1665983538

Lazy arrays and linear algebra in Julia

This package supports lazy analogues of array operations like `vcat`

, `hcat`

, and multiplication. This helps with the implementation of matrix-free methods for iterative solvers.

The package has been designed with high-performance in mind, so should outperform the non-lazy analogues from Base for many operations like `copyto!`

and broadcasting. Some operations will be inherently slower due to extra computation, like `getindex`

. Please file an issue for any examples that are significantly slower than their the analogue in Base.

To construct a lazy representation of a function call `f(x,y,z...)`

, use the command `applied(f, x, y, z...)`

. This will return an unmaterialized object typically of type `Applied`

that represents the operation. To realize that object, call `materialize`

, which will typically be equivalent to calling `f(x,y,z...)`

. A macro `@~`

is available as a shorthand:

```
julia> using LazyArrays, LinearAlgebra
julia> applied(exp, 1)
Applied(exp,1)
julia> materialize(applied(exp, 1))
2.718281828459045
julia> materialize(@~ exp(1))
2.718281828459045
julia> exp(1)
2.718281828459045
```

Note that `@~`

causes sub-expressions to be wrapped in an `applied`

call, not just the top-level expression. This can lead to `MethodError`

s when lazy application of sub-expressions is not yet implemented. For example,

```
julia> @~ Vector(1:10) .* ones(10)'
ERROR: MethodError: ...
julia> A = Vector(1:10); B = ones(10); (@~ sum(A .* B')) |> materialize
550.0
```

The benefit of lazy operations is that they can be materialized in-place, possible using simplifications. For example, it is possible to do BLAS-like Matrix-Vector operations of the form `α*A*x + β*y`

as implemented in `BLAS.gemv!`

using a lazy applied object:

```
julia> A = randn(5,5); b = randn(5); c = randn(5); d = similar(c);
julia> d .= @~ 2.0 * A * b + 3.0 * c # Calls gemv!
5-element Array{Float64,1}:
-2.5366335879717514
-5.305097174484744
-9.818431932350942
2.421562605495651
0.26792916096572983
julia> 2*(A*b) + 3c
5-element Array{Float64,1}:
-2.5366335879717514
-5.305097174484744
-9.818431932350942
2.421562605495651
0.26792916096572983
julia> function mymul(A, b, c, d) # need to put in function for benchmarking
d .= @~ 2.0 * A * b + 3.0 * c
end
mymul (generic function with 1 method)
julia> @btime mymul(A, b, c, d) # calls gemv!
77.444 ns (0 allocations: 0 bytes)
5-element Array{Float64,1}:
-2.5366335879717514
-5.305097174484744
-9.818431932350942
2.421562605495651
0.26792916096572983
julia> @btime 2*(A*b) + 3c; # does not call gemv!
241.659 ns (4 allocations: 512 bytes)
```

This also works for inverses, which lower to BLAS calls whenever possible:

```
julia> A = randn(5,5); b = randn(5); c = similar(b);
julia> c .= @~ A \ b
5-element Array{Float64,1}:
-2.5366335879717514
-5.305097174484744
-9.818431932350942
2.421562605495651
0.26792916096572983
```

Often we want lazy realizations of matrices, which are supported via `ApplyArray`

. For example, the following creates a lazy matrix exponential:

```
julia> E = ApplyArray(exp, [1 2; 3 4])
2×2 ApplyArray{Float64,2,typeof(exp),Tuple{Array{Int64,2}}}:
51.969 74.7366
112.105 164.074
```

A lazy matrix exponential is useful for, say, in-place matrix-exponential*vector:

```
julia> b = Vector{Float64}(undef, 2); b .= @~ E*[4,4]
2-element Array{Float64,1}:
506.8220830628333
1104.7145995988594
```

While this works, it is not actually optimised (yet).

Other options do have special implementations that make them fast. We now give some examples.

Lazy `vcat`

and `hcat`

allow for representing the concatenation of vectors without actually allocating memory, and support a fast `copyto!`

for allocation-free population of a vector.

```
julia> using BenchmarkTools
julia> A = ApplyArray(vcat,1:5,2:3) # allocation-free
7-element ApplyArray{Int64,1,typeof(vcat),Tuple{UnitRange{Int64},UnitRange{Int64}}}:
1
2
3
4
5
2
3
julia> Vector(A) == vcat(1:5, 2:3)
true
julia> b = Array{Int}(undef, length(A)); @btime copyto!(b, A);
26.670 ns (0 allocations: 0 bytes)
julia> @btime vcat(1:5, 2:3); # takes twice as long due to memory creation
43.336 ns (1 allocation: 144 bytes)
```

Similar is the lazy analogue of `hcat`

:

```
julia> A = ApplyArray(hcat, 1:3, randn(3,10))
3×11 ApplyArray{Float64,2,typeof(hcat),Tuple{UnitRange{Int64},Array{Float64,2}}}:
1.0 1.16561 0.224871 -1.36416 -0.30675 0.103714 0.590141 0.982382 -1.50045 0.323747 -1.28173
2.0 1.04648 1.35506 -0.147157 0.995657 -0.616321 -0.128672 -0.671445 -0.563587 -0.268389 -1.71004
3.0 -0.433093 -0.325207 -1.38496 -0.391113 -0.0568739 -1.55796 -1.00747 0.473686 -1.2113 0.0119156
julia> Matrix(A) == hcat(A.args...)
true
julia> B = Array{Float64}(undef, size(A)...); @btime copyto!(B, A);
109.625 ns (1 allocation: 32 bytes)
julia> @btime hcat(A.args...); # takes twice as long due to memory creation
274.620 ns (6 allocations: 560 bytes)
```

