Book Image

Scientific Computing with Python - Second Edition

By : Claus Führer, Jan Erik Solem, Olivier Verdier
Book Image

Scientific Computing with Python - Second Edition

By: Claus Führer, Jan Erik Solem, Olivier Verdier

Overview of this book

Python has tremendous potential within the scientific computing domain. This updated edition of Scientific Computing with Python features new chapters on graphical user interfaces, efficient data processing, and parallel computing to help you perform mathematical and scientific computing efficiently using Python. This book will help you to explore new Python syntax features and create different models using scientific computing principles. The book presents Python alongside mathematical applications and demonstrates how to apply Python concepts in computing with the help of examples involving Python 3.8. You'll use pandas for basic data analysis to understand the modern needs of scientific computing, and cover data module improvements and built-in features. You'll also explore numerical computation modules such as NumPy and SciPy, which enable fast access to highly efficient numerical algorithms. By learning to use the plotting module Matplotlib, you will be able to represent your computational results in talks and publications. A special chapter is devoted to SymPy, a tool for bridging symbolic and numerical computations. By the end of this Python book, you'll have gained a solid understanding of task automation and how to implement and test mathematical algorithms within the realm of scientific computing.

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Table of Contents (23 chapters)
Preface
Preface
Who this book is for
What this book covers
To get the most out of this book
Download the example code files
Download the color images
Conventions used
Get in touch
Reviews
1
Getting Started
Getting Started
1.1 Installation and configuration instructions
1.1.1 Installation
1.1.2 Anaconda
1.1.3 Spyder
1.1.4 Configuration
1.1.5 Python shell
1.1.6 Executing scripts
1.1.7 Getting help
1.1.8 Jupyter – Python notebook
1.2 Program and program flow
1.2.1 Comments
1.2.2 Line joining
1.3 Basic data types in Python
1.3.1 Numbers
1.3.2 Strings
1.3.3 Variables
1.3.4 Lists
Operations on lists
1.3.6 Boolean expressions
1.4 Repeating statements with loops
1.4.1 Repeating a task
1.4.2 break and else
1.5 Conditional statements
1.6 Encapsulating code with functions
1.7 Understanding scripts and modules
1.7.1 Simple modules – collecting functions
1.7.2 Using modules and namespaces
1.8 Python interpreter
Summary
Free Chapter
2
Variables and Basic Types
Variables and Basic Types
2.1 Variables
2.2 Numeric types
2.2.1 Integers
Plain integers
2.2.2 Floating-point numbers
Floating-point representation
Infinite and not a number
Underflow – Machine epsilon
Other float types in NumPy
2.2.3 Complex numbers
Complex numbers in mathematics
The j notation
Real and imaginary parts
2.3 Booleans
2.3.1 Boolean operators
2.3.2 Boolean casting
Automatic Boolean casting
2.3.3 Return values of and and or
2.3.4 Booleans and integers
2.4 Strings
2.4.1 Escape sequences and raw strings
2.4.2 Operations on strings and string methods
2.4.3 String formatting
2.5 Summary
2.6 Exercises
3
Container Types
Container Types
3.1 Lists
3.1.1 Slicing
Strides
3.1.2 Altering lists
3.1.3 Belonging to a list
3.1.4 List methods
In-place operations
3.1.5 Merging lists – zip
3.1.6 List comprehension
3.2 A quick glance at the concept of arrays
3.3 Tuples
3.3.1 Packing and unpacking variables
3.4 Dictionaries
3.4.1 Creating and altering dictionaries
3.4.2 Looping over dictionaries
3.5 Sets
3.6 Container conversions
3.7 Checking the type of a variable
3.8 Summary
3.9 Exercises
4
Linear Algebra - Arrays
Linear Algebra - Arrays
4.1 Overview of the array type
4.1.1 Vectors and matrices
4.1.2 Indexing and slices
4.1.3 Linear algebra operations
Solving a linear system
4.2 Mathematical preliminaries
4.2.1 Arrays as functions
4.2.2 Operations are elementwise
4.2.3 Shape and number of dimensions
4.2.4 The dot operations
4.3 The array type
4.3.1 Array properties
4.3.2 Creating arrays from lists
Array and Python parentheses
4.4 Accessing array entries
4.4.1 Basic array slicing
4.4.2 Altering an array using slices
4.5 Functions to construct arrays
4.6 Accessing and changing the shape
4.6.1 The function shape
4.6.2 Number of dimensions
4.6.3 Reshape
Transpose
4.7 Stacking
4.7.1 Stacking vectors
4.8 Functions acting on arrays
4.8.1 Universal functions
Built-in universal functions
Creation of universal functions
