Numerical Methods with Applications
Front Cover
Information
License
Dedication
Principal Author
Preface
Chapter 01.01: Introduction to Numerical Methods
Lesson: Why Numerical Methods?
Learning Objectives
Introduction
Lesson: Steps of Solving an Engineering Problem
Learning Objectives
Introduction
Problem Description
Simple Mathematical Model
Solution to Simple Mathematical Model
Accurate Mathematical Model
Solution to More Accurate Mathematical Model
Implementing the Solution
Lesson: Overview of Mathematical Processes Covered in This Course
Learning Objectives
Introduction
Roots of a Nonlinear Equation
Simultaneous Linear Equations
Curve Fitting by Interpolation
Numerical Differentiation
Curve Fitting by Regression
Numerical Integration
Numerical Solution of Ordinary Differential Equations
Multiple Choice Test
Problem Set
Chapter 01.02: Quantifying Errors
Lesson: True Error
Learning Objectives
Introduction
What is true error?
Example 1
What is the relative true error?
Example 2
Lesson: Approximate Errors
Learning Objectives
Introduction
What is approximate error?
Example 1
What is the relative approximate error?
Example 2
Lesson: Review of Significant Digits
Learning Objectives
What do we mean by significant digits?
Example
Lesson: Relationship Between Significant Digits Correct and Relative Approximate Error
Learning Objectives
While solving a mathematical model using numerical methods, how can we use the value of relative approximate error to figure out if we have achieved an acceptable answer?
Example 1
Multiple Choice Test
Problem Set
Chapter 01.03: Sources of Error
Lesson: Round-Off Errors
Learning Objectives
Introduction
What is a round-off error?
What problems can be created by round-off errors?
Show an example of how not carrying enough significant digits can affect the solution to a problem?
References
Lesson: Truncation error
Learning Objectives
Introduction
Examples of Truncation Error
Example 1
Example 2
Lesson: Truncation error Extended Examples
Learning Objectives
Can you give me other examples of truncation errors?
Example 1
Example 2
Multiple Choice Test
Problem Set
Chapter 01.04: Fixed-Point Binary Representation of Numbers
Lesson: Introduction to Binary Representation of Numbers
Learning Objectives
Introduction
Example 1
Example 2
Example 3
Lesson: Fixed-Point Binary Representation of Numbers
Learning Objectives
Introduction
Example 1
But what is the mathematics behinds this process of converting a decimal number to binary format?
Example 2
Lesson: Writing a Fixed-Point Binary Number in Given Word Length
Learning Objectives
Introduction
Example 1
Multiple Choice Test
Problem Set
Chapter 01.05: Floating-Point Binary Representation of Numbers
Lesson: Floating-Point Representation Background
Learning Objectives
Introduction
Fixed point representation
Example 1
Floating-point representation
Example 2
Lesson: Introduction to Binary Floating-Point Representation of Numbers
Learning Objectives
How does the floating-point format work for binary representation?
Example 1
Example 2
Example 3
Example 4
Lesson: Binary Floating-Point Representation of Numbers with Biased Exponent
Learning Objectives
What is the floating-point format with a
biased
exponent for binary format? How is it different from the floating-point format with an
unbiased
exponent?
Example 1
Example 2
Appendix
How are numbers represented in floating-point in double precision in a computer?
Lesson: Accuracy of Binary Floating-Point Representation of Numbers
Learning Objectives
How do you determine the accuracy of a floating-point binary format number?
What is the significance of machine epsilon for a student in an introductory course in numerical methods?
Example 1
Appendix
Multiple Choice Test
Problem Set
Chapter 01.07: Taylor Theorem Revisited
Lesson: Introduction to Taylor Series
Learning Objectives
Introduction
Why are applications of Taylor’s theorem important for numerical methods?
Example 1
What does this mean in plain English?