We can represent Kronecker products of arrays without constructing the full array:

```
julia> A = randn(2,2); B = randn(3,3);
julia> K = ApplyArray(kron,A,B)
6×6 ApplyArray{Float64,2,typeof(kron),Tuple{Array{Float64,2},Array{Float64,2}}}:
-1.08736 -0.19547 -0.132824 1.60531 0.288579 0.196093
0.353898 0.445557 -0.257776 -0.522472 -0.657791 0.380564
-0.723707 0.911737 -0.710378 1.06843 -1.34603 1.04876
1.40606 0.252761 0.171754 -0.403809 -0.0725908 -0.0493262
-0.457623 -0.576146 0.333329 0.131426 0.165464 -0.0957293
0.935821 -1.17896 0.918584 -0.26876 0.338588 -0.26381
julia> C = Matrix{Float64}(undef, 6, 6); @btime copyto!(C, K);
61.528 ns (0 allocations: 0 bytes)
julia> C == kron(A,B)
true
```

Base includes a lazy broadcast object called `Broadcasting`

, but this is not a subtype of `AbstractArray`

. Here we have `BroadcastArray`

which replicates the functionality of `Broadcasting`

while supporting the array interface.

```
julia> A = randn(6,6);
julia> B = BroadcastArray(exp, A);
julia> Matrix(B) == exp.(A)
true
julia> B = BroadcastArray(+, A, 2);
julia> B == A .+ 2
true
```

Such arrays can also be created using the macro `@~`

which acts on ordinary broadcasting expressions combined with `LazyArray`

:

```
julia> C = rand(1000)';
julia> D = LazyArray(@~ exp.(C))
julia> E = LazyArray(@~ @. 2 + log(C))
julia> @btime sum(LazyArray(@~ C .* C'); dims=1) # without `@~`, 1.438 ms (5 allocations: 7.64 MiB)
74.425 μs (7 allocations: 8.08 KiB)
```

Author: JuliaArrays

Source Code: https://github.com/JuliaArrays/LazyArrays.jl

License: MIT license

1670560264

## Learn how to use Python arrays. Create arrays in Python using the array module. You'll see how to define them and the different methods commonly used for performing operations on them.

The artcile covers arrays that you create by importing the `array module`

. We won't cover NumPy arrays here.

- Introduction to Arrays
- The differences between Lists and Arrays
- When to use arrays

- How to use arrays
- Define arrays
- Find the length of arrays
- Array indexing
- Search through arrays
- Loop through arrays
- Slice an array

- Array methods for performing operations
- Change an existing value
- Add a new value
- Remove a value

- Conclusion

Let's get started!

Arrays are a fundamental data structure, and an important part of most programming languages. In Python, they are containers which are able to store more than one item at the same time.

Specifically, they are an ordered collection of elements with every value being of the same data type. That is the most important thing to remember about Python arrays - the fact that they can only hold a sequence of multiple items that are of the same type.

Lists are one of the most common data structures in Python, and a core part of the language.

Lists and arrays behave similarly.

Just like arrays, lists are an ordered sequence of elements.

They are also mutable and not fixed in size, which means they can grow and shrink throughout the life of the program. Items can be added and removed, making them very flexible to work with.

However, lists and arrays are **not** the same thing.

**Lists** store items that are of **various data types**. This means that a list can contain integers, floating point numbers, strings, or any other Python data type, at the same time. That is not the case with arrays.

As mentioned in the section above, **arrays** store only items that are of the **same single data type**. There are arrays that contain only integers, or only floating point numbers, or only any other Python data type you want to use.

Lists are built into the Python programming language, whereas arrays aren't. Arrays are not a built-in data structure, and therefore need to be imported via the `array module`

in order to be used.

Arrays of the `array module`

are a thin wrapper over C arrays, and are useful when you want to work with homogeneous data.

They are also more compact and take up less memory and space which makes them more size efficient compared to lists.

If you want to perform mathematical calculations, then you should use NumPy arrays by importing the NumPy package. Besides that, you should just use Python arrays when you really need to, as lists work in a similar way and are more flexible to work with.

In order to create Python arrays, you'll first have to import the `array module`

which contains all the necassary functions.

There are three ways you can import the `array module`

:

- By using
`import array`

at the top of the file. This includes the module`array`

. You would then go on to create an array using`array.array()`

.

```
import array
#how you would create an array
array.array()
```

- Instead of having to type
`array.array()`

all the time, you could use`import array as arr`

at the top of the file, instead of`import array`

alone. You would then create an array by typing`arr.array()`

. The`arr`

acts as an alias name, with the array constructor then immediately following it.

```
import array as arr
#how you would create an array
arr.array()
```

- Lastly, you could also use
`from array import *`

, with`*`

importing all the functionalities available. You would then create an array by writing the`array()`

constructor alone.

```
from array import *
#how you would create an array
array()
```

Once you've imported the `array module`

, you can then go on to define a Python array.