4.8.2 Array functions
4.9 Linear algebra methods in SciPy
4.9.1 Solving several linear equation systems with LU
4.9.2 Solving a least square problem with SVD
4.9.3 More methods
4.10 Summary
4.11 Exercises
5
Advanced Array Concepts
Advanced Array Concepts
5.1 Array views and copies
5.1.1 Array views
5.1.2 Slices as views
5.1.3 Generating views by transposing and reshaping
5.1.4 Array copies
5.2 Comparing arrays
5.2.1 Boolean arrays
5.2.2 Checking for array equality
5.2.3 Boolean operations on arrays
5.3 Array indexing
5.3.1 Indexing with Boolean arrays
5.3.2 Using the command where
5.4 Performance and vectorization
5.4.1 Vectorization
5.5 Broadcasting
5.5.1 Mathematical views
Constant functions
Functions of several variables
General mechanism
Conventions
5.5.2 Broadcasting arrays
The broadcasting problem
Shape mismatch
5.5.3 Typical examples
Rescale rows
Rescale columns
Functions of two variables
5.6. Sparse matrices
5.6.1 Sparse matrix formats
Compressed sparse row format (CSR)
Compressed sparse column format (CSC)
Row-based linked list format (LIL)
Altering and slicing matrices in LIL format
5.6.2 Generating sparse matrices
5.6.3 Sparse matrix methods
5.7 Summary
6
Plotting
Plotting
6.1 Making plots with basic plotting commands
6.1.1 Using the plot command and some of its variants
6.1.2 Formatting
6.1.3 Working with meshgrid and contours
6.1.4 Generating images and contours
6.2 Working with Matplotlib objects directly
6.2.1 Creating axes objects
6.2.2 Modifying line properties
6.2.3 Making annotations
6.2.4 Filling areas between curves
6.2.5 Defining ticks and tick labels
6.2.6 Setting spines makes your plot more instructive – a comprehensive example
6.3 Making 3D plots
6.4 Making movies from plots
6.5 Summary
6.6 Exercises
7
Functions
Functions
7.1 Functions in mathematics and functions in Python
7.2 Parameters and arguments
7.2.1 Passing arguments – by position and by keyword
7.2.2 Changing arguments
7.2.3 Access to variables defined outside the local namespace
7.2.4 Default arguments
Beware of mutable default arguments
7.2.5 Variable number of arguments
7.3 Return values
7.4 Recursive functions
7.5 Function documentation
7.6 Functions are objects
7.6.1 Partial application
7.6.2 Using closures
7.7 Anonymous functions – the keyword lambda
7.7.1 The lambda construction is always replaceable
7.8 Functions as decorators
7.9 Summary
7.10 Exercises
8
Classes
Classes
8.1 Introduction to classes
8.1.1 A guiding example: Rational numbers
8.1.2 Defining a class and making an instance
8.1.3 The __init__ method
8.1.4 Attributes and methods
8.1.5 Special methods
Reverse operations
Methods mimicking function calls and iterables
8.2 Attributes that depend on each other
8.2.1 The function property
8.3 Bound and unbound methods
8.4 Class attributes and class methods
8.4.1 Class attributes
8.4.2 Class methods
8.5 Subclasses and inheritance
8.6 Encapsulation
8.7 Classes as decorators
8.8 Summary
8.9 Exercises
9
Iterating
Iterating
9.1 The for statement
9.2 Controlling the flow inside the loop
9.3 Iterable objects
9.3.1 Generators
9.3.2 Iterators are disposable
9.3.3 Iterator tools
9.3.4 Generators of recursive sequences
Arithmetic geometric mean
Convergence acceleration
9.4 List-filling patterns
9.4.1 List filling with the append method
9.4.2 List from iterators
9.4.3 Storing generated values
9.5 When iterators behave as lists
9.5.1 Generator expressions
9.5.2 Zipping iterators
9.6 Iterator objects
9.7 Infinite iterations
9.7.1 The while loop
9.7.2 Recursion
9.8 Summary
9.9 Exercises
10
Series and Dataframes - Working with Pandas
Series and Dataframes - Working with Pandas
10. 1 A guiding example: Solar cells
10.2 NumPy arrays and pandas dataframes
10.2.1 Indexing rules
10.3 Creating and modifying dataframes
10.3.1 Creating a dataframe from imported data
10.3.2 Setting the index
10.3.3 Deleting entries
10.3.4 Merging dataframes
10.3.5 Missing data in a dataframe
10.4 Working with dataframes
10.4.1 Plotting from dataframes
10.4.2 Calculations within dataframes
10.4.3 Grouping data
10.5 Summary
11
Communication by a Graphical User Interface
Communication by a Graphical User Interface
11.1 A guiding example to widgets
11.1.1 Changing a value with a slider bar
An example with two sliders
11.2 The button widget and mouse events
11.2.1 Updating curve parameters with a button
11.2.2 Mouse events and textboxes
11.3 Summary
12
Error and Exception Handling
Error and Exception Handling
12.1 What are exceptions?
12.1.1 Basic principles
Raising exceptions
Catching exceptions
12.1.2 User-defined exceptions