More Examples of Taylor Series
Example 2
Lesson: Application of Taylor Series
Learning Objectives
What You Have Learned So Far
Example 1
Example 2
Example 3
Multiple Choice Test
Chapter 02.00: Physical Problem for Numerical Differentiation
Lesson: General Engineering
Summary
Modeling
Questions
Lesson: Chemical Engineering
Summary
Modeling
Questions
Chapter 02.01: Prerequisites to Numerical Differentiation
Lesson: Prerequisites to Numerical Differentiation
Learning Objectives
Introduction
Example 1
Example 2
Derivative of a Function
Example 3
Second Definition of Derivatives
Example 4
Higher-Order Derivatives
Example 5
Other Notations of Derivatives
Lesson: Equation of a Tangent Line
Learning Objectives
Finding equations of a tangent line
Example 1
Multiple Choice Test
Chapter 02.02: Numerical Differentiation of Continuous Functions
Lesson: Numerical Differentiation of Continuous Functions - First Derivative
Learning Objectives
Introduction
Forward Difference Approximation of the First Derivative
Example 1
Backward Difference Approximation of the First Derivative
Example 2
Central Divided Difference Approximation of the First Derivative
Example 3
Order of Accuracy of Divided Difference Schemes
Numerical Differentiation of Continuous Functions - Second Derivative
Learning Objectives
Introduction
Example 1
Error Analysis of Finite Difference Approximations
Learning Objectives
Introduction
Example 1
Example 2
Example 3
Richardson’s Extrapolation Formula for Central Divided Difference Approximation of First Derivative
Example 4
Example 5
Appendix
Multiple Choice Test
Problem Set
Chapter 02.03: Numerical Differentiation of Functions Given at Discrete Data Points
Lesson: Numerical Differentiation of Functions Given as Discrete Data Points - First Derivative
Lesson Objectives
Introduction
Example 1
Lesson: Numerical Differentiation of Functions Given as Discrete Data Points - Second Derivative
Learning Objectives
Introduction
Example 1
Direct Fit Polynomials to Find Derivatives of Functions Given as Discrete Data Points
Learning Objectives {-}
Introduction {-}
Vandermonde Polynomial {-}
Example 1 {-}
Example 2 {-}
Multiple Choice Test
Problem Set
Chapter 03.00: Physical Problem for Nonlinear Equations
Lesson: General Engineering
Summary
Modeling
Lesson: Mechanical Engineering
Summary
Modeling
Chapter 03.01: Prerequisites to Numerical Methods for Solving Nonlinear Equations
Lesson: Review of Quadratic Equations
Learning Objectives
What are quadratic equations, and how do we solve them?
Example 1
Example 2
Multiple Choice Test
Chapter 03.03: Bisection Method for Solving a Nonlinear Equation
Lesson: Background of Bisection Method
Learning Objectives
Introduction
Theorem
Lesson: Bisection Method Algorithm
Learning Objectives
What is the bisection method, and what is it based on?
Theorem
Bisection method
Algorithm for the bisection method
Lesson: Application of Bisection Method
Learning Objectives
Applications
Example 1
Example 2
Lesson: Advantages and Pitfalls of Bisection Method
Learning Objectives
Advantages of the bisection method
Drawbacks of bisection method
Multiple Choice Test
Problem Set
Chapter 03.04: Newton-Raphson Method for Solving a Nonlinear Equation
Lesson: Tangent to a Curve
Learning Objectives
Introduction
Example
Lesson: Graphical Derivation of Newton-Raphson Method
Learning Objectives
Introduction
Derivation
Algorithm
Lesson: Derivation of Newton-Raphson Method from Taylor Series
Learning Objectives
Introduction
Lesson: Application of Newton-Raphson Method
Learning Objectives
Applications
Example 1
Example 2
Lesson: Advantages and Pitfalls of Newton-Raphson Method
Learning Objectives
Advantages of the Newton-Raphson Method
Drawbacks of the Newton-Raphson Method
Appendix A. What is an inflection point?
Multiple Choice Test
Problem Set
Chapter 04.00: Physical Problem for Simultaneous Linear Equations
Lesson: General Engineering
Summary
Modeling
Questions
Lesson: Civil Engineering
Summary
Modeling
Questions
Lesson: Mechanical Engineering
Summary
Modeling
Questions
Chapter 04.01: Prerequisites to Simultaneous Linear Equations
Lesson: Definition of Matrices and Special Matrices
Learning Objectives
What does a matrix look like?
So, what is a matrix?
What are the special types of matrices?
Row Vector
Example 1
Column vector
Example 2
Submatrix
Square matrix
Example 3
Upper triangular matrix
Example 4
Lower triangular matrix
Example 5
Diagonal matrix
Example 6
Identity matrix
Example 7
Lesson: Binary Matrix Operations
Learning Objectives
How do you add two matrices?