The general syntax for creating an array looks like this:

```
variable_name = array(typecode,[elements])
```

Let's break it down:

`variable_name`

would be the name of the array.- The
`typecode`

specifies what kind of elements would be stored in the array. Whether it would be an array of integers, an array of floats or an array of any other Python data type. Remember that all elements should be of the same data type. - Inside square brackets you mention the
`elements`

that would be stored in the array, with each element being separated by a comma. You can also create an*empty*array by just writing`variable_name = array(typecode)`

alone, without any elements.

Below is a typecode table, with the different typecodes that can be used with the different data types when defining Python arrays:

TYPECODE | C TYPE | PYTHON TYPE | SIZE |
---|---|---|---|

'b' | signed char | int | 1 |

'B' | unsigned char | int | 1 |

'u' | wchar_t | Unicode character | 2 |

'h' | signed short | int | 2 |

'H' | unsigned short | int | 2 |

'i' | signed int | int | 2 |

'I' | unsigned int | int | 2 |

'l' | signed long | int | 4 |

'L' | unsigned long | int | 4 |

'q' | signed long long | int | 8 |

'Q' | unsigned long long | int | 8 |

'f' | float | float | 4 |

'd' | double | float | 8 |

Tying everything together, here is an example of how you would define an array in Python:

```
import array as arr
numbers = arr.array('i',[10,20,30])
print(numbers)
#output
#array('i', [10, 20, 30])
```

Let's break it down:

- First we included the array module, in this case with
`import array as arr`

. - Then, we created a
`numbers`

array. - We used
`arr.array()`

because of`import array as arr`

. - Inside the
`array()`

constructor, we first included`i`

, for signed integer. Signed integer means that the array can include positive*and*negative values. Unsigned integer, with`H`

for example, would mean that no negative values are allowed. - Lastly, we included the values to be stored in the array in square brackets.

Keep in mind that if you tried to include values that were not of `i`

typecode, meaning they were not integer values, you would get an error:

```
import array as arr
numbers = arr.array('i',[10.0,20,30])
print(numbers)
#output
#Traceback (most recent call last):
# File "/Users/dionysialemonaki/python_articles/demo.py", line 14, in <module>
# numbers = arr.array('i',[10.0,20,30])
#TypeError: 'float' object cannot be interpreted as an integer
```

In the example above, I tried to include a floating point number in the array. I got an error because this is meant to be an integer array only.

Another way to create an array is the following:

```
from array import *
#an array of floating point values
numbers = array('d',[10.0,20.0,30.0])
print(numbers)
#output
#array('d', [10.0, 20.0, 30.0])
```

The example above imported the `array module`

via `from array import *`

and created an array `numbers`

of float data type. This means that it holds only floating point numbers, which is specified with the `'d'`

typecode.

To find out the exact number of elements contained in an array, use the built-in `len()`

method.

It will return the integer number that is equal to the total number of elements in the array you specify.

```
import array as arr
numbers = arr.array('i',[10,20,30])
print(len(numbers))
#output
# 3
```

In the example above, the array contained three elements – `10, 20, 30`

– so the length of `numbers`

is `3`

.

Each item in an array has a specific address. Individual items are accessed by referencing their *index number*.

Indexing in Python, and in all programming languages and computing in general, starts at `0`

. It is important to remember that counting starts at `0`

and **not** at `1`

.

To access an element, you first write the name of the array followed by square brackets. Inside the square brackets you include the item's index number.

The general syntax would look something like this:

```
array_name[index_value_of_item]
```

Here is how you would access each individual element in an array:

```
import array as arr
numbers = arr.array('i',[10,20,30])
print(numbers[0]) # gets the 1st element
print(numbers[1]) # gets the 2nd element
print(numbers[2]) # gets the 3rd element
#output
#10
#20
#30
```

Remember that the index value of the last element of an array is always one less than the length of the array. Where `n`

is the length of the array, `n - 1`

will be the index value of the last item.

Note that you can also access each individual element using negative indexing.

With negative indexing, the last element would have an index of `-1`

, the second to last element would have an index of `-2`

, and so on.

Here is how you would get each item in an array using that method:

```
import array as arr
numbers = arr.array('i',[10,20,30])
print(numbers[-1]) #gets last item
print(numbers[-2]) #gets second to last item
print(numbers[-3]) #gets first item
#output
#30
#20
#10
```

You can find out an element's index number by using the `index()`

method.

You pass the value of the element being searched as the argument to the method, and the element's index number is returned.

```
import array as arr
numbers = arr.array('i',[10,20,30])
#search for the index of the value 10
print(numbers.index(10))
#output
#0
```

If there is more than one element with the same value, the index of the first instance of the value will be returned:

```
import array as arr
numbers = arr.array('i',[10,20,30,10,20,30])
#search for the index of the value 10
#will return the index number of the first instance of the value 10
print(numbers.index(10))
#output
#0
```

You've seen how to access each individual element in an array and print it out on its own.

You've also seen how to print the array, using the `print()`

method. That method gives the following result:

```
import array as arr
numbers = arr.array('i',[10,20,30])
print(numbers)
#output
#array('i', [10, 20, 30])
```

What if you want to print each value one by one?

This is where a loop comes in handy. You can loop through the array and print out each value, one-by-one, with each loop iteration.

For this you can use a simple `for`

loop:

```
import array as arr
numbers = arr.array('i',[10,20,30])
for number in numbers:
print(number)
#output
#10
#20
#30
```

You could also use the `range()`

function, and pass the `len()`

method as its parameter. This would give the same result as above:

```
import array as arr
values = arr.array('i',[10,20,30])
#prints each individual value in the array
for value in range(len(values)):
print(values[value])
#output
#10
#20
#30
```

To access a specific range of values inside the array, use the slicing operator, which is a colon `:`

.

