Notebook

Note: Click on "Kernel" > "Restart Kernel and Clear All Outputs" in JupyterLab before reading this notebook to reset its output. If you cannot run this file on your machine, you may want to open it in the cloud .

Chapter 11: Classes & Instances (continued)¶

In this fourth part of the chapter, we finalize our Vector and Matrix classes. As both class definitions have become rather lengthy, we learn how we to organize them into a Python package and import them in this Jupyter notebook.

Packages vs. Modules¶

In Chapter 2 , we introduce the concept of a Python module that is imported with the import statement. Essentially, a module is a single plain text *.py file on disk that contains Python code (e.g., sample_module.py in Chapter 2's folder ).

Conceptually, a package is a generalization of a module whose code is split across several *.py to achieve a better organization of the individual parts. The *.py files are stored within a folder (e.g., sample_package in Chapter 11's folder ). In addition to that, a "__init__.py" file that may be empty must be put inside the folder. The latter is what the Python interpreter looks for to decide if a folder is a package or not.

Let's look at an example with the final version of our Vector and Matrix classes.

!pwd shows the location of this Jupyter notebook on the computer you are running JupyterLab on: It is the local equivalent of Chapter 11's folder in this book's GitHub repository .

In [1]:

!pwd

/home/alexander/Repositories/intro-to-python/11_classes

!ls lists all the files and folders in the current location: These are Chapter 11's Jupyter notebooks (i.e., the *.ipynb files) and the sample_package folder.

In [2]:

!ls

00_content.ipynb    02_content.ipynb  04_content.ipynb	06_review.ipynb
01_exercises.ipynb  03_content.ipynb  05_summary.ipynb	sample_package

If we run !ls with the sample_package folder as the argument, we see the folder's contents: Four *.py files. Alternatively, you can use JupyterLab' File Browser on the left to navigate into the package.

In [3]:

!ls sample_package

__init__.py  matrix.py	utils.py  vector.py

The package is organized such that the matrix.py and vector.py modules each define just one class, Matrix and Vector. That is intentional as both classes consist of several hundred lines of code and comments.

The utils.py module contains code that is shared by both classes. Such code snippets are commonly called "utilities" or "helpers," which explains the module's name.

Finally, the __init__.py file contains mostly meta information and defines what objects should be importable from the package's top level.

With the import statement, we can import the entire package just as we would import a module from the standard library .

In [4]:

import sample_package as pkg

The above cell runs the code in the __init__.py file from top to bottom, which in turn runs the matrix.py , utils.py , and vector.py modules (cf., look at the import statements in the four *.py files to get the idea). As both matrix.py and vector.py depend on each other (i.e., the Matrix class needs the Vector class to work and vice versa), understanding the order in that the modules are executed is not trivial. Without going into detail, we mention that Python guarantees that each *.py file is run only once and figures out the order on its own. If Python is unable to do that, for example, due to unresolvable cirular imports, it aborts with an ImportError.

Below, pkg is an object of type module ...

In [5]:

pkg

Out[5]:

<module 'sample_package' from '/home/alexander/Repositories/intro-to-python/11_classes/sample_package/__init__.py'>

In [6]:

type(pkg)

Out[6]:

module

... and we use the built-in dir() function to check what attributes pkg comes with.

In [7]:

dir(pkg)

Out[7]:

['Matrix',
 'Vector',
 '__all__',
 '__author__',
 '__builtins__',
 '__cached__',
 '__doc__',
 '__file__',
 '__loader__',
 '__name__',
 '__package__',
 '__path__',
 '__spec__',
 '__version__',
 'matrix',
 'utils',
 'vector']

The package's meta information and documentation are automatically parsed from the __init__.py file.

In [8]:

help(pkg)

Help on package linear_algebra_tools:

NAME
    linear_algebra_tools - This package provides linear algebra functionalities.

DESCRIPTION
    The package is split into three modules:
    - matrix: defines the Matrix class
    - vector: defines the Vector class
    - utils: defines the norm() function that is shared by Matrix and Vector
             and package-wide constants

    The classes implement arithmetic operations involving vectors and matrices.

    See the docstrings in the modules and classes for further info.

PACKAGE CONTENTS
    matrix
    utils
    vector

CLASSES
    builtins.object
        sample_package.matrix.Matrix
        sample_package.vector.Vector

