book

Introduction to Engineering and Scientific Computing with Python

Name: Introduction to Engineering and Scientific Computing with Python
ISBN: 9781000648300

by David E. Clough, Steven C. Chapra

September 2022

Intermediate to advanced

420 pages

10h 1m

English

CRC Press

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Cover
Half Title
Title Page
Copyright Page
Dedication Page
Table of Contents
List of Examples
Preface
Acknowledgments
Authors

Chapter 1 Engineering and Scientific Calculations
Chapter Objectives1.1 Numerical Quantities1.1.1 Positional and Scientific Notation1.1.2 Accuracy and Precision1.1.3 Significant Figures1.1.4 Rounding1.2 Mathematical Functions1.2.1 Absolute Value and Sign Functions1.2.2 Exponents and Logarithms1.2.3 Trigonometric Functions1.2.4 Hyperbolic Functions1.3 Complex Numbers1.4 Engineering Units1.5 Organizing and Planning Solutions to ProblemsProblems
Chapter 2 Computer-Based Calculations
Chapter Objectives2.1 Numerical Quantities as Stored in the Computer2.1.1 Integer Numbers2.1.2 Real Numbers2.2 How the Computer Stores Text2.3 Boolean True/False Information2.4 Computer Storage Evolution and TerminologyProblems
Chapter 3 Python Basics
Chapter Objectives3.1 The Spyder/IPython Environment3.2 Mathematical Functions3.3 Variables and Assignment3.4 Objects, Attributes, Methods, and Data Types3.4.1 Boolean Type3.4.2 Character Type3.5 Collections of Data3.6 Creating Plots3.7 The Spyder Editor3.8 Input and Output3.8.1 Console Input and Output3.8.2 File Input and Output3.8.3 Formatting Output3.9 Obtaining HelpProblems
Chapter 4 Structured Programming with Python
Chapter Objectives4.1 An Overview of Program Structure4.2 Implementing Decision Structures with Python4.3 Implementing Repetition Structures with Python4.3.1 The General Loop Structure4.3.2 The List-Driven and Count-Controlled Loop Structures4.3.3 The break and Continue Statements with the for Loop4.4 User-Defined Functions in Python4.4.1 lambda Functions4.4.2 Function Arguments4.4.3 Variable ScopeProblems
Chapter 5 Graphics—Matplotlib
Chapter Objectives5.1 Introduction to Matplotlib5.2 Customizing Line and Scatter Plots5.3 Using Figure Window Objects5.4 Creating Bar Plots Including Histograms5.5 Creating Other Plots of Interest5.6 Contour and Surface PlotsProblems
Chapter 6 Array and Matrix Operations
Chapter Objectives6.1 Creating Arrays in Python6.1.1 Creating Special Arrays6.1.2 Combining, Stacking, and Splitting Arrays6.1.3 Reshaping Arrays6.2 Indexing: Array Subscripts6.3 Array Operations6.4 Vector/Matrix Operations6.4.1 Matrix/Vector Multiplication6.4.2 Transpose6.4.3 Matrix InversionProblems
Chapter 7 Solving Single Algebraic Equations
Chapter Objectives7.1 The Nature of Single, Nonlinear Equations in One Unknown7.2 Bracketing Methods—Bisection7.3 Bracketing Methods—False Position7.4 Open Methods—Newton-Raphson7.5 Open Methods—Modified Secant7.6 Circular Methods—Fixed-Point Iteration7.7 Circular Methods—The Wegstein Method7.8 A Hybrid Approach—Brent’s Method7.9 Solving for the Roots of Polynomials7.10 Case Study: Trajectories of Projectiles in AirProblems
Chapter 8 Solving Sets of Algebraic Equations
Chapter Objectives8.1 Systems of Linear Algebraic Equations8.2 Solving Small Numbers of Linear Algebraic Equations8.2.1 Graphical Method8.2.2 Determinants and Cramer’s Rule8.2.2.1 Determinants8.2.2.2 Cramer’s Rule8.2.3 Elimination of Unknowns8.3 Gaussian Elimination8.3.1 Naive Gaussian Elimination8.3.2 Gaussian Elimination Computer Algorithm8.3.2.1 Naive Gaussian Elimination Algorithm8.3.2.2 Adding Determinant Evaluation8.3.2.3 Partial Pivoting8.3.2.4 Detecting Singular and Ill-Conditioned Systems8.4 Solving Sets of Linear Equations with the NumPy linalg Module8.5 Solving Sets of Nonlinear Algebraic Equations8.5.1 Solution of Nonlinear Algebraic Equations by Successive Substitution8.5.2 The Newton-Raphson Method for Nonlinear Systems of Equations8.6 Use of the root Function from the SciPy optimize Module to Solve Nonlinear EquationsProblems
Chapter 9 Solving Differential Equations
Chapter Objectives9.1 Describing Differential Equations9.2 Quadrature — Finding the Area under the Curve9.2.1 Pre-computer Methods9.2.2 Quadrature for Continuous Functions9.2.3 The quad Function from SciPy’s integrate Module9.2.4 Quadrature for Discrete Data9.3 Solving Differential Equations with Initial Conditions9.3.1 Euler’s Method9.3.2 Heun’s Method9.3.3 Systems of Differential Equations9.4 Solving Differential Equations with the solve _ ivp Function from SciPy’s integrate ModuleProblems
Chapter 10 Working with Data
Chapter Objectives10.1 Characterizing Data Sets: Initial Observations and Sample Statistics10.1.1 General Data Concepts10.1.2 Sample Statistics: Central Tendency and Dispersion10.1.2.1 Central Tendency10.1.2.2 Spread or Dispersion10.1.3 Using Boxplots to Diagnose Outliers10.2 Distributions10.2.1 Several Important Distributions10.2.1.1 Uniform Distribution10.2.1.2 Normal Distribution10.2.1.3 Weibull Distribution10.2.2 Python and Distributions10.2.3 Random Numbers10.3 Making Claims Based on Data10.3.1 Comparison of Data with a Standard10.3.2 Comparison between Two Samples10.3.3 Determining Whether Data Are Normally Distributed10.4 Fitting Mathematical Models to Data10.4.1 Straight-line Linear Regression10.4.2 Fitting Polynomials10.4.3 General Issues and PrecautionsProblems
References
Index
Index of Python Terminology

Content preview from Introduction to Engineering and Scientific Computing with Python

7 Solving Single Algebraic Equations

DOI: 10.1201/9781003256861-7

Chapter Objectives

Learn bracketing methods for solving single equations, both bisection and false position.
Implement the bracketing methods by developing Python functions.
Understand the background behind open methods for solving single equations, including the Newton-Raphson method and the secant method.
Develop Python functions to implement the open methods.
Learn about a hybrid, bracketing/open method, Brent’s method, and how to use the built-in brentq function from the SciPy optimize module.
See how some equations occur naturally in a successive substitution or circular format, x = g(x), and understand the characteristics that govern convergence of fixed-point iteration. ...

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Introduction to Engineering and Scientific Computing with Python

by David E. Clough, Steven C. Chapra

7 Solving Single Algebraic Equations

Chapter Objectives

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
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