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Programming for Computations - Python : A Gentle Introduction to Numerical Simulations with Python 3. 6.

Bibliographic Details
Main Author: Linge, Svein.
Other Authors: Langtangen, Hans Petter.
Format: eBook
Language:English
Published: Cham : Springer International Publishing AG, 2019.
Edition:2nd ed.
Series:Texts in Computational Science and Engineering Series
Subjects:
Electronic books.
Online Access:Click to View
Table of Contents:
  • Intro
  • Preface
  • Abstract
  • Contents
  • List of Exercises
  • 1 The First Few Steps
  • 1.1 What Is a Program? And What Is Programming?
  • 1.1.1 Installing Python
  • 1.2 A Python Program with Variables
  • 1.2.1 The Program
  • 1.2.2 Dissecting the Program
  • 1.2.3 Why Use Variables?
  • 1.2.4 Mathematical Notation Versus Coding
  • 1.2.5 Write and Run Your First Program
  • 1.3 A Python Program with a Library Function
  • 1.4 Importing from Modules and Packages
  • 1.4.1 Importing for Use Without Prefix
  • 1.4.2 Importing for Use with Prefix
  • 1.4.3 Imports with Name Change
  • 1.4.4 Importing from Packages
  • 1.4.5 The Modules/Packages Used in This Book
  • 1.5 A Python Program with Vectorization and Plotting
  • 1.6 Plotting, Printing and Input Data
  • 1.6.1 Plotting with Matplotlib
  • 1.6.2 Printing: The String Format Method
  • 1.6.3 Printing: The f-String
  • 1.6.4 User Input
  • 1.7 Error Messages and Warnings
  • 1.8 Concluding Remarks
  • 1.8.1 Programming Demands You to Be Accurate!
  • 1.8.2 Write Readable Code
  • 1.8.3 Fast Code or Slower and Readable Code?
  • 1.8.4 Deleting Data No Longer in Use
  • 1.8.5 Code Lines That Are Too Long
  • 1.8.6 Where to Find More Information?
  • 1.9 Exercises
  • Exercise 1.1: Error Messages
  • Exercise 1.2: Volume of a Cube
  • Exercise 1.3: Area and Circumference of a Circle
  • Exercise 1.4: Volumes of Three Cubes
  • Exercise 1.5: Average of Integers
  • Exercise 1.6: Formatted Print to Screen
  • 2 A Few More Steps
  • 2.1 Using Python Interactively
  • 2.1.1 The IPython Shell
  • 2.1.2 Command History
  • 2.1.3 TAB Completion
  • 2.2 Variables, Objects and Expressions
  • 2.2.1 Choose Descriptive Variable Names
  • 2.2.2 Reserved Words
  • 2.2.3 Assignment
  • 2.2.4 Object Type and Type Conversion
  • 2.2.5 Automatic Type Conversion
  • 2.2.6 Operator Precedence
  • 2.2.7 Division-Quotient and Remainder.
  • 2.2.8 Using Parentheses
  • 2.2.9 Round-Off Errors
  • 2.2.10 Boolean Expressions
  • 2.3 Numerical Python Arrays
  • 2.3.1 Array Creation and Array Elements
  • 2.3.2 Indexing an Array from the End
  • 2.3.3 Index Out of Bounds
  • 2.3.4 Copying an Array
  • 2.3.5 Slicing an Array
  • 2.3.6 Two-Dimensional Arrays and Matrix Computations
  • 2.4 Random Numbers
  • 2.5 Exercises
  • Exercise 2.1: Interactive Computing of Volume
  • Exercise 2.2: Interactive Computing of Circumference and Area
  • Exercise 2.3: Update Variable at Command Prompt
  • Exercise 2.4: Multiple Statements on One Line
  • Exercise 2.5: Boolean Expression-Even or Odd Number?