    class Matrix(builtins.object)
     |  Matrix(data)
     |
     |  An m-by-n-dimensional matrix from linear algebra.
     |
     |  All entries are converted to floats, or whatever is set in the typing attribute.
     |
     |  Attributes:
     |      storage (callable): data type used to store the entries internally;
     |          defaults to tuple
     |      typing (callable): type casting applied to all entries upon creation;
     |          defaults to float
     |      vector_cls (vector.Vector): a reference to the Vector class to work with
     |      zero_threshold (float): max. tolerance when comparing an entry to zero;
     |          defaults to 1e-12
     |
     |  Methods defined here:
     |
     |  __abs__(self)
     |      The Frobenius norm of a Matrix.
     |
     |  __add__(self, other)
     |      Handle `self + other` and `other + self`.
     |
     |      This may be either matrix addition or broadcasting addition.
     |
     |      Example Usage:
     |          >>> Matrix([(1, 2), (3, 4)]) + Matrix([(2, 3), (4, 5)])
     |          Matrix(((3.0, 5.0,), (7.0, 9.0,)))
     |
     |          >>> Matrix([(1, 2), (3, 4)]) + 5
     |          Matrix(((6.0, 7.0,), (8.0, 9.0,)))
     |
     |          >>> 10 + Matrix([(1, 2), (3, 4)])
     |          Matrix(((11.0, 12.0,), (13.0, 14.0,)))
     |
     |  __bool__(self)
     |      A Matrix is truthy if its Frobenius norm is strictly positive.
     |
     |  __eq__(self, other)
     |      Handle `self == other`.
     |
     |      Compare two Matrix instances for equality.
     |
     |      Example Usage:
     |          >>> Matrix([(1, 2), (3, 4)]) == Matrix([(1, 2), (3, 4)])
     |          True
     |
     |          >>> Matrix([(1, 2), (3, 4)]) == Matrix([(5, 6), (7, 8)])
     |          False
     |
     |  __float__(self)
     |      Cast a Matrix as a scalar.
     |
     |      Returns:
     |          scalar (float)
     |
     |      Raises:
     |          RuntimeError: if the Matrix has more than one entry
     |
     |  __getitem__(self, index)
     |      Obtain an individual entry of a Matrix.
     |
     |      Args:
     |          index (int / tuple of int's): if index is an integer,
     |              the Matrix is viewed as a sequence in row-major order;
     |              if index is a tuple of integers, the first one refers to
     |              the row and the second one to the column of the entry
     |
     |      Returns:
     |          entry (Matrix.typing)
     |
     |      Example Usage:
     |          >>> m = Matrix([(1, 2), (3, 4)])
     |          >>> m[0]
     |          1.0
     |          >>> m[-1]
     |          4.0
     |          >>> m[0, 1]
     |          2.0
     |
     |  __init__(self, data)
     |      Create a new matrix.
     |
     |      Args:
     |          data (sequence of sequences): the matrix's entries;
     |              viewed as a sequence of the matrix's rows (i.e., row-major order);
     |              use the .from_columns() class method if the data come as a sequence
     |              of the matrix's columns (i.e., column-major order)
     |
     |      Raises:
     |          ValueError:
     |              - if no entries are provided
     |              - if the number of columns is inconsistent across the rows
     |
     |      Example Usage:
     |          >>> Matrix([(1, 2), (3, 4)])
     |          Matrix(((1.0, 2.0,), (3.0, 4.0,)))
     |
     |  __iter__(self)
     |      Loop over a Matrix's entries.
     |
     |      See .entries() for more customization options.
     |
     |  __len__(self)
     |      Number of entries in a Matrix.
     |
     |  __mul__(self, other)
     |      Handle `self * other` and `other * self`.
     |
     |      This may be either scalar multiplication, matrix-vector multiplication,
     |      vector-matrix multiplication, or matrix-matrix multiplication.
     |
     |      Example Usage:
     |          >>> Matrix([(1, 2), (3, 4)]) * Matrix([(1, 2), (3, 4)])
     |          Matrix(((7.0, 10.0,), (15.0, 22.0,)))
     |
     |          >>> 2 * Matrix([(1, 2), (3, 4)])
     |          Matrix(((2.0, 4.0,), (6.0, 8.0,)))
     |
     |          >>> Matrix([(1, 2), (3, 4)]) * 3
     |          Matrix(((3.0, 6.0,), (9.0, 12.0,)))
     |
     |          Matrix-vector and vector-matrix multiplication are not commutative.
     |
     |          >>> from sample_package import Vector
     |
     |          >>> Matrix([(1, 2), (3, 4)]) * Vector([5, 6])
     |          Vector((17.0, 39.0))
     |
     |          >>> Vector([5, 6]) * Matrix([(1, 2), (3, 4)])
     |          Vector((23.0, 34.0))
     |
     |  __neg__(self)
     |      Handle `-self`.
     |
     |      Negate all entries of a Matrix.
     |
     |  __pos__(self)
     |      Handle `+self`.
     |
     |      This is simply an identity operator returning the Matrix itself.
     |
     |  __radd__(self, other)
     |      See docstring for .__add__().
     |
     |  __repr__(self)
     |      Text representation of a Matrix.
     |
     |  __reversed__(self)
     |      Loop over a Matrix's entries in reverse order.
     |
     |      See .entries() for more customization options.
     |
     |  __rmul__(self, other)
     |      See docstring for .__mul__().
     |
     |  __rsub__(self, other)
     |      See docstring for .__sub__().
     |
     |  __str__(self)
     |      Human-readable text representation of a Matrix.
     |
     |  __sub__(self, other)
     |      Handle `self - other` and `other - self`.
     |
     |      This may be either matrix subtraction or broadcasting subtraction.
     |
     |      Example Usage:
     |          >>> Matrix([(2, 3), (4, 5)]) - Matrix([(1, 2), (3, 4)])
     |          Matrix(((1.0, 1.0,), (1.0, 1.0,)))
     |
     |          >>> Matrix([(1, 2), (3, 4)]) - 1
     |          Matrix(((0.0, 1.0,), (2.0, 3.0,)))
     |
     |          >>> 10 - Matrix([(1, 2), (3, 4)])
     |          Matrix(((9.0, 8.0,), (7.0, 6.0,)))
     |
     |  __truediv__(self, other)
     |      Handle `self / other`.
     |
     |      Divide a Matrix by a scalar.
     |
     |      Example Usage:
     |          >>> Matrix([(1, 2), (3, 4)]) / 4
     |          Matrix(((0.25, 0.5,), (0.75, 1.0,)))
     |
     |  as_vector(self)
     |      Get a Vector representation of a Matrix.
     |
     |      Returns:
     |          vector (vector.Vector)
     |
     |      Raises:
     |          RuntimeError: if one of the two dimensions, .n_rows or .n_cols, is not 1
     |
     |      Example Usage:
     |          >>> Matrix([(1, 2, 3)]).as_vector()
     |          Vector((1.0, 2.0, 3.0))
     |
     |  cols(self)
     |      Loop over a Matrix's columns.
     |
     |      Returns:
     |          columns (generator): produces a Matrix's columns as Vectors
     |
     |  entries(self, *, reverse=False, row_major=True)
     |      Loop over a Matrix's entries.
     |
     |      Args:
     |          reverse (bool): flag to loop backwards; defaults to False
     |          row_major (bool): flag to loop in row-major order; defaults to True
     |
     |      Returns:
     |          entries (generator): produces a Matrix's entries
     |
     |  rows(self)
     |      Loop over a Matrix's rows.
     |
     |      Returns:
     |          rows (generator): produces a Matrix's rows as Vectors
     |
     |  transpose(self)
     |      Switch the rows and columns of a Matrix.
     |
     |      Returns:
     |          matrix (Matrix)
     |
     |      Example Usage:
     |          >>> m = Matrix([(1, 2), (3, 4)])
     |          >>> m
     |          Matrix(((1.0, 2.0,), (3.0, 4.0,)))
     |          >>> m.transpose()
     |          Matrix(((1.0, 3.0,), (2.0, 4.0,)))
     |
     |  ----------------------------------------------------------------------
     |  Class methods defined here:
     |
     |  from_columns(data) from builtins.type
     |      Create a new matrix.
     |
     |      This is an alternative constructor for data provided in column-major order.
     |
     |      Args:
     |          data (sequence of sequences): the matrix's entries;
     |              viewed as a sequence of the matrix's columns (i.e., column-major order);
     |              use the normal constructor method if the data come as a sequence
     |              of the matrix's rows (i.e., row-major order)
     |
     |      Raises:
     |          ValueError:
     |              - if no entries are provided
     |              - if the number of rows is inconsistent across the columns
     |
     |      Example Usage:
     |          >>> Matrix.from_columns([(1, 2), (3, 4)])
     |          Matrix(((1.0, 3.0,), (2.0, 4.0,)))
     |
     |  from_rows(data) from builtins.type
     |      See docstring for .__init__().
     |
     |  ----------------------------------------------------------------------
     |  Readonly properties defined here:
     |
     |  n_cols
     |      Number of columns in a Matrix.
     |
     |  n_rows
     |      Number of rows in a Matrix.
     |
     |  ----------------------------------------------------------------------
     |  Data descriptors defined here:
     |
     |  __dict__
     |      dictionary for instance variables
     |
     |  __weakref__
     |      list of weak references to the object
     |
     |  ----------------------------------------------------------------------
     |  Data and other attributes defined here:
     |
     |  __hash__ = None
     |
     |  storage = <class 'tuple'>
     |      Built-in immutable sequence.
     |
     |      If no argument is given, the constructor returns an empty tuple.
     |      If iterable is specified the tuple is initialized from iterable's items.
     |
     |      If the argument is a tuple, the return value is the same object.
     |
     |
     |  typing = <class 'float'>
     |      Convert a string or number to a floating point number, if possible.
     |
     |
     |  vector_cls = <class 'sample_package.vector.Vector'>
     |      A one-dimensional vector from linear algebra.
     |
     |      All entries are converted to floats, or whatever is set in the typing attribute.
     |
     |      Attributes:
     |          matrix_cls (matrix.Matrix): a reference to the Matrix class to work with
     |          storage (callable): data type used to store the entries internally;
     |              defaults to tuple
     |          typing (callable): type casting applied to all entries upon creation;
     |              defaults to float
     |          zero_threshold (float): max. tolerance when comparing an entry to zero;
     |              defaults to 1e-12
     |
     |
     |  zero_threshold = 1e-12