Example 1
How do you subtract two matrices?
Example 2
How do I multiply two matrices?
Example 3
Lesson: Setting Up Problems in Matrix Form
Learning Objectives
Matrix algebra is used for solving systems of equations. Can you illustrate this concept?
Example 1
Lesson: Inverse of a Square Matrix
Learning Objectives
Can you divide two matrices?
Example 1
Can I use the concept of the inverse of a matrix to find the solution of a set of equations [A][X] = [C]?
How do I find the inverse of a matrix?
Example 2
Multiple Choice Test
Chapter 04.06: Gaussian Elimination Method for Solving Simultaneous Linear Equations
Lesson: Theory of Naive Gauss Elimination Method
Learning Objectives
How is a set of equations solved numerically by Gaussian elimination method?
Forward Elimination of Unknowns:
Back Substitution:
Lesson: Application of Naive Gauss Elimination Method
Learning Objectives
Applications
Example 1
Lesson: Finding Determinant of a Square Matrix Using Gaussian Elimination
Learning Objectives
Can we use Naive Gauss elimination methods to find the determinant of a square matrix?
Example 1
What if I cannot find the determinant of the matrix using the Naive Gauss elimination method, for example, if I get division by zero problems during the Naive Gauss elimination method?
Example 2
Example 3
Section: Pitfalls of Naive Gauss Elimination Method
Learning Objectives
Are there any pitfalls of the Naive Gauss elimination method?
Example 1
Example 2
Lesson: Theory of Gaussian Elimination with Partial Pivoting Method
Learning Objectives
What are some techniques for improving the Naïve Gauss elimination method?
How does Gaussian elimination with partial pivoting differ from Naïve Gauss elimination?
Lesson: Application of Gaussian Elimination with Partial Pivoting Method
Learning Objectives
Applications
Example 1
Example 2
Multiple Choice Test
Problem Set
Chapter 04.07: LU Decomposition Method for Solving Simultaneous Linear Equations
Lesson: Theory of LU Decomposition Method
Learning Objectives
I hear about LU decomposition used as a method to solve a set of simultaneous linear equations. What is it?
So how do I decompose a nonsingular matrix [A], that is, how do I find [A] = [L][U]?
Lesson: Application of LU Decomposition Method
Learning Objectives
Applications
Example 1
Example 2
Lesson: Finding Inverse of a Matrix Using LU Decomposition Method
Learning Objectives
How do I find the inverse of a square matrix using LU decomposition?
Example 1
Lesson: Computational Efficiency
Learning Objectives
LU decomposition looks more complicated than Gaussian elimination. Do we use LU decomposition because it is computationally more efficient than Gaussian elimination to solve a set of n equations given by
\([A][X]=[C]\)
?
This has confused me further! Why should I learn LU decomposition method when it takes the same computational time as Gaussian elimination, and that too when the two methods are closely related in the procedure? Please convince me that LU decomposition has its place in solving linear equations!
Appendix
Multiple Choice Test
Problem Set
Chapter 05.00: Physical Problem for Interpolation
Lesson: General Engineering
Summary
Problem Statement
Lesson: Mechanical Engineering
Summary
Problem Statement
Solution
Questions
Chapter 05.01: Prerequisites to Interpolation
Lesson: Prerequisites to Interpolation
Learning Objectives
What is interpolation?
Lesson: Uniqueness of Interpolating Polynomials
Learning Objectives
Proof for a unique polynomial of degree
\(n\)
or less passes through
\(n+1\)
data points.
Example
Multiple Choice Test
Chapter 05.02: Direct Method of Interpolation
Lesson: Direct Method of Interpolation
Learning Objectives
Direct Method
Example 1
Example 2
Example 3
Multiple Choice Test
Problem Set
Chapter 05.05: Spline Method of Interpolation
Lesson: Why Do We Need Spline Interpolation?