  • Exercise 2.6: Plotting Array Data
  • Exercise 2.7: Switching Values
  • Exercise 2.8: Drawing Random Numbers
  • 3 Loops and Branching
  • 3.1 The for Loop
  • 3.1.1 Example: Printing the 5 Times Table
  • 3.1.2 Characteristics of a Typical for Loop
  • 3.1.3 Combining for Loop and Array
  • 3.1.4 Using the range Function
  • 3.1.5 Using break and continue
  • 3.2 The while Loop
  • 3.2.1 Example: Finding the Time of Flight
  • 3.2.2 Characteristics of a Typical while Loop
  • 3.3 Branching (if, elif and else)
  • 3.3.1 Example: Judging the Water Temperature
  • 3.3.2 The Characteristics of Branching
  • 3.3.3 Example: Finding the Maximum Height
  • 3.3.4 Example: Random Walk in Two Dimensions
  • 3.4 Exercises
  • Exercise 3.1: A for Loop with Errors
  • Exercise 3.2: The range Function
  • Exercise 3.3: A while Loop with Errors
  • Exercise 3.4: while Loop Instead of for Loop
  • Exercise 3.5: Compare Integers a and b
  • Exercise 3.6: Area of Rectangle Versus Circle
  • Exercise 3.7: Frequency of Random Numbers
  • Exercise 3.8: Game 21
  • Exercise 3.9: Simple Search: Verification
  • Exercise 3.10: Sort Array with Numbers
  • Exercise 3.11: Compute
  • 4 Functions and the Writing of Code
  • 4.1 Functions: How to Write Them?.
  • 4.1.1 Example: Writing Our First Function
  • 4.1.2 Characteristics of a Function Definition
  • 4.1.3 Functions and the Main Program
  • 4.1.4 Local Versus Global Variables
  • 4.1.5 Calling a Function Defined with Positional Parameters
  • 4.1.6 A Function with Two Return Values
  • 4.1.7 Calling a Function Defined with Keyword Parameters
  • 4.1.8 A Function with Another Function as Input Argument
  • 4.1.9 Lambda Functions
  • 4.1.10 A Function with Several Return Statements
  • 4.2 Programming as a Step-Wise Strategy
  • 4.2.1 Making a Times Tables Test
  • 4.2.2 The 1st Version of Our Code
  • 4.2.3 The 2nd Version of Our Code
  • 4.2.4 The 3rd Version of Our Code
  • 4.3 Exercises
  • Exercise 4.1: Errors with Colon, Indent, etc.
  • Exercise 4.2: Reading Code 1
  • Exercise 4.3: Reading Code 2
  • Exercise 4.4: Functions for Circumference and Area of a Circle
  • Exercise 4.5: Function for Adding Vectors
  • Exercise 4.6: Function for Area of a Rectangle
  • Exercise 4.7: Average of Integers
  • Exercise 4.8: When Does Python Check Function Syntax?
  • Exercise 4.9: Find Crossing Points of Two Graphs
  • Exercise 4.10: Linear Interpolation
  • Exercise 4.11: Test Straight Line Requirement
  • Exercise 4.12: Fit Straight Line to Data
  • Exercise 4.13: Fit Sines to Straight Line
  • 5 Some More Python Essentials
  • 5.1 Lists and Tuples: Alternatives to Arrays
  • 5.2 Exception Handling
  • 5.2.1 The Fourth Version of Our Times Tables Program
  • 5.3 Symbolic Computations
  • 5.3.1 Numerical Versus Symbolic Computations
  • 5.3.2 SymPy: Some Basic Functionality
  • 5.3.3 Symbolic Calculations with Some Other Tools
  • 5.4 Making Our Own Module
  • 5.4.1 A Naive Import
  • 5.4.2 A Module for Vertical Motion
  • 5.4.3 Module or Program?
  • 5.5 Files: Read and Write
  • 5.6 Measuring Execution Time
  • 5.6.1 The timeit Module
  • 5.7 Exercises
  • Exercise 5.1: Nested for Loops and Lists.
  • Exercise 5.2: Exception Handling: Divisions in a Loop
  • Exercise 5.3: Taylor Series, sympy and Documentation
  • Exercise 5.4: Fibonacci Numbers
  • Exercise 5.5: Read File: Total Volume of Boxes
  • Exercise 5.6: Area of a Polygon
  • Exercise 5.7: Count Occurrences of a String in a String
  • Exercise 5.8: Compute Combinations of Sets
  • 6 Computing Integrals and Testing Code
  • 6.1 Basic Ideas of Numerical Integration
  • 6.2 The Composite Trapezoidal Rule
  • 6.2.1 The General Formula
  • 6.2.2 A General Implementation
  • 6.2.3 A Specific Implementation: What's the Problem?