    class Vector(builtins.object)
     |  Vector(data)
     |
     |  A one-dimensional vector from linear algebra.
     |
     |  All entries are converted to floats, or whatever is set in the typing attribute.
     |
     |  Attributes:
     |      matrix_cls (matrix.Matrix): a reference to the Matrix class to work with
     |      storage (callable): data type used to store the entries internally;
     |          defaults to tuple
     |      typing (callable): type casting applied to all entries upon creation;
     |          defaults to float
     |      zero_threshold (float): max. tolerance when comparing an entry to zero;
     |          defaults to 1e-12
     |
     |  Methods defined here:
     |
     |  __abs__(self)
     |      The Euclidean norm of a vector.
     |
     |  __add__(self, other)
     |      Handle `self + other` and `other + self`.
     |
     |      This may be either vector addition or broadcasting addition.
     |
     |      Example Usage:
     |          >>> Vector([1, 2, 3]) + Vector([2, 3, 4])
     |          Vector((3.0, 5.0, 7.0))
     |
     |          >>> Vector([1, 2, 3]) + 4
     |          Vector((5.0, 6.0, 7.0))
     |
     |          >>> 10 + Vector([1, 2, 3])
     |          Vector((11.0, 12.0, 13.0))
     |
     |  __bool__(self)
     |      A Vector is truthy if its Euclidean norm is strictly positive.
     |
     |  __eq__(self, other)
     |      Handle `self == other`.
     |
     |      Compare two Vectors for equality.
     |
     |      Example Usage:
     |          >>> Vector([1, 2, 3]) == Vector([1, 2, 3])
     |          True
     |
     |          >>> Vector([1, 2, 3]) == Vector([4, 5, 6])
     |          False
     |
     |  __float__(self)
     |      Cast a Vector as a scalar.
     |
     |      Returns:
     |          scalar (float)
     |
     |      Raises:
     |          RuntimeError: if the Vector has more than one entry
     |
     |  __getitem__(self, index)
     |      Obtain an individual entry of a Vector.
     |
     |  __init__(self, data)
     |      Create a new vector.
     |
     |      Args:
     |          data (sequence): the vector's entries
     |
     |      Raises:
     |          ValueError: if no entries are provided
     |
     |      Example Usage:
     |          >>> Vector([1, 2, 3])
     |          Vector((1.0, 2.0, 3.0))
     |
     |          >>> Vector(range(3))
     |          Vector((0.0, 1.0, 2.0))
     |
     |  __iter__(self)
     |      Loop over a Vector's entries.
     |
     |  __len__(self)
     |      Number of entries in a Vector.
     |
     |  __mul__(self, other)
     |      Handle `self * other` and `other * self`.
     |
     |      This may be either the dot product of two vectors or scalar multiplication.
     |
     |      Example Usage:
     |          >>> Vector([1, 2, 3]) * Vector([2, 3, 4])
     |          20.0
     |
     |          >>> 2 * Vector([1, 2, 3])
     |          Vector((2.0, 4.0, 6.0))
     |
     |          >>> Vector([1, 2, 3]) * 3
     |          Vector((3.0, 6.0, 9.0))
     |
     |  __neg__(self)
     |      Handle `-self`.
     |
     |      Negate all entries of a Vector.
     |
     |  __pos__(self)
     |      Handle `+self`.
     |
     |      This is simply an identity operator returning the Vector itself.
     |
     |  __radd__(self, other)
     |      See docstring for .__add__().
     |
     |  __repr__(self)
     |      Text representation of a Vector.
     |
     |  __reversed__(self)
     |      Loop over a Vector's entries in reverse order.
     |
     |  __rmul__(self, other)
     |      See docstring for .__mul__().
     |
     |  __rsub__(self, other)
     |      See docstring for .__sub__().
     |
     |  __str__(self)
     |      Human-readable text representation of a Vector.
     |
     |  __sub__(self, other)
     |      Handle `self - other` and `other - self`.
     |
     |      This may be either vector subtraction or broadcasting subtraction.
     |
     |      Example Usage:
     |          >>> Vector([7, 8, 9]) - Vector([1, 2, 3])
     |          Vector((6.0, 6.0, 6.0))
     |
     |          >>> Vector([1, 2, 3]) - 1
     |          Vector((0.0, 1.0, 2.0))
     |
     |          >>> 10 - Vector([1, 2, 3])
     |          Vector((9.0, 8.0, 7.0))
     |
     |  __truediv__(self, other)
     |      Handle `self / other`.
     |
     |      Divide a Vector by a scalar.
     |
     |      Example Usage:
     |          >>> Vector([9, 6, 12]) / 3
     |          Vector((3.0, 2.0, 4.0))
     |
     |  as_matrix(self, *, column=True)
     |      Get a Matrix representation of a Vector.
     |
     |      Args:
     |          column (bool): if the vector is interpreted as a
     |              column vector or a row vector; defaults to True
     |
     |      Returns:
     |          matrix (matrix.Matrix)
     |
     |      Example Usage:
     |          >>> v = Vector([1, 2, 3])
     |          >>> v.as_matrix()
     |          Matrix(((1.0,), (2.0,), (3.0,)))
     |          >>> v.as_matrix(column=False)
     |          Matrix(((1.0, 2.0, 3.0,)))
     |
     |  ----------------------------------------------------------------------
     |  Data descriptors defined here:
     |
     |  __dict__
     |      dictionary for instance variables
     |
     |  __weakref__
     |      list of weak references to the object
     |
     |  ----------------------------------------------------------------------
     |  Data and other attributes defined here:
     |
     |  __hash__ = None
     |
     |  matrix_cls = <class 'sample_package.matrix.Matrix'>
     |      An m-by-n-dimensional matrix from linear algebra.
     |
     |      All entries are converted to floats, or whatever is set in the typing attribute.
     |
     |      Attributes:
     |          storage (callable): data type used to store the entries internally;
     |              defaults to tuple
     |          typing (callable): type casting applied to all entries upon creation;
     |              defaults to float
     |          vector_cls (vector.Vector): a reference to the Vector class to work with
     |          zero_threshold (float): max. tolerance when comparing an entry to zero;
     |              defaults to 1e-12
     |
     |
     |  storage = <class 'tuple'>
     |      Built-in immutable sequence.
     |
     |      If no argument is given, the constructor returns an empty tuple.
     |      If iterable is specified the tuple is initialized from iterable's items.
     |
     |      If the argument is a tuple, the return value is the same object.
     |
     |
     |  typing = <class 'float'>
     |      Convert a string or number to a floating point number, if possible.
     |
     |
     |  zero_threshold = 1e-12