When using the slicing operator and you only include one value, the counting starts from `0`

by default. It gets the first item, and goes up to but not including the index number you specify.

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30])
#get the values 10 and 20 only
print(numbers[:2]) #first to second position
#output
#array('i', [10, 20])
```

When you pass two numbers as arguments, you specify a range of numbers. In this case, the counting starts at the position of the first number in the range, and up to but not including the second one:

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30])
#get the values 20 and 30 only
print(numbers[1:3]) #second to third position
#output
#rray('i', [20, 30])
```

Arrays are mutable, which means they are changeable. You can change the value of the different items, add new ones, or remove any you don't want in your program anymore.

Let's see some of the most commonly used methods which are used for performing operations on arrays.

You can change the value of a specific element by speficying its position and assigning it a new value:

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30])
#change the first element
#change it from having a value of 10 to having a value of 40
numbers[0] = 40
print(numbers)
#output
#array('i', [40, 20, 30])
```

To add one single value at the end of an array, use the `append()`

method:

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30])
#add the integer 40 to the end of numbers
numbers.append(40)
print(numbers)
#output
#array('i', [10, 20, 30, 40])
```

Be aware that the new item you add needs to be the same data type as the rest of the items in the array.

Look what happens when I try to add a float to an array of integers:

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30])
#add the integer 40 to the end of numbers
numbers.append(40.0)
print(numbers)
#output
#Traceback (most recent call last):
# File "/Users/dionysialemonaki/python_articles/demo.py", line 19, in <module>
# numbers.append(40.0)
#TypeError: 'float' object cannot be interpreted as an integer
```

But what if you want to add more than one value to the end an array?

Use the `extend()`

method, which takes an iterable (such as a list of items) as an argument. Again, make sure that the new items are all the same data type.

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30])
#add the integers 40,50,60 to the end of numbers
#The numbers need to be enclosed in square brackets
numbers.extend([40,50,60])
print(numbers)
#output
#array('i', [10, 20, 30, 40, 50, 60])
```

And what if you don't want to add an item to the end of an array? Use the `insert()`

method, to add an item at a specific position.

The `insert()`

function takes two arguments: the index number of the position the new element will be inserted, and the value of the new element.

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30])
#add the integer 40 in the first position
#remember indexing starts at 0
numbers.insert(0,40)
print(numbers)
#output
#array('i', [40, 10, 20, 30])
```

To remove an element from an array, use the `remove()`

method and include the value as an argument to the method.

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30])
numbers.remove(10)
print(numbers)
#output
#array('i', [20, 30])
```

With `remove()`

, only the first instance of the value you pass as an argument will be removed.

See what happens when there are more than one identical values:

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30,10,20])
numbers.remove(10)
print(numbers)
#output
#array('i', [20, 30, 10, 20])
```

Only the first occurence of `10`

is removed.

You can also use the `pop()`

method, and specify the position of the element to be removed:

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30,10,20])
#remove the first instance of 10
numbers.pop(0)
print(numbers)
#output
#array('i', [20, 30, 10, 20])
```

And there you have it - you now know the basics of how to create arrays in Python using the `array module`

. Hopefully you found this guide helpful.

You'll start from the basics and learn in an interacitve and beginner-friendly way. You'll also build five projects at the end to put into practice and help reinforce what you learned.

Thanks for reading and happy coding!

Original article source athttps://www.freecodecamp.org

#python

1666082925

This tutorialvideo on 'Arrays in Python' will help you establish a strong hold on all the fundamentals in python programming language. Below are the topics covered in this video:

1:15 What is an array?

2:53 Is python list same as an array?

3:48 How to create arrays in python?

7:19 Accessing array elements

9:59 Basic array operations

- 10:33 Finding the length of an array

- 11:44 Adding Elements

- 15:06 Removing elements

- 18:32 Array concatenation

- 20:59 Slicing

- 23:26 Looping

**Python Array Tutorial – Define, Index, Methods**

In this article, you'll learn how to use Python arrays. You'll see how to define them and the different methods commonly used for performing operations on them.

The artcile covers arrays that you create by importing the `array module`

. We won't cover NumPy arrays here.

- Introduction to Arrays
- The differences between Lists and Arrays
- When to use arrays

- How to use arrays
- Define arrays
- Find the length of arrays
- Array indexing
- Search through arrays
- Loop through arrays
- Slice an array

- Array methods for performing operations
- Change an existing value
- Add a new value
- Remove a value

- Conclusion

Let's get started!

Arrays are a fundamental data structure, and an important part of most programming languages. In Python, they are containers which are able to store more than one item at the same time.

Specifically, they are an ordered collection of elements with every value being of the same data type. That is the most important thing to remember about Python arrays - the fact that they can only hold a sequence of multiple items that are of the same type.

Lists are one of the most common data structures in Python, and a core part of the language.

Lists and arrays behave similarly.

Just like arrays, lists are an ordered sequence of elements.

They are also mutable and not fixed in size, which means they can grow and shrink throughout the life of the program. Items can be added and removed, making them very flexible to work with.

However, lists and arrays are **not** the same thing.

**Lists** store items that are of **various data types**. This means that a list can contain integers, floating point numbers, strings, or any other Python data type, at the same time. That is not the case with arrays.