12.1.3 Context managers – the with statement
12.2 Finding errors: debugging
12.2.1 Bugs
12.2.2 The stack
12.2.3 The Python debugger
12.2.4 Overview – debug commands
12.2.5 Debugging in IPython
12.3 Summary
13
Namespaces, Scopes, and Modules
Namespaces, Scopes, and Modules
13.1 Namespaces
13.2 The scope of a variable
13.3 Modules
13.3.1 Introduction
13.3.2 Modules in IPython
The IPython magic command – run
13.3.3 The variable __name__
13.3.4 Some useful modules
13.4 Summary
14
Input and Output
Input and Output
14.1 File handling
14.1.1 Interacting with files
14.1.2 Files are iterables
14.1.3 File modes
14.2 NumPy methods
14.2.1 savetxt
14.2.3 loadtxt
14.3 Pickling
14.4 Shelves
14.5 Reading and writing Matlab data files
14.6 Reading and writing images
14.7 Summary
15
Testing
Testing
15.1 Manual testing
15.2 Automatic testing
15.2.1 Testing the bisection algorithm
15.2.2 Using the unittest module
15.2.3 Test setUp and tearDown methods
Setting up testdata when a test case is created
15.2.4 Parameterizing tests
15.2.5 Assertion tools
15.2.6 Float comparisons
15.2.7 Unit and functional tests
15.2.8 Debugging
15.2.9 Test discovery
15.3 Measuring execution time
15.3.1 Timing with a magic function
15.3.2 Timing with the Python module timeit
15.3.3 Timing with a context manager
15.4 Summary
15.5 Exercises
16
Symbolic Computations - SymPy
Symbolic Computations - SymPy
16.1 What are symbolic computations?
16.1.1 Elaborating an example in SymPy
16.2 Basic elements of SymPy
16.2.1 Symbols – the basis of all formulas
16.2.2 Numbers
16.2.3 Functions
Undefined functions
16.2.4 Elementary functions
16.2.5 Lambda functions
16.3 Symbolic linear algebra
16.3.1 Symbolic matrices
16.3.2 Examples for linear algebra methods in SymPy
16.4 Substitutions
16. 5 Evaluating symbolic expressions
16.5.1 Example: A study on the convergence order of Newton's method
16.5.2 Converting a symbolic expression into a numeric function
A study on the parameter dependency of polynomial coefficients
16.6 Summary
17
Interacting with the Operating System
Interacting with the Operating System
17.1 Running a Python program in a Linux shell
17.2 The module sys
17.2.1 Command-line arguments
17.2.2 Input and output streams
Redirecting streams
Building a pipe between a Linux command and a Python script
17.3 How to execute Linux commands from Python
17.3.1 The modules subprocess and shlex
A complete process: subprocess.run
Creating processes: subprocess.Popen
17.4 Summary
18
Python for Parallel Computing
Python for Parallel Computing
18.1 Multicore computers and computer clusters
18.2 Message passing interface (MPI)
18.2.1 Prerequisites
18.3 Distributing tasks to different cores
18.3.1 Information exchange between processes
18.3.2 Point-to-point communication
18.3.3 Sending NumPy arrays
18.3.4 Blocking and non-blocking communication
18.3.5 One-to-all and all-to-one communication
Preparing the data for communication
The commands – scatter and gather
A final data reduction operation – the command reduce
Sending the same message to all
Buffered data
18.4 Summary
19
Comprehensive Examples
Comprehensive Examples
19.1 Polynomials
19.1.1 Theoretical background
19.1.2 Tasks
19.1.3 The polynomial class
19.1.4 Usage examples of the polynomial class
19.1.5 Newton polynomial
19.2 Spectral clustering
19.3 Solving initial value problems
19.4 Summary
19.5 Exercises
20
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About Packt
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21
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References
References

16.1 What are symbolic computations?

All computations we did so far in this book were so-called numeric computations. These were a sequence of operations mainly on floating-point numbers. It is the nature of numeric computations that the result is an approximation of the exact solution.

Symbolic computations operate on formulas or symbols by transforming them as taught in algebra or calculus into other formulas. The last step of these transformations might then require that numbers are inserted and a numeric evaluation is performed.

We illustrate the difference by computing this definite integral:

Symbolically this expression can be transformed by considering the primitive function of the integrand:

We now obtain a formula for the definite integral by inserting the integral bounds:

This is called a closed-form expression for the integral. Very few mathematical problems have a solution that can be given in a closed-form expression. It is the exact value...

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