DATA
    __all__ = ['Matrix', 'Vector']

VERSION
    0.1.0

AUTHOR
    Alexander Hess

FILE
    /home/alexander/Repositories/intro-to-python/11_classes/sample_package/__init__.py

The meta information could also be accessed separately and individually.

In [9]:

pkg.__name__

Out[9]:

'linear_algebra_tools'

In [10]:

pkg.__version__  # follows the semantic versioning format

Out[10]:

'0.1.0'

We create Vector and Matrix instances in the usual way by calling the Vector and Matrix classes from the package's top level.

In [11]:

pkg.Vector([1, 2, 3])

Out[11]:

Vector((1.0, 2.0, 3.0))

In [12]:

pkg.Matrix([(1, 2, 3), (4, 5, 6), (7, 8, 9)])

Out[12]:

Matrix(((1.0, 2.0, 3.0,), (4.0, 5.0, 6.0,), (7.0, 8.0, 9.0,)))

A common practice by package authors is to put all the objects on the package's top level that they want the package users to work with directly. That is achieved via the import statements in the __init__.py file.

However, users can always reach into a package and work with its internals.

For example, the Vector and Matrix classes are also available via their qualified name (cf., PEP 3155 ): First, we access the vector and matrix modules on pkg, and then the Vector and Matrix classes on the modules.

In [13]:

pkg.vector.Vector

Out[13]:

sample_package.vector.Vector

In [14]:

pkg.matrix.Matrix

Out[14]:

sample_package.matrix.Matrix

Also, let's import the utils.py module with the norm() function into the global scope. As this function is integrated into the Vector.__abs__() and Matrix.__abs__() methods, there is actually no need to work with it explicitly.

In [15]:

from sample_package import utils

In [16]:

help(utils.norm)

Help on function norm in module sample_package.utils:

norm(vec_or_mat)
    Calculate the Frobenius or Euclidean norm of a matrix or vector.

    Find more infos here: https://en.wikipedia.org/wiki/Matrix_norm#Frobenius_norm

    Args:
        vec_or_mat (Vector / Matrix): object whose entries are squared and summed up

    Returns:
        norm (float)

    Example Usage:
        As Vector and Matrix objects are by design non-empty sequences,
        norm() may be called, for example, with `[3, 4]` as the argument:
        >>> norm([3, 4])
        5.0

Many tutorials on the internet begin by importing "everything" from a package into the global scope with from ... import *.

That is commonly considered a bad practice as it may overwrite already existing variables. However, if the package's __init__.py file defines an __all__ attribute, a list with all the names to be "exported," the star import is safe to be used, in particular, in interactive sessions like Jupyter notebooks. We emphasize that the star import should not be used within packages and modules as then it is not directly evident from a name where the corresponding object is defined.

For more best practices regarding importing we refer to, among others, Google's Python Style Guide.

The following import statement makes the Vector and Matrix classes available in the global scope.

In [17]:

from sample_package import *

In [18]:

Vector

Out[18]:

sample_package.vector.Vector

In [19]:

Matrix

Out[19]:

sample_package.matrix.Matrix

For further information on modules and packages, we refer to the official tutorial .

The final `Vector` and `Matrix` Classes¶

The final implementations of the Vector and Matrix classes are in the matrix.py and vector.py files: They integrate all of the functionalities introduced in this chapter. In addition, the code is cleaned up and fully documented, including examples of common usages.

We strongly suggest the eager student go over the files in the sample_package in detail at some point to understand what well-written and (re-)usable code looks like.

In [20]:

v = Vector([1, 2, 3])

In [21]:

Out[21]:

Vector((1.0, 2.0, 3.0))

In [22]:

type(v)

Out[22]:

sample_package.vector.Vector

In [23]:

m = Matrix([(1, 2, 3), (4, 5, 6), (7, 8, 9)])

In [24]:

Out[24]:

Matrix(((1.0, 2.0, 3.0,), (4.0, 5.0, 6.0,), (7.0, 8.0, 9.0,)))

In [25]:

type(m)

Out[25]:

sample_package.matrix.Matrix

Furthermore, the classes are designed for easier maintenence in the long-run.

