ ans = [A(1, 1, 1), A(2, 1, 2), A(1, 2, 1)]
```

An array with ‘nd’ dimensions can be indexed by an index expression which has from 1 to ‘nd’ components. For the ordinary and most common case, the number of components ‘M’ matches the number of dimensions ‘nd’. In this case the ordinary indexing rules apply and each component corresponds to the respective dimension of the array.

However, if the number of indexing components exceeds the number of dimensions (`M > nd`) then the excess components must all be singletons (`1`). Moreover, if `M < nd`, the behavior is equivalent to reshaping the input object so as to merge the trailing `nd - M` dimensions into the last index dimension `M`. Thus, the result will have the dimensionality of the index expression, and not the original object. This is the case whenever dimensionality of the index is greater than one (`M > 1`), so that the special rules for linear indexing are not applied. This is easiest to understand with an example:

```A = reshape (1:8, 2, 2, 2)  # Create 3-D array
A =

ans(:,:,1) =

1   3
2   4

ans(:,:,2) =

5   7
6   8

## 2-D indexing causes third dimension to be merged into second dimension.
## Equivalent array for indexing, Atmp, is now 2x4.
Atmp = reshape (A, 2, 4)
Atmp =

1   3   5   7
2   4   6   8

A(2,1)   # Reshape to 2x4 matrix, second entry of first column: ans = 2
A(2,4)   # Reshape to 2x4 matrix, second entry of fourth column: ans = 8
A(:,:)   # Reshape to 2x4 matrix, select all rows & columns, ans = Atmp
```

Note here the elegant use of the double colon to replace the call to the `reshape` function.

Another advanced use of linear indexing is to create arrays filled with a single value. This can be done by using an index of ones on a scalar value. The result is an object with the dimensions of the index expression and every element equal to the original scalar. For example, the following statements

```a = 13;
a(ones (1, 4))
```

produce a row vector whose four elements are all equal to 13.

Similarly, by indexing a scalar with two vectors of ones it is possible to create a matrix. The following statements

```a = 13;
a(ones (1, 2), ones (1, 3))
```

create a 2x3 matrix with all elements equal to 13. This could also have been written as

```13(ones (2, 3))
```

It is more efficient to use indexing rather than the code construction `scalar * ones (M, N, …)` because it avoids the unnecessary multiplication operation. Moreover, multiplication may not be defined for the object to be replicated whereas indexing an array is always defined. The following code shows how to create a 2x3 cell array from a base unit which is not itself a scalar.

```{"Hello"}(ones (2, 3))
```

It should be noted that `ones (1, n)` (a row vector of ones) results in a range object (with zero increment). A range is stored internally as a starting value, increment, end value, and total number of values; hence, it is more efficient for storage than a vector or matrix of ones whenever the number of elements is greater than 4. In particular, when ‘r’ is a row vector, the expressions

```  r(ones (1, n), :)
```
```  r(ones (n, 1), :)
```

will produce identical results, but the first one will be significantly faster, at least for ‘r’ and ‘n’ large enough. In the first case the index is held in compressed form as a range which allows Octave to choose a more efficient algorithm to handle the expression.

A general recommendation for users unfamiliar with these techniques is to use the function `repmat` for replicating smaller arrays into bigger ones, which uses such tricks.

A second use of indexing is to speed up code. Indexing is a fast operation and judicious use of it can reduce the requirement for looping over individual array elements, which is a slow operation.

Consider the following example which creates a 10-element row vector a containing the values a(i) = sqrt (i).

```for i = 1:10
a(i) = sqrt (i);
endfor
```

It is quite inefficient to create a vector using a loop like this. In this case, it would have been much more efficient to use the expression

```a = sqrt (1:10);
```

which avoids the loop entirely.

In cases where a loop cannot be avoided, or a number of values must be combined to form a larger matrix, it is generally faster to set the size of the matrix first (pre-allocate storage), and then insert elements using indexing commands. For example, given a matrix `a`,

```[nr, nc] = size (a);
x = zeros (nr, n * nc);
for i = 1:n
x(:,(i-1)*nc+1:i*nc) = a;
endfor
```

is considerably faster than

```x = a;
for i = 1:n-1
x = [x, a];
endfor
```

because Octave does not have to repeatedly resize the intermediate result.

: ind = sub2ind (dims, i, j)
: ind = sub2ind (dims, s1, s2, …, sN)

Convert subscripts to linear indices.

The input dims is a dimension vector where each element is the size of the array in the respective dimension (see `size`). The remaining inputs are scalars or vectors of subscripts to be converted.

The output vector ind contains the converted linear indices.

Background: Array elements can be specified either by a linear index which starts at 1 and runs through the number of elements in the array, or they may be specified with subscripts for the row, column, page, etc. The functions `ind2sub` and `sub2ind` interconvert between the two forms.

The linear index traverses dimension 1 (rows), then dimension 2 (columns), then dimension 3 (pages), etc. until it has numbered all of the elements. Consider the following 3-by-3 matrices:

```[(1,1), (1,2), (1,3)]     [1, 4, 7]
[(2,1), (2,2), (2,3)] ==> [2, 5, 8]
[(3,1), (3,2), (3,3)]     [3, 6, 9]
```

The left matrix contains the subscript tuples for each matrix element. The right matrix shows the linear indices for the same matrix.

The following example shows how to convert the two-dimensional indices `(2,1)` and `(2,3)` of a 3-by-3 matrix to linear indices with a single call to `sub2ind`.

```s1 = [2, 2];
s2 = [1, 3];
ind = sub2ind ([3, 3], s1, s2)
⇒ ind =  2   8
```

: [s1, s2, …, sN] = ind2sub (dims, ind)

Convert linear indices to subscripts.

The input dims is a dimension vector where each element is the size of the array in the respective dimension (see `size`). The second input ind contains linear indices to be converted.

The outputs s1, …, sN contain the converted subscripts.

Background: Array elements can be specified either by a linear index which starts at 1 and runs through the number of elements in the array, or they may be specified with subscripts for the row, column, page, etc. The functions `ind2sub` and `sub2ind` interconvert between the two forms.

The linear index traverses dimension 1 (rows), then dimension 2 (columns), then dimension 3 (pages), etc. until it has numbered all of the elements. Consider the following 3-by-3 matrices:

```[1, 4, 7]     [(1,1), (1,2), (1,3)]
[2, 5, 8] ==> [(2,1), (2,2), (2,3)]
[3, 6, 9]     [(3,1), (3,2), (3,3)]
```

The left matrix contains the linear indices for each matrix element. The right matrix shows the subscript tuples for the same matrix.

The following example shows how to convert the linear indices `2` and `8` to appropriate subscripts of a 3-by-3 matrix.

```ind = [2, 8];
[r, c] = ind2sub ([3, 3], ind)
⇒ r =  2   2
⇒ c =  1   3
```

If the number of output subscripts exceeds the number of dimensions, the exceeded dimensions are set to `1`. On the other hand, if fewer subscripts than dimensions are provided, the exceeding dimensions are merged into the final requested dimension. For clarity, consider the following examples:

```ind  = [2, 8];
dims = [3, 3];
## same as dims = [3, 3, 1]
[r, c, s] = ind2sub (dims, ind)
⇒ r =  2   2
⇒ c =  1   3
⇒ s =  1   1
## same as dims = 
r = ind2sub (dims, ind)
⇒ r =  2   8
```