Learning Objectives
Introduction
Lesson: Linear Spline Interpolation
Learning Objectives
Linear Spline Interpolation
Example 1
Lesson: Quadratic Spline Interpolation
Learning Objectives
Interpolating Quadratic Spline
Lesson: Application of Quadratic Spline Interpolation
Learning Objectives
Applications
Example 1
Lesson: Outline of Cubic Spline Interpolation
Learning Objectives
Introduction
Interpolating Cubic Spline
Multiple Choice Test
Problem Set
Chapter 05.06: Extrapolation is a Bad Idea
Lesson: Extrapolation is a Bad Idea
Learning Objectives
Description
Chapter 05.10: Length of a Curve
Lesson: Length of a Curve
Learning Objectives
Introduction
Example 1
Lesson: Comparing Lengths of Curves
Learning Objectives
Introduction
Example 1
Multiple Choice Test
Chapter 06.00: Physical Problem for Regression
Lesson: General Engineering
Summary
Modeling
Questions
Lesson: Mechanical Engineering
Summary
Modeling
Solution
Questions
Chapter 06.01: Prerequisites to Regression
Lesson: Simple Statistics
Learning Objectives
Introduction
Example 1.
Lesson: Absolute Minimum of a Function of One Variable
Learning Objectives
Minimum of a twice differentiable continuous function
Example 1
Example 2
Lesson: Partial Derivatives
Learning Objectives
Introduction
Example 1
Lesson: Absolute Minimum of a Function of Multiple Variables
Learning Objectives
Recap: Local Minimum of a Single-Variant Function
Local Minimum of a Multivariant Function
First-Order Optimality Condition
Second-Order Optimality Condition:
Critical Points of a Function of Two Variables
Example 1
Appendix
Multiple Choice Test
Chapter 06.03: Linear Regression
Lesson: Introduction to Linear Regression
Learning Objectives
Introduction
Lesson: Straight-Line Regression Model without an Intercept
Learning Objectives
Introduction
Example 1
Lesson: Theory of General Straight-Line Regression Model
Learning Objectives
Introduction
Appendix
Lesson: Application of General Straight-Line Regression Model
Learning Objectives
Recap
Example 1
Multiple Choice Test
Problem Set
Chapter 06.04: Nonlinear Regression
Lesson: Introduction to Nonlinear Regression
Learning Objectives
Introduction
Exponential model
Power model
Saturation growth model
Harmonic decline model
Lesson: Exponential Regression Model with Transformation
Learning Objectives
Derivation of nonlinear regression models
Exponential model
Exponential Model through Transformation of Data
Example 1
Lesson: Nonlinear Regression Model Without Transformation
Learning Objectives
Nonlinear models using least squares
Exponential model
Example 1
Example 2
Lesson: Polynomial Regression Model
Learning Objectives
Polynomial Models
Example 1
Lesson: Optimum Order of a Polynomial Regression Model
Learning Objectives
Introduction
Lesson: Other Nonlinear Regression Models
Learning Objectives
Introduction
Growth model
Example 1
Logistic Growth Model
Logarithmic Functions
Example 2
Power Functions
Example 3
Multiple Choice Test
Problem Set
Chapter 06.05: Adequacy of Linear Regression Models
Lesson: Introduction to Adequacy of Linear Regression Models
Learning Objectives
Introduction
Lesson: Criteria for Adequacy of Linear Regression Models
Learning Objectives
Check 1. Plot the data and the regression model.
Check 2. Is the standard estimate of error within bounds?
Check 3. How close is the coefficient of determination to one?
Check 4. Find if the model meets the assumptions of random errors.
Lesson: Abuses of Regression
Learning Objectives
Introduction
Extrapolation
Generalization
Misidentification
Multiple Choice Test
Problem Set
Chapter 07.00: Physical Problem for Integration
Lesson: General Engineering
Summary
Modeling
Lesson: Industrial Engineering
Summary
Modeling
Questions
Lesson: Mechanical Engineering
Summary
Modeling
Questions
Chapter 07.01: Prerequisites to Numerical Integration
Lesson: Prerequisites to Numerical Integration
Learning Objectives
What is integration?