For example, the Matrix/Vector.storage and Matrix/Vector.typing class attributes replace the "hard coded" tuple() and float() built-ins in the .__init__() methods: As self.storage and self.typing are not defined on the instances, Python automatically looks them up on the classes.

In [26]:

Vector.storage

Out[26]:

tuple

In [27]:

Vector.typing

Out[27]:

float

In [28]:

Vector.__init__??

Signature: Vector.__init__(self, data)
Source:   
    def __init__(self, data):
        """Create a new vector.

        Args:
            data (sequence): the vector's entries

        Raises:
            ValueError: if no entries are provided

        Example Usage:
            >>> Vector([1, 2, 3])
            Vector((1.0, 2.0, 3.0))

            >>> Vector(range(3))
            Vector((0.0, 1.0, 2.0))
        """
        self._entries = self.storage(self.typing(x) for x in data)
        if len(self) == 0:
            raise ValueError("a vector must have at least one entry")
File:      ~/Repositories/intro-to-python/11_classes/sample_package/vector.py
Type:      function

Both Matrix/Vector.storage and Matrix/Vector.typing themselves reference the DEFAULT_ENTRIES_STORAGE and DEFAULT_ENTRY_TYPE constants in the utils.py module. This way, we could, for example, change only the constants and thereby also change how the ._entries are stored internally in both classes. Also, this single single source of truth ensures that both classes are consistent with each other at all times.

In [29]:

utils.DEFAULT_ENTRIES_STORAGE

Out[29]:

tuple

In [30]:

utils.DEFAULT_ENTRY_TYPE

Out[30]:

float

For the same reasons, we also replace the "hard coded" references to the Vector and Matrix classes within the various methods.

Every instance object has an automatically set .__class__ attribute referencing its class.

In [31]:

m.__class__

Out[31]:

sample_package.matrix.Matrix

Of course, we could also use the type() built-in instead.

In [32]:

type(m)

Out[32]:

sample_package.matrix.Matrix

So, for example, the Matrix.transpose() method makes a self.__class__(...) instead of a Matrix(...) call.

In [33]:

Matrix.transpose??

Signature: Matrix.transpose(self)
Source:   
    def transpose(self):
        """Switch the rows and columns of a Matrix.

        Returns:
            matrix (Matrix)

        Example Usage:
            >>> m = Matrix([(1, 2), (3, 4)])
            >>> m
            Matrix(((1.0, 2.0,), (3.0, 4.0,)))
            >>> m.transpose()
            Matrix(((1.0, 3.0,), (2.0, 4.0,)))
        """
        return self.__class__(zip(*self._entries))
File:      ~/Repositories/intro-to-python/11_classes/sample_package/matrix.py
Type:      function

Whenever we need a str representation of a class's name, we use the .__name__ attribute on the class, ...

In [34]:

Matrix.__name__

Out[34]:

'Matrix'

... or access it via the .__class__ attribute on an instance.

In [35]:

m.__class__.__name__

Out[35]:

'Matrix'

For example, the .__repr__() and .__str__() methods make use of that.

In [36]:

Matrix.__repr__??

Signature: Matrix.__repr__(self)
Source:   
    def __repr__(self):
        """Text representation of a Matrix."""
        name = self.__class__.__name__
        args = ", ".join(
            "(" + ", ".join(repr(c) for c in r) + ",)" for r in self._entries
        )
        return f"{name}(({args}))"
File:      ~/Repositories/intro-to-python/11_classes/sample_package/matrix.py
Type:      function

In order to not have to "hard code" the name of another class (e.g., the Vector.as_matrix() method references the Matrix class), we apply the following "hack:" First, we store a reference to the other class as a class attribute (e.g., Matrix.vector_cls and Vector.matrix_cls), and then reference that attribute within the methods, just like .storage and .typing above.

In [37]:

Matrix.vector_cls

Out[37]:

sample_package.vector.Vector

In [38]:

Vector.matrix_cls

Out[38]:

sample_package.matrix.Matrix

As an example, the Vector.as_matrix() method makes a self.matrix_cls(...) instead of a Matrix(...) call.

In [39]:

Vector.as_matrix??

Signature: Vector.as_matrix(self, *, column=True)
Source:   
    def as_matrix(self, *, column=True):
        """Get a Matrix representation of a Vector.

        Args:
            column (bool): if the vector is interpreted as a
                column vector or a row vector; defaults to True

        Returns:
            matrix (matrix.Matrix)

        Example Usage:
            >>> v = Vector([1, 2, 3])
            >>> v.as_matrix()
            Matrix(((1.0,), (2.0,), (3.0,)))
            >>> v.as_matrix(column=False)
            Matrix(((1.0, 2.0, 3.0,)))
        """
        if column:
            return self.matrix_cls([x] for x in self)
        return self.matrix_cls([(x for x in self)])
File:      ~/Repositories/intro-to-python/11_classes/sample_package/vector.py
Type:      function

For completeness sake, we mention that in the final Vector and Matrix classes, the .__sub__() and .__rsub__() methods use the negation operator implemented in .__neg__() and then dispatch to .__add__() instead of implementing the subtraction logic themselves.

"Real-life" Experiment¶

Let's do some math with bigger Matrix and Vector instances.

In [40]:

import random

In [41]:

random.seed(42)

We initialize m as a $100x50$ dimensional Matrix with random numbers in the range between 0 and 1_000.

In [42]:

m = Matrix((1_000 * random.random() for _ in range(50)) for _ in range(100))

We quickly lose track with all the numbers in the Matrix, which is why we implemented the __str__() method as a summary representation.