As mentioned in the section above, **arrays** store only items that are of the **same single data type**. There are arrays that contain only integers, or only floating point numbers, or only any other Python data type you want to use.

Lists are built into the Python programming language, whereas arrays aren't. Arrays are not a built-in data structure, and therefore need to be imported via the `array module`

in order to be used.

Arrays of the `array module`

are a thin wrapper over C arrays, and are useful when you want to work with homogeneous data.

They are also more compact and take up less memory and space which makes them more size efficient compared to lists.

If you want to perform mathematical calculations, then you should use NumPy arrays by importing the NumPy package. Besides that, you should just use Python arrays when you really need to, as lists work in a similar way and are more flexible to work with.

In order to create Python arrays, you'll first have to import the `array module`

which contains all the necassary functions.

There are three ways you can import the `array module`

:

- By using
`import array`

at the top of the file. This includes the module`array`

. You would then go on to create an array using`array.array()`

.

```
import array
#how you would create an array
array.array()
```

- Instead of having to type
`array.array()`

all the time, you could use`import array as arr`

at the top of the file, instead of`import array`

alone. You would then create an array by typing`arr.array()`

. The`arr`

acts as an alias name, with the array constructor then immediately following it.

```
import array as arr
#how you would create an array
arr.array()
```

- Lastly, you could also use
`from array import *`

, with`*`

importing all the functionalities available. You would then create an array by writing the`array()`

constructor alone.

```
from array import *
#how you would create an array
array()
```

Once you've imported the `array module`

, you can then go on to define a Python array.

The general syntax for creating an array looks like this:

`variable_name = array(typecode,[elements])`

Let's break it down:

`variable_name`

would be the name of the array.- The
`typecode`

specifies what kind of elements would be stored in the array. Whether it would be an array of integers, an array of floats or an array of any other Python data type. Remember that all elements should be of the same data type. - Inside square brackets you mention the
`elements`

that would be stored in the array, with each element being separated by a comma. You can also create an*empty*array by just writing`variable_name = array(typecode)`

alone, without any elements.

Below is a typecode table, with the different typecodes that can be used with the different data types when defining Python arrays:

TYPECODE | C TYPE | PYTHON TYPE | SIZE |
---|---|---|---|

'b' | signed char | int | 1 |

'B' | unsigned char | int | 1 |

'u' | wchar_t | Unicode character | 2 |

'h' | signed short | int | 2 |

'H' | unsigned short | int | 2 |

'i' | signed int | int | 2 |

'I' | unsigned int | int | 2 |

'l' | signed long | int | 4 |

'L' | unsigned long | int | 4 |

'q' | signed long long | int | 8 |

'Q' | unsigned long long | int | 8 |

'f' | float | float | 4 |

'd' | double | float | 8 |

Tying everything together, here is an example of how you would define an array in Python:

```
import array as arr
numbers = arr.array('i',[10,20,30])
print(numbers)
#output
#array('i', [10, 20, 30])
```

Let's break it down:

- First we included the array module, in this case with
`import array as arr`

. - Then, we created a
`numbers`

array. - We used
`arr.array()`

because of`import array as arr`

. - Inside the
`array()`

constructor, we first included`i`

, for signed integer. Signed integer means that the array can include positive*and*negative values. Unsigned integer, with`H`

for example, would mean that no negative values are allowed. - Lastly, we included the values to be stored in the array in square brackets.

Keep in mind that if you tried to include values that were not of `i`

typecode, meaning they were not integer values, you would get an error:

```
import array as arr
numbers = arr.array('i',[10.0,20,30])
print(numbers)
#output
#Traceback (most recent call last):
# File "/Users/dionysialemonaki/python_articles/demo.py", line 14, in <module>
# numbers = arr.array('i',[10.0,20,30])
#TypeError: 'float' object cannot be interpreted as an integer
```

In the example above, I tried to include a floating point number in the array. I got an error because this is meant to be an integer array only.

Another way to create an array is the following:

```
from array import *
#an array of floating point values
numbers = array('d',[10.0,20.0,30.0])
print(numbers)
#output
#array('d', [10.0, 20.0, 30.0])
```

The example above imported the `array module`

via `from array import *`

and created an array `numbers`

of float data type. This means that it holds only floating point numbers, which is specified with the `'d'`

typecode.

To find out the exact number of elements contained in an array, use the built-in `len()`

method.

It will return the integer number that is equal to the total number of elements in the array you specify.

```
import array as arr
numbers = arr.array('i',[10,20,30])
print(len(numbers))
#output
# 3
```

In the example above, the array contained three elements – `10, 20, 30`

– so the length of `numbers`

is `3`

.

Each item in an array has a specific address. Individual items are accessed by referencing their *index number*.

Indexing in Python, and in all programming languages and computing in general, starts at `0`

. It is important to remember that counting starts at `0`

and **not** at `1`

.

To access an element, you first write the name of the array followed by square brackets. Inside the square brackets you include the item's index number.

The general syntax would look something like this:

`array_name[index_value_of_item]`

Here is how you would access each individual element in an array:

```
import array as arr
numbers = arr.array('i',[10,20,30])
print(numbers[0]) # gets the 1st element
print(numbers[1]) # gets the 2nd element
print(numbers[2]) # gets the 3rd element
#output
#10
#20
#30
```

Remember that the index value of the last element of an array is always one less than the length of the array. Where `n`

is the length of the array, `n - 1`

will be the index value of the last item.