Riemann Sum
Example 1
Example 2
Mean Value of a Function
Example 3
Multiple Choice Test
Chapter 07.02: Trapezoidal Rule of Integration
Lesson: Single-Application Trapezoidal Rule
Learning Objectives
Introduction
Example 1
Example 2
Appendix
Lesson: Composite Trapezoidal Rule
Learning Objectives
Introduction
Derivation
Example 1
Example 2
Error in Composite Trapezoidal Rule
Appendix
Example A.1
Example A.2
Lesson: Error Analysis of Trapezoidal Rule
Learning Objective
Error in Composite Trapezoidal Rule
Richardson’s Extrapolation Formula for Trapezoidal Rule
Example 1
Multiple Choice Test
Problem Set
Chapter 07.05: Gauss Quadrature Rule of Integration
Lesson: Theory of Gauss Quadrature Rule
Learning Objective
Introduction
Trapezoidal Rule Derivation Using Method of Undetermined Coefficients
Two-point Gaussian Quadrature Rule Using Method of Undetermined Coefficients
One-point Gaussian Quadrature Rule Using Method of Undetermined Coefficients
Lesson: Applications of Gauss Quadrature Rule
Learning Objectives
Recap
Example 1
Example 2
Example 3
Lesson: Higher-Point Gauss Quadrature Rule
Learning Objectives
Introduction
Arguments and weighing factors for
n
-point Gauss quadrature rules
So if the table is given for integrals with [-1,1] integration limits, how does one solve for integrals with [a,b] integration limits.
Example 1
Example 2
So does Gaussian quadrature require that the integral must be transformed to the integral limit of [-1,1]?
Appendix
Example A.1
Example A.2
Multiple Choice Test
Problem Set
Chapter 07.06: Integrating Functions Given as Discrete Data Points
Lesson: Trapezoidal Rule for Functions Given as Discrete Points
Learning Objectives
Introduction
Example 1
Example 2
Lesson: Alternative Methods of Integrating Functions Given as Discrete Points
Learning Objectives
Introduction
Example 1
Example 2
Multiple Choice Test
Problem Set
Chapter 08.00: Physical Problem for Ordinary Differential Equations
Lesson: General Engineering
Summary
Modeling
Questions
Lesson: Mechanical Engineering
Summary
Modeling
Chapter 08.01: Prerequisites to Numerical Methods for Solving Ordinary Differential Equations
Lesson: Why Do We Need Ordinary Differential Equations?
Learning Objectives
Introduction
Formulation of differential equations
Lesson: Solving First-Order Ordinary Differential Equations Exactly
Learning Objectives
Introduction
Classical Technique
Homogenous Part of the Solution
Particular Part of the Solution
Example 1
Example 2
Lesson: Solving Second-Order Differential Equations Exactly
Learning Objectives
Applications
Example 1
Example 2
Example 3
Lesson: Solving an Integral as a First-Order Ordinary Differential Equation
Learning Objectives
Introduction
Example 1
Multiple Choice Test
Chapter 08.02: Euler’s Method for Solving Ordinary Differential Equations
Lesson: Derivation of Euler’s Method
Learning Objectives
What is Euler’s method?
Example 1
Example 2
Derivation of Euler’s method
Derivation of Euler’s Method from Taylor Series
Lesson: Application of Euler’s Method
Learning Objectives
Recap of Euler’s Method
Example 1
Lesson: Using Euler’s Method to Solve Integrals
Learning Objectives
Introduction
Example 1
Multiple Choice Test
Problem Set
Chapter 08.03: Runge-Kutta 2nd-Order Method for Solving Ordinary Differential Equations
Lesson: Theory of Runge-Kutta 2nd-Order Method
Learning Objectives
What is the Runge-Kutta 2nd order method?
Runge-Kutta 2
nd
order method
Heun’s Method
Midpoint Method
Ralston’s Method
Lesson: Application of Runge-Kutta 2nd-Order Method
Learning Objectives
Recap
Example 1
How do these three methods compare with results obtained if we found
\(\mathbf{f^\prime (x,y)}\)
directly?
Lesson: Derivation of Runge-Kutta 2nd-Order Method
Learning Objectives
How do we get the 2nd order Runge-Kutta method equations?
Multiple Choice Test
Problem Set
Chapter 08.05: On Solving Higher-Order and Coupled Ordinary Differential Equations
Lesson: Solving Higher-Order Ordinary Differential Equations
Learning Objectives
Description
Example 1
Example 2
Example 3
Lesson: State-Space Modeling and Higher-Order and Coupled Ordinary Differential Equations
Learning Objectives
Introduction
Example 1
Example 2
Example 3
Multiple Choice Test
Problem Set
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Numerical Methods with Applications
Dedication
To Sherrie, Candace, Angelie, and Bucky J Barks