In [43]:

Out[43]:

Matrix(((639.4267984578837, 25.010755222666937, 275.02931836911927, 223.21073814882274, 736.4712141640124, 676.6994874229113, 892.1795677048455, 86.93883262941615, 421.9218196852704, 29.797219438070343, 218.63797480360336, 505.3552881033624, 26.535969683863627, 198.8376506866485, 649.8844377795232, 544.9414806032166, 220.4406220406967, 589.2656838759087, 809.4304566778267, 6.498759678061017, 805.8192518328079, 698.1393949882269, 340.25051651799185, 155.47949981178155, 957.2130722067812, 336.59454511262675, 92.7458433801479, 96.71637683346401, 847.4943663474597, 603.7260313668911, 807.1282732743801, 729.7317866938179, 536.2280914547007, 973.1157639793706, 378.5343772083535, 552.040631273227, 829.4046642529948, 618.5197523642461, 861.7069003107772, 577.352145256762, 704.5718362149235, 45.82438365566222, 227.89827565154687, 289.38796360210716, 79.79197692362749, 232.79088636103018, 101.00142940972911, 277.97360311009214, 635.6844442644002, 364.8321789700842,), (370.18096711688264, 209.50703077148768, 266.97782204911334, 936.6545877124939, 648.0353852465936, 609.1310056669882, 171.13864819809697, 729.1267979503492, 163.40249376192838, 379.4554417576478, 989.5233506365953, 639.9997598540929, 556.9497437746462, 684.6142509898746, 842.8519201898096, 775.9999115462448, 229.04807196410437, 32.10024390403776, 315.45304805908194, 267.74087597570275, 210.98284358632645, 942.9097143350544, 876.3676264726688, 314.67788079847793, 655.43866529488, 395.63190106066423, 914.5475897405435, 458.8518525873988, 264.88016649805246, 246.62750769398346, 561.3681341631508, 262.7416085229353, 584.5859902235405, 897.822883602477, 399.4005051403973, 219.32075915728333, 997.5376064951103, 509.5262936764645, 90.9094121737939, 47.11637542473457, 109.64913035065915, 627.44604170309, 792.0793643629642, 422.159966799684, 63.52770615195713, 381.61928650653675, 996.1213802400968, 529.114345099137, 971.0783776136182, 860.779702234498,), (11.481021942819636, 720.7218193601947, 681.7103690265748, 536.9703304087951, 266.82518995254276, 640.961798579808, 111.55217359587644, 434.765250669105, 453.72370632920644, 953.8159275210801, 875.852940378194, 263.38905075109074, 500.5861130502983, 178.65188053013136, 912.6278393448205, 870.5185698367669, 298.4447914486329, 638.9494948660051, 608.9702114381723, 152.83926854963482, 762.5108000751512, 539.3790301196258, 778.6264786305583, 530.3536721951775, 0.5718961279435053, 324.1560570046731, 19.476742385832303, 929.0986162646171, 878.7218778231842, 831.6655293611794, 307.5141254026614, 57.92516649418755, 878.0095992040406, 946.9494452979941, 85.65345206787877, 485.9904633166138, 69.2125184683836, 760.6021652572316, 765.8344293069878, 128.3914644997628, 475.28237809873133, 549.8035934949439, 265.0566289400591, 872.4330410852574, 423.13794020088693, 211.79820544208206, 539.2960887794584, 729.9310690899762, 201.1510633896959, 311.71629130089497,), (995.1493566608947, 649.8780576394535, 438.10008391450407, 517.5758410355907, 121.00419586826573, 224.69733703155737, 338.08556214745533, 588.3087184572333, 230.114732596577, 220.21738445155947, 70.99308600903254, 631.102957270099, 228.94178381115438, 905.420013006128, 859.6354002537465, 70.85734988865345, 238.00463436899523, 668.9777782962806, 214.2368073704386, 132.311848725025, 935.514240580671, 571.0430933252844, 472.67102631179415, 784.6194242907534, 807.4969977666434, 190.4099143618777, 96.93081422882332, 431.0511824063775, 423.57862301992077, 467.024668036675, 729.0758494598506, 673.3645472933015, 984.1652113659661, 98.41787115195888, 402.62128210226876, 339.30260539496317, 861.6725363527911, 248.65633392028565, 190.2089084408115, 448.6135478331319, 421.8816398344042, 278.5451446669405, 249.80644788210049, 923.2655992760128, 443.13074505345696, 861.3491047618306, 550.3253124498481, 50.58832952488124, 999.2824684127266, 836.0275850799519,), (968.9962572847512, 926.3669830081276, 848.6957344143054, 166.31111060391402, 485.64112545071845, 213.74729919918167, 401.0402925494526, 58.635399972178924, 378.9731189769161, 985.308843779726, 265.20305817215194, 784.0706019485693, 455.0083673391433, 423.00748599016293, 957.3176408596732, 995.4226894927139, 555.7683234056182, 718.408275296326, 154.79682527406413, 296.7078254945642, 968.7093649691589, 579.1802908162562, 542.1952013742742, 747.9755603790641, 57.16527290748308, 584.1775944589713, 502.85038291951355, 852.7198920482854, 157.43272793948327, 960.7789032744504, 80.11146524058688, 185.8249609807232, 595.0351064500277, 675.2125536040902, 235.2038950009312, 119.88661394712419, 890.2873141294375, 246.21534778862485, 594.5191535334412, 619.3815103321031, 419.2249153358725, 583.6722892912247, 522.7827155319588, 934.7062577364272, 204.25919942353644, 716.1918007894149, 238.68595261584602, 395.7858467912545, 671.6902229599713, 299.99707979876223,), (316.17719627185403, 751.8644924144021, 72.54311449315732, 458.2855226185861, 998.4544408544424, 996.0964478550944, 73.260721099633, 213.1543122670404, 265.20041475040136, 933.2593779937091, 880.8641736864396, 879.2702424845428, 369.52708873888395, 157.74683235723197, 833.744954639807, 703.539925087371, 611.6777657259502, 987.2330636315044, 653.9763177107326, 7.823107152157949, 817.1041351154616, 299.3787521999779, 663.3887149660774, 938.9300039271039, 134.2911143933677, 115.4286704191022, 107.03597770941764, 553.223640884816, 272.3482123148163, 604.8298270302239, 717.6121871387979, 203.59731232745293, 634.2379588850797, 263.98390163040943, 488.53185214937656, 905.3364910793232, 846.1037132948554, 