  • 6.3 The Composite Midpoint Method
  • 6.3.1 The General Formula
  • 6.3.2 A General Implementation
  • 6.3.3 Comparing the Trapezoidal and the Midpoint Methods
  • 6.4 Vectorizing the Functions
  • 6.4.1 Vectorizing the Midpoint Rule
  • 6.4.2 Vectorizing the Trapezoidal Rule
  • 6.4.3 Speed up Gained with Vectorization
  • 6.5 Rate of Convergence
  • 6.6 Testing Code
  • 6.6.1 Problems with Brief Testing Procedures
  • 6.6.2 Proper Test Procedures
  • 6.6.3 Finite Precision of Floating-Point Numbers
  • 6.6.4 Constructing Unit Tests and Writing Test Functions
  • 6.7 Double and Triple Integrals
  • 6.7.1 The Midpoint Rule for a Double Integral
  • 6.7.2 The Midpoint Rule for a Triple Integral
  • 6.7.3 Monte Carlo Integration for Complex-Shaped Domains
  • 6.8 Exercises
  • Exercise 6.1: Hand Calculations for the Trapezoidal Method
  • Exercise 6.2: Hand Calculations for the Midpoint Method
  • Exercise 6.3: Compute a Simple Integral
  • Exercise 6.4: Hand-Calculations with Sine Integrals
  • Exercise 6.5: Make Test Functions for the Midpoint Method
  • Exercise 6.6: Explore Rounding Errors with Large Numbers
  • Exercise 6.7: Write Test Functions for 04xdx
  • Exercise 6.8: Rectangle Methods
  • Exercise 6.9: Adaptive Integration
  • Exercise 6.10: Integrating x Raised to x.
  • Exercise 6.11: Integrate Products of Sine Functions
  • Exercise 6.12: Revisit Fit of Sines to a Function
  • Exercise 6.13: Derive the Trapezoidal Rule for a Double Integral
  • Exercise 6.14: Compute the Area of a Triangle by Monte Carlo Integration
  • 7 Solving Nonlinear Algebraic Equations
  • 7.1 Brute Force Methods
  • 7.1.1 Brute Force Root Finding
  • 7.1.2 Brute Force Optimization
  • 7.1.3 Model Problem for Algebraic Equations
  • 7.2 Newton's Method
  • 7.2.1 Deriving and Implementing Newton's Method
  • 7.2.2 Making a More Efficient and Robust Implementation
  • 7.3 The Secant Method
  • 7.4 The Bisection Method
  • 7.5 Rate of Convergence
  • 7.6 Solving Multiple Nonlinear Algebraic Equations
  • 7.6.1 Abstract Notation
  • 7.6.2 Taylor Expansions for Multi-Variable Functions
  • 7.6.3 Newton's Method
  • 7.6.4 Implementation
  • 7.7 Exercises
  • Exercise 7.1: Understand Why Newton's Method Can Fail
  • Exercise 7.2: See If the Secant Method Fails
  • Exercise 7.3: Understand Why the Bisection Method Cannot Fail
  • Exercise 7.4: Combine the Bisection Method with Newton's Method
  • Exercise 7.5: Write a Test Function for Newton's Method
  • Exercise 7.6: Halley's Method and the Decimal Module
  • Exercise 7.7: Fixed Point Iteration
  • Exercise 7.8: Solve Nonlinear Equation for a Vibrating Beam
  • 8 Solving Ordinary Differential Equations
  • 8.1 Filling a Water Tank: Two Cases
  • 8.1.1 Case 1: Piecewise Constant Rate
  • 8.1.2 Case 2: Continuously Increasing Rate
  • 8.1.3 Reformulating the Problems as ODEs
  • 8.2 Population Growth: A First Order ODE
  • 8.2.1 Derivation of the Model
  • 8.2.2 Numerical Solution: The Forward Euler (FE) Method
  • 8.2.3 Programming the FE Scheme
  • the Special Case
  • 8.2.4 Understanding the Forward Euler Method
  • 8.2.5 Programming the FE Scheme
  • the General Case
  • 8.2.6 A More Realistic Population Growth Model.
  • 8.2.7 Verification: Exact Linear Solution of the Discrete Equations.

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