# Arrays

## Contents

# Arrays#

Lists are very “general” in the sense that they can contain objects of multiple different types. However, this generality is a trade-off against efficiency. For many applications, we instead use * arrays*. These are a central feature of

`numpy`

, and can only contain elements all of the same type (usually `int`

or `float`

, though `numpy`

makes finer distinctions to both of these types). Arrays make operations with large amounts of numeric data much faster.## Creating Arrays#

There are a number of ways to create arrays.

### Conversion from `list`

#

Given a `list`

of objects, you can easily create an `array`

from it:

```
import numpy as np
my_array = np.array([1, 3, 2, 5, 10]) # make an array from a list
my_array
```

```
array([ 1, 3, 2, 5, 10])
```

If you try and convert a list of mixed types then `numpy`

does its best to convert it to a sensible array - but you may not get what you expect!

```
my_list = [1, 2, 45.3, "hello"]
my_array2 = np.array(my_list)
my_array2
```

```
array(['1', '2', '45.3', 'hello'], dtype='<U32')
```

You can also create higher-dimensional arrays by converting lists of lists:

```
np.array([[1, 2], [3, 4]])
```

```
array([[1, 2],
[3, 4]])
```

`arange`

and `linspace`

#

When we are dealing with large arrays, it is impractical to type in each entry by hand. There are a number of helpful functions provided by `numpy`

to create arrays with certain properties. To create an array of evenly-spaced values between a `start`

and `stop`

value, we can use the `arange`

or `linspace`

functions from `numpy`

.

With both functions we specify the start and stop values (like with `range`

), but with `arange`

we specify the spacing between the values, and with `linspace`

we specify the number of points we want in the array (with even spacing).

```
np.arange(0.0, 4.2, 0.2)
```

```
array([0. , 0.2, 0.4, 0.6, 0.8, 1. , 1.2, 1.4, 1.6, 1.8, 2. , 2.2, 2.4,
2.6, 2.8, 3. , 3.2, 3.4, 3.6, 3.8, 4. ])
```

```
np.linspace(0.0, 4.2, num = 10)
```

```
array([0. , 0.46666667, 0.93333333, 1.4 , 1.86666667,
2.33333333, 2.8 , 3.26666667, 3.73333333, 4.2 ])
```

You might have expected the last array to contain `0, 0.42, ..., 4.2`

given the argument `num=10`

. However, the first point `0`

is included as one of these ten points; this means that the spacing is given by `(b-a)/(num-1)`

where `a`

is the start point and `b`

is the end point. If we wanted a spacing of `0.42`

, we should use `arange`

or give the argument `num=11`

:

```
np.linspace(0.0, 4.2, num = 11)
```

```
array([0. , 0.42, 0.84, 1.26, 1.68, 2.1 , 2.52, 2.94, 3.36, 3.78, 4.2 ])
```

### All-zero arrays#

Another common way to create arrays is by creating an array with all entries 0, then setting the entries to be the correct value. To create an all-zero array, use `np.zeros`

:

```
np.zeros(10)
```

```
array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0.])
```

```
# For multi-dimensional arrays, the dimensions have to be given
# as a tuple (dim1, dim2,...) - hence the extra pair of brackets in the line below.
np.zeros((3, 5))
```

```
array([[0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0.]])
```

In the following example, we create a \(3 \times 4\) matrix \(A\) with entries $\(a_{i,j} = \begin{cases}2i + j & i, j > 1 \\ 0 &\text{otherwise.} \end{cases}\)\( for \)1 \leq i \leq 3\( and \)1 \leq j \leq 4$.

Notice that we index matrices from 1 in mathematics, but from 0 in programming - you have to be very careful with this, and it’s best to clarify which you mean.

```
# Create a 3x4 matrix of zeros; notice the argument is a pair (dim1, dim2)
dim1 = 3
dim2 = 4
mat = np.zeros((dim1, dim2))
# now set the entries
for i in range(1,dim1):
for j in range(1,dim2):
# We use (i + 1) and (j + 1) to agree with the mathematical definition.
mat[i, j] = 2 * (i + 1) + (j + 1)
# Note we don't need multiple square brackets with numpy arrays.
# You could use mat[i][j] instead, but mat[i,j] is more efficient.
# view mat
mat
```

```
array([[ 0., 0., 0., 0.],
[ 0., 6., 7., 8.],
[ 0., 8., 9., 10.]])
```

### Diagonal arrays#

We often want to create arrays with entries only on the main diagonal. There are several ways to do this in `numpy`

; here are some examples.

```
# create the N by N identity matrix with np.identity(N)
np.identity(3)
```

```
array([[1., 0., 0.],
[0., 1., 0.],
[0., 0., 1.]])
```

```
# You can also use np.eye(N).
# This is more flexible than np.identity - you make extra columns
# containing zeros using the "M" argument
np.eye(3, M = 5)
```

```
array([[1., 0., 0., 0., 0.],
[0., 1., 0., 0., 0.],
[0., 0., 1., 0., 0.]])
```

```
# Create an array with specified diagonal entries.
# Note the extra brackets to make the argument a single tuple.
np.diag((1, 4, 2, 6))
```

```
array([[1, 0, 0, 0],
[0, 4, 0, 0],
[0, 0, 2, 0],
[0, 0, 0, 6]])
```

## Working with `arrays`

#

`Numpy`

arrays are extremely powerful and we will not attempt to discuss all (or even a small portion) of their features in this Notebook. One of the most useful features is that many `numpy`

functions can be applied to all elements of an array at once:

```
np.sin(mat)
```

```
array([[ 0. , 0. , 0. , 0. ],
[ 0. , -0.2794155 , 0.6569866 , 0.98935825],
[ 0. , 0.98935825, 0.41211849, -0.54402111]])
```

This has many uses, like generating the \(y\)-coordinates for plots. Applying functions like this to arrays is not only more convenient than using lists and `for`

-loops, but often much faster. Unlike lists, we can also do arithmetic with arrays:

```
2 * mat + np.sin(mat)
```

```
array([[ 0. , 0. , 0. , 0. ],
[ 0. , 11.7205845 , 14.6569866 , 16.98935825],
[ 0. , 16.98935825, 18.41211849, 19.45597889]])
```

You can access elements in the same way as for lists:

```
# the second element of mat is the row [0, 6, 7, 8]
mat[1]
```

```
array([0., 6., 7., 8.])
```

```
# We could get the third element of the second row using mat[1][2],
# but it is more efficient to do mat[1, 2].
# This syntax does not work for lists, only for arrays.
mat[1,2]
```

```
7.0
```

Multiplication using `*`

means * component-wise multiplication*, using the formula \(c_{ij} = a_{ij}b_{ij}\).

```
mat = np.array([[1, 2], [3, 4]])
mat
```

```
array([[1, 2],
[3, 4]])
```

```
mat * mat
```

```
array([[ 1, 4],
[ 9, 16]])
```

For * standard matrix multiplication*, use

`@`

instead:```
mat @ mat
```

```
array([[ 7, 10],
[15, 22]])
```