Note that you can also access each individual element using negative indexing.

With negative indexing, the last element would have an index of `-1`

, the second to last element would have an index of `-2`

, and so on.

Here is how you would get each item in an array using that method:

```
import array as arr
numbers = arr.array('i',[10,20,30])
print(numbers[-1]) #gets last item
print(numbers[-2]) #gets second to last item
print(numbers[-3]) #gets first item
#output
#30
#20
#10
```

You can find out an element's index number by using the `index()`

method.

You pass the value of the element being searched as the argument to the method, and the element's index number is returned.

```
import array as arr
numbers = arr.array('i',[10,20,30])
#search for the index of the value 10
print(numbers.index(10))
#output
#0
```

If there is more than one element with the same value, the index of the first instance of the value will be returned:

```
import array as arr
numbers = arr.array('i',[10,20,30,10,20,30])
#search for the index of the value 10
#will return the index number of the first instance of the value 10
print(numbers.index(10))
#output
#0
```

You've seen how to access each individual element in an array and print it out on its own.

You've also seen how to print the array, using the `print()`

method. That method gives the following result:

```
import array as arr
numbers = arr.array('i',[10,20,30])
print(numbers)
#output
#array('i', [10, 20, 30])
```

What if you want to print each value one by one?

This is where a loop comes in handy. You can loop through the array and print out each value, one-by-one, with each loop iteration.

For this you can use a simple `for`

loop:

```
import array as arr
numbers = arr.array('i',[10,20,30])
for number in numbers:
print(number)
#output
#10
#20
#30
```

You could also use the `range()`

function, and pass the `len()`

method as its parameter. This would give the same result as above:

```
import array as arr
values = arr.array('i',[10,20,30])
#prints each individual value in the array
for value in range(len(values)):
print(values[value])
#output
#10
#20
#30
```

To access a specific range of values inside the array, use the slicing operator, which is a colon `:`

.

When using the slicing operator and you only include one value, the counting starts from `0`

by default. It gets the first item, and goes up to but not including the index number you specify.

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30])
#get the values 10 and 20 only
print(numbers[:2]) #first to second position
#output
#array('i', [10, 20])
```

When you pass two numbers as arguments, you specify a range of numbers. In this case, the counting starts at the position of the first number in the range, and up to but not including the second one:

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30])
#get the values 20 and 30 only
print(numbers[1:3]) #second to third position
#output
#rray('i', [20, 30])
```

Arrays are mutable, which means they are changeable. You can change the value of the different items, add new ones, or remove any you don't want in your program anymore.

Let's see some of the most commonly used methods which are used for performing operations on arrays.

You can change the value of a specific element by speficying its position and assigning it a new value:

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30])
#change the first element
#change it from having a value of 10 to having a value of 40
numbers[0] = 40
print(numbers)
#output
#array('i', [40, 20, 30])
```

To add one single value at the end of an array, use the `append()`

method:

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30])
#add the integer 40 to the end of numbers
numbers.append(40)
print(numbers)
#output
#array('i', [10, 20, 30, 40])
```

Be aware that the new item you add needs to be the same data type as the rest of the items in the array.

Look what happens when I try to add a float to an array of integers:

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30])
#add the integer 40 to the end of numbers
numbers.append(40.0)
print(numbers)
#output
#Traceback (most recent call last):
# File "/Users/dionysialemonaki/python_articles/demo.py", line 19, in <module>
# numbers.append(40.0)
#TypeError: 'float' object cannot be interpreted as an integer
```

But what if you want to add more than one value to the end an array?

Use the `extend()`

method, which takes an iterable (such as a list of items) as an argument. Again, make sure that the new items are all the same data type.

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30])
#add the integers 40,50,60 to the end of numbers
#The numbers need to be enclosed in square brackets
numbers.extend([40,50,60])
print(numbers)
#output
#array('i', [10, 20, 30, 40, 50, 60])
```

And what if you don't want to add an item to the end of an array? Use the `insert()`

method, to add an item at a specific position.

The `insert()`

function takes two arguments: the index number of the position the new element will be inserted, and the value of the new element.

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30])
#add the integer 40 in the first position
#remember indexing starts at 0
numbers.insert(0,40)
print(numbers)
#output
#array('i', [40, 10, 20, 30])
```

To remove an element from an array, use the `remove()`

method and include the value as an argument to the method.

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30])
numbers.remove(10)
print(numbers)
#output
#array('i', [20, 30])
```

With `remove()`

, only the first instance of the value you pass as an argument will be removed.

See what happens when there are more than one identical values:

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30,10,20])
numbers.remove(10)
print(numbers)
#output
#array('i', [20, 30, 10, 20])
```

Only the first occurence of `10`

is removed.

You can also use the `pop()`

method, and specify the position of the element to be removed:

```
import array as arr
#original array
numbers = arr.array('i',[10,20,30,10,20])
#remove the first instance of 10
numbers.pop(0)
print(numbers)
#output
#array('i', [20, 30, 10, 20])
```

And there you have it - you now know the basics of how to create arrays in Python using the `array module`

. Hopefully you found this guide helpful.

Thanks for reading and happy coding!

#python #programming

1665983538

Lazy arrays and linear algebra in Julia

This package supports lazy analogues of array operations like `vcat`

, `hcat`

, and multiplication. This helps with the implementation of matrix-free methods for iterative solvers.

The package has been designed with high-performance in mind, so should outperform the non-lazy analogues from Base for many operations like `copyto!`

and broadcasting. Some operations will be inherently slower due to extra computation, like `getindex`

. Please file an issue for any examples that are significantly slower than their the analogue in Base.

To construct a lazy representation of a function call `f(x,y,z...)`

, use the command `applied(f, x, y, z...)`

. This will return an unmaterialized object typically of type `Applied`

that represents the operation. To realize that object, call `materialize`

, which will typically be equivalent to calling `f(x,y,z...)`

. A macro `@~`

is available as a shorthand:

```
julia> using LazyArrays, LinearAlgebra
julia> applied(exp, 1)
Applied(exp,1)
julia> materialize(applied(exp, 1))
2.718281828459045
julia> materialize(@~ exp(1))
2.718281828459045
julia> exp(1)
2.718281828459045
```

Note that `@~`

causes sub-expressions to be wrapped in an `applied`

call, not just the top-level expression. This can lead to `MethodError`

s when lazy application of sub-expressions is not yet implemented. For example,

```
julia> @~ Vector(1:10) .* ones(10)'
ERROR: MethodError: ...
julia> A = Vector(1:10); B = ones(10); (@~ sum(A .* B')) |> materialize
550.0
```

The benefit of lazy operations is that they can be materialized in-place, possible using simplifications. For example, it is possible to do BLAS-like Matrix-Vector operations of the form `α*A*x + β*y`

as implemented in `BLAS.gemv!`

using a lazy applied object:

```
julia> A = randn(5,5); b = randn(5); c = randn(5); d = similar(c);
julia> d .= @~ 2.0 * A * b + 3.0 * c # Calls gemv!
5-element Array{Float64,1}:
-2.5366335879717514
-5.305097174484744
-9.818431932350942
2.421562605495651
0.26792916096572983
julia> 2*(A*b) + 3c
5-element Array{Float64,1}:
-2.5366335879717514
-5.305097174484744
-9.818431932350942
2.421562605495651
0.26792916096572983
julia> function mymul(A, b, c, d) # need to put in function for benchmarking
d .= @~ 2.0 * A * b + 3.0 * c
end
mymul (generic function with 1 method)
julia> @btime mymul(A, b, c, d) # calls gemv!
77.444 ns (0 allocations: 0 bytes)
5-element Array{Float64,1}:
-2.5366335879717514
-5.305097174484744
-9.818431932350942
2.421562605495651
0.26792916096572983
julia> @btime 2*(A*b) + 3c; # does not call gemv!
241.659 ns (4 allocations: 512 bytes)
```

This also works for inverses, which lower to BLAS calls whenever possible:

```
julia> A = randn(5,5); b = randn(5); c = similar(b);
julia> c .= @~ A \ b
5-element Array{Float64,1}:
-2.5366335879717514
-5.305097174484744
-9.818431932350942
2.421562605495651
0.26792916096572983
```

Often we want lazy realizations of matrices, which are supported via `ApplyArray`

. For example, the following creates a lazy matrix exponential:

```
julia> E = ApplyArray(exp, [1 2; 3 4])
2×2 ApplyArray{Float64,2,typeof(exp),Tuple{Array{Int64,2}}}:
51.969 74.7366
112.105 164.074
```

A lazy matrix exponential is useful for, say, in-place matrix-exponential*vector:

```
julia> b = Vector{Float64}(undef, 2); b .= @~ E*[4,4]
2-element Array{Float64,1}:
506.8220830628333
1104.7145995988594
```

While this works, it is not actually optimised (yet).

Other options do have special implementations that make them fast. We now give some examples.

Lazy `vcat`

and `hcat`

allow for representing the concatenation of vectors without actually allocating memory, and support a fast `copyto!`

for allocation-free population of a vector.

```
julia> using BenchmarkTools
julia> A = ApplyArray(vcat,1:5,2:3) # allocation-free
7-element ApplyArray{Int64,1,typeof(vcat),Tuple{UnitRange{Int64},UnitRange{Int64}}}:
1
2
3
4
5
2
3
julia> Vector(A) == vcat(1:5, 2:3)
true
julia> b = Array{Int}(undef, length(A)); @btime copyto!(b, A);
26.670 ns (0 allocations: 0 bytes)
julia> @btime vcat(1:5, 2:3); # takes twice as long due to memory creation
43.336 ns (1 allocation: 144 bytes)
```

Similar is the lazy analogue of `hcat`

:

```
julia> A = ApplyArray(hcat, 1:3, randn(3,10))
3×11 ApplyArray{Float64,2,typeof(hcat),Tuple{UnitRange{Int64},Array{Float64,2}}}:
1.0 1.16561 0.224871 -1.36416 -0.30675 0.103714 0.590141 0.982382 -1.50045 0.323747 -1.28173
2.0 1.04648 1.35506 -0.147157 0.995657 -0.616321 -0.128672 -0.671445 -0.563587 -0.268389 -1.71004
3.0 -0.433093 -0.325207 -1.38496 -0.391113 -0.0568739 -1.55796 -1.00747 0.473686 -1.2113 0.0119156
julia> Matrix(A) == hcat(A.args...)
true
julia> B = Array{Float64}(undef, size(A)...); @btime copyto!(B, A);
109.625 ns (1 allocation: 32 bytes)
julia> @btime hcat(A.args...); # takes twice as long due to memory creation
274.620 ns (6 allocations: 560 bytes)
```

We can represent Kronecker products of arrays without constructing the full array:

```
julia> A = randn(2,2); B = randn(3,3);
julia> K = ApplyArray(kron,A,B)
6×6 ApplyArray{Float64,2,typeof(kron),Tuple{Array{Float64,2},Array{Float64,2}}}:
-1.08736 -0.19547 -0.132824 1.60531 0.288579 0.196093
0.353898 0.445557 -0.257776 -0.522472 -0.657791 0.380564
-0.723707 0.911737 -0.710378 1.06843 -1.34603 1.04876
1.40606 0.252761 0.171754 -0.403809 -0.0725908 -0.0493262
-0.457623 -0.576146 0.333329 0.131426 0.165464 -0.0957293
0.935821 -1.17896 0.918584 -0.26876 0.338588 -0.26381
julia> C = Matrix{Float64}(undef, 6, 6); @btime copyto!(C, K);
61.528 ns (0 allocations: 0 bytes)
julia> C == kron(A,B)
true
```

Base includes a lazy broadcast object called `Broadcasting`

, but this is not a subtype of `AbstractArray`

. Here we have `BroadcastArray`

which replicates the functionality of `Broadcasting`

while supporting the array interface.

```
julia> A = randn(6,6);
julia> B = BroadcastArray(exp, A);
julia> Matrix(B) == exp.(A)
true
julia> B = BroadcastArray(+, A, 2);
julia> B == A .+ 2
true
```

Such arrays can also be created using the macro `@~`

which acts on ordinary broadcasting expressions combined with `LazyArray`

:

```
julia> C = rand(1000)';
julia> D = LazyArray(@~ exp.(C))
julia> E = LazyArray(@~ @. 2 + log(C))
julia> @btime sum(LazyArray(@~ C .* C'); dims=1) # without `@~`, 1.438 ms (5 allocations: 7.64 MiB)
74.425 μs (7 allocations: 8.08 KiB)
```

Author: JuliaArrays

Source Code: https://github.com/JuliaArrays/LazyArrays.jl

License: MIT license

1624447260

Because I am continuously endeavouring to improve my knowledge and skill of the Python programming language, I decided to take some free courses in an attempt to improve upon my knowledge base. I found one such course on linear algebra, which I found on YouTube. I decided to watch the video and undertake the course work because it focused on the Python programming language, something that I wanted to improve my skill in. Youtube video this course review was taken from:- (4) Python for linear algebra (for absolute beginners) — YouTube

The course is for absolute beginners, which is good because I have never studied linear algebra and had no idea what the terms I would be working with were.

Linear algebra is the branch of mathematics concerning linear equations, such as linear maps and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics.

Whilst studying linear algebra, I have learned a few topics that I had not previously known. For example:-

A scalar is simply a number, being an integer or a float. Scalers are convenient in applications that don’t need to be concerned with all the ways that data can be represented in a computer.

A vector is a one dimensional array of numbers. The difference between a vector is that it is mutable, being known as dynamic arrays.

A matrix is similar to a two dimensional rectangular array of data stored in rows and columns. The data stored in the matrix can be strings, numbers, etcetera.

In addition to the basic components of linear algebra, being a scalar, vector and matrix, there are several ways the vectors and matrix can be manipulated to make it suitable for machine learning.

I used Google Colab to code the programming examples and the assignments that were given in the 1 hour 51 minute video. It took a while to get into writing the code of the various subjects that were studied because, as the video stated, it is a course for absolute beginners.

The two main libraries that were used for this course were numpy and matplotlib. Numpy is the library that is used to carry out algebraic operations and matplotlib is used to graphically plot the points that are created in the program.

#numpy #matplotlib #python #linear-algebra #course review: python for linear algebra #linear algebra

1665744840

The purpose of this package is partly to extend linear algebra functionality in base to cover generic element types, e.g. `BigFloat`

and `Quaternion`

, and partly to be a place to experiment with fast linear algebra routines written in Julia (except for optimized BLAS). It is my hope that it is possible to have implementations that are generic, fast, and readable.

So far, this has mainly been my playground but you might find some of the functionality here useful. The package has a generic implementation of a singular value solver which will make it possible to compute `norm`

and `cond`

of matrices of `BigFloat`

. Hence

```
julia> using GenericLinearAlgebra
julia> A = big.(randn(10,10));
julia> cond(A)
1.266829904721752610946505846921202851190952179974780602509001252204638657237828e+03
julia> norm(A)
6.370285271475041598951769618847832429030388948627697440637424244721679386430589
```

The package also includes functions for the blocked Cholesky and QR factorization, the self-adjoint (symmetric) and the general eigenvalue problem. These routines can be accessed by fully qualifying the names

```
julia> using GenericLinearAlgebra
julia> A = randn(1000,1000); A = A'A;
julia> cholesky(A);
julia> @time cholesky(A);
0.013036 seconds (16 allocations: 7.630 MB)
julia> GenericLinearAlgebra.cholRecursive!(copy(A), Val{:L});
julia> @time GenericLinearAlgebra.cholRecursive!(copy(A), Val{:L});
0.012098 seconds (7.00 k allocations: 7.934 MB)
```

Author: JuliaLinearAlgebra

Source Code: https://github.com/JuliaLinearAlgebra/GenericLinearAlgebra.jl

License: MIT license