92.29846771273343, 423.57577256372633, 276.68022397225167, 3.5456890877823, 771.119223019627, 637.1133773013796, 261.9552624343482, 741.2309083479308, 551.6804211263913, 427.68691898067937, 9.669699608339965, 75.24386007376704, 883.1063933001429,), (903.9285715598932, 545.5902892055224, 834.5950198860166, 582.509566489794, 148.09378556748266, 127.44551928213876, 308.2583499301337, 898.9814887425899, 796.1223048880418, 860.7025820009028, 898.9246365264746, 210.07653833975405, 249.52973922292443, 102.79362167178563, 780.1162418714426, 884.1347014510089, 406.3773898321168, 620.6615101507128, 154.55333833220465, 929.8810156936744, 864.605696219964, 976.2060329309629, 810.7717199403969, 881.4162046633244, 24.786361898188723, 736.5644717550821, 332.1854679464287, 930.8158860483255, 802.2351389371389, 864.0640283752793, 810.749316574389, 266.80570959447203, 787.3745091354712, 108.09562640295711, 872.1667829060897, 858.5932513377817, 222.43371754566442, 816.586605596929, 460.3032346789421, 305.1908673386006, 795.3454991528617, 227.59548740777035, 23.66443470145152, 193.12978832770867, 328.26195119770654, 864.3529420302864, 966.8891040483611, 279.1249927218714, 641.4817386076277, 399.6783843600609,), (981.1496871982602, 536.2157324787219, 939.2371403247157, 115.34175185142759, 970.400611022228, 178.56781617246364, 962.5343157615555, 265.4663625229686, 108.40254721471109, 434.56375856464433, 728.5450606527043, 313.67731419499125, 606.2088533061433, 511.4230596694781, 385.1954333447272, 576.5880434965995, 254.72250613858193, 708.7852838341706, 1.6912782186294661, 925.5751654990827, 538.4519970927919, 719.4299991448455, 741.9500778394765, 670.6285044329995, 364.2214717812642, 69.97381112631018, 664.2376849112724, 330.2000360425964, 313.91564505835964, 848.0152795063354, 719.7542630139502, 300.3222682112642, 309.28466220865323, 408.3929086192168, 402.40038705772463, 295.655202525947, 127.28779905915322, 420.4463337729083, 940.363670730183, 677.3179452727329, 902.8055457325827, 615.5149159513805, 300.9498745655653, 547.9372131356982, 0.4059396972875273, 286.91371686892717, 429.8881499898346, 579.984781195682, 654.7056237030716, 464.9881902470142,), (442.1597993048074, 213.70140098910028, 473.18618590932624, 901.1808258282542, 796.0247601267803, 169.69139619805475, 84.79553672512175, 515.4520099152164, 632.9408557657957, 335.1882554098009, 818.4234645366643, 751.1381375407323, 672.795670557167, 224.64066599728827, 199.12993272657664, 24.425387726826344, 244.84254407835016, 475.1363442188051, 849.737694624732, 72.82822918456911, 414.44101099771933, 629.7653807377137, 194.4352367397093, 696.3542504905049, 494.37716901043694, 243.98443957843884, 656.058011111784, 5.54481813803176, 750.9644766184729, 770.0461885740251, 106.58729656353894, 425.1461939427341, 175.88668170653165, 957.9660422795397, 517.9577504437408, 50.21838514064092, 249.19827965997166, 848.3363473516597, 456.4618254701725, 801.4166017222644, 667.5777325863531, 987.8924530664481, 595.4523184694197, 950.0396084431559, 891.425925810437, 612.6523227617629, 719.273961275967, 504.778164824402, 830.569169721415, 547.8719506108284,), (897.2081032332621, 743.6554421595849, 474.6744368230533, 259.19154846501937, 247.23973750965956, 637.6614367761563, 765.8136842971654, 521.2998128279821, 626.7484369817813, 274.59744691753826, 77.48335386473582, 285.7281508631525, 271.7151070821846, 319.7095684187623, 540.1522225184564, 138.37406151615573, 231.26147972818677, 693.9498122990523, 706.4191416945522, 64.22885071387807, 407.5993696665867, 542.6111405039153, 415.77423410315595, 206.83438951441025, 420.14351777342563, 904.838478340177, 584.0794142042251, 695.5229864979681, 856.7320323039343, 765.5945761180694, 380.38102892771167, 5.8960835839930725, 351.7588026718247, 753.4751250593858, 853.4479505691046, 953.4303384701062, 419.0212826298219, 747.5156689780508, 546.1323097338391, 603.2525889412414, 220.5386943238189, 219.42163462143617, 435.8359760466366, 29.02481994671524, 336.12954369838246, 679.1418850283497, 404.31666913763706, 165.04473120350883, 467.39014923231014, 127.6277972811607,), (622.2569609740647, 26.96645190513769, 394.0202563397047, 564.3919830247742, 27.102046340312434, 642.7496480093357, 135.69948723056424, 461.69844405151173, 50.28463348862755, 379.1038641881396, 211.66028421148152, 326.84580488130854, 761.2297078940271, 379.12621556413626, 752.0098235547848, 831.9242851552726, 252.2715317823806, 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868.7523516449686, 820.1339564299961, 13.733292693431377, 693.7952250561176, 111.27799231646985, 450.0615706383459, 22.748115151202008, 209.00954112420212, 538.0054579069861, 203.80135287660516, 523.2658814665313, 258.6580034255985, 483.02633034469176, 729.9235835097888, 141.34742092394183, 698.7552320082528, 18.38851255630447, 583.0049433264157, 663.5283419996432, 43.482007499458966, 170.31964704476331, 284.0135410479907, 789.1884945074221, 617.9647763913978, 53.08524865627895, 654.757871815387, 8.334175248427611, 388.6451472369792, 271.3103790803375, 852.0829447572587, 660.1003733293647, 864.2822735599534, 19.078323934194174, 867.4104428544812, 649.4111447522284, 231.16862013781702, 380.7012029628479, 976.6119398898542, 99.60371184253735, 315.45596930877974, 866.7727455281198, 531.5606451632132, 186.41738741751902, 500.65082286887883, 457.98626214778074, 926.3499730264335,), (21.488458214345528, 247.39213974975127, 529.4153638551198, 333.56963463077574, 393.3998178272747, 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In [44]:

print(m)

Matrix((639.4267984578837, ...), ..., (..., 353.85334437419004))[100x50]

Similarily, v is now a Vector with $50$ entries.

In [45]:

v = Vector(1_000 * random.random() for _ in range(50))

In [46]:

Out[46]:

Vector((129.7131375543026, 562.6343376666077, 519.7058001844938, 631.8576074968361, 492.50445371851595, 179.90723165619127, 609.4057577306154, 708.5870675033931, 979.2576595660704, 1.580917225831202, 23.986797518176004, 625.4607423896532, 117.92572392127187, 848.0697885701867, 799.5643512555062, 998.9870072501308, 414.04113952437183, 333.7922569338235, 560.4155049178265, 637.5035435880566, 11.297267978029769, 201.1871389300902, 281.62670325053864, 790.1955028462116, 307.77254924389706, 506.6896270960092, 323.9238393830701, 6.131262435292828, 685.8357886469579, 341.36158523000273, 724.3966428110118, 615.9933429964842, 29.117381811460618, 175.62908823799773, 330.51483209301, 337.936855273543, 672.4729573660214, 916.1630531735751, 797.2543838289249, 645.6522206645749, 481.4955231203607, 627.2004877076889, 892.0583267899182, 536.9675449723645, 335.10965453438513, 783.9890332085388, 413.95306533418574, 742.5846461655369, 835.1057359656187, 299.3437466393607))

In [47]:

print(v)

Vector(129.7131375543026, ..., 299.3437466393607)[50]

The arithmetic works as before.

In [48]:

w = m * v

In [49]:

print(w)

Vector(11378937.310589302, ..., 13593029.305789862)[100]

We can multiply m with its transpose or the other way round.

In [50]:

n = m * m.transpose()

In [51]:

print(n)

Matrix((14370711.26526542, ...), ..., (..., 16545418.239505697))[100x100]

In [52]:

o = m.transpose() * m

In [53]:

print(o)

Matrix((32618511.507031415, ...), ..., (..., 32339164.778032355))[50x50]

Comparison with numpy ¶

We started out in this chapter by realizing that Python provides us no good data type to model a vector $\vec{x}$ or a matrix $\bf{A}$ . Then, we built up two custom data types, Vector and Matrix, that wrap a simple tuple object for $\vec{x}$ and a tuple of tuples for $\bf{A}$ so that we can interact with their ._entries in a "natural" way, which is similar to how we write linear algebra tasks by hand. By doing this, we extend Python with our own little "dialect" or domain-specific language (DSL).

If we feel like sharing our linear algebra library with the world, we could easily do so on either GitHub or PyPI. However, for the domain of linear algebra this would be rather pointless as there is already a widely adopted library with numpy that not only has a lot more features than ours but also is implemented in C, which makes it a lot faster with big data.

Let's model the example in the first part with both numpy and our own DSL and compare them.

In [54]:

x = (1, 2, 3)
A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]

In [55]:

A * x

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
Cell In[55], line 1
----> 1 A * x

TypeError: can't multiply sequence by non-int of type 'tuple'

The creation of vectors and matrices is similar to our DSL. However, numpy uses the more general concept of an n-dimensional array (i.e., the ndarray type) where a vector is only a special case of a matrix and a matrix is yet another special case of an even higher dimensional structure.

In [56]:

import numpy as np

In [57]:

x_arr = np.array(x)
A_arr = np.array(A)

In [58]:

x_vec = Vector(x)
A_mat = Matrix(A)

The text representations are very similar. However, numpy's ndarrays keep the entries as ints while our Vector and Matrix objects contain floats.

In [59]:

x_arr

Out[59]:

array([1, 2, 3])

In [60]:

x_vec

Out[60]:

Vector((1.0, 2.0, 3.0))

In [61]:

A_arr

Out[61]:

array([[1, 2, 3],
       [4, 5, 6],
       [7, 8, 9]])

In [62]:

A_mat

Out[62]:

Matrix(((1.0, 2.0, 3.0,), (4.0, 5.0, 6.0,), (7.0, 8.0, 9.0,)))

numpy's ndarrays come with a .shape instance attribute that returns a tuple with the dimensions ...

In [63]:

x_arr.shape

Out[63]:

(3,)

In [64]:

A_arr.shape

Out[64]:

(3, 3)

... while Matrix objects come with .n_rows and .n_cols properties.

In [65]:

A_mat.n_rows, A_mat.n_cols

Out[65]:

(3, 3)

The built-in len() function does not return the number of entries in an ndarray but the number of the rows instead. This is equivalent to the first element in the .shape attribute.

In [66]:

len(x_arr)

Out[66]:

In [67]:

len(x_vec)

Out[67]:

In [68]:

len(A_arr)

Out[68]:

In [69]:

len(A_mat)

Out[69]:

The .transpose() method also exists for ndarrays.

In [70]:

A_arr.transpose()

Out[70]:

array([[1, 4, 7],
       [2, 5, 8],
       [3, 6, 9]])

In [71]:

A_mat.transpose()

Out[71]:

Matrix(((1.0, 4.0, 7.0,), (2.0, 5.0, 8.0,), (3.0, 6.0, 9.0,)))

To perform matrix-matrix, matrix-vector, or vector-matrix multiplication in numpy, we use the .dot() method. If we use the * operator with ndarrays, an entry-wise multiplication is performed.

In [72]:

A_arr.dot(x_arr)

Out[72]:

array([14, 32, 50])

In [73]:

A_arr * x_arr

Out[73]:

array([[ 1,  4,  9],
       [ 4, 10, 18],
       [ 7, 16, 27]])

In [74]:

A_mat * x_vec

Out[74]:

Vector((14.0, 32.0, 50.0))

Scalar multiplication, however, works as expected.

In [75]:

10 * x_arr

Out[75]:

array([10, 20, 30])

In [76]:

10 * x_vec

Out[76]:

Vector((10.0, 20.0, 30.0))

Because we implemented our classes to support the sequence protocol, numpy's one-dimensional ndarrays are actually able to work with them: The * operator is applied on a per-entry basis.

In [77]:

x_arr + x_vec

Out[77]:

array([2., 4., 6.])

In [78]:

x_arr * x_vec

Out[78]:

array([1., 4., 9.])

In [79]:

A_arr + A_mat

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
Cell In[79], line 1
----> 1 A_arr + A_mat

ValueError: operands could not be broadcast together with shapes (3,3) (9,)

We conclude that it is rather easy to extend Python in a way that makes the resulting application code read like core Python again. As there are many well established third-party packages out there, it is unlikely that we have to implement a fundamental library ourselves. Yet, we can apply the concepts introduced in this chapter to organize the code in the applications we write.

Chapter 11: Classes & Instances (continued)¶

Packages vs. Modules¶

The final Vector and Matrix Classes¶

"Real-life" Experiment¶

Comparison with numpy¶

The final `Vector` and `Matrix` Classes¶

Comparison with